Design Baseline Computed Torque Controller

Farzin Piltan, Mina Mirzaei, Forouzan Shahriari, Iman Nazari & Sara Emamzadeh
International Journal of Engineering (IJE), Volume (6) : Issue (3) : 2012 129
Design Baseline Computed Torque Controller
Farzin Piltan SSP.ROBOTIC@yahoo.com
Industrial Electrical and Electronic
Engineering SanatkadeheSabze
Pasargad. CO (S.S.P. Co), NO:16
,PO.Code 71347-66773, Fourth floor
Dena Apr , Seven Tir Ave , Shiraz , Iran
Mina Mirzaei SSP.ROBOTIC@yahoo.com
Forouzan Shahriari SSP.ROBOTIC@yahoo.com
Iman Nazari SSP.ROBOTIC@yahoo.com
Sara Emamzadeh SSP.ROBOTIC@yahoo.com
Abstract
The application of design nonlinear controller such as computed torque controller in control of 6
degrees of freedom (DOF) robot arm will be investigated in this research. One of the significant
challenges in control algorithms is a linear behavior controller design for nonlinear systems (e.g.,
robot manipulator). Some of robot manipulators which work in industrial processes are controlled
by linear PID controllers, but the design of linear controller for robot manipulators is extremely
difficult because they are hardly nonlinear and uncertain. To reduce the above challenges, the
nonlinear robust controller is used to control of robot manipulator. Computed torque controller is a
powerful nonlinear controller under condition of partly uncertain dynamic parameters of system.
This controller is used to control of highly nonlinear systems especially for robot manipulators. To
adjust this controller’s coefficient baseline methodology is used and applied to CTC.
Keywords: Baseline Tuning Computed Torque Controller, Computed Torque Controller,
Unstructured Model Uncertainties, Adaptive Method.

1. INTRODUCTION and MOTIVATION
PUMA 560 robot manipulator is a 6 DOF serial robot manipulator. From the control point of view,
robot manipulator divides into two main parts i.e. kinematics and dynamic parts. Controller is a
device which can sense information from linear or nonlinear system (e.g., robot manipulator) to
improve the systems performance [1-4]. The main targets in designing control systems are
stability, good disturbance rejection, and small tracking error[5-6]. Several industrial robot
manipulators are controlled by linear methodologies (e.g., Proportional-Derivative (PD) controller,
Proportional- Integral (PI) controller or Proportional- Integral-Derivative (PID) controller), but when
robot manipulator works with various payloads and have uncertainty in dynamic models this
technique has limitations. In some applications robot manipulators are used in an unknown and
unstructured environment, therefore strong mathematical tools used in new control
methodologies to design nonlinear robust controller with an acceptable performance (e.g.,
minimum error, good trajectory, disturbance rejection). Computed torque controller (CTC) is an
influential nonlinear controller to certain systems which it is based on feedback linearization and
computes the required arm torques using the nonlinear feedback control law. When all dynamic
and physical parameters are known the controller works superbly; practically a large amount of
systems have uncertainties and sliding mode controller reduce this kind of limitation [7]. This
controller is used to control of highly nonlinear systems especially for robot manipulators. In
various dynamic parameters systems that need to be training on-line adaptive control
methodology is used.
Background
Computed torque controller (CTC) is a powerful nonlinear controller which it widely used in
control robot manipulator. It is based on Feed-back linearization and computes the required arm
torques using the nonlinear feedback control law. This controller works very well when all
dynamic and physical parameters are known but when the robot manipulator has variation in
dynamic parameters, in this situation the controller has no acceptable performance[14]. In
practice, most of physical systems (e.g., robot manipulators) parameters are unknown or time
variant, therefore, computed torque like controller used to compensate dynamic equation of robot
manipulator[1, 6]. Research on computed torque controller is significantly growing on robot
manipulator application which has been reported in [1, 6, 15-16]. Vivas and Mosquera [15]have
proposed a predictive functional controller and compare to computed torque controller for tracking
response in uncertain environment. However both controllers have been used in Feed-back
linearization, but predictive strategy gives better result as a performance. A computed torque
control with non parametric regression models have been presented for a robot arm[16]. This
controller also has been problem in uncertain dynamic models. Based on [1, 6]and [15-
16]Computed torque controller is a significant nonlinear controller to certain systems which it is
based on feedback linearization and computes the required arm torques using the nonlinear
feedback control law. When all dynamic and physical parameters are known the controller works
fantastically; practically a large amount of systems have uncertainