Mtrn9224 robot design
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This document provides an introduction and history of robot design and mechatronics. It discusses how the term "robot" was coined in a 1921 play and how the field of robotics has evolved from industrial robots in the 1950s to modern applications. The document also summarizes Karel Capek's introduction of the term "robot", Isaac Asimov's popularization of robotics in science fiction, and Asimov's Three Laws of Robotics. Finally, it discusses the development of early industrial robots and modern capabilities in sensing, control, and manufacturing applications.
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2 MECHATRONICS
2.1 Difinition
Interdisciplinary field or engineering dealing with the design of products whose function relies on
the integration of mechanical and electronic components coordinated by a control architecture.
Originally introduced in Japan in early 1960’s, spread in Europe in late 1960’s and then in USA
in early 1970’s.
Primary disciplines – Mechanics, electronics, controls and computer engineering
Examples of Mechatronics systems
Aercraft flight control and navigation system
Automobile electronic fuel injection and antilock brake systems
Numerically controlled (NC) machine tools
Robots
Smart kitchen
Toys, etc
Figure 1.1 illustrates all the components in a typical mechatronic system.
Actuators – produce motion or cause some action
Sensors – detect the state of the system parameters, inputs and outputs
Digital devices – control the system
Conditioning and interfacing circuits – provide connections between the control
circuits and the input/output devices
Graphical displays – provide visual feedback to users
4
3 ROBOTS
3.1 Difinition
A reprogrammable, multifunctional manipulator designed to move material, parts, tools, or
specialized devices through various programmed motions for the performance of a variety of tasks.
A more inspiring definition can be found in Webster. According to Webster a robot is:
An automatic device that performs functions normally ascribed to humans or a machine in the
form of a human.
3.2 History
First use of the word 'robot'
The acclaimed Czech playwright Karel Capek (1890-1938) made the first use of the word ‘robot’,
from the Czech word for forced labor or serf. Capek was reportedly several times a candidate for the
Nobel prize for his works and very influential and prolific as a writer and playwright.
The use of the word Robot was introduced into his play R.U.R. (Rossum's Universal Robots) which
opened in Prague in January 1921.
In R.U.R., Capek poses a paradise, where the machines initially bring so many benefits but in the
end bring an equal amount of blight in the form of unemployment and social unrest.
The play was an enormous success and productions soon opened throughout Europe and the U.S.
R.U.R's theme, in part, was the dehumanization of man in a technological civilization.
You may find it surprising that the robots were not mechanical in nature but were created through chemical means. In
fact, in an essay written in 1935, Capek strongly fought that this idea was at all possible and, writing in the third person,
said:
"It is with horror, frankly, that he rejects all responsibility for the idea that metal contraptions could ever replace human
beings, and that by means of wires they could awaken something like life, love, or rebellion. He would deem this dark
prospect to be either an overestimation of machines, or a grave offence against life."
[The Author of Robots Defends Himself - Karl Capek, Lidove noviny, June 9, 1935, translation: Bean Comrada]
There is some evidence that the word robot was actually coined by Karl's brother Josef, a writer in his own right. In a
short letter, Capek writes that he asked Josef what he should call the artificial workers in his new play.
Karel suggests Labori, which he thinks too 'bookish' and his brother mutters "then call them Robots" and turns back to
his work, and so from a curt response we have the word robot.
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![MTRN9224 Robot Design
Lecture 3: Kinematic Fundamentals
Tomonari Furukawa
School of Mechanical and Manufacturing Engineering
University of New South Wales
SUMMARY
Gear ratio
1
s
w
R
R
η = >>
Ackerman steering type: Kinematic equations
cos
sin
tan
w
w
w
w
w w
R
x
R
y
R
L
ω θ
η
ω θ
η
ω
θ γ
η
=
=
=
Differential steering type: Kinematic equations
( )
( )
( )
cos
2
sin
2
w
l r
w
l r
w
l r
R
x
R
y
R
l
η
ω ω θ
η
ω ω θ
η
θ ω ω
= +
= +
= −
1 STATE SPACE REPRESENTATION
1.1 Equations
In MTRN3212 Principles of Control, we learned that a linear dynamical system could be
represented in the state space form.
=
x Ax + Bu
(3.1)
where x and u are state variables and control inputs, respectively. Most of the robot systems are
non-linear. We expand Equation (3.1) to a more general form:
2
( , )
=
x f x u
(3.2)
The robot on a three-dimensional plane often has state variables of position [ ]
,
x y and orientation
θ , but the control variables depend upon the robot.
1.2 Feedforward Control
The algorithms to simulate the motion of the robot are shown below.
/*
Definition:
k: Iteration
x(k): State variable vector
u(k): Control input vector
func(x(k), u(k)): Robot model
xdot(k): Rate of change of state vector
Initially given:
u(k) for all k=0,…,K
x0: Initial state variable vector
dt: Time step interval
*/
k = 0; // Set initial time step to 0
x(k) = x0; // Substitute initial state x0 into current state
do{
xdot(k) = func(x(k), u(k));
x(k+1) = x(k) + dt * xdot(k);
k = k+1;
}while(until terminal condition is satisfied);
2 GEAR RATIO
DC motors, used for driving wheels of a wheeled robot, are designed to rotate fast. The angular
velocity of the motor, m
ω , should not direcly become the velocity of the robot. Speed reduction is
accomplished by coupling two wheels together through physical contact on their outside diameters.
Gears are common speed reducers.
How does the speed of the driving gear relate to the speed of the output gear connected to the
shaft? Driving gear’s pitch radius m
R is smaller than the output gear’s radius w
R . The output gear
thus spins faster ( w
ω ) than the driving gear m
ω , and the relationship is given by
m
w m
w
R
R
ω ω
= (3.1)
The actual motors have a more complicated internal structure, but the following gear ratio,
1
w
m
R
R
η = (3.2)
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is given as the most important property of the motor gear.
3 ACKERMAN STEERING TYPE
We will derive the equations of motion of a tricycle for simplicity.
3.1 Control Inputs
Figure 3.1 illustrates a tricycle. The control inputs of the tricycle are:
Steering angle: γ [rad]
Angular velocity of motor: m
ω [rad/s]
Note that the angular velocity of the motor directly leads to the velocity of the vehicle in terms of
the following equation:
w
w w m
R
v R ω ω
η
= = (3.3)
where w
R is the radius of a rear wheel (We assume that the two rear wheels have the same radius)
and η is the gear ratio.
x
y
0
y
x,
θ
γ
v
Figure 3.1 Tricycle model
3.2 Instantaneous Curvature
3.2.1 Representation
We will first learn the idea of instantaneous curvature, as it makes us easier to derive and
understand the orientation of the tricycle. Figure 3.2 shows the instantaneous curvature whose
radius is c
R , as well as variables and parameters of the model when the tricycle is turning left.
4
x
y
0
c
R
γ
v
c
θ
L
Figure 3.2 Instantaneous curvature
The angle c
θ in the figure firstly becomes the steering angle of the tricycle:
c
θ γ
= (3.4)
The radius c
R is then given in terms of the length of the vehicle L and steering angle γ by
tan
c
L
R
γ
= (3.5)
3.2.2 Alternative representation for radius
We have considered that γ is positive when the vehicle turns left. Therefore, the vehicle is
meant to turn right when c
R is negative. However,
0 0
0
0
tan
0
0 0
c
L
R Singular
γ
γ
→ + → +∞
→ =
= =
→ − → −∞
(3.6)
It is not easy to represent the instantaneous curvature in terms of c
R . Curvature rate κ , which is
defined as
1 tan
c
R L
γ
κ ≡ = , (3.7)
is often used as an alternative variable to represent the instantaneous curvature. Continuity is
guaranteed in this representation:
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0 0
0 0
tan
0 0
0 0
0 0
L
γ
γ κ
→ + → +
→ =
= =
→ − → −
(3.8)
3.3 Orientation
The tricycle moves along the tangent of the instantaneous curvature. The velocity of the tricycle
v is given in terms of the rate of change of the curvature by
c
v R θ
= (3.9)
The substitution of Equation (3.5) into Equation (3.9) yields the equation for the orientation of the
tricycle
tan tan
w m
c
R
v v
R L L
ω
θ γ γ
η
= = =
(3.10)
3.4 Position
The velocity of the vehicle can be given as a vector in Cartesian coordinates
( [ ]
, ,
T
x y
v v x y
≡ =
v , 2 2
v x y
≡ = +
v ). The tricycle moves tangentially to the circumference of
the instantaneous curvature, so that the position of the tricycle is expressed in state form as
cos cos
sin sin
w
m
w
m
R
x v
R
y v
θ ω θ
η
θ ω θ
η
= =
= =
(3.11)
4 DIFFERENTIAL STEERING TYPE
We will consider in this section the differetial steering type of wheeled robot where two wheels
are each driven by a motor.
Question 2
Does the attachment of a support wheel influence the motion of the differential steering type of
wheeled robot?
4.1 Control Inputs
Figure 3.3 illustrates the robot. Each wheel has the same radius of w
R , and the wheels have a
distance of l . The control inputs of the robot are:
6
Angular velocity of motor on left wheel: l
ω [rad/s]
Angular velocity of motor on right wheel: r
ω [rad/s]
x
y
0
y
x,
θ
l
ω
v
r
ω
c
R
ω
l
Figure 3.3 Differential steering robot
4.2 Instantaneous Curvature
Let us find an instantaneous curvature when l
ω r
ω . Figure 3.3 shows the instantaneous
curvature of such a robot. The linear and angular velocities of the center of the robot are
( )
2
w
l r
R
v ω ω
η
= + (3.12)
( )
w
l r
R
l
ω ω ω
η
= − (3.13)
The radius of the instantaneous curvature is given by
c
v
R
ω
= (3.14)
4.3 Orientation
State space equation describing the orientation of the robot is given from Equation (3.13) by
( )
w
l r
R
l
θ ω ω
η
= −
(3.15)
4.4 Position
From Equation (3.12), state space equations describing the position of the robot is given by
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y = z(2);
th = z(3);
% Control variables
v = u(1);
st = u(2);
% Robot model
xdot = v * cos(th);
ydot = v * sin(th);
thdot = v / L * tan(st);
% Outputs
zdot(1) = xdot;
zdot(2) = ydot;
zdot(3) = thdot;
5.3.6 model_config.m
% -----------------------------------------------------
% model_config.m
% This M-function acquires the configuration of the robot
% Programmed by Tomonari Furukawa for MTRN9224
% March 11, 2004
%
% -----------------------------------------------------
function [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z, u)
global L;
global d;
global R_w;
x = z(1); % Length of the vehicle
y = z(2);
th = z(3);
v = u(1);
st = u(2);
xfc = x + L*cos(th);
yfc = y + L*sin(th);
xfl = xfc - 0.5*d*sin(th);
yfl = yfc + 0.5*d*cos(th);
xfr = xfc + 0.5*d*sin(th);
yfr = yfc - 0.5*d*cos(th);
xrl = x - 0.5*d*sin(th);
yrl = y + 0.5*d*cos(th);
xrr = x + 0.5*d*sin(th);
yrr = y - 0.5*d*cos(th);
xfwf = xfc + R_w*cos(th+st);
yfwf = yfc + R_w*sin(th+st);
xfwr = xfc - R_w*cos(th+st);
yfwr = yfc - R_w*sin(th+st);
xrlwf = xrl + R_w*cos(th);
yrlwf = yrl + R_w*sin(th);
xrlwr = xrl - R_w*cos(th);
yrlwr = yrl - R_w*sin(th);
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xrrwf = xrr + R_w*cos(th);
yrrwf = yrr + R_w*sin(th);
xrrwr = xrr - R_w*cos(th);
yrrwr = yrr - R_w*sin(th);
zfc = [xfc, yfc];
xb = [xfl, xfr, xrr, xrl, xfl];
yb = [yfl, yfr, yrr, yrl, yfl];
xfw = [xfwf, xfwr];
yfw = [yfwf, yfwr];
xrlw = [xrlwf, xrlwr];
yrlw = [yrlwf, yrlwr];
xrrw = [xrrwf, xrrwr];
yrrw = [yrrwf, yrrwr];
5.3.7 plot_rectangular.m
% -----------------------------------------------------
% plot_rectangular.m
% This M-function plots an rectangular object
% Programmed by Tomonari Furukawa for MTRN9224
% March 11, 2004
%
% -----------------------------------------------------
function r = plot_rectangular(x, y, c)
xtl = x(1);
xtr = x(2);
ytb = y(1);
ytt = y(2);
xt = [xtl, xtr, xtr, xtl, xtl];
yt = [ytb, ytb, ytt, ytt, ytb];
plot(xt, yt, c);
5.3.8 simulate.m
% -----------------------------------------------------
% simulate.m
% Main M-script file that simulates the robot movement
% Programmed by Tomonari Furukawa for MTRN9224
% March 11, 2004
%
% -----------------------------------------------------
clear all; % Clear all variables
close all; % Close all figures
global L;
global d;
global R_w;
% Physical parameters of the robot
L = 0.25; % Length between the front wheel axis and rear wheel axis [m]
d = 0.18; % Distance between the rear wheels [m]
m_max_rpm = 10000; % Motor max speed [rpm]
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gratio = 20; % Gear ratio
R_w = 0.05; % Radius of wheel [m]
st_max_deg = 26; % Maximum steering angle of the robot [deg]
% Initial location of the robot
x0 = 0; % Initial x coodinate of robot [m]
y0 = 0; % Initial y coodinate of robot [m]
th_deg0 = -26; % Initial orientation of the robot (theta [deg])
% Target and obstacle locations
xt = [3.5,4]; % Target
yt = [-0.25,0.25];
xo1 = [2.5,2.8]; % Obstacle 1
yo1 = [0,1];
xo2 = [1,1.3]; % Obstacle 2
yo2 = [-1,0];
% Parameters related to simulations
t_max = 2; % Simulation time [s]
n = 50; % Number of iterations
dt = t_max/n; % Time step interval
% Derivation of parameters related to performance of the robot
m_max_rads = rpm2rads(m_max_rpm); % Motor max speed [m/s]
w_max_rads = m_max_rads / gratio; % Wheel max speed [m/s]
v_max = w_max_rads * R_w; % Max robot speed [m/s]
st_max_rad = deg2rad(st_max_deg); % Maximum steering angle [rad]
t = [0:dt:t_max]; % Time vector (n+1 components)
v = v_max * ones(1, n+1); % Velocity vector (n+1 components)
st_rad = st_max_rad * sin(5*t); % Steering angle vector (n+1 components) [rad]
th_rad0 = deg2rad(th_deg0); % Initial orientation [rad]
v0 = v(1); % Initial velocity [m/s]
st_rad0 = st_rad(1); % Initial steering angle [rad]
z0 = [x0, y0, th_rad0]; % Initial state vector
u0 = [v0, st_rad0]; % Initial control vector
fig1 = figure(1); % Figure set-up (fig1)
plot_rectangular(xt, yt, 'r');
axis([-0.2 4.6 -2 2]);
hold on;
plot_rectangular(xo1, yo1, 'b');
plot_rectangular(xo2, yo2, 'b');
% Acquire the configuration of robot for plot
% [xb, yb]: Vertices of rectangular robot base
% [xfw, yfw]: Front wheel position vector
% [xrlw, yrlw]: Rear left wheel position vector
% [xrrw, yrrw]: Rear right wheel position vector
[xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z0, u0);
% Plot robot and define plot id
plotzb = plot(xb, yb); % Plot robot base
plotzfw = plot(xfw, yfw, 'r'); % Plot front wheel
plotzrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel
plotzrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel
% Draw fast and erase fast
set(gca, 'drawmode','fast');
12
set(plotzb, 'erasemode', 'xor');
set(plotzfw, 'erasemode', 'xor');
set(plotzrlw, 'erasemode', 'xor');
set(plotzrrw, 'erasemode', 'xor');
z1 = z0; % Set initial state to z1 for simulation
% Biginning of simulation
for i = 1:n+1
if v(i) (v_max * cos(2*st_rad(i)))
v(i) = v_max * cos(2*st_rad(i));
end
u = [v(i), st_rad(i)]; % Set control input
zdot = robot_model(z1, u); % Derive zdot using robot model
z2 = z1 + zdot * dt; % Update the state of robot
z1 = z2; % Substitute z2 to z1
% Acquire the configuration of robot for plot
[xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z1, u);
% Plot robot
set(plotzb,'xdata',xb);
set(plotzb,'ydata',yb);
set(plotzfw,'xdata',xfw);
set(plotzfw,'ydata',yfw);
set(plotzrlw,'xdata',xrlw);
set(plotzrlw,'ydata',yrlw);
set(plotzrrw,'xdata',xrrw);
set(plotzrrw,'ydata',yrrw);
pause(0.2); % Pause by 0.2s for slower simulation
end
% Plot the resultant velocity and steering angle configurations
fig2 = figure(2); % Figure set-up (fig2)
subplot(2,1,1); % Upper half of fig1
plot(t, v); % Plot velocity-time curve
xlabel('Time [s]');
ylabel('Velocity [m/s]');
subplot(2,1,2); % Lower half of fig1
std = rad2deg(st_rad); % Steering angle vector (n+1 components) [deg]
plot(t,std); % Plot steering angle-time curve
xlabel('Time [s]');
ylabel('Steering angle [deg]');
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8. Use Equations (4.22) and (4.23) to find if the robot can achieve the desired maximum speed
and acceleration.
Table 4.1 shows some sample motors.
Table 4.1 Sample motors
Manufacturer Part No. Volts No-load
(RPM)
Torque (oz-
in.)
Stall current
(amps)
Tamiya 3633k36 7.2 196 173 5.4
Jameco 161381 12 200 50 1.4
Lynxmotion GHM-01 6 186 75 2.1
Servo Systems BC-101 12 180 144 1.4
Maxon 118742 9 218 227 7.3
1 oz-in. = 7.06e-3 Nm
4 SIMULATION PROGRAM
4.1 Program Files
The simulation program uses the following files:
simulate_linear.m
dynamic_model.m
compute_torque.m
compute_acceleration.m
4.2 Running the Program
To run the program, save all the M-files in the directory where the M-files for Lecture 3 were
stored. Then, type “simulate_linear” to run the program.
4.3 Source Codes
4.3.1 simulate_linear.m
% -----------------------------------------------------
% simulate_linear.m
% Main M-script file that simulates the dynamic motion
% of the robot
% Programmed by Tomonari Furukawa for MTRN9224
% March 20, 2004
%
10
% -----------------------------------------------------
clear all; % Clear all variables
close all; % Close all figures
global M;
global gratio;
global I_0;
global R_m;
global k_v;
global k_T;
global V_max;
global k_d;
global R_w;
% Physical parameters of the robot
M = 5; % [kg]
gratio = 20; %
I_0 = 0.340; % [A]
R_m = 0.821; % [Ohm]
k_v = rpm2rads(2385); % [m/s/V]
k_T = 0.567 * 7.06e-3; % [Nm/A]
V_max = 7.2; % [V]
k_d = 0; % [N.s/m]
R_w = 0.05; % [m]
% Parameters related to simulations
t_max = 10; % Simulation time [s]
n = 100; % Number of iterations
dt = t_max / n; % Time step interval
t = [0:dt:t_max]; % Time vector (n+1 components)
V_in = V_max * ones(1, n+1); % Velocity vector (n+1 components)
% Initial conditions
V_in_1 = V_in(1);
w_m_rpm_1 = 16000;
w_m_ms(1) = rpm2rads(w_m_rpm_1);
%w_w_ms(1) = w_m_ms_1 / gratio;
for i = 1:n+1
torque(i) = compute_torque(V_in(i), w_m_ms(i));
w_m_acc(i) = compute_acceleration(torque(i), w_m_ms(i));
if i ~= n+1
w_m_ms(i+1) = w_m_ms(i) + dt * w_m_acc(i);
end
end
v = R_w/gratio*w_m_ms;
figure(1);
subplot(2,2,1);
plot(t,v,'.');
xlabel('Time [s]');
ylabel('Velocity [m/s]');
subplot(2,2,2);
plot(t,w_m_ms*60/(2*pi),'.');
xlabel('Time [s]');
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ylabel('Motor Velocity [rpm]');
subplot(2,2,3);
plot(t,torque,'.');
xlabel('Time [s]');
ylabel('Torque [Nm]');
subplot(2,2,4);
plot(t,w_m_acc,'.');
xlabel('Time [s]');
ylabel('Acceleration [rad/s^2]');
4.3.2 dynamic_model.m
% -----------------------------------------------------
% dynamic_model.m
% This M-function defines the dynamic model of the robot
% Programmed by Tomonari Furukawa for MTRN9224
% March 20, 2004
%
% -----------------------------------------------------
function zdot = dynamic_model(z, u)
global M;
global gratio;
global rw;
w = z(2);
torque = u(1);
zdot(1) = w;
zdot(2) = (gratio*torque/rw)/M - kv*w/gratio*rw;
4.3.3 compute_torque.m
% -----------------------------------------------------
% compute_torque.m
% This M-function returns a torque value, given
% motor voltage and speed
% Programmed by Tomonari Furukawa for MTRN9224
% March 20, 2004
%
% -----------------------------------------------------
function torque = compute_torque(V_in, w_m)
global I_0;
global R_m;
global k_v;
global k_T;
torque = k_T*(V_in/R_m - w_m/(R_m*k_v) - I_0);
4.3.4 compute_acceleration.m
% -----------------------------------------------------
% compute_acceleration.m
% This M-function returns an accelation value, given
% motortorque and speed
% Programmed by Tomonari Furukawa for MTRN9224
% March 20, 2004
12
%
% -----------------------------------------------------
function acc_m = compute_acceleration(T_m, w_m)
global M;
global gratio;
global k_d;
global R_w;
acc_m = (gratio/R_w)^2/M*T_m - k_d/M*w_m;
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![MTRN9224 Robot Design
Lecture 7: Sensor Technologies
Tomonari Furukawa
School of Mechanical and Manufacturing Engineering
University of New South Wales
1 COORDINATE SYSTEMS AND SENSORS
1.1 Coordinate Systems
Figure 7.1 illustrates coordinate systems concerned with the navigation of the wheeled robot. As
shown in the figure, they include:
Global coordinate system { }
G
Robot coordinate system { }
R
Sensor coordinate system { }
S
Object coordinate system { }
O
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
R
{ }
G
{ }
S
[ ]
, ,
s s s
x y θ
[ ]
, ,
o o o
x y θ
Figure 7.1 Global and local coordinate systems of wheeled robot
1.2 Types of Sensor with respect to Coordinate Systems
Sensors can be classified in terms of which coordinate system they are attached to:
• Global sensor – Attached to the global coordinate system,
• Local (passive) sensor – Attached to the robot coordinate system,
• Local (active) sensor – Attached to a sensor coordinate system attached to the robot
coordinate system.
2
Figure 7.2 shows these sensors.
{ }
A
S
Passive sensor
{ } { }
P
S R
=
x
{ }
O
0
y
{ } { }
G
S G
=
Global sensor
Active sensor
Figure 7.2 Sensors
2 KINEMATIC TRANSFORMATION
2.1 Problem Formulation
For simplicity, we will deal with a case where the sensor is mounted at the robot coordinate
{ } { }
S R
= , as shown in Figure 7.3.
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ } { }
S R
=
{ }
G
[ ]
, ,
o o o
x y θ
Figure 7.3 Coordinate systems of concern
2.2 Representation of Position of an Object
First, we do not take the orientation of an object into account. Suppose that the position of an
object in two-dimensional space is measured by the sensor mounted on the wheeled robot. The
position is given with respect to the robot coordinate system
{ }
[ ]
,
R
o o
x y .
If the robot coordinate system { }
R has the same orientation as the global coordinate system { }
G
but dislocated by [ ]
,
r r
x y , the representation of the point with respect to the global coordinate
system { }
G can be given by
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![3
{ } { } { }
{ }
G R G
o o r
o o r
R
x x x
y y x
= +
, (7.1)
If the robot coordinate system { }
R has the same position as the global coordinate system { }
G but
located by an angle r
θ , the representation of the point with respect to the global coordinate system
{ }
G can be given by
{ } { }
{ }
{ }
cos sin
sin cos
G R
G
o o
r r
o o
r r
R
x x
y y
θ θ
θ θ
−
=
, (7.2)
The position of the point with respect to the global coordinate { }
G through the combination of
linear and rotational transformations can be thus given by
{ } { } { }
{ }
cos sin
sin cos
1 0 0 1 1
G G R
o r r r o
o r r r o
R
x x x
y y y
θ θ
θ θ
−
=
, (7.3a)
or
( )
{ } { } { }
{ } , ,
G G R
o R r r r o
x y θ
=
z T z . (7.3b)
2.3 Representation of Position and Orientation of an Object
If the orientation of the object can also be measured by the sensor, the rotational transformation
from the robot coordinate system { }
R to the global coordinate system { }
G can be represented as
{ } { } { }
R G R
o r o
θ θ θ
= + . (7.4)
The position and orientation of the object can be thus given by
{ } { } { }
{ }
cos sin 0
sin cos 0
0 0 1
1 0 0 0 1 1
G G R
o r r r o
o r r r o
o r o
R
x x x
y y y
θ θ
θ θ
θ θ θ
−
=
, (7.5a)
or
( )
{ } { } { }
{ } , ,
G G R
o R r r r o
x y θ
=
z T z . (7.5b)
3 SENSORS FOR POSITION/ORIENTATION CONTROL
3.1 Global Sensors
Global sensors are used to measure the position/orientation of a robot in the global coordinate
system { }
G
o
z . Typical sensors to measure the position and orientation include:
{ }
G
r
x , { }
G
r
y : GPS
4
{ }
G
r
θ : Compass
Both: Global vision
Figure 7.4: GPS (Garmin GPSMAP 60C) Figure 7.5: Electronic digital compass (Co-Pilot
V1000)
Question 1
In what situation can the global vision be used to measure the position/orientation of the robot?
3.2 Relative Sensors
Relative sensors measures the position/orientation of the object in the sensor coordinate system
{ }
S
o
z and are used for one of the following three objectives:
1) To detect obstacles (how far they are from the robot).
2) To estimate the position/orientation of an obstacle in the global coordinate system (Mapping).
In 2), the position/orientation of the obstacle in the global coordinate system can be obtained by
( ) ( )
{ } { } { } { }
{ } { }
, , , ,
G G R S
o R GR r r r S RS s s s o
x y x y
θ θ
=
z T T z . (7.6)
Example
Consider that the position of an obstacle in the sensor coordinate system is obtained. The
position of the obstacle in the global coordinate system can be calculated as
{ } { } { } { }
{ } { }
cos sin cos sin
sin cos sin cos
1 0 0 1 0 0 1 1
G G R S
o r r r s s s o
o r r r s s s o
R S
x x x x
y y y y
θ θ θ θ
θ θ θ θ
− −
=
. (7.7)
3.2.1 Range (Distance) and bearing (angle) sensors
This class of sensors can detect an object and measure its range and bearing from the sensor
coordinate system, { } { }
,
S S
o o
d θ
when the object is within the sensor range { }
[ ]
min max
,
S
o
d d d
∈ ,
{ }
[ ]
max max
,
S
o
θ θ θ
∈ − . They can be used (at least) to navigate the robot such that the robot does not
hit obstacles. Such sensors include:
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• Laser range finder: min
d = 1 m, max
d = 10,000 m, max
θ = 90° (Figure 7.6).
• Stereovision: min
d = ? m, max
d = ? m, max
θ 90° (Depends on the camera).
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
G
[ ]
, ,
s s s
x y θ
[ ]
,
o o
x y
{ }
R
{ }
S
R
θ
L
θ
R
p
L
p
Figure 7.6: Laser range finder (Sick LMS-200) Figure 7.7: Stereovision
Question 2
What would be differences between the laser range finder and the stereovision?
In addition to the detection of obstacles, this class of sensors can be used to localize the robot in
the global coordinate system. By measuring { } { }
,
S S
o o
d θ
, the position/orientation of the robot in
the sensor coordinate system can be derived as follows.
{ } { } { }
cos
S S S
o o o
x d θ
= , (7.8a)
{ } { } { }
sin
S S S
o o o
y d θ
= , (7.8b)
Transformations of the position/orientation of the robot in the sensor coordinate system into that
in the robot coordinate system and the global coordinate system can be conducted using Equation
(7.5).
Figure 7.7 shows how to derive { } { }
,
S S
o o
x y
from { } { }
,
S S
o o
d θ
using a stereovision. Initially
known is the distance between the two cameras, L R
p p p
= + . As the vision provides bearing
information ( L
θ from the left camera and R
θ from the right camera) on the object, the given values
can be related to { }
S
o
x as
{ } { }
{ }
( )
tan tan
tan tan
L R
S S
o L o R
S
o L R
p p p
x x
x
θ θ
θ θ
= +
= +
= +
(7.9)
{ }
S
o
x is therefore,
{ }
tan tan
S
o
L R
p
x
θ θ
=
+
(7.10)
6
{ }
S
o
y can also be calculated as follows:
{ }
( )
( )
2
2
tan tan
2 tan tan
S
o L
R L
R L
L R
p
y p
p p
p θ θ
θ θ
= −
−
=
−
=
+
(7.11)
3.2.2 Range (Distance) sensors in one direction
This class of sensors can detect an object and measure its distance from the origin of the sensor
coordinate system, { }
S
o
d , only when the sensor is directed to the object. The sensors include:
• Ultrasonic : min
d = 0.15 m, max
d = 3 m (Figure 7.8)
• Infrared sensor : min
d = 0.1 m, max
d = 1.5 m (Figure 7.9)
Figure 7.8: Ultrasonic sensor (Devantech SRF04
Ranger)
Figure 7.9: Infrared range sensor (Sharp
GP2Y0D02YK)
Question 3
In which situation will ultrasonic and infrared sensors each fail in measuring distance?
Many of these sensors can also be used as proximity sensors.
Question 4
What is the difference between the distance sensor and proximity sensor?
If the sensor is used to search in all directions, there are fundamentally two designs:
• Attach many sensors around the robot (Figure 7.10).
• Connect the sensor by a revolute joint (Figure 7.11).
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Provided that the position and orientation of the sensor are known with respect to the robot
coordinate system, the position of the object in the robot coordinate system can be calculated.
These set-ups can also be used for localization, but, because of inaccuracy, they are used only for
detecting obstacles in general.
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
G
[ ]
, ,
s s s
x y θ
[ ]
,
o o
x y
{ }
R
{ }
S
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
G
[ ]
, ,
s s s
x y θ
[ ]
,
o o
x y
{ }
R
{ }
S
Figure 7.10: Robot with many distance sensors Figure 7.11: Robot with an active distance
sensor
3.2.3 Bearing sensors
This class of sensors can detect an object and measure its bearing in the sensor coordinate system,
{ }
S
o
θ , when the object is within the sensor range { }
[ ]
max max
,
S
o
θ θ θ
∈ − . Such sensors include:
• Single camera: max
θ 90° (Depends on the camera).
• Omni-directional camera: All directions.
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
G
[ ]
, ,
s s s
x y θ
[ ]
,
o o
x y
{ }
R
{ }
S
x
{ }
O
0
[ ]
, ,
r r r
x y θ
y
{ }
G
[ ]
, ,
s s s
x y θ
[ ]
,
o o
x y
{ }
R
{ }
S
Figure 7.4: Single camera Figure 7.5: Omni-directional camera
3.3 Joint Use of Global and Relative Sensors
In addition to the two benefits of using relative sensors described in Section 3.2, the use of global
sensor(s) together with relative sensor(s) brings the following advantage:
3) To estimate the position/orientation of the robot in the global coordinate system if the obstacle
position/orientation is known in the global coordinate system (Localization).
8
In 3), the position/orientation of the robot in the global coordinate system can be obtained by
equating the following two equations and solve for
{ }
[ ]
, ,
G
r r r
x y θ .
( )
{ } { } { }
{ } , ,
R R S
o S RS s s s o
x y θ
=
z T z (7.12a)
( )
1
{ } { } { }
{ } , ,
R G G
o R GR r r r o
x y θ
−
=
z T z (7.12b)
Example
Consider a robot equipped with a compass (able to measure { }
G
r
θ ) and a laser range finder (able to
measure { } { }
,
S S
o o
x y
). If the position of an obstacle in the sensor coordinate system is obtained
by the laser range finder, the position of the robot in the global coordinate system can be
calculated by equating the following two equations:
{ } { } { }
{ }
{ } { } { } { } { }
{ } { } { } { } { }
cos sin
sin cos
1 0 0 1 1
cos sin
sin cos
1
R R S
o s s s o
o s s s o
S
S R S R R
o s o s s
S R S R R
o s o s s
x x x
y y y
x y x
x y y
θ θ
θ θ
θ θ
θ θ
−
=
− +
= + +
. (7.13a)
{ } { } { } { } { } { }
{ } { } { } { } { } { }
{ } { } { }
{ } { } 1 { }
{ }
{ }
{ }
{ }
cos sin
sin cos
1 0 0 1 1
cos sin sin cos
sin cos sin cos
0 0 1 1
cos
R G G
o r r r o
o r r r o
R
R
G G G G G G G
r r r r r r o
G G G G G G
r r r r r r o
G
G G G
o r
x x x
y y y
x x x
x y y
x
θ θ
θ θ
θ θ θ θ
θ θ θ θ
θ
−
−
=
− − −
= − −
−
=
{ } { } { } { } { }
{ } { } { } { } { } { } { } { }
sin sin cos
sin cos sin cos
1
G G G G G
o r r r r r
G G G G G G G G
o r o r r r r r
y x x
x y x y
θ θ θ
θ θ θ θ
− −
− + + −
. (7.13b)
Known in these equations are
• { }
G
r
θ : Measurement from the compass
• { } { }
,
S S
o o
x y
: Measurement from the laser
• { } { }
,
G G
o o
x y
: Global map
• { } { } { }
, ,
R R R
s s s
x y θ
: Measurable from the location of the laser
The position/orientation of the robot can be derived by going through the following steps:
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1) In Equation (7.13b), substitute { }
G
r
θ and { } { }
,
G G
o o
x y
, and represent { } { }
,
G G
r r
x y
each as a
function of { } { }
,
R R
o o
x y
.
2) Substitute { } { } { }
, ,
R R R
s s s
x y θ
into Equation (7.13a) and derive { } { }
,
R R
o o
x y
.
3) Substitute { } { }
,
R R
o o
x y
into the functions derived in 1) and determine { } { }
,
G G
r r
x y
.
3.4 Odometry Sensors and their Joint Use with Other Relative Sensors
Typical odometry sensors used for wheeled robots include angular encoders. They are mounted
on the wheel axel and/or the steering to ultimately derive the position and orientation of the robot
{ } { } { }
, ,
G G G
r r r
x y θ
by path integration or dead reckoning. For instance, in the Ackerman steering,
an encoder is attached to the rear shaft and the steering to measure the wheel speed w
ω and the
steering angle γ , respectively. The measurements then derive the rate of change of the position and
orientation of the robot:
{ }
{ }
{ }
cos
sin
tan
G w
r w r
G w
r w r
G w w
r
R
x
R
y
R
L
ω θ
η
ω θ
η
ω
θ γ
η
=
=
=
, (7.14)
and the integration of the state space equations gives the position and orientation of the robot
{ } { } { }
, ,
G G G
r r r
x y θ
.
Information that can be derived by the odometry sensors is the same as that by the global sensors,
though dead-reckon sensors are not as accurate as global sensors. The dead-reckon sensors can be
useful in an environment where global sensors are not available.
3.5 Velocity Sensors
When the dynamics of the robot are concerned, linear and angular velocities also become state
variables. Velocity sensors are used to measure the rate of change of the robot’s position and
orientation in the robot coordinate system, and the following are used as velocity sensors:
{ }
R
r
x
, { }
R
r
y
: Encoders (as described in the previous section), Accelerometers
{ }
R
r
θ : Encoders (as desceibed in the previous section), Gyroscope (accelerometer)
Both: Inertia Measurement Unit – Collection of accelerometers (and encoders).
If the global coordinate system is concerned, measurements can be transformed to information in
the global coordinate system as follows:
( )
{ } { } { }
{ } , ,
G G R
r R GR r r r r
x y θ
=
z T z
(7.12a)
10
4 OTHER SENSORS
The previous chapters have dealt with sensors measuring position/orientation to control the
position/orientation of the robot. There are sensors that extract other types of information. We
shall not discuss such sensors in detail, but they include:
• Force sensors – measure external forces x
F and/or y
F .
• Torque sensors – measure external torques Tθ .
• Optical sensors – measure the lightness.
• Magnet sensors – measure the magnetic field.
• Temperature sensors – measure the temperature.
• Humidity sensors – measure the humidity.
• Gas sensors – detect hydrogen (H2), hydrocarbons (CxHy), nitrogen oxide (NOx), carbon
monoxide (CO), oxygen (O2) and carbon dioxide (CO2)
[http://www.grc.nasa.gov/WWW/chemsensors/].
• Chemical sensors – measure various chemical components.
• Smell sensors
5 PROGRAMS
5.1 simulate_sensor.m
% -----------------------------------------------------
% simulate_sensor.m
% Main M-script file that simulates the robot movement
% Programmed by Tomonari Furukawa for MTRN9224
% May 11, 2004
%
% -----------------------------------------------------
clear all;
close all;
global L;
global d;
global R_w;
% Define obstacles
obstacles = 2;
obstacle(1).x = 3;
obstacle(1).y = 0;
%obstacle(1).th = atan(obstacle(1).y/obstacle(1).x);
obstacle(2).x = 2.5;
obstacle(2).y = 1;
%obstacle(2).th = atan(obstacle(2).y/obstacle(2).x);
fig1 = figure(1); % Figure set-up (fig1)
for i = 1:obstacles
plotobstacle(i) = plot(obstacle(i).x, obstacle(i).y, 'gx');
hold on;
end
axis([-0.2 4.6 -2 2]);
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% Physical parameters of the robot
L = 0.25; % Length between the front wheel axis and rear wheel axis [m]
d = 0.18; % Distance between the rear wheels [m]
R_w = 0.05; % Radius of wheel [m]
% Physical parameters of the sensor
d_max = 1; % Maximum distance that the sensor can detect an obstacle [m]
th_deg_max = 70; % Maximum angle that the sensor can detect an obstacle [deg]
th_rad_max = deg2rad(th_deg_max);
partitions = 10;
dth = 2 * th_rad_max / partitions;
th_inc = [-th_rad_max:dth:th_rad_max];
% Sensor position/orientation in robot coordinate system
R_x_s = L;
R_y_s = 0;
R_th_s = 0;
R_z_s = [R_x_s, R_y_s, R_th_s];
% Initial location of the robot
x0 = 2; % Initial x coodinate of robot [m]
y0 = 0; % Initial y coodinate of robot [m]
th_deg0 = 30; % Initial orientation of the robot (theta [deg])
th_rad0 = deg2rad(th_deg0);
% Initial position/orientation of the robot in global coordinate system
G_x_r = x0;
G_y_r = y0;
G_th_r = th_rad0;
G_z_r = [G_x_r, G_y_r, G_th_r];
z0 = [G_x_r, G_y_r, G_th_r]; % Initial state vector
u0 = [1, 0]; % Initial control vector
% Acquire the configuration of robot for plot
% [xb, yb]: Vertices of rectangular robot base
% [xfw, yfw]: Front wheel position vector
% [xrlw, yrlw]: Rear left wheel position vector
% [xrrw, yrrw]: Rear right wheel position vector
[xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z0, u0);
% Plot robot and define plot id
plotzb = plot(xb, yb); % Plot robot base
plotzfw = plot(xfw, yfw, 'r'); % Plot front wheel
plotzrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel
plotzrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel
% Measure the configuration of the sensor for plot
G_z_s = transform_po(R_z_s, G_z_r);
G_x_s = G_z_s(1);
G_y_s = G_z_s(2);
G_th_s = G_z_s(3);
for i = 1:obstacles
d_x = obstacle(i).x - G_x_s;
d_y = obstacle(i).y - G_y_s;
d(i) = (d_x^2 + d_y^2)^0.5;
th(i) = atan(d_y/d_x) - G_th_s;
end
x_range = [G_x_s, G_x_s + d_max*cos(th_inc+G_th_s), G_x_s];
y_range = [G_y_s, G_y_s + d_max*sin(th_inc+G_th_s), G_y_s];
plotrange = plot(x_range, y_range, 'r');
12
% Detect
for i = 1:obstacles
if d(i) d_max
obstacle(i).d_observableFlag = 1;
else
obstacle(i).d_observableFlag = 0;
end
if (th(i) -th_rad_max) (th(i) th_rad_max)
obstacle(i).th_observableFlag = 1;
else
obstacle(i).th_observableFlag = 0;
end
if (obstacle(i).d_observableFlag == 1) (obstacle(i).th_observableFlag == 1)
obstacle(i).observableFlag = 1;
set(plotobstacle(i), 'color', 'r');
S_x_o = d(i) * cos(th(i));
S_y_o = d(i) * sin(th(i));
S_z_o = [S_x_o, S_y_o];
R_z_o = transform_p(S_z_o, R_z_s);
G_z_o = transform_p(R_z_o, G_z_r)
else
obstacle(i).observableFlag = 0;
set(plotobstacle(i), 'color', 'g');
end
end
% Start simulation
% Draw fast and erase fast
set(gca, 'drawmode','fast');
set(plotzb, 'erasemode', 'xor');
set(plotzfw, 'erasemode', 'xor');
set(plotzrlw, 'erasemode', 'xor');
set(plotzrrw, 'erasemode', 'xor');
set(plotrange, 'erasemode', 'xor');
for i = 1:obstacles
set(plotobstacle(i), 'erasemode', 'xor');
end
% Plot
set(plotzb,'xdata',xb);
set(plotzb,'ydata',yb);
set(plotzfw,'xdata',xfw);
set(plotzfw,'ydata',yfw);
set(plotzrlw,'xdata',xrlw);
set(plotzrlw,'ydata',yrlw);
set(plotzrrw,'xdata',xrrw);
set(plotzrrw,'ydata',yrrw);
set(plotrange,'xdata',x_range);
set(plotrange,'ydata',y_range);
5.2 transform_p.m
% -----------------------------------------------------
% transform_p.m
% This M-function transforms position in
% one coordinate system to another
% Programmed by Tomonari Furukawa for MTRN9224
% May 11, 2004
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![13
%
% -----------------------------------------------------
function y = transform_p(x, a)
x_t = a(1);
y_t = a(2);
th_t = a(3);
z_in = [x(1); x(2); 1];
T = [cos(th_t), -sin(th_t), x_t;
sin(th_t), cos(th_t), y_t;
0, 0, 1];
z_out = T * z_in;
y(1) = z_out(1);
y(2) = z_out(2);
5.3 transform_po.m
% -----------------------------------------------------
% transform_po.m
% This M-function transforms position/orientation in
% one coordinate system to another
% Programmed by Tomonari Furukawa for MTRN9224
% May 11, 2004
%
% -----------------------------------------------------
function y = transform_po(x, a)
x_t = a(1);
y_t = a(2);
th_t = a(3);
z_in = [x(1); x(2); x(3); 1];
T = [cos(th_t), -sin(th_t), 0, x_t;
sin(th_t), cos(th_t), 0, y_t;
0, 0, 1, th_t;
0, 0, 0, 1];
z_out = T * z_in;
y(1) = z_out(1);
y(2) = z_out(2);
y(3) = z_out(3);
14
TUTORIAL QUESTIONS
1. Design a wheeled robot with sensor feedback control.
2. Implement sensors into your program.
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![MTRN9224 Robot Design
Lecture 8: Multi-Body Systems
Tomonari Furukawa
School of Mechanical and Manufacturing Engineering
University of New South Wales
1 WHEELED ROBOTS
Typical multi-body wheeled robots include
Car-like robot with trailers
Snake robot
These robots are effectively used when they
Are driven on a narrow road or path
Require carrying bulky items
Need to be long in dimension
1.1 Transformation
Kinematics of multi-body wheeled robots can be easily understood by the transformation of
coordinate systems learned in Lecture 7. As an example, we will consider a robot constituted by a
car (body 0) and n trailers (bodies 1,…,n ). The midpoint between the rear wheels is taken as the
reference for each body; its coordinates are [ ]
, ,
i i i
x y θ , { }
0,...,
i n
∀ ∈ , as shown in Figure 8.1.
The position and orientation of an object in coordinate system { }
I can be converted into those in
the neighboring coordinate system { }
1
I + using the following transformation matrix:
{ 1} { 1} { }
{ }
cos sin 0
sin cos 0
0 0 1
1 0 0 0 1 1
I I I
o i i i o
o i i i o
o i o
I
x y x
y y y
θ θ
θ θ
θ θ θ
+ +
−
=
. (8.1)
2
x
{ }
O
0
y
{ }
G
[ ]
, ,
o o o
x y θ
[ ]
0 0 0
, ,
x y θ
{ }
0
0
l
[ ]
1 1 1
, ,
x y θ
{ }
1
[ ]
1 1 1
, ,
n n n
x y θ
− − −
{ }
1
N −
[ ]
, ,
n n n
x y θ
{ }
N
1
l
γ
Figure 8.1: Car-like robot with trailers
In order to form a convoy (the car pulls all the other trailers which are hooked up to the next),
each trailer is subject to holonomic constraints
{ } { }
{ } { }
1 1
1
1 1
cos
2 2
sin
2
I I
i i
i i
I I
i
i i
l l
x
l
y
θ
θ
+ +
+
+ +
= +
=
, (8.2)
The transformation matrix resultantly becomes
{ 1}
1
{ 1} { }
{ }
cos sin 0 cos
2 2
sin cos 0 sin
2
0 0 1
1 1
0 0 0 1
I
i i
I I
i i i
o o
o i o
i i i
o o
i
I
l l
x x
y l y
θ θ θ
θ θ θ
θ θ
θ
+
+
+
− +
=
, (8.3a)
or
( )
{ 1} { 1} { }
{ }
I I I
o I i o
θ
+ +
=
z T z . (8.3b)
Generally, the position and orientation of an object in coordinate system { }
I can be transformed
into those in { }
K
( ) ( ) ( ) ( )
{ } { } { 1} { 2} { 1} { }
{ 1} { 2} 1 { 1} 1 { }
K K K I I I
o K k K k I i I i o
θ θ θ θ
− + +
− − − + +
=
z T T T T z
. (8.4)
1.2 State space equations
Because of the dependence of [ ]
,
i i
x y on i
θ as described in Equation (8.2), state variables of the
robot are [ ]
0 0 0 1 2 1
, , , , ,..., ,
n n
x y θ θ θ θ θ
− . In general, the state-space equations of a multi-body robot
are accordingly given by
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Mtrn9224 robot design
- 1. MTRN9224 Robot Design Lecture 1: Introduction Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales 1 HISTORY 19th century – Industrial Revolution (Birth of Mechanical Engineering) Mines needed large pumps never before seen to keep their shafts dry Iron and steel mills required pressures and temperatures beyond levels used commercially until then Transportation systems needed more than real horse power to move goods Structures began to stretch across ever wider abysses and to climb to dizzying heights Manufacturing moved from the shop bench to large factories ↓ Engineering disciplines People began to specialize and build bodies of knowledge. Early 20th century – Growth of individual engineering disciplines Each discipline (body of knowledge) has grown individually Mechanical engineering Electrical engineering Civil engineering Chemical engineering 2 Mid 20th century – Divergence of individual engineering disciplines Mechanical engineering Aeronautical engineering Space engineering Naval engineering Precision engineering Manufacturing, etc Mechanical engineering Electrical engineering Civil engineering Chemical engineering Electrical engineering Electronical engineering, etc Civil engineering Structural engineering, etc Chemical engineering Biochemical engineering, etc Late 20th century – Information revolution (Semiconductor and information technologies) Mechanical engineering Aeronautical engineering Space engineering Naval engineering Precision engineering Manufacturing, etc Electrical engineering Electronical engineering, etc Civil engineering Structural engineering, etc Chemical engineering Biochemical engineering, etc Semiconductor technology Information technology CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 2. 3 2 MECHATRONICS 2.1 Difinition Interdisciplinary field or engineering dealing with the design of products whose function relies on the integration of mechanical and electronic components coordinated by a control architecture. Originally introduced in Japan in early 1960’s, spread in Europe in late 1960’s and then in USA in early 1970’s. Primary disciplines – Mechanics, electronics, controls and computer engineering Examples of Mechatronics systems Aercraft flight control and navigation system Automobile electronic fuel injection and antilock brake systems Numerically controlled (NC) machine tools Robots Smart kitchen Toys, etc Figure 1.1 illustrates all the components in a typical mechatronic system. Actuators – produce motion or cause some action Sensors – detect the state of the system parameters, inputs and outputs Digital devices – control the system Conditioning and interfacing circuits – provide connections between the control circuits and the input/output devices Graphical displays – provide visual feedback to users 4 3 ROBOTS 3.1 Difinition A reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices through various programmed motions for the performance of a variety of tasks. A more inspiring definition can be found in Webster. According to Webster a robot is: An automatic device that performs functions normally ascribed to humans or a machine in the form of a human. 3.2 History First use of the word 'robot' The acclaimed Czech playwright Karel Capek (1890-1938) made the first use of the word ‘robot’, from the Czech word for forced labor or serf. Capek was reportedly several times a candidate for the Nobel prize for his works and very influential and prolific as a writer and playwright. The use of the word Robot was introduced into his play R.U.R. (Rossum's Universal Robots) which opened in Prague in January 1921. In R.U.R., Capek poses a paradise, where the machines initially bring so many benefits but in the end bring an equal amount of blight in the form of unemployment and social unrest. The play was an enormous success and productions soon opened throughout Europe and the U.S. R.U.R's theme, in part, was the dehumanization of man in a technological civilization. You may find it surprising that the robots were not mechanical in nature but were created through chemical means. In fact, in an essay written in 1935, Capek strongly fought that this idea was at all possible and, writing in the third person, said: "It is with horror, frankly, that he rejects all responsibility for the idea that metal contraptions could ever replace human beings, and that by means of wires they could awaken something like life, love, or rebellion. He would deem this dark prospect to be either an overestimation of machines, or a grave offence against life." [The Author of Robots Defends Himself - Karl Capek, Lidove noviny, June 9, 1935, translation: Bean Comrada] There is some evidence that the word robot was actually coined by Karl's brother Josef, a writer in his own right. In a short letter, Capek writes that he asked Josef what he should call the artificial workers in his new play. Karel suggests Labori, which he thinks too 'bookish' and his brother mutters "then call them Robots" and turns back to his work, and so from a curt response we have the word robot. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 3. 5 First use of the word 'robotics' The word 'robotics' was first used in Runaround, a short story published in 1942, by Isaac Asimov (born Jan. 2, 1920, died Apr. 6, 1992). I, Robot, a collection of several of these stories, was published in 1950. One of the first robots Asimov wrote about was a robotherapist. A modern counterpart to Asimov's fictional character is Eliza. Eliza was born in 1966 by a Massachusetts Institute of Technology Professor Joseph Weizenbaum who wrote Eliza -- a computer program for the study of natural language communication between man and machine. She was initially programmed with 240 lines of code to simulate a psychotherapist by answering questions with questions. Three Laws of Robotics Asimov also proposed his three "Laws of Robotics", and he later added a 'zeroth law'. Law Zero: A robot may not injure humanity, or, through inaction, allow humanity to come to harm. Law One: A robot may not injure a human being, or, through inaction, allow a human being to come to harm, unless this would violate a higher order law. Law Two: A robot must obey orders given it by human beings, except where such orders would conflict with a higher order law. Law Three: A robot must protect its own existence as long as such protection does not conflict with a higher order law. The First Robot: 'Unimate' After the technology explosion during World War II, in 1956, a historic meeting occurs between George C. Devol, a successful inventor and entrepreneur, and engineer Joseph F. Engelberger, over cocktails the two discuss the writings of Isaac Asimov. Together they made a serious and commercially successful effort to develop a real, working robot. They persuaded Norman Schafler of Condec Corporation in Danbury that they had the basis of a commercial success. Engelberger started a manufacturing company 'Unimation' which stood for universal automation and so the first commercial company to make robots was formed. Devol wrote the necessary patents. Their first robot nicknamed the 'Unimate'. As a result, Engelberger has been called the 'father of robotics.' The first Unimate was installed at a General Motors plant to work with heated die-casting machines. In fact most Unimates were sold to extract die castings from die casting machines and to perform spot welding on auto bodies, both tasks being particularly hateful jobs for people. Both applications were commercially successful, i.e., the robots worked reliably and saved money by replacing people. An industry was spawned and a variety of other tasks were also performed by robots, such as loading and unloading machine tools. Ultimately Westinghouse acquired Unimation and the entrepreneurs' dream of wealth was achieved. Unimation is still in production today, with robots for sale. The robot idea was hyped to the skies and became high fashion in the Boardroom. Presidents of large corporations bought them, for about $100,000 each, just to put into laboratories to "see what they could do;" in fact these sales constituted a large part of the robot market. Some companies even reduced their ROI (Return On Investment criteria for investment) for robots to encourage their use. 6 Modern Industrial Robots The image of the "electronic brain" as the principal part of the robot was pervasive. Computer scientists were put in charge of robot departments of robot customers and of factories of robot makers. Many of these people knew little about machinery or manufacturing but assumed that they did. (There is a common delusion of electrical engineers that mechanical phenomena are simple because they are visible. Variable friction, the effects of burrs, minimum and redundant constraints, nonlinearities, variations in workpieces, accommodation to hostile environments and hostile people, etc. are like the "Purloined Letter" in Poe's story, right in front of the eye, yet unseen.) They also had little training in the industrial engineer's realm of material handling, manufacturing processes, manufacturing economics and human behavior in factories. As a result, many of the experimental tasks in those laboratories were made to fit their robot's capabilities but had little to do with the real tasks of the factory. Modern industrial arms have increased in capability and performance through controller and language development, improved mechanisms, sensing, and drive systems. In the early to mid 80's the robot industry grew very fast primarily due to large investments by the automotive industry. The quick leap into the factory of the future turned into a plunge when the integration and economic viability of these efforts proved disastrous. The robot industry has only recently recovered to mid-80's revenue levels. In the meantime there has been an enormous shakeout in the robot industry. In the US, for example, only one US company, Adept, remains in the production industrial robot arm business. Most of the rest went under, consolidated, or were sold to European and Japanese companies. In the research community the first automata were probably Grey Walter's machina (1940's) and the John's Hopkins beast. Teleoperated or remote controlled devices had been built even earlier with at least the first radio controlled vehicles built by Nikola Tesla in the 1890's. Tesla is better known as the inventor of the induction motor, AC power transmission, and numerous other electrical devices. Tesla had also envisioned smart mechanisms that were as capable as humans. An excellent biography of Tesla is Margaret Cheney's Tesla, Man Out of Time, Published by Prentice-Hall, c1981. SRI's Shakey navigated highly structured indoor environments in the late 60's and Moravec's Stanford Cart was the first to attempt natural outdoor scenes in the late 70's. From that time there has been a proliferation of work in autonomous driving machines that cruise at highway speeds and navigate outdoor terrains in commercial applications. Fully functioning androids (robots that look like human beings) are many years away due to the many problems that must be solved. However, real, working, sophisticated robots are in use today and they are revolutionizing the workplace. These robots do not resemble the romantic android concept of robots. They are industrial manipulators and are really computer controlled "arms and hands". Industrial robots are so different to the popular image that it would be easy for the average person not to recognize one. Benefits Robots offer specific benefits to workers, industries and countries. If introduced correctly, industrial robots can improve the quality of life by freeing workers from dirty, boring, dangerous and heavy labor. it is true that robots can cause unemployment by replacing human workers but robots also create jobs: robot technicians, salesmen, engineers, programmers and supervisors. The benefits of robots to industry include improved management control and productivity and consistently high quality products. Industrial robots can work tirelessly night and day on an assembly line without an loss in performance. Consequently, they can greatly reduce the costs of manufactured goods. As a result of these industrial benefits, countries that effectively use robots in their industries will have an economic advantage on world market. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 4. 7 3.3 Classification of robots Robots has been classified according to their generation, at their level of intelligence, its level of control, and its level of programming language These classifications reflect the power of software in the controller, in individual, the sophisticated interaction of the sensors. The generation of a robot is determined by the historical order of developments in the robotics. Five generations normally are assigned robots industrialists. According to its Generation 1. - Robots Play-back, which regenerate a sequence of recorded instructions, like a robot used in covering by spray or arc welding. These robots commonly have a control of open loop. 2. - Robots controlled by sensors, these have a control in loopback of manipulated movements, and make decisions based on data collected by sensors. 3. - Robots controlled by vision, where robots information can manipulate an object when tilizar from a vision system. 4. - Robots controlled adaptably, where robots can automatically reprogramar their actions on the base of the data collected by the sensors. 5. - Robots with artificial intelligence, where robots uses the techniques of artificial intelligence to make their own decisions and to solve problems. According to its Level of Intelligence 1. - Devices of manual handling, controlled by a person. 2. - Robots of neat sequence. 3. - Robots of variable sequence, where an operator can modify the sequence easily. 4. - Robots regenerative, where the human operator leads the robot through the task. 5. - Robots of numerical control, where the operator feeds the programming on the movement, until the task is taught manually. 6. - Robots intelligent, which can understand and interact with changes in the medio.ambiente. According to its Level of Control 1. - Level of artificial intelligence, where the program will accept a commando like " raising the product " and disturbing it within a sequence of commandos of low level based on a strategic model of the tasks. 2. - Level of way of control where the movements of the system are modeled, for which includes the dynamic interaction between the different mechanisms, planned trajectories, and the selected points of allocation. 3. - Levels of servosystems, where the actuators control the parameters of the mechanisms with the use of an internal feedback of the data collected by the sensors, and the route is modified on the base of the data that are obtained from external sensors. All the detections of faults and mechanisms of correction are implemented in this level. 8 According to its Level of Programming language The key for an effective application of robots for an ample variety of tasks, is the development of high-level languages. Many systems of programming of robots exist, although most of advanced software more it is in the research laboratories. The systems of programming of robots fall within three classes: 1. - Guided systems, in which the user leads the robot through the movements to be made. 2. - Systems of programming of level-robot, in which the user writes a program of computer when specifying the movement and the sensado one. 3. - Systems of programming of level-task, in which the user specifies the operation by his actions on the objects that the robot manipulates. 3.4 Applications Robots are used in a diversity of applications, from robots turtles in the halls classes, robots soldering irons in the automotive industry, to arms teleoperados in the space shuttle. Each robot takes with problematic himself its own and its compatible solutions; even though much people consider that the automatization of processes through robots is in its beginnings, is an undeniable fact that the introduction of the robotic technology in the industry, already has caused a great impact. In this sense the Automotive industry plays a role preponderant. It is necessary to even mention the problems of social type, economic and politician, who can generate a bad direction of robotization of the industry. One becomes indispensable that the planning of the human, technological and financial resources is made of an intelligent way. The first electronic arms were used in the Industry and were called robots industrialists. An industrial robot is a programmable machine of general use that has some anthropomorphic characteristics or “humanoids”. Present the more typical humanoids characteristics of robots are the one of their movable arms, those that will move by means of sequences of movements that are programmed for the execution of utility tasks. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 5. MTRN9224 Robot Design Lecture 2: Design of a Base Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales 1 NUMER OF WHEELS AND ROBOT TYPES Question 1 How many wheels do we need at least to have a wheeled robot driven on a plane? Actions to be taken by the wheeled robot: Go straight at a different speed Turn to a different direction Major components of the wheeled robot: Base Batteries Motor(s) Wheels Driving wheels – connected to motors Support wheels Steering mechanism Question 2 How many wheels should be equipped with a motor? 2 There are a variety of types of wheeled robot. There are summarized in Figure 2.1. One-wheeled vehicles, two-wheeled vehicles (bicycles) and omni-directional vehicles are not included. Figure 2.1 Types of wheeled robots (a): FR. Front steering wheels for steering, Rear driving wheels. (b): FF. Front wheels for both driving and steering. Rear support wheels. (c): 4WD/4WS (d): Articulated steering. Steering mechanism of a wheel loader used in construction and mining industry. (e) – (f): Typical three-wheeled vehicles (tricycles) (g) – (i): Two driving wheels are independently controlled to turn. Classification in terms of turning a vehicle: The use of a servo motor to steer the robot (Ackerman Steering): (a) – (f) The use of two independent driving wheels (Differential Steering, also called tank steering): (g) – (i) CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 6. 3 2 ACKERMAN STEERING 2.1 Introduction Question 3 What are the advantages and disadvantages of Ackerman steering? Figure 2.2 shows slips occurring in each wheel. The following facts can be derived: The wheel rotates in a direction orthogonal to the wheel axis. Slips occur in this direction only if the wheel is motored (This is explained in Section 2.2). The wheel slips in a direction parallel to the wheel axis. Wheel axis Slip Rotate Figure 2.2 Directions of slip and rotation In order to turn without slips, all the wheels must have the same centre of rotation as shown in Figure 2.3. Center of rotation Figure 2.3 Centre of rotation 4 2.2 Differential Gear When a vehicle is turning, each wheel of the vehicle travels a different driving distance, i.e., Outer wheels travels longer than inner wheels. The angular velocity of each driving wheel that is connected to a common motor must be controlled such that the no slip occurs. Differential gear controls the angular velocity of each driving wheel such that no slip occurs. The differential gear is illustrated in Figures 2.4-2.7. P ω L ω R ω r ω Figure 2.4 Differential gear Figure 2.5 Differential gear (LEGO) Figure 2.5 Differential gear without the square set of four gears. Both wheels turn at the same angular velocity. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 7. 5 Figure 2.6 Differential "square" alone. It should be apparent that turning one wheel results in the opposite wheel turning in the opposite direction at the same rate. This is how the automobile differential works. It only comes into play when one wheel needs to rotate differentially with respect to its counterpart. When the car is moving in a straight line, the differential gears do not rotate with respect to their axes. When the car negotiates a turn, however, the differential allows the two wheels to rotate differentially with respect to each other. The role of the differential gears is mathematically represented as ( ) 1 2 P r L R ω ω ω ω γ = = + (2.1) where γ is the reduction ratio between the driving pinion and the ring gear. 2.3 Ackerman Link for Four-wheeled vehicles In order to yield no slip in a four-wheeled vehicle in a direction parallel to the wheel axis, there must be a mechanism that brings the same centre of rotation to all the wheels when the vehicle is turning. Ackerman link can achieve this requirement as shown in Figures 2.7-2.9. L θ R θ L R θ θ > Figure 2.7 Ackerman link 6 Figure 2.8 Ackerman link Figure 2.9 Wheeled robot with Ackerman steering 3 DIFFERENTIAL STEERING The turning of the differential steering robot can be controlled by providing a different angular velocity to each wheel (changing the speed of each motor). The term differential steering comes from the fact that the turning radius of the robot is a function of the ratios (or differences) of the wheel or tank-tread speeds on both sides of the robot. The important note is that one support wheel must be at least attached to this robot to enable three-point contact. Possible differential steering robots are illustrated in Figure 2.10. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 8. 7 Figure 2.10 Possible differential steering robots. 4 HOLONOMIC MOTION CONTROL A robot based on this steering method uses omni-directional roller wheels as shown in Figure 2.11. Figure 2.11 Omni-directional wheel A robot using holonomic motion control can move in any direction even without changing the orientation of the robot, and you can precisely control the rotational speed of the robot. 5 CENTER OF GRAVITY Figure 2.8 shows where the center-of-gravity can be located when the robot is stationary. 8 Figure 2.8 Safety region of center-of-gravity location when robot is stationary. 6 WEIGHT AND BASE MATERIALS 6.1 Weight Weight is often a premium in a robot. For most robots, the heaviest components are the batteries, the motor(s), and the base: Six-cell nickel-cadmium (NiCd) battery pack: 310 g. Each small gear motor: 150 g – 300 g. Base: ?? It is normally important to use the strongest and lightest materials. 6.2 Base Materials 6.2.1 Wood Easy to cut, drill, glue, bolt and screw. It is also the cheapest material. It is an ideal material for prototyping. 6.2.2 Plastic Hundreds of different types of plastics. Commonly sold as flat sheets with some thickness (1.5mm, 3mm, 4.5mm, 6mm). Typical ones for robots are: Acrylic – Hard, strong, rigid and clear in colour. Higher tensile strength than polycarbonate. Can be tapped, so that you can attach threaded fasteners directly to it. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 9. 9 Polycarbonate – Hard, strong, rigid and clear in colour. Has a very high-impact resistance. Easier to machine than acrylics. Expanded foam polyvinyl chloride (PVC) – The surface of this material is hard, but the inner core is like dense foam. It is not soft like a sponge, but rather hard. Expanded foam PVC is lighter and easier to machine than Acrylic and Polycarbonate. It would be the best plastic material for robots. 6.2.3 Aluminium Aluminium would be the best material for robots. It is very strong and has the lowest density of all of the common metals available. It is one of the easiest metals to machine. It is easy to obtain and cheap compared to magnesium and titanium. 6.2.4 Aluminium alloys Aluminium alloys include silicon, iron, copper, magnesium, chromium, and zinc as elements added to aluminium. They have different material properties such as strength, ductility, hardness and corrosion resistance. 6.2.5 Brass Brass is a strong metal that is easy to machine, but it is very heavy. Its density is three times greater than aluminium and almost ten times greater than the plastics described. The advantage is the types of shapes that can be obtained at hobby and hardware stores. 6.2.6 Comparison Material Density (lb/in^3) Yield stress (lb/in^2) Extended PVC (Sintra) 0.025 2,000 Alminium, 1100 0.098 4,000 Polycarbonate 0.045 9,000 Acrylic 0.043 11,000 Steel, 1018 0.283 32,000 Alminium, 6061-T6 0.098 40,000 Alminium, 2024-T4 0.098 43,000 Brass, 260 0.310 52,000 Alminium, 7075-T6 0.098 48,000 Titanium 6Al-4V 0.17 145,000 1 lb/in^3 = 27679.904 kg/m^3 1 lb/in^2 = 6894.757 Pa 10 7 FASTENERS FOR ROBOT ASSEMBLY There are four different ways to assemble robots: screws/bolts, adhesives, rivets and welding. 7.1 Adhesives There are permanent and temporary adhesives. Permanent adhesives, superglue and epoxies, should be used for parts that you do not intend to take apart in the future. Acrylic, polycarbonate, and expanded foam PVC can be bonded together with two-part epoxies and superglues. These adhesives cannot be used alone when the robot is subject to a lot of impacts. 7.2 Screws and Bolts The best way to assemble parts. You can assemble and disassemble your robot many times. Plastic materials do not have the same strength for holding screws as do aluminium or other metals. Question 3 How can you fasten plastic materials? 7.3 Welding and Riveting Welding and riveting are for permanent assembly. They may not be adequate for robot assembly. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 10. 11 8 TUTORIAL QUESTIONS 1. Determine a robot competition you are going to participate in. Examples of competition are: Soccer Rugby Combat game Sumo Search-and-rescue Stair and slope climbing Racing Maze exploration Narrow passage passing, etc. 2. Describe the basic rules of the robot competition. 3. Design the robot base, specifying the items below. Describe reasons for each item. Geometry, material and dimension of the base Drive train configuration (Ex. 4-wheel, 2-motor, differential steering, etc.) Geometry and material of the wheel CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 11. MTRN9224 Robot Design Lecture 3: Kinematic Fundamentals Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales SUMMARY Gear ratio 1 s w R R η = >> Ackerman steering type: Kinematic equations cos sin tan w w w w w w R x R y R L ω θ η ω θ η ω θ γ η = = = Differential steering type: Kinematic equations ( ) ( ) ( ) cos 2 sin 2 w l r w l r w l r R x R y R l η ω ω θ η ω ω θ η θ ω ω = + = + = − 1 STATE SPACE REPRESENTATION 1.1 Equations In MTRN3212 Principles of Control, we learned that a linear dynamical system could be represented in the state space form. = x Ax + Bu (3.1) where x and u are state variables and control inputs, respectively. Most of the robot systems are non-linear. We expand Equation (3.1) to a more general form: 2 ( , ) = x f x u (3.2) The robot on a three-dimensional plane often has state variables of position [ ] , x y and orientation θ , but the control variables depend upon the robot. 1.2 Feedforward Control The algorithms to simulate the motion of the robot are shown below. /* Definition: k: Iteration x(k): State variable vector u(k): Control input vector func(x(k), u(k)): Robot model xdot(k): Rate of change of state vector Initially given: u(k) for all k=0,…,K x0: Initial state variable vector dt: Time step interval */ k = 0; // Set initial time step to 0 x(k) = x0; // Substitute initial state x0 into current state do{ xdot(k) = func(x(k), u(k)); x(k+1) = x(k) + dt * xdot(k); k = k+1; }while(until terminal condition is satisfied); 2 GEAR RATIO DC motors, used for driving wheels of a wheeled robot, are designed to rotate fast. The angular velocity of the motor, m ω , should not direcly become the velocity of the robot. Speed reduction is accomplished by coupling two wheels together through physical contact on their outside diameters. Gears are common speed reducers. How does the speed of the driving gear relate to the speed of the output gear connected to the shaft? Driving gear’s pitch radius m R is smaller than the output gear’s radius w R . The output gear thus spins faster ( w ω ) than the driving gear m ω , and the relationship is given by m w m w R R ω ω = (3.1) The actual motors have a more complicated internal structure, but the following gear ratio, 1 w m R R η = (3.2) CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 12. 3 is given as the most important property of the motor gear. 3 ACKERMAN STEERING TYPE We will derive the equations of motion of a tricycle for simplicity. 3.1 Control Inputs Figure 3.1 illustrates a tricycle. The control inputs of the tricycle are: Steering angle: γ [rad] Angular velocity of motor: m ω [rad/s] Note that the angular velocity of the motor directly leads to the velocity of the vehicle in terms of the following equation: w w w m R v R ω ω η = = (3.3) where w R is the radius of a rear wheel (We assume that the two rear wheels have the same radius) and η is the gear ratio. x y 0 y x, θ γ v Figure 3.1 Tricycle model 3.2 Instantaneous Curvature 3.2.1 Representation We will first learn the idea of instantaneous curvature, as it makes us easier to derive and understand the orientation of the tricycle. Figure 3.2 shows the instantaneous curvature whose radius is c R , as well as variables and parameters of the model when the tricycle is turning left. 4 x y 0 c R γ v c θ L Figure 3.2 Instantaneous curvature The angle c θ in the figure firstly becomes the steering angle of the tricycle: c θ γ = (3.4) The radius c R is then given in terms of the length of the vehicle L and steering angle γ by tan c L R γ = (3.5) 3.2.2 Alternative representation for radius We have considered that γ is positive when the vehicle turns left. Therefore, the vehicle is meant to turn right when c R is negative. However, 0 0 0 0 tan 0 0 0 c L R Singular γ γ → + → +∞ → = = = → − → −∞ (3.6) It is not easy to represent the instantaneous curvature in terms of c R . Curvature rate κ , which is defined as 1 tan c R L γ κ ≡ = , (3.7) is often used as an alternative variable to represent the instantaneous curvature. Continuity is guaranteed in this representation: CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 13. 5 0 0 0 0 tan 0 0 0 0 0 0 L γ γ κ → + → + → = = = → − → − (3.8) 3.3 Orientation The tricycle moves along the tangent of the instantaneous curvature. The velocity of the tricycle v is given in terms of the rate of change of the curvature by c v R θ = (3.9) The substitution of Equation (3.5) into Equation (3.9) yields the equation for the orientation of the tricycle tan tan w m c R v v R L L ω θ γ γ η = = = (3.10) 3.4 Position The velocity of the vehicle can be given as a vector in Cartesian coordinates ( [ ] , , T x y v v x y ≡ = v , 2 2 v x y ≡ = + v ). The tricycle moves tangentially to the circumference of the instantaneous curvature, so that the position of the tricycle is expressed in state form as cos cos sin sin w m w m R x v R y v θ ω θ η θ ω θ η = = = = (3.11) 4 DIFFERENTIAL STEERING TYPE We will consider in this section the differetial steering type of wheeled robot where two wheels are each driven by a motor. Question 2 Does the attachment of a support wheel influence the motion of the differential steering type of wheeled robot? 4.1 Control Inputs Figure 3.3 illustrates the robot. Each wheel has the same radius of w R , and the wheels have a distance of l . The control inputs of the robot are: 6 Angular velocity of motor on left wheel: l ω [rad/s] Angular velocity of motor on right wheel: r ω [rad/s] x y 0 y x, θ l ω v r ω c R ω l Figure 3.3 Differential steering robot 4.2 Instantaneous Curvature Let us find an instantaneous curvature when l ω r ω . Figure 3.3 shows the instantaneous curvature of such a robot. The linear and angular velocities of the center of the robot are ( ) 2 w l r R v ω ω η = + (3.12) ( ) w l r R l ω ω ω η = − (3.13) The radius of the instantaneous curvature is given by c v R ω = (3.14) 4.3 Orientation State space equation describing the orientation of the robot is given from Equation (3.13) by ( ) w l r R l θ ω ω η = − (3.15) 4.4 Position From Equation (3.12), state space equations describing the position of the robot is given by CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 14. 7 ( ) ( ) cos 2 sin 2 w l r w l r R x R y ω ω θ η ω ω θ η = + = + (3.11) 5 SIMULATION PROGRAM 5.1 Program Files The simulation program uses the following files: rad2deg.m deg2rad.m rpm2rads.m rads2rpm.m robot_model.m model_config.m plot_rectangular.m simulate.m 5.2 Running the Program To run the program, take the following three steps: 1. Start MATLAB by double-clicking MATLAB icon. 2. Change the working directory to where your programs are. For instance, if your working directory is C:Documents and SettingsstudentMy Documents Program then type cd C:’Documents and Settings’student’My Documents’ Program 3. Run the program by typing simulate 5.3 Source Codes 5.3.1 rad2deg.m % ----------------------------------------------------- % rad2deg.m % This M-function converts radian to degree % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- 8 function deg = rad2deg(rad) deg = rad / pi * 180; 5.3.2 deg2rad.m % ----------------------------------------------------- % deg2rad.m % This M-function converts degree to radian % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function rad = deg2rad(deg) rad = deg / 180 * pi; 5.3.3 rpm2rads.m % ----------------------------------------------------- % rpm2rads.m % This M-function converts rpm to rad/s % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function rads = rpm2rads(rpm) rads = rpm * 2*pi / 60; 5.3.4 rads2rpm.m % ----------------------------------------------------- % rads2rpm.m % This M-function converts rad/s to rpm % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function rpm = rads2rpm(rads) rpm = rads * 2*pi / 60; 5.3.5 robot_model.m % ----------------------------------------------------- % robot_model.m % This M-function defines the robot model % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function zdot = robot_model(z, u) global L; % State variables x = z(1); CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 15. 9 y = z(2); th = z(3); % Control variables v = u(1); st = u(2); % Robot model xdot = v * cos(th); ydot = v * sin(th); thdot = v / L * tan(st); % Outputs zdot(1) = xdot; zdot(2) = ydot; zdot(3) = thdot; 5.3.6 model_config.m % ----------------------------------------------------- % model_config.m % This M-function acquires the configuration of the robot % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z, u) global L; global d; global R_w; x = z(1); % Length of the vehicle y = z(2); th = z(3); v = u(1); st = u(2); xfc = x + L*cos(th); yfc = y + L*sin(th); xfl = xfc - 0.5*d*sin(th); yfl = yfc + 0.5*d*cos(th); xfr = xfc + 0.5*d*sin(th); yfr = yfc - 0.5*d*cos(th); xrl = x - 0.5*d*sin(th); yrl = y + 0.5*d*cos(th); xrr = x + 0.5*d*sin(th); yrr = y - 0.5*d*cos(th); xfwf = xfc + R_w*cos(th+st); yfwf = yfc + R_w*sin(th+st); xfwr = xfc - R_w*cos(th+st); yfwr = yfc - R_w*sin(th+st); xrlwf = xrl + R_w*cos(th); yrlwf = yrl + R_w*sin(th); xrlwr = xrl - R_w*cos(th); yrlwr = yrl - R_w*sin(th); 10 xrrwf = xrr + R_w*cos(th); yrrwf = yrr + R_w*sin(th); xrrwr = xrr - R_w*cos(th); yrrwr = yrr - R_w*sin(th); zfc = [xfc, yfc]; xb = [xfl, xfr, xrr, xrl, xfl]; yb = [yfl, yfr, yrr, yrl, yfl]; xfw = [xfwf, xfwr]; yfw = [yfwf, yfwr]; xrlw = [xrlwf, xrlwr]; yrlw = [yrlwf, yrlwr]; xrrw = [xrrwf, xrrwr]; yrrw = [yrrwf, yrrwr]; 5.3.7 plot_rectangular.m % ----------------------------------------------------- % plot_rectangular.m % This M-function plots an rectangular object % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- function r = plot_rectangular(x, y, c) xtl = x(1); xtr = x(2); ytb = y(1); ytt = y(2); xt = [xtl, xtr, xtr, xtl, xtl]; yt = [ytb, ytb, ytt, ytt, ytb]; plot(xt, yt, c); 5.3.8 simulate.m % ----------------------------------------------------- % simulate.m % Main M-script file that simulates the robot movement % Programmed by Tomonari Furukawa for MTRN9224 % March 11, 2004 % % ----------------------------------------------------- clear all; % Clear all variables close all; % Close all figures global L; global d; global R_w; % Physical parameters of the robot L = 0.25; % Length between the front wheel axis and rear wheel axis [m] d = 0.18; % Distance between the rear wheels [m] m_max_rpm = 10000; % Motor max speed [rpm] CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 16. 11 gratio = 20; % Gear ratio R_w = 0.05; % Radius of wheel [m] st_max_deg = 26; % Maximum steering angle of the robot [deg] % Initial location of the robot x0 = 0; % Initial x coodinate of robot [m] y0 = 0; % Initial y coodinate of robot [m] th_deg0 = -26; % Initial orientation of the robot (theta [deg]) % Target and obstacle locations xt = [3.5,4]; % Target yt = [-0.25,0.25]; xo1 = [2.5,2.8]; % Obstacle 1 yo1 = [0,1]; xo2 = [1,1.3]; % Obstacle 2 yo2 = [-1,0]; % Parameters related to simulations t_max = 2; % Simulation time [s] n = 50; % Number of iterations dt = t_max/n; % Time step interval % Derivation of parameters related to performance of the robot m_max_rads = rpm2rads(m_max_rpm); % Motor max speed [m/s] w_max_rads = m_max_rads / gratio; % Wheel max speed [m/s] v_max = w_max_rads * R_w; % Max robot speed [m/s] st_max_rad = deg2rad(st_max_deg); % Maximum steering angle [rad] t = [0:dt:t_max]; % Time vector (n+1 components) v = v_max * ones(1, n+1); % Velocity vector (n+1 components) st_rad = st_max_rad * sin(5*t); % Steering angle vector (n+1 components) [rad] th_rad0 = deg2rad(th_deg0); % Initial orientation [rad] v0 = v(1); % Initial velocity [m/s] st_rad0 = st_rad(1); % Initial steering angle [rad] z0 = [x0, y0, th_rad0]; % Initial state vector u0 = [v0, st_rad0]; % Initial control vector fig1 = figure(1); % Figure set-up (fig1) plot_rectangular(xt, yt, 'r'); axis([-0.2 4.6 -2 2]); hold on; plot_rectangular(xo1, yo1, 'b'); plot_rectangular(xo2, yo2, 'b'); % Acquire the configuration of robot for plot % [xb, yb]: Vertices of rectangular robot base % [xfw, yfw]: Front wheel position vector % [xrlw, yrlw]: Rear left wheel position vector % [xrrw, yrrw]: Rear right wheel position vector [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z0, u0); % Plot robot and define plot id plotzb = plot(xb, yb); % Plot robot base plotzfw = plot(xfw, yfw, 'r'); % Plot front wheel plotzrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel plotzrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel % Draw fast and erase fast set(gca, 'drawmode','fast'); 12 set(plotzb, 'erasemode', 'xor'); set(plotzfw, 'erasemode', 'xor'); set(plotzrlw, 'erasemode', 'xor'); set(plotzrrw, 'erasemode', 'xor'); z1 = z0; % Set initial state to z1 for simulation % Biginning of simulation for i = 1:n+1 if v(i) (v_max * cos(2*st_rad(i))) v(i) = v_max * cos(2*st_rad(i)); end u = [v(i), st_rad(i)]; % Set control input zdot = robot_model(z1, u); % Derive zdot using robot model z2 = z1 + zdot * dt; % Update the state of robot z1 = z2; % Substitute z2 to z1 % Acquire the configuration of robot for plot [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z1, u); % Plot robot set(plotzb,'xdata',xb); set(plotzb,'ydata',yb); set(plotzfw,'xdata',xfw); set(plotzfw,'ydata',yfw); set(plotzrlw,'xdata',xrlw); set(plotzrlw,'ydata',yrlw); set(plotzrrw,'xdata',xrrw); set(plotzrrw,'ydata',yrrw); pause(0.2); % Pause by 0.2s for slower simulation end % Plot the resultant velocity and steering angle configurations fig2 = figure(2); % Figure set-up (fig2) subplot(2,1,1); % Upper half of fig1 plot(t, v); % Plot velocity-time curve xlabel('Time [s]'); ylabel('Velocity [m/s]'); subplot(2,1,2); % Lower half of fig1 std = rad2deg(st_rad); % Steering angle vector (n+1 components) [deg] plot(t,std); % Plot steering angle-time curve xlabel('Time [s]'); ylabel('Steering angle [deg]'); CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 17. 13 TUTORIAL QUESTIONS 1. Design a wheeled robot. The items to be designed include: Type of robot (should not be a tricycle) Dimensions of base Dimensions and location of wheels Physical limitations of the robot (Maximum steering angle, maximum motor speed, etc.) 2. Derive the kinematic model of the robot. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 18. MTRN9224 Robot Design Lecture 4: Robot Dynamics and Selection of Motors Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales SUMMARY 1 MOUNTING WHEELS TO DRIVE SHAFTS For most wheels, you can use a flange-style adapter to attach a wheel to a shaft. The flange portion of the adapter consists of a set of holes or lugs for bolting the wheel onto the adapter, and the hub portion of the adapter is used to secure the adapter to the shaft. 1.1 Set Screw Mounting Many different types of shaft adapters use set screws to secure the adapter to the shaft (Figure 4.1). On one side of the adapter’s hub, there is a threaded hole for the set screw. The proper method for using set screws with shafts is to machine a flat on the side of the shaft. Figure 4.1 Set screw mounting 1.2 Pinned Shaft Method Pinning the adapter to the shaft is another popular method for securing a wheel on a shaft (Figure 4.2). The pin is used to prevent the wheel from twisting on the shaft. The diameter of the pin should not be greater than one-third the diameter of the shaft, and the length of the pin should be approximately equal to the diameter to the hub that is mounted on the shaft. 2 Figure 4.2 Pinned shaft method 1.3 Quick-Release Pinning With the quick-release pinning approach, the pin is pressed only into the shaft. The hub has a slot cut into the end that will slip around the pin when the wheel adapter is attached to the shaft (Figure 4.3). Figure 4.3 Quick-release pinning 2 DYNAMICS 2.1 Pushing force Pushing force created by the motor(s) controls the robot movement. In the selection of a motor, we must make sure that the motor selected is able to (1) Provide a pushing force large enough to make the robot start moving (overcoming the sum of static friction force s F and other external forces), CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 19. 3 (2) Provide a pushing force to yield desired maximum velocity and acceleration (overcoming the sum of dynamic friction force d F and other external forces). Question 1 Which of the above friction forces is larger in general? Figure 4.4 illustrates the free body diagrams of a wheeled robot. x 0 M m m a F F Figure 4.4 Free body diagram of a wheeled robot. Question 2 If the motor torque is τ and attached to a gear, will the output torque increase? The torque m τ is generated by the motor to rotate each wheel. This creates the force a F to drive the robot: m w a R F ητ = (4.1) where η is the gear ratio. Hence, m a w F R ητ = (4.2) This means that the we can increase the driving force a F by choosing wheels of a small radius or the gear of a high gear ratio. 2.2 Equation of motion Let masses of the robot base and each wheel M and m respectively. The equation of motion for three-wheeled Ackerman steering robot (3 wheels, 1 motor) is given by ( ) 3 m a w M m x F F F R ητ + = − = − (4.3) where F is the sum of friction forces and the acceleration of the robot is, by definition, 4 w w R x ω η = (4.4) . We will consider a case where the wheel weight is small compared to the base mass ( M m ) for simplicity; i.e., the equation can be simplified to m a w Mx F F F R ητ = − = − (4.5) For three-wheeled differential steering robot (2 driving wheels, 1 support wheel, 2 motors), if the two motors rotate in the same direction, the equation of motion is given by 2 m w Mx F R ητ = − (4.6) Question 3 What is the control input for dynamical equations (4.5) and (4.6)? 2.3 Friction forces Typical friction forces include Friction with the ground Friction with the rotor shaft Friction of the differential gear Air friction The largest would be the ground friction. The friction force can can be static (static friction force s F ) and dynamic (dynamic friction force d F ). s F depends on the mass of the robot and takes a maximum value just before the robot starts moving. This force is called maximum static friction force. If the robot is located on the horizontal surface parallel to the ground, the maximum static friction is given by ( )max s F Mg µ = (4.7) where µ is a friction coefficient. In classical physics, the friction coefficient has a value that is never greater than 1 and is not a function of the contact surface area. This is generally true when considering rigid materials sliding on rigid surfaces, such as steel on steel. But when it comes to elastic materials such as rubber, the coefficient can vary greatly (0.5 – 3.0). For estimating purposes, you can use 1.5 for safety. The dynamic friction force is normally proportional to the velocity of the robot: d d F k x = (4.8) s F is often much larger than d F . However, both the forces must be considered in the design process as follows (the case of Ackerman steering robot): The driving torque is large enough to make the robot start moving: 0 m w Mx Mg R ητ µ = − (4.9) CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 20. 5 The robot can reach desired maximum speed max x and acceleration max x by solving the following dynamical equation: m d w Mx k x R ητ = − (4.10) 2.4 Other external forces Typical other external forces are Gravitational force (when the robot is on a slope) Contact force (when the robot requires contact with other objects) When the robot is on a slope of angle α , Equation (4.4) needs to be modified as sin m w Mx F mg R ητ α = − − (4.11) 3 MOTORS 3.1 Motor Types Various types of motors are available for wheeled robots. Typical motors for small-size robots can be classified as follows. 3.1.1 PMDC motors PMDC motors are fairly low in cost and relatively easy to control. These motors are found in many electrical devices – hobby models, medical instruments, cordless toothbrushes, high-powerd cordless drills, etc. They typicall use ferrite magnets and have running speeds from 5,000 to 20,000 rpm. Figure 4.5 shows several different PMDC motors. Figure 4.5 PMDC motors 3.1.2 Radio-controlled (R/C) car motors These ultra-high-velocity motors offer a lot of power in a small package. Many of these motors can draw over 30 amps in normal nonstalled operation. Since these motors typically run at speeds up to 20,000 to 40,000 rpm, fairly large gear reductions will be required. If you use an R/C car 6 motor in your robot, it is highly recommended that you use an R/C car electronic speed controller to control the motors. 3.1.3 Rare-earth and ferrite magnet motors High-performance motors use rare-earth magnets and have efficiencies up to 80 to 90 %. Typical ferrite magnet motors have efficiencies between 50 to 70%. Motors with rare-earth magnets generally run much cooler than ferrite motors do. The major drawback of using rare-earth magnet high-performance motors is that they are significantly more expensive than regular motors. 3.1.4 Brushless PMDC motors The brushes in an ordinary motor can be the source of several problems: They spark and cause radio interference, they are a source of friction, and they wear out. Brushless motors have sensors to detect the position of the roor relative to the windings, and this information is sent to a special motor controller that energizes the windings at the optimum moment. Like rare-earth magnet motors, brushless motors are more expensive than regular motors. 3.2 Motor Basics Knowing motor performance is important as electric motors directly affect the speed and driving capability of the robot. All PMDC motors have the following two characteristics: • The motor speed is proportional to the applied voltage, i.e., m V m k V ω = (4.12) • The motor’s output torque is proportional to the amount of current the motor is drawing from the batteries: m m k I τ τ = (4.13) 3.2.1 Motor speed The motor speed is only a function of the applied voltage, but, in reality, the motor slows down because of the internal voltage losses due to the current going through the motor: m in in m V V I R = − (4.14) Thus, the motor speed is given by ( ) m V in in m k V I R ω = − (4.15) 3.2.2 Motor torque When there is no load on the motor, you will notice that the motor is drawing a small amount of current from the batteries. This loss includes that required to overcome the internal “frictional and inertial” losses inside the motor, known as no-load current, 0 I . The torque is resultantly written as ( ) 0 m m in k I k I I τ τ τ = = − (4.16) By substituting Equation (4.16) to Equation (4.15), the motor speed is given by CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 21. 7 0 m m V in m k V R I kτ τ ω = − + (4.17) These mean that • m τ is determined by in I , • m ω is determined by in V and m τ . 3.2.3 Power and efficiency The other set of relationships that should be considered is power and motor efficiency. They are defined as Input power: in in in P V I = (4.18) Output power: ( )( ) 0 out in in in m P I I V I R = − − (4.19) Efficiency: ( )( ) 0 in in in m out in in in I I V I R P P V I − − = (4.20) 3.2.4 Motor performance chart Many motor manufacturers provide the motor constants V k , kτ , 0 I and m R and the motor performance chart derived from the constants. Figure 4.6 shows a typical motor performance chart for a Mabuchi RK-370SD motor. Figure 4.6 Mabuchi RK-370SD motor 3.3 Control by motor input voltage We will first write a dynamical movement of an Ackerman steering robot given by Equation (4.10) in state space form. As w m R x ω η = , the equation is rewritten as w d w m m m w MR k R R η ω τ ω η η = − (4.21) 8 or 2 2 d m m m w k M MR η ω τ ω = − (4.22) The control input of the robot at a glance in this dynamical equation is the torque m τ , but the actual control input is the input voltage to the motor, i.e., in V . The problem is that m τ is not controllable only by in I , not by in V (See Equation (4.16)). in V is used to control the motor speed m ω only, and m ω is given as a function of in V and m τ . In order to control the robot by in V , we will first rewrite equation (4.17) for m τ : 0 in m m m m V V k I R R k τ ω τ = − − (4.23) m τ has been given as a function of in V and m ω : The robot motion can be controlled with in V by the following process: 1. Set k = 0. 2. Given the initial ( ) in k V and ( ) m k ω , derive the initial ( ) m k τ using Equation (4.23). 3. Derive motor acceleration ( ) m k ω from ( ) m k τ and ( ) m k ω using Equation (4.22). 4. Compute m ω at k +1 using ( ) ( ) ( ) 1 m m m k k k t ω ω ω + = + + . 5. As ( ) 1 m k ω + is known, specify ( ) 1 in k V + and derive ( ) 1 m k τ + using Equation (4.23). 6. Set k = k + 1. 7. Go to 2 unless a terminal condition is satisfied. By applying the maximum input voltage ( )max in V to the process, we can estimate what the maximum velocity max x and acceleration max x will be. 3.4 Selection of a Motor To figure out what you need to look for in a motor, you need to define how you want your robot to perform. The procedure for selecting an appropriate motor may be as follows: 1. Decide the environment in which the robot is driven. Model possible external forces. 2. Decide the maximum speed of the robot you want to achieve. 3. Decide the maximum acceleration of the robot you want to achieve. 4. Design a robot. 5. Estimate all the parameters of your robot. 6. Construct a dynamic equation for your robot (ex., Equation (4.22)). 7. Find if the robot is able to start moving by using Equation (4.9). CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 22. 9 8. Use Equations (4.22) and (4.23) to find if the robot can achieve the desired maximum speed and acceleration. Table 4.1 shows some sample motors. Table 4.1 Sample motors Manufacturer Part No. Volts No-load (RPM) Torque (oz- in.) Stall current (amps) Tamiya 3633k36 7.2 196 173 5.4 Jameco 161381 12 200 50 1.4 Lynxmotion GHM-01 6 186 75 2.1 Servo Systems BC-101 12 180 144 1.4 Maxon 118742 9 218 227 7.3 1 oz-in. = 7.06e-3 Nm 4 SIMULATION PROGRAM 4.1 Program Files The simulation program uses the following files: simulate_linear.m dynamic_model.m compute_torque.m compute_acceleration.m 4.2 Running the Program To run the program, save all the M-files in the directory where the M-files for Lecture 3 were stored. Then, type “simulate_linear” to run the program. 4.3 Source Codes 4.3.1 simulate_linear.m % ----------------------------------------------------- % simulate_linear.m % Main M-script file that simulates the dynamic motion % of the robot % Programmed by Tomonari Furukawa for MTRN9224 % March 20, 2004 % 10 % ----------------------------------------------------- clear all; % Clear all variables close all; % Close all figures global M; global gratio; global I_0; global R_m; global k_v; global k_T; global V_max; global k_d; global R_w; % Physical parameters of the robot M = 5; % [kg] gratio = 20; % I_0 = 0.340; % [A] R_m = 0.821; % [Ohm] k_v = rpm2rads(2385); % [m/s/V] k_T = 0.567 * 7.06e-3; % [Nm/A] V_max = 7.2; % [V] k_d = 0; % [N.s/m] R_w = 0.05; % [m] % Parameters related to simulations t_max = 10; % Simulation time [s] n = 100; % Number of iterations dt = t_max / n; % Time step interval t = [0:dt:t_max]; % Time vector (n+1 components) V_in = V_max * ones(1, n+1); % Velocity vector (n+1 components) % Initial conditions V_in_1 = V_in(1); w_m_rpm_1 = 16000; w_m_ms(1) = rpm2rads(w_m_rpm_1); %w_w_ms(1) = w_m_ms_1 / gratio; for i = 1:n+1 torque(i) = compute_torque(V_in(i), w_m_ms(i)); w_m_acc(i) = compute_acceleration(torque(i), w_m_ms(i)); if i ~= n+1 w_m_ms(i+1) = w_m_ms(i) + dt * w_m_acc(i); end end v = R_w/gratio*w_m_ms; figure(1); subplot(2,2,1); plot(t,v,'.'); xlabel('Time [s]'); ylabel('Velocity [m/s]'); subplot(2,2,2); plot(t,w_m_ms*60/(2*pi),'.'); xlabel('Time [s]'); CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 23. 11 ylabel('Motor Velocity [rpm]'); subplot(2,2,3); plot(t,torque,'.'); xlabel('Time [s]'); ylabel('Torque [Nm]'); subplot(2,2,4); plot(t,w_m_acc,'.'); xlabel('Time [s]'); ylabel('Acceleration [rad/s^2]'); 4.3.2 dynamic_model.m % ----------------------------------------------------- % dynamic_model.m % This M-function defines the dynamic model of the robot % Programmed by Tomonari Furukawa for MTRN9224 % March 20, 2004 % % ----------------------------------------------------- function zdot = dynamic_model(z, u) global M; global gratio; global rw; w = z(2); torque = u(1); zdot(1) = w; zdot(2) = (gratio*torque/rw)/M - kv*w/gratio*rw; 4.3.3 compute_torque.m % ----------------------------------------------------- % compute_torque.m % This M-function returns a torque value, given % motor voltage and speed % Programmed by Tomonari Furukawa for MTRN9224 % March 20, 2004 % % ----------------------------------------------------- function torque = compute_torque(V_in, w_m) global I_0; global R_m; global k_v; global k_T; torque = k_T*(V_in/R_m - w_m/(R_m*k_v) - I_0); 4.3.4 compute_acceleration.m % ----------------------------------------------------- % compute_acceleration.m % This M-function returns an accelation value, given % motortorque and speed % Programmed by Tomonari Furukawa for MTRN9224 % March 20, 2004 12 % % ----------------------------------------------------- function acc_m = compute_acceleration(T_m, w_m) global M; global gratio; global k_d; global R_w; acc_m = (gratio/R_w)^2/M*T_m - k_d/M*w_m; CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 24. 13 TUTORIAL QUESTIONS 1. Decide the environment in which the robot is driven. Model possible external forces. 2. Estimate all the parameters of your robot. 3. Construct a dynamic equation for your robot (ex., Equation (4.22)). 4. Find if the robot is able to start moving by using Equation (4.9). 5. Run the programs in #306. Find the maximum velocity and maximum acceleration of your robot by running the programs. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 25. MTRN9224 Robot Design Lecture 5: Motor Control Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales SUMMARY In Lecture 4, dynamical behavior of a robot has been related to input voltages to motors. Important roles of the motor are to (1) change the direction and (2) control the speed of the robot. Direction – H-bridge is used to make the robot move forward and backward and stop. Speed – One of two methods, i.e., variable resistors and Pulse Width Modulation, is used to change speed of the robot. 1 MOTOR-DIRECTION CONTROL 1.1 H-bridge One of the features about PMDC motors is that their rotational direction can be changed by simply reversing the direction of the current flowing through the motors. To control the motor direction, you can use a simple circuit called an H-bridge. Figure 6.1 shows a simple schematic of an H-bridge that is commonly used to control the current direction going through a motor. From the figure, you can see why it is called an H-bridge. Figure 6.1 H-bridge 2 Table 6.1 shows what the motor will do based on which switch is closed. There are a total of 16 different switch combinations that result in only five different motor actions: Break is when the motor rapidly slows down and resist turning. This does not mean that the motor will be mechanically prevented from turning, as when you apply the brakes on your car. One pair of AB or CD must be on while the other pair is off (Motor is steady at +V or 0). Free wheel is when the motor freely slows down and does not resist any applied torque on the motor shaft. No or only one switch is on (No current goes through the motor). Forward results in the motor shaft rotating clockwise. Only AD are on (+V goes to 0). Reverse results in the motor shaft rotating counter-clockwise. Only BC are on (–V goes to 0). Short circuit switch positions should be avoided at all costs, since they will directly short the battery to the ground (+V or –V are directly connected to 0). Question 1 Complete Table 6.1. Table 6.1 Logic table for H-bridge (0-Open, 1-Closed) A B C D Motor Result 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1.2 Relay-controlled H-bridge Relay-controlled motors are relatively easy and inexpensive to build. Instead of using switches as shown in Figure 6.1, relays are used to control the motor. With a relay, you can use a low-voltage control circuit to control a high-voltage motor. A low voltage is used to turn a relay on or off, and then the relay can be used to pass a higher voltage and/or higher current from the batteries to the motor. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 26. 3 Figure 6.2 shows how to wire a set of four relays and two switches to make a motor control circuit. Figure 6.2 Relay-controlled H-bridge Note that the H-bridge is composed inside the circuit. With this circuit, you can control the forward and reverse directions of the motor, and you can apply motor braking. Question 2 Complete Table 6.2. Table 6.2 Logic table for Relay-controlled H-bridge (0-Open, 1-Closed) S1 S2 A B C D Motor Result NO NO NO NC NC NO NC NC 1.3 Relay-controlled H-bridge with transistors For autonomous and most remote-control robots, the relays are controlled by other electrical circuits. This is usually done by using a transistor instead of a mechanical switch to direct current to the relay coils. Typical miniature relay coils require from 20 to 100 mA of current to close. Since most microcontrollers cannot apply this much current, a transistor must be used to control the relay. Figure 6.3 shows two simple schematics for using a bipolar-style transistor to control a relay. The resistor that is connected to the transistor’s base is there to limit the input current to the transistor. 4 Figure 6.3 Bipolar transistor-controlled relay With transistors, you can use a simple microprocessor to control a relay-controlled H-bridge. Figure 6.4 shows a schematic of this type of a control system. Figure 6.4 Relay-controlled H-bridge with transistors Necessity for diodes A flyback diode is placed across the coil terminals. With all inductors (relay coils and motor coils are inductors), a voltage is induced across the inductor that is proportional to the rate of change in the current. When a switch is used to turn off a relay coil, the current does not instantaneously go to zero, but it does go to zero in an extremely short period of time. Since the high rate of change is decreasing, the inductor will induce a voltage that can be several hundred volts, flowing in the opposite direction. For a transistor, the voltage can be high enough to damage the transistor. The flyback diode provides a current path away from the transistor during these shutoff periods. Question 3 Complete Table 6.3. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 27. 5 Table 6.3 Logic table for Relay-controlled H-bridge with transistors (0-Open(Off), 1-Closed(On)) X Y A B C D Motor Result 0 0 0 1 1 0 1 1 1.4 Solid-state H-bridge An H-bridge using relays is more of a mechanical switching circuit, since relays use mechanical contacts to switch the current paths. Another class of H-bridge is called the solid-state H-bridge. A solid-state H-bridge uses transistors instead of relays. Whether you should use a solid-state H-bridge or relay H-bridge depends on the application. For constant-speed applications, both systems will work fine. When you need to switch high currents or use high-frequency switching, transistors are preferred over relays. Figure 6.5 shows two different types of transistor-based H-bridges: one uses MOSFET transistors, and the other uses bipolar transistors. Figure 6.5 Bipolar and MOSFET transistor-based H-bridge Notice that they look similar to a relay H-bridge (see Figure 6.2). The biggest differences between these two types of H-bridges are their voltage and current-handling capabilities. Generally speaking, MOSFET transistors have higher current-handling capability than bipolar transistors do. Bipolar transistors can operate at lower voltages than MOSFET transistors. A bipolar transistor’s current-handling capability is proportional to the current going into the transistor’s base. A MOSFET transistor’s current-handling capability is proportional to the voltage going into the transistor’s gate. In other words, bipolar transistors are controlled by current, and MOSFET transistors are controlled by voltage. A bipolar transistor may require more current than what can be supplied directly from a microcontroller. This will require a second transistor to be placed between the microcontroller and the H-bridge transistor. Most MOSFET transistors require the gate voltage to be 10 volts higher than the source voltage to be fully turned on. Thus, MOSFET H-bridges should be used with motor controllers that have a 12-volt power supply to control the transistors, although the motors can be 6 operated at a lower voltage. There are some H-bridge motor controllers that operate at less than 12 volts, but they have special voltage boosting circuits to control the MOSFETs. MOSFET transistors are very sensitive to static electricity and short circuits. They short out very easily when misused. On the other hand, bipolar transistors are very tolerant of mistakes. Whether you should use a bipolar or MOSFET transistor H-bridge depends on the current and voltage requirements for the motor controller. Bipolar H-bridges are well-suited for low-voltage and low- current motors. MOSFET transistor H-bridges should be used for all other motor controllers that need more than 5A. 1.5 Integrated-circuit There are several types of integrated circuit-based motor controllers that work well for low- and medium-powered robots. These circuits include the L293D and the L298N chips from ST Microeletcronics. Both of these integrated circuits have two complete internal H-bridge circuits that can be used to independently control two separate motors. The L293D can apply 0.6A continuously and 1.2A momentarily. The L298N can supply 2A continuously and 4A momentarily. Figure 6.6 shows a schematic using the L293D. The 293D has the flyback diodes inside the chip, which makes it an easy circuit to build for a low-powered motor controller. The logic table in the figure shows how the motor direction is controlled based on the input signals from a microcontroller. Figure 6.6 Simple dual-motor controller using L293D and SN754410 dual H-bridge CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 28. 7 Shown in Figure 6.7 is a low-cost L298 board from HVW Technologies (www.hvwtech.com), which can be used to control two motors at the same time. Figure 6.7 L298 board 2 VARIABLE-SPEED CONTROL For some robots, speed control is an important feature. Some robot builders want to control the speed of their robot, so that, for example, the robot moves at a slower speed when searching for its opponent and then attacks at full speed. Lecture 4 showed how motor speed is a function of the applied voltage. So, to make a variable- speed motor, you will need a way to vary the applied voltage to the motor. There are two methods that you can use to accomplish this: a variable resistor or pulse with modulation (PWM). 2.1 Variable Resistors Controlling the motor speed using a variable resistor is an old-fashioned but very effective method. A variable resistor is placed in series between the battery and the motor. All of the motor power will go through this resistor. An arm on the resistor moves to change the resistance in the circuit, which results in a voltage drop across the variable resistor and reduces the applied voltage to the motor. Figure 6.8 shows a variable-resistor motor speed controller used in older R/C cars and boats. This particular circuit can also reverse the direction of the motor. The resistor arm is moved using a traditional R/C servo motor. Figure 6.8 Variable-resistor motor speed controller 8 Question 4 What is the disadvantage of the variable-resistor motor speed controller? 2.2 Pulse Width Modulation Since electric motors are inductors, cutting off the power does not immediately stop the current from flowing through the motor. The induced voltage from the collapsing magnetic field actually helps keep the motor running for a moment. When the voltage is applied as a square wave, the motor will time-average the applied voltage. To the motor, the time-averaged voltage has the same effect as operating at a constant voltage that is equal to the time-averaged voltage. Figure 6.9 shows a Pulse Width Modulation (PWM) signal and the average voltage. Figure 6.9 Typical PWM control signal The duty cycle of a PWM signal is a percent of the amount of time the applied voltage is on during the frequency cycle: Duty Cycle ON ON OFF t t t = + (6.1) When the frequency of the square wave is high enough (at least 1,000Hz for most motors), the duty-cycle percentage will be equal to the average voltage percentage the motor will see: ON average in ON OFF t V V t t = + (6.2) The duty cycle will range from 0 to 100%. The minimum PWM frequency to ‘smooth’ the average voltage depends on the motor. Some motors require a PWM frequency higher than 20,000Hz; some require a PWM frequency as low as 100Hz. Frequencies about 1,000Hz will work for most motors used in small-size robots. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 29. 9 In order to change the speed of a motor, one of the active transistors in the H-bridge must be oscillated at the PWM frequency at some duty cycle. It does not matter which transistor is oscillated, as long as it is one of the transistors that is used to conduct current through the motor. Figure 6.6, shown earlier, illustrates an integrated circuit version of an H-bridge. To make one of these circuits a variable-speed motor controller, apply the PWM signal to the enable signal line. The two input control lines set the direction of the motor, and the enable line turns on the power to the motor. One of the drawbacks to using a relay-based H-bridge is that rapidly turning on and off the relay will shorten its life. Also, this type of H-bridge cannot switch fast enough to take full advantage of the voltage averaging being equal to the duty cycle. If a transistor is placed between the H-bridge and the batteries, the relays can be used to set the motor direction, and the transistor can be used to turn on and off the power to the motor. This is analogous to how the integrated circuit versions of the H-bridges work. Figure 6.10 shows a schematic of this type of a circuit using an N-channel MOSFET. Figure 6.10 A variable-speed motor controller using relays for the H-bridge and a MOSFET for the variable-speed control Since the gate voltage on the MOSFET transistor must be 10 volts higher than the source voltage to be completely turned on, a bipolar NPN transistor can be used to supply the correct voltage to the MOSFET. With this circuit, you can build a simple high-current, variable-speed motor controller. 2.3 Electronic Speed Controller Probably the best choice for a motor speed controller is a commercially made Electronic Speed Controller (ESC). Commercial ESCs fall into: H-bridge-only motor controllers, R/C ESCs, 10 Complete variable-speed motor controllers. 2.3.1 H-bridge-only motor controller Several companies sell complete H-bridge circuits that can be used to directly control the speed and direction of your motors. These controller boards use relays, transistors, MOSFETs, or single- chip motor controllers. The motor and power supply directly connect to the motor controller, and a microprocessor is used to set the motor speed and direction. The control signals set the motor direction control pines high or low and send a PWM signal. Table 6.4 lists several popular commercially available H-bridges. Figure 6.11 shows L298 H- bridge from HVW Technologies (www.hvwtech.com), the Maxi Dual H-bridge from Mondo- tronics (www.robotstore.com), and ICON H-bridge from Solutions Cubed (www.solutions- cubed.com). Table 6.4 Some commercial H-bridges Name Voltage Current (A) Circuits Size (in.) Type Manufacturer ICON H- Bridge 12-46 12 1 1.9 x 2.5 MOSFET Solution Cubed L298 6-46 2 2 1.5 x 2.7 L298 HVW Technologies Maxi Dual H- Bridge 12-40 10 2 2.1 x 2.3 MOSFET Mondo-tronics Mini Dual H- Bridge 5 0.3 2 1.0 x 1.1 Transistor Mondo-tronics Magnavision 12-55 3 2 2.0 x 3.5 LMD18200 Acroname Figure 6.11 Commercial H-bridge motor controllers from Mondo-tronics, Solutions Cubed and HVW Technologies 2.3.2 R/C ESCs The ESCs that are used for R/C off-road and racing cars are excellent off-the-shelf technology that can be used in sumo robots. Instead of needing to use two to four wires to set the motor direction and one wire for motor speed, an R/C ESC needs only one wire to control both direction and speed. This simplifies some of the wiring and control requirements in the robot. Although there are some complications and special considerations when using R/C car ESCs, they still make great choices for small-size robots. They are small, lightweight, reliable, easy to CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 30. 11 control with a microcontroller, and reasonably priced. Their main shortcoming is their documentation. 2.3.3 Complete variable-speed motor controllers Because of the popularity of robot combat events such as BattleBots and Robot Wars, several companies have developed specialized motor controllers that use the standard R/C 1 to 2 ms pulse width control method. Suppliers include Sozbots (www.sozbots.com), Vantec’s (www.vantec.com) and Innovation First (www.innovationfirst.com). 12 TUTORIAL QUESTIONS 1. Complete the tutorial question 5 in #306. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 31. MTRN9224 Robot Design Lecture 7: Sensor Technologies Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales 1 COORDINATE SYSTEMS AND SENSORS 1.1 Coordinate Systems Figure 7.1 illustrates coordinate systems concerned with the navigation of the wheeled robot. As shown in the figure, they include: Global coordinate system { } G Robot coordinate system { } R Sensor coordinate system { } S Object coordinate system { } O x { } O 0 [ ] , , r r r x y θ y { } R { } G { } S [ ] , , s s s x y θ [ ] , , o o o x y θ Figure 7.1 Global and local coordinate systems of wheeled robot 1.2 Types of Sensor with respect to Coordinate Systems Sensors can be classified in terms of which coordinate system they are attached to: • Global sensor – Attached to the global coordinate system, • Local (passive) sensor – Attached to the robot coordinate system, • Local (active) sensor – Attached to a sensor coordinate system attached to the robot coordinate system. 2 Figure 7.2 shows these sensors. { } A S Passive sensor { } { } P S R = x { } O 0 y { } { } G S G = Global sensor Active sensor Figure 7.2 Sensors 2 KINEMATIC TRANSFORMATION 2.1 Problem Formulation For simplicity, we will deal with a case where the sensor is mounted at the robot coordinate { } { } S R = , as shown in Figure 7.3. x { } O 0 [ ] , , r r r x y θ y { } { } S R = { } G [ ] , , o o o x y θ Figure 7.3 Coordinate systems of concern 2.2 Representation of Position of an Object First, we do not take the orientation of an object into account. Suppose that the position of an object in two-dimensional space is measured by the sensor mounted on the wheeled robot. The position is given with respect to the robot coordinate system { } [ ] , R o o x y . If the robot coordinate system { } R has the same orientation as the global coordinate system { } G but dislocated by [ ] , r r x y , the representation of the point with respect to the global coordinate system { } G can be given by CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 32. 3 { } { } { } { } G R G o o r o o r R x x x y y x = + , (7.1) If the robot coordinate system { } R has the same position as the global coordinate system { } G but located by an angle r θ , the representation of the point with respect to the global coordinate system { } G can be given by { } { } { } { } cos sin sin cos G R G o o r r o o r r R x x y y θ θ θ θ − = , (7.2) The position of the point with respect to the global coordinate { } G through the combination of linear and rotational transformations can be thus given by { } { } { } { } cos sin sin cos 1 0 0 1 1 G G R o r r r o o r r r o R x x x y y y θ θ θ θ − = , (7.3a) or ( ) { } { } { } { } , , G G R o R r r r o x y θ = z T z . (7.3b) 2.3 Representation of Position and Orientation of an Object If the orientation of the object can also be measured by the sensor, the rotational transformation from the robot coordinate system { } R to the global coordinate system { } G can be represented as { } { } { } R G R o r o θ θ θ = + . (7.4) The position and orientation of the object can be thus given by { } { } { } { } cos sin 0 sin cos 0 0 0 1 1 0 0 0 1 1 G G R o r r r o o r r r o o r o R x x x y y y θ θ θ θ θ θ θ − = , (7.5a) or ( ) { } { } { } { } , , G G R o R r r r o x y θ = z T z . (7.5b) 3 SENSORS FOR POSITION/ORIENTATION CONTROL 3.1 Global Sensors Global sensors are used to measure the position/orientation of a robot in the global coordinate system { } G o z . Typical sensors to measure the position and orientation include: { } G r x , { } G r y : GPS 4 { } G r θ : Compass Both: Global vision Figure 7.4: GPS (Garmin GPSMAP 60C) Figure 7.5: Electronic digital compass (Co-Pilot V1000) Question 1 In what situation can the global vision be used to measure the position/orientation of the robot? 3.2 Relative Sensors Relative sensors measures the position/orientation of the object in the sensor coordinate system { } S o z and are used for one of the following three objectives: 1) To detect obstacles (how far they are from the robot). 2) To estimate the position/orientation of an obstacle in the global coordinate system (Mapping). In 2), the position/orientation of the obstacle in the global coordinate system can be obtained by ( ) ( ) { } { } { } { } { } { } , , , , G G R S o R GR r r r S RS s s s o x y x y θ θ = z T T z . (7.6) Example Consider that the position of an obstacle in the sensor coordinate system is obtained. The position of the obstacle in the global coordinate system can be calculated as { } { } { } { } { } { } cos sin cos sin sin cos sin cos 1 0 0 1 0 0 1 1 G G R S o r r r s s s o o r r r s s s o R S x x x x y y y y θ θ θ θ θ θ θ θ − − = . (7.7) 3.2.1 Range (Distance) and bearing (angle) sensors This class of sensors can detect an object and measure its range and bearing from the sensor coordinate system, { } { } , S S o o d θ when the object is within the sensor range { } [ ] min max , S o d d d ∈ , { } [ ] max max , S o θ θ θ ∈ − . They can be used (at least) to navigate the robot such that the robot does not hit obstacles. Such sensors include: CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 33. 5 • Laser range finder: min d = 1 m, max d = 10,000 m, max θ = 90° (Figure 7.6). • Stereovision: min d = ? m, max d = ? m, max θ 90° (Depends on the camera). x { } O 0 [ ] , , r r r x y θ y { } G [ ] , , s s s x y θ [ ] , o o x y { } R { } S R θ L θ R p L p Figure 7.6: Laser range finder (Sick LMS-200) Figure 7.7: Stereovision Question 2 What would be differences between the laser range finder and the stereovision? In addition to the detection of obstacles, this class of sensors can be used to localize the robot in the global coordinate system. By measuring { } { } , S S o o d θ , the position/orientation of the robot in the sensor coordinate system can be derived as follows. { } { } { } cos S S S o o o x d θ = , (7.8a) { } { } { } sin S S S o o o y d θ = , (7.8b) Transformations of the position/orientation of the robot in the sensor coordinate system into that in the robot coordinate system and the global coordinate system can be conducted using Equation (7.5). Figure 7.7 shows how to derive { } { } , S S o o x y from { } { } , S S o o d θ using a stereovision. Initially known is the distance between the two cameras, L R p p p = + . As the vision provides bearing information ( L θ from the left camera and R θ from the right camera) on the object, the given values can be related to { } S o x as { } { } { } ( ) tan tan tan tan L R S S o L o R S o L R p p p x x x θ θ θ θ = + = + = + (7.9) { } S o x is therefore, { } tan tan S o L R p x θ θ = + (7.10) 6 { } S o y can also be calculated as follows: { } ( ) ( ) 2 2 tan tan 2 tan tan S o L R L R L L R p y p p p p θ θ θ θ = − − = − = + (7.11) 3.2.2 Range (Distance) sensors in one direction This class of sensors can detect an object and measure its distance from the origin of the sensor coordinate system, { } S o d , only when the sensor is directed to the object. The sensors include: • Ultrasonic : min d = 0.15 m, max d = 3 m (Figure 7.8) • Infrared sensor : min d = 0.1 m, max d = 1.5 m (Figure 7.9) Figure 7.8: Ultrasonic sensor (Devantech SRF04 Ranger) Figure 7.9: Infrared range sensor (Sharp GP2Y0D02YK) Question 3 In which situation will ultrasonic and infrared sensors each fail in measuring distance? Many of these sensors can also be used as proximity sensors. Question 4 What is the difference between the distance sensor and proximity sensor? If the sensor is used to search in all directions, there are fundamentally two designs: • Attach many sensors around the robot (Figure 7.10). • Connect the sensor by a revolute joint (Figure 7.11). CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 34. 7 Provided that the position and orientation of the sensor are known with respect to the robot coordinate system, the position of the object in the robot coordinate system can be calculated. These set-ups can also be used for localization, but, because of inaccuracy, they are used only for detecting obstacles in general. x { } O 0 [ ] , , r r r x y θ y { } G [ ] , , s s s x y θ [ ] , o o x y { } R { } S x { } O 0 [ ] , , r r r x y θ y { } G [ ] , , s s s x y θ [ ] , o o x y { } R { } S Figure 7.10: Robot with many distance sensors Figure 7.11: Robot with an active distance sensor 3.2.3 Bearing sensors This class of sensors can detect an object and measure its bearing in the sensor coordinate system, { } S o θ , when the object is within the sensor range { } [ ] max max , S o θ θ θ ∈ − . Such sensors include: • Single camera: max θ 90° (Depends on the camera). • Omni-directional camera: All directions. x { } O 0 [ ] , , r r r x y θ y { } G [ ] , , s s s x y θ [ ] , o o x y { } R { } S x { } O 0 [ ] , , r r r x y θ y { } G [ ] , , s s s x y θ [ ] , o o x y { } R { } S Figure 7.4: Single camera Figure 7.5: Omni-directional camera 3.3 Joint Use of Global and Relative Sensors In addition to the two benefits of using relative sensors described in Section 3.2, the use of global sensor(s) together with relative sensor(s) brings the following advantage: 3) To estimate the position/orientation of the robot in the global coordinate system if the obstacle position/orientation is known in the global coordinate system (Localization). 8 In 3), the position/orientation of the robot in the global coordinate system can be obtained by equating the following two equations and solve for { } [ ] , , G r r r x y θ . ( ) { } { } { } { } , , R R S o S RS s s s o x y θ = z T z (7.12a) ( ) 1 { } { } { } { } , , R G G o R GR r r r o x y θ − = z T z (7.12b) Example Consider a robot equipped with a compass (able to measure { } G r θ ) and a laser range finder (able to measure { } { } , S S o o x y ). If the position of an obstacle in the sensor coordinate system is obtained by the laser range finder, the position of the robot in the global coordinate system can be calculated by equating the following two equations: { } { } { } { } { } { } { } { } { } { } { } { } { } { } cos sin sin cos 1 0 0 1 1 cos sin sin cos 1 R R S o s s s o o s s s o S S R S R R o s o s s S R S R R o s o s s x x x y y y x y x x y y θ θ θ θ θ θ θ θ − = − + = + + . (7.13a) { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } 1 { } { } { } { } { } cos sin sin cos 1 0 0 1 1 cos sin sin cos sin cos sin cos 0 0 1 1 cos R G G o r r r o o r r r o R R G G G G G G G r r r r r r o G G G G G G r r r r r r o G G G G o r x x x y y y x x x x y y x θ θ θ θ θ θ θ θ θ θ θ θ θ − − = − − − = − − − = { } { } { } { } { } { } { } { } { } { } { } { } { } sin sin cos sin cos sin cos 1 G G G G G o r r r r r G G G G G G G G o r o r r r r r y x x x y x y θ θ θ θ θ θ θ − − − + + − . (7.13b) Known in these equations are • { } G r θ : Measurement from the compass • { } { } , S S o o x y : Measurement from the laser • { } { } , G G o o x y : Global map • { } { } { } , , R R R s s s x y θ : Measurable from the location of the laser The position/orientation of the robot can be derived by going through the following steps: CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 35. 9 1) In Equation (7.13b), substitute { } G r θ and { } { } , G G o o x y , and represent { } { } , G G r r x y each as a function of { } { } , R R o o x y . 2) Substitute { } { } { } , , R R R s s s x y θ into Equation (7.13a) and derive { } { } , R R o o x y . 3) Substitute { } { } , R R o o x y into the functions derived in 1) and determine { } { } , G G r r x y . 3.4 Odometry Sensors and their Joint Use with Other Relative Sensors Typical odometry sensors used for wheeled robots include angular encoders. They are mounted on the wheel axel and/or the steering to ultimately derive the position and orientation of the robot { } { } { } , , G G G r r r x y θ by path integration or dead reckoning. For instance, in the Ackerman steering, an encoder is attached to the rear shaft and the steering to measure the wheel speed w ω and the steering angle γ , respectively. The measurements then derive the rate of change of the position and orientation of the robot: { } { } { } cos sin tan G w r w r G w r w r G w w r R x R y R L ω θ η ω θ η ω θ γ η = = = , (7.14) and the integration of the state space equations gives the position and orientation of the robot { } { } { } , , G G G r r r x y θ . Information that can be derived by the odometry sensors is the same as that by the global sensors, though dead-reckon sensors are not as accurate as global sensors. The dead-reckon sensors can be useful in an environment where global sensors are not available. 3.5 Velocity Sensors When the dynamics of the robot are concerned, linear and angular velocities also become state variables. Velocity sensors are used to measure the rate of change of the robot’s position and orientation in the robot coordinate system, and the following are used as velocity sensors: { } R r x , { } R r y : Encoders (as described in the previous section), Accelerometers { } R r θ : Encoders (as desceibed in the previous section), Gyroscope (accelerometer) Both: Inertia Measurement Unit – Collection of accelerometers (and encoders). If the global coordinate system is concerned, measurements can be transformed to information in the global coordinate system as follows: ( ) { } { } { } { } , , G G R r R GR r r r r x y θ = z T z (7.12a) 10 4 OTHER SENSORS The previous chapters have dealt with sensors measuring position/orientation to control the position/orientation of the robot. There are sensors that extract other types of information. We shall not discuss such sensors in detail, but they include: • Force sensors – measure external forces x F and/or y F . • Torque sensors – measure external torques Tθ . • Optical sensors – measure the lightness. • Magnet sensors – measure the magnetic field. • Temperature sensors – measure the temperature. • Humidity sensors – measure the humidity. • Gas sensors – detect hydrogen (H2), hydrocarbons (CxHy), nitrogen oxide (NOx), carbon monoxide (CO), oxygen (O2) and carbon dioxide (CO2) [http://www.grc.nasa.gov/WWW/chemsensors/]. • Chemical sensors – measure various chemical components. • Smell sensors 5 PROGRAMS 5.1 simulate_sensor.m % ----------------------------------------------------- % simulate_sensor.m % Main M-script file that simulates the robot movement % Programmed by Tomonari Furukawa for MTRN9224 % May 11, 2004 % % ----------------------------------------------------- clear all; close all; global L; global d; global R_w; % Define obstacles obstacles = 2; obstacle(1).x = 3; obstacle(1).y = 0; %obstacle(1).th = atan(obstacle(1).y/obstacle(1).x); obstacle(2).x = 2.5; obstacle(2).y = 1; %obstacle(2).th = atan(obstacle(2).y/obstacle(2).x); fig1 = figure(1); % Figure set-up (fig1) for i = 1:obstacles plotobstacle(i) = plot(obstacle(i).x, obstacle(i).y, 'gx'); hold on; end axis([-0.2 4.6 -2 2]); CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 36. 11 % Physical parameters of the robot L = 0.25; % Length between the front wheel axis and rear wheel axis [m] d = 0.18; % Distance between the rear wheels [m] R_w = 0.05; % Radius of wheel [m] % Physical parameters of the sensor d_max = 1; % Maximum distance that the sensor can detect an obstacle [m] th_deg_max = 70; % Maximum angle that the sensor can detect an obstacle [deg] th_rad_max = deg2rad(th_deg_max); partitions = 10; dth = 2 * th_rad_max / partitions; th_inc = [-th_rad_max:dth:th_rad_max]; % Sensor position/orientation in robot coordinate system R_x_s = L; R_y_s = 0; R_th_s = 0; R_z_s = [R_x_s, R_y_s, R_th_s]; % Initial location of the robot x0 = 2; % Initial x coodinate of robot [m] y0 = 0; % Initial y coodinate of robot [m] th_deg0 = 30; % Initial orientation of the robot (theta [deg]) th_rad0 = deg2rad(th_deg0); % Initial position/orientation of the robot in global coordinate system G_x_r = x0; G_y_r = y0; G_th_r = th_rad0; G_z_r = [G_x_r, G_y_r, G_th_r]; z0 = [G_x_r, G_y_r, G_th_r]; % Initial state vector u0 = [1, 0]; % Initial control vector % Acquire the configuration of robot for plot % [xb, yb]: Vertices of rectangular robot base % [xfw, yfw]: Front wheel position vector % [xrlw, yrlw]: Rear left wheel position vector % [xrrw, yrrw]: Rear right wheel position vector [xb, yb, xfw, yfw, xrlw, yrlw, xrrw, yrrw] = model_config(z0, u0); % Plot robot and define plot id plotzb = plot(xb, yb); % Plot robot base plotzfw = plot(xfw, yfw, 'r'); % Plot front wheel plotzrlw = plot(xrlw, yrlw, 'r'); % Plot rear left wheel plotzrrw = plot(xrrw, yrrw, 'r'); % Plot rear right wheel % Measure the configuration of the sensor for plot G_z_s = transform_po(R_z_s, G_z_r); G_x_s = G_z_s(1); G_y_s = G_z_s(2); G_th_s = G_z_s(3); for i = 1:obstacles d_x = obstacle(i).x - G_x_s; d_y = obstacle(i).y - G_y_s; d(i) = (d_x^2 + d_y^2)^0.5; th(i) = atan(d_y/d_x) - G_th_s; end x_range = [G_x_s, G_x_s + d_max*cos(th_inc+G_th_s), G_x_s]; y_range = [G_y_s, G_y_s + d_max*sin(th_inc+G_th_s), G_y_s]; plotrange = plot(x_range, y_range, 'r'); 12 % Detect for i = 1:obstacles if d(i) d_max obstacle(i).d_observableFlag = 1; else obstacle(i).d_observableFlag = 0; end if (th(i) -th_rad_max) (th(i) th_rad_max) obstacle(i).th_observableFlag = 1; else obstacle(i).th_observableFlag = 0; end if (obstacle(i).d_observableFlag == 1) (obstacle(i).th_observableFlag == 1) obstacle(i).observableFlag = 1; set(plotobstacle(i), 'color', 'r'); S_x_o = d(i) * cos(th(i)); S_y_o = d(i) * sin(th(i)); S_z_o = [S_x_o, S_y_o]; R_z_o = transform_p(S_z_o, R_z_s); G_z_o = transform_p(R_z_o, G_z_r) else obstacle(i).observableFlag = 0; set(plotobstacle(i), 'color', 'g'); end end % Start simulation % Draw fast and erase fast set(gca, 'drawmode','fast'); set(plotzb, 'erasemode', 'xor'); set(plotzfw, 'erasemode', 'xor'); set(plotzrlw, 'erasemode', 'xor'); set(plotzrrw, 'erasemode', 'xor'); set(plotrange, 'erasemode', 'xor'); for i = 1:obstacles set(plotobstacle(i), 'erasemode', 'xor'); end % Plot set(plotzb,'xdata',xb); set(plotzb,'ydata',yb); set(plotzfw,'xdata',xfw); set(plotzfw,'ydata',yfw); set(plotzrlw,'xdata',xrlw); set(plotzrlw,'ydata',yrlw); set(plotzrrw,'xdata',xrrw); set(plotzrrw,'ydata',yrrw); set(plotrange,'xdata',x_range); set(plotrange,'ydata',y_range); 5.2 transform_p.m % ----------------------------------------------------- % transform_p.m % This M-function transforms position in % one coordinate system to another % Programmed by Tomonari Furukawa for MTRN9224 % May 11, 2004 CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 37. 13 % % ----------------------------------------------------- function y = transform_p(x, a) x_t = a(1); y_t = a(2); th_t = a(3); z_in = [x(1); x(2); 1]; T = [cos(th_t), -sin(th_t), x_t; sin(th_t), cos(th_t), y_t; 0, 0, 1]; z_out = T * z_in; y(1) = z_out(1); y(2) = z_out(2); 5.3 transform_po.m % ----------------------------------------------------- % transform_po.m % This M-function transforms position/orientation in % one coordinate system to another % Programmed by Tomonari Furukawa for MTRN9224 % May 11, 2004 % % ----------------------------------------------------- function y = transform_po(x, a) x_t = a(1); y_t = a(2); th_t = a(3); z_in = [x(1); x(2); x(3); 1]; T = [cos(th_t), -sin(th_t), 0, x_t; sin(th_t), cos(th_t), 0, y_t; 0, 0, 1, th_t; 0, 0, 0, 1]; z_out = T * z_in; y(1) = z_out(1); y(2) = z_out(2); y(3) = z_out(3); 14 TUTORIAL QUESTIONS 1. Design a wheeled robot with sensor feedback control. 2. Implement sensors into your program. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 38. MTRN9224 Robot Design Lecture 8: Multi-Body Systems Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales 1 WHEELED ROBOTS Typical multi-body wheeled robots include Car-like robot with trailers Snake robot These robots are effectively used when they Are driven on a narrow road or path Require carrying bulky items Need to be long in dimension 1.1 Transformation Kinematics of multi-body wheeled robots can be easily understood by the transformation of coordinate systems learned in Lecture 7. As an example, we will consider a robot constituted by a car (body 0) and n trailers (bodies 1,…,n ). The midpoint between the rear wheels is taken as the reference for each body; its coordinates are [ ] , , i i i x y θ , { } 0,..., i n ∀ ∈ , as shown in Figure 8.1. The position and orientation of an object in coordinate system { } I can be converted into those in the neighboring coordinate system { } 1 I + using the following transformation matrix: { 1} { 1} { } { } cos sin 0 sin cos 0 0 0 1 1 0 0 0 1 1 I I I o i i i o o i i i o o i o I x y x y y y θ θ θ θ θ θ θ + + − = . (8.1) 2 x { } O 0 y { } G [ ] , , o o o x y θ [ ] 0 0 0 , , x y θ { } 0 0 l [ ] 1 1 1 , , x y θ { } 1 [ ] 1 1 1 , , n n n x y θ − − − { } 1 N − [ ] , , n n n x y θ { } N 1 l γ Figure 8.1: Car-like robot with trailers In order to form a convoy (the car pulls all the other trailers which are hooked up to the next), each trailer is subject to holonomic constraints { } { } { } { } 1 1 1 1 1 cos 2 2 sin 2 I I i i i i I I i i i l l x l y θ θ + + + + + = + = , (8.2) The transformation matrix resultantly becomes { 1} 1 { 1} { } { } cos sin 0 cos 2 2 sin cos 0 sin 2 0 0 1 1 1 0 0 0 1 I i i I I i i i o o o i o i i i o o i I l l x x y l y θ θ θ θ θ θ θ θ θ + + + − + = , (8.3a) or ( ) { 1} { 1} { } { } I I I o I i o θ + + = z T z . (8.3b) Generally, the position and orientation of an object in coordinate system { } I can be transformed into those in { } K ( ) ( ) ( ) ( ) { } { } { 1} { 2} { 1} { } { 1} { 2} 1 { 1} 1 { } K K K I I I o K k K k I i I i o θ θ θ θ − + + − − − + + = z T T T T z . (8.4) 1.2 State space equations Because of the dependence of [ ] , i i x y on i θ as described in Equation (8.2), state variables of the robot are [ ] 0 0 0 1 2 1 , , , , ,..., , n n x y θ θ θ θ θ − . In general, the state-space equations of a multi-body robot are accordingly given by CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 39. 3 ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 1 0 1 1 2 1 2 0 1 2 1 1 1 1 cos cos sin sin cos sin cos n n n n i i i n x y d d d α θ α θ θ β α θ θ θ α θ θ θ θ θ α θ θ θ θ θ − − − = = = = = − = − − = − − ∏ # . (8.5) where α and β are any two reals. 2 LEGGED ROBOTS Legged robots are extremely complicated. The complexity can be understood from Figure 8.2, which shows that leg kinematics and dynamics differ depending on how many contact points legs have on the ground. 2 2 , l l θ τ 2 2 , r r θ τ 1 1 , l l θ τ 1 1 , r r θ τ , lr lr θ τ b L 1 l L 2 l L 1 r L 2 r L , b b θ τ (a) Two-point contact (b) One-point contact (c) No-point contact Figure 8.2: Legged robot 2.1 Kinematics Transformation matrices presented for multi-body wheeled robots in the previous section can be also used for legged robots. The significant difference appears when two legs are in contact with the ground (Figure 8.2a), where two legs form a closed-loop. While four links ( 1 2 1 2 , , , l l r r L L L L ) form a closed-loop, there are five joint angles ( 1 2 1 1 , , , , l l r r lr θ θ θ θ θ ) to adjust. This means that an infinite number of configurations are possible to allow two-point contact. Joint angles are chosen such that the robot is statically stable. 4 2.2 Dynamics The dynamics of the manipulator are written in the form: ( ) ( ) ( ) , = + + τ M θ θ V θ θ G θ . (8.6) where ( ) M θ is the n n × mass matrix, ( ) , V θ θ is an 1 n× vector of centrifugal and Coriolis terms, and ( ) G θ is an 1 n× vector of gravity terms. In the case of a two-link manipulator, each term is expressed as ( ) ( ) 2 2 2 2 2 1 2 2 2 1 1 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 l m l l m c l m m l m l l m c l m l l m c l m + + + + = + M θ . (8.7a) ( ) 2 2 1 2 2 2 2 1 2 2 2 1 2 2 2 1 2 2 1 2 m l l s m l l s c m l l s θ θ θ θ − − = V θ . (8.7b) ( ) ( ) 2 2 12 1 2 1 1 2 2 12 m l gc m m l c m l gc + + = G θ . (8.7c) where 1 1 sin s θ = , 2 2 sin s θ = , 1 1 cos c θ = , 2 2 cos c θ = , 12 1 2 cos cos c θ θ = . The formulation directly applies to the one-point contact case (Figure 8.2b). CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 40. MTRN9224 Robot Design Lecture 9: Vision Sensors Tomonari Furukawa School of Mechanical and Manufacturing Engineering University of New South Wales 1 OVERVIEW There is a very large body of work associated with vision systems, image processing and pattern recognition that addresses many different hardware- and software-related topics. 1.1 Vision Sensing Process Figure 9.1 shows the fundamental process of vision sensing for mobile robot navigation. Image processing Image analysis – object identification Type Pose Size (Shape) Mapping Coordinate transformation Labeling Figure 9.1: Vision sensing process Image processing relates to the preparation of an image for later analysis and use such as mapping. Image captured by a camera are not necessarily in a form that can be used by image analysis routines. Some may need improvement to reduce noise, others may need to be simplified, and still others may need to be enhanced, altered, segmented, filtered, etc. • Image processing is the collection of routines and techniques that improve, simplify, enhance or otherwise alter an image. • Image analysis is the collection of processes in which a captured image that is prepared by image processing is analyzed in order to extract information about the image and to identify objects or facts about the objects such as type, pose, size and shape. • Mapping is the collection of processes to locate objects in the environment (global coordinate system). The processes include coordinate transformation and labeling of objects. 1.2 Image Area An image is a representation of a real scene, either in black and white or in color, and either in print form or in a digital form. In the digital form, digital images are divided into small sections called picture cells, or pixels (in three-dimensional images, they are called volume cells or voxels) 2 where the size of all pixels are the same, while the intensity of light in each pixel is varied to create the gray images. 1.3 Acquisition of Images 1.3.1 Solid-state silicon wafer There are two types of vision cameras, analog and digital. Analog cameras are not very common anymore, but are still around. A digital camera is based on solid-state technology. As with other cameras, a set of lenses is used to project the area of interest onto the image area of the camera. The main part of the camera is a solid-state silicon wafer image area that has hundreds of thousands of extremely small photosensitive areas called photosites printed on it. Each small area of the wafer is a pixel. As the image is projected onto the image area, at each pixel location of the wafer, a charge is developed that is proportional to the intensity of light at that location (Thus, a digital camera is also called a charge coupled device, or CCD camera). The collection of charges, if read sequentially, would be a representation of the image pixels. 1.3.2 Acquisition The wafer may have as many as 520,000 pixels in an area with dimensions of a fraction of an inch. Obviously, it is impossible to have direct wire connections to all of these pixels to measure the charge in each one. To read such an enormous number of pixels, 30 times a second, the charges are moved to optically isolated shift registers next to each photosite, are moved down to an output line, and then read. The result is that every thirtieth of a second, the charges in all pixel locations are read sequentially and stored or recorded. The output is a discrete representation of the image – a voltage sampled in time. 1.4 Digital Images The sampled voltages from the aforementioned process are first digitized through an analog-to- digital (A/D) converter and then either stored in the computer storage unit in an image format such as TIFF, JPG, Bitmap, etc., or displayed on a monitor. A digitized image is nothing more than a computer file that contains a collection of 0’s and 1’s that represent the intensity of light at each pixel. An image that has different gray levels at each pixel location is called a gray image. The gray values are digitized by a digitizer, yielding strings of 0’s and 1’s that are subsequently displayed or stored. A color image is obtained by superimposing three images of red, green, and blue hues, each with a varying intensity and each equivalent to a gray image (but in a colored state). Thus, when the image is digitized, it will similarly have strings of 0’s and 1’s for each hue. 2 IMAGE PROCESSING 2.1 Frequency Domain vs. Spatial Domain Many processes that are used in image processing and analysis are based on the frequency domain or the spatial domain. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 41. 3 In frequency-domain processing, the frequency spectrum of the image is used to alter, analyze or process the image. In this case, the individual pixels and their contents are not used. Instead, a frequency representation of the whole image is used for the process. In spatial-domain processing, the process is applied to the individual pixels of the image. As a result, each pixel is affected directly by the process. However, the two techniques are equally important and powerful and are used for different purposes. Note that, although spatial- and frequency-domain techniques are used differently, they are related. For example, suppose that a spatial filter is used to reduce noise in an image. As a result, noise level in the image will be reduced, but at the same time, the frequency spectrum of the image will also be affected, due to the reduction in noise. 2.2 Filtering in Frequency Domain 2.2.1 Fourier Transform and frequency contents of a signal As you may remember from your mathematics and other course, any periodic signal may be decomposed into a number of sines and cosines of different amplitudes and frequencies as a Fourier series: 0 1 ( ) cos sin 2 n n n a f t a n t a n t ω ω ∞ = = + + ∑ . (9.1) Since a square wave functions represents a sharp change, this means that rapid changes such as an impulse, a pulse, and a square wave have a large number of frequencies. Thus, any video (or other) signal that contains shape changes (noise, high contrasts or an impulse or step function) or that has detailed information (high-resolution signals with fast, varying changes) will have a large number of frequencies in its frequency spectrum. A similar analysis can be applied to non-repeating signals as well (The equation used is a Fourier transform or, sometimes a fast Fourier transform (FFT)). In the transformation, major frequencies, or harmonics, are used to identifying and labeling a signal, including recognizing voices, shapes, objects, etc. Time Amplitude Frequency Amplitude 1 3 5 7 9 Figure 9.2: Time-domain and freqnency-domain plots of a function 2.2.2 Frequency contents of an image: noise, edges Consider sequentially plotting the gray values of the pixels of an image (on the y-axis) against time or pixel location (on the x-axis) as the image is scanned. The result will be a discrete time plot of varying amplitudes showing the intensity of light at each pixel as indicated in Figure 9.3. Let’s 4 say that we are on the ninth row and a re looking at pixels 129-144. The intensity of pixel 136 is very different from the intensities of the pixels around it and may be considered to be noise (Generally, noise is information that does not belong to the surrounding environment.). The intensities of pixels 134 and 141 are also different from the neighboring pixels and may indicate a transition between the object and the background. Thus, these pixels can be construed as representing the edges of the object. Both noises and edges are instances in which one pixel value is very different from the neighboring ones. Thus, noises and edges create the larger frequencies of a typical frequency spectrum, whereas slowly varying gray level sets of pixels, representing the object, contribute to the lower frequencies of the spectrum. If a high-frequency signal is passed through a low-pass filter, the filter will reduce the influence of all high frequencies, including the noises and edges. 1 17 129 129 144 Pixel # Intensity Figure 9.3: Noise and edge information in an intensity diagram of an image Figure 9.4 shows an intensity diagram after a simple first-approximation Fourier transform of the gray values was performed for the first four harmonic frequencies. 129 144 Pixel # Intensity 129 144 Pixel # Intensity Figure 9.4: Intensity diagram after Fourier transform using only the first four freqencies 2.3 Filtering in Spatial Domain 2.3.1 Convolution mask Spatial-domain processes access and operate on the information contained in an individual pixel. As a result, the image is directly affected by the operation. Most processes used in vision systems CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 42. 5 are in the spatial domain. One of the most popular and most common techniques in the spatial domain is the convolution mask, which can be adapted to many different tasks from filtering to edge detection. Imagine that an image is composed of pixels, each with a particular gray level or color information, that collectively constitute the image. (Suppose that the gray level is not digitized into 0’s and 1’s, but the analog value is indicated). If we consider part of a large image with pixel values shown symbolically as 11 x ,…, 33 x shown in Figure 9.5a and apply a mask that has the values indicated by 11 m ,…, 33 m as shown in Figure 9.5b, the new value of 22 x , 22 ' x , is given by 3 3 1 1 22 3 3 1 1 ' ij ij i j ij i j m x x m = = = = = ∑ ∑ ∑ ∑ . (9.2) Note that, if the denominator is 0, 1 is used. x11 x12 x13 x1m x21 x21 x23 x2m x31 x31 x33 x3m xn1 xn2 xn3 xnm m11 m12 m13 m21 m21 m23 m31 m31 m33 Figure 9.5: Convolution mask 2.3.2 Noise reduction 2.3.2.1 Neighborhood averaging Neighborhood averaging can be used to reduce noise in an image, but it also reduces the sharpness of the image. 1 1 1 1 1 1 1 1 1 Figure 9.6: Neighborhood-averaging mask 2.3.2.2 Gaussian averaging Gaussian averaging filter, also called the mild isotropic low-pass filter, improve an image similarly to the neighborhood averaging filter, but with a slightly different result. 6 1 4 6 4 1 4 16 24 16 4 6 24 36 24 6 4 16 24 16 4 1 4 6 4 1 1 2 1 2 4 2 1 2 1 Figure 9.7: Gaussian averaging filters 2.3.2.3 Image averaging In this technique, a number of images of the exact same scene are averaged together. Since the camera has to acquire multiple images of the same scene, all actions in the scene must be halted. Question What are disadvantages of the image averaging technique? 2.3.2.4 Median filters One of the main problems in using neighborhood averaging is that, along with removing noises, the filter will blur edges. A variation of this technique is to use a median filter, in which the value of the pixel is replaced by the median of the values of the pixels in a mask around the given pixel, sorted in ascending order. A median is the value such that half of the values in the set are below and half are above the median. The median is independent of the value of any single pixel in the set, the median filter will be much stronger in eliminating spike-like noises without blurring the object or decreasing the overall sharpness of the image. Figure 9.8 illustrates an example. The gray values, in ascending order, are 1, 2, 3, 4, 5, 6, 7, 8, 9. The middle value of 5 is thus inserted. 2 1 3 8 9 4 7 5 6 2 1 3 8 5 4 7 5 6 (a) Before application (b) After application Figure 9.8: Median filter CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 43. 7 2.4 Filtering through Histogram A histogram is a representation of the total number of pixels of an image at each gray level. Histogram information is used in a number of different processes. Thresholding – Histogram determines a cutoff point when an image is to be transformed into binary values. Noise reduction – Suppose a systematic source of noise in an image causes many pixels to have one noisy gray level. Then a histogram can be used to determine what the noisy gray level is in order to attempt to remove or neutralize the noise. Improvement of image – Histogram can improve image when an image has all its pixel gray levels clustered between two relatively close values. The improvement can be achieved by equalizing the histogram such that the gray levels are spread out between the minimum and maximum gray levels. 2.5 Segmentation Segmentation is a generic name for a number of different techniques that divide the image into segments of constituents. The purpose of segmentation is to separate the information contained in the image into smaller entities that can be used for other purposes. For example, an image can be segmented by the edges in the scene, by small areas, etc. Each of these entities can then be used for further processing, representation or identification. Segmentation includes, but is not limited to, edge detection, region growing and texture analysis. 2.5.1 Edge detection Edge detection is a general name for a class of routines and techniques that operate on an image and result in a line drawing of the image. The lines represent changes in values such as cross sections of planes, intersections of planes, textures, lines, and colors, as well as differences in shading and textures. As was mentioned earlier, like noise, edges create higher frequencies in the spectrum and hence can be separated by high-pass filters. Masks can be designed to behave like a high-pass filter, reducing the amplitude of the lower frequencies while not affecting the amplitudes of the higher frequencies as much. -1 0 -1 0 4 0 -1 0 -1 Figure 9.9: Typical high-pass edge detector mask (Laplacian1) 8 -1 0 -1 0 8 0 -1 0 -1 -1 0 -1 0 6 0 -1 0 -1 -1 0 -1 0 5 0 -1 0 -1 (a) Laplacian2 (b) Sharpen, low (c) Sharpen, medium Figure 9.9: Other high-pass filters 2.5.2 Region growing and region splitting Region growing and image splitting are techniques of segmentation, as are edge detection routines. Through these techniques, an attempt is made to separate the different parts of an image into segments or components with similar characteristics that can be used in further analysis, such as object detection. While an edge detection will find the separation lines of textures, colors, planes, and gray levels, segmentation by regions will naturally result in complete and closed boundaries. Two approaches are used for region segmentation; region growing, which relates regions by their average similarities or spatial relationships, and region splitting, which will split images into smaller areas by using their finer differences. Region splitting – One technique of region splitting is thresholding. The image is split into closed areas of neighboring pixels by comparing them with thresholding value or range. Any pixel that falls below a threshold (or between a range of values) will belong to a given region; otherwise, it belongs to another region. Generally, although it is simple, it is not very effective, because choosing an appropriate threshold is difficult. Region growing – First nuclei regions, small clusters of pixels that are formed at the beginning of segmentation, are formed on the basis of some specific selection law. The result is a large number of little regions. Successively, these regions are combined into larger regions on the basis of some other attributes or rules. Although these rules will merge many smaller regions together to create a smoother set of regions, they may unnecessarily merge certain features that should not be merged, such as holes, smaller but distinct areas or distinct areas with similar intensities. 2.6 Binary Morphology Operations Morphology operations refer to a family of operations that are performed on the shape (and thus the morphology) of objects in an image. Some of them are based on the connectivity information, which establishes whether neighboring pixels are related to each other. The morphology operations include many different operations, both for binary and gray images, such as thickening, dilation, erosion, skeletonization, opening, closing and filling. These operations are performed on an image in order to aid in its analysis, as well as to reduce the extra information that may be present in the image. 2.6.1 Connectivity +4-connectivity – (x+1, y), (x-1, y), (x, y+1), (y, y-1) ×4-connectivity – (x+1, y+1), (x+1, y-1), (x-1, y+1), (x-1, y-1) H6-connectivity – (x-1, y+1), (x, y+1), (x+1, y+1), (x-1, y-1), (x, y-1), (x+1, y-1) CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 44. 9 V6-connectivity – (x-1, y+1), (x-1, y), (x-1, y-1), (x+1, y+1), (x+1, y), (x+1, y-1) 8-connectivity – (x-1, y-1), (x, y-1), (x+1, y-1), (x-1, y), (x+1, y), (x-1, y+1), (x, y+1), (x+1, y+1) 2.6.2 Thickening operation The thickening operation fills the small holes and cracks on the boundary of and object and can be used to smooth the boundary. 2.6.3 Dilation In dilation, the background pixels that are 8-connected to the foreground (object) are changed to foreground pixels. As a result, a layer is effectively added to the object every time the process is implemented. 2.6.4 Erosion In erosion, foreground pixels that are 8-connected to a background pixel are eliminated. This effectively eats away a layer of the foreground each time the operation is performed. 2.6.5 Skeletonization A skeleton is a stick representative of an object in which all thickness have been reduced to one pixel at any location. Skeletonization is a variation of erosion. Whereas in erosion the thickness of an object may go to zero and the object may be totally lost, in skeletonization, as soon as the thickness of the object becomes one pixel, the operation at that location stops. 2.6.6 Open operation Opening is erosion followed by dilation and causes a limited smoothing of convex parts of the object. Opening can be used as an intermediate operation before skeletonization. 2.6.7 Close operation Closing is dilation followed by erosion and causes a limited smoothing of convex parts of the object. Like opening, closing can be used as an intermediate operation before skeletonization. 2.6.8 Fill operation The fill operation fills the holes in the foreground. 3 IMAGE ANALYSIS 3.1 Object Recognition by Features Objects in an image may be recognized by their features, which may include • Gray-level histograms • Morphological features such as area, perimeter, number of holes, etc. • Aspect ratio • Thinness • Moment • Template matching 10 • Discrete Fourier Descriptors • Computed tomography In many cases, the information extracted is compared with a priori information about the object, which may be in a lookup table. For example, suppose that two objects are present in the image, one with two holes and one with one hole. By using previously discussed routines, it is possible to determine how many holes each part has, and by comparing the two parts (let’s say they are assigned to regions 1 and 2) with information about them in a lookup table, it is possible to determine what each of the two parts is. As another example, suppose that a moment analysis of a known part is performed at different angles, that the moment of the part relative to an axis is calculated for these angles, and that the resulting data are collected in a lookup table. Later, when the moment of the part in the image is calculated relative to the same axis is compared with the information in the lookup table, the angle of the part in the image can be estimated. 3.2 Gray-level histograms The average, maximum or minimum gray levels may be used to identify different parts or objects in an image. As an example, assume that the image is divided into three parts, each with a different color or texture that will create different gray levels in the image. If the average, maximum or minimum gray levels of the objects are found, say, through histogram mapping, the object can be recognized by comparing them with this information. In other cases, even the presence of one particular gray level may be enough to recognize a part. 3.3 Morphological features The perimeter, area and diameter of an object, as well as the number of holes it has and other morphological characteristics, may be used to identify the object. The perimeter of an object may be found by first applying an edge detection routine and then counting the number of pixels on the perimeter. The area of the object can be calculated by region-growing techniques. Moment equations can also be used, as discussed later. The diameter of a non-circular object is defined as the maximum distance between any two points on any line that crosses the identified area of the object. 3.4 Aspect ratio An object’s aspect ratio is the ratio of the width to the length of a rectangle enclosed about the object as shown in Figure 9.?. All aspect ratio, except for the minimum aspect ratio, are sensitive to orientation. Thus, the minimum aspect ratio is generally used to identify objects. 3.5 Thinness Thinness h is defined as one of the following ratios: 2 p h A = . (9.3) d h A = . (9.4) CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m
- 45. 11 where p , A and d are the perimeter, area and diameter of the object. 3.6 Moment Imagine an object within a binary image. The object is represented by pixels that are turned on, and the background is represented by pixels that are turned off. This effect can be achieved either by backlighting or by rendering the image in binary form. Now consider the general moment equation , a b a b i j i j M x y = ∑∑ . (9.5) where , a b M is the moment of the object within the image with indices a and b , and i x and j y are the coordinate of each pixel that is turned on within the image. Therefore, 0,0 M is the summation of as many 1’s as there are “on”-pixels, which is equivalent to the area of the object. All combinations of a and b values between 0 and 3 are common. The locations of the center of the area relative to the x-axis and y-axis can be respectively calculated by 0,1 0,0 j j y M y A M = = ∑ . (9.6a) 1,0 0,0 i i x M x A M = = ∑ . (9.6b) Now consider the second moment of the area relative to the x-axis and y axis, i.e., 0,2 M and 2,0 M . As you can imagine, the moment of inertia of an object will vary significantly as the object rotates around its center. Suppose that one would calculate the moments of the area about, say, the x-axis, at different orientations. Since each orientation creates a unique value, a lookup table that contains these values can later be used to identify the orientation of the object. Thus, if a lookup table containing the values of the moments of inertia of the known object at different orientations is prepared, the subsequent orientation of the object can be estimated by comparing its second moment with the values in the table. As a result, with the use of the moment equations, an object, its location and orientation can be identified. 3.7 Template Matching Another technique for object recognition is model (template) matching. If a suitable line drawing of a scene is found, the topological or structural elements, such as the total number of lines (sides), vertices, and interconnections can be matched to a model. Coordinate transformations can be performed to eliminate the differences between the model and the object resulting from differences in position, orientation, or depth between them. This technique is limited by the fact that a priori knowledge of the models is needed for matching. Thus, if the object is different from the models, they will not match, and the object will not be recognized. Another major limitation is that if one object is occluded by other objects, it will not match a model. 12 3.8 Discrete Fourier Descriptors In a manner similar to a Fourier transform that is calculated for an analog signal, a discrete Fourier transform (DFT) of a set of discrete points (such as pixels) can be calculated. The result of a DFT calculation is a set of frequencies and amplitudes in the frequency domain that describe the spatial relationship of the points in question. 3.9 Computed Tomography (CT) Tomography is a technique of determining the distribution of material density in the part being examined. In computed tomography (CT), a three-dimensional image of the density distribution of the object is reconstructed from a large number of two-dimensional images of the material density taken by different scanning techniques, such as X-rays or ultrasonics. In computed tomography, it is assumed that the part consists of a sequence of overlaying slices. Images of the density distribution of each slice are taken repeatedly around the object. Although partial coverage of the part has been used as well, a complete coverage of 360 deg is preferred. The data are stored in a computer and subsequently reduced to a three-dimensional image of the part’s density distribution, which is shown on a CRT. Tomography is the only efficient technique for mapping the internals of objects. CuuDuongThanCong.com https://fb.com/tailieudientucntt c u u d u o n g t h a n c o n g . c o m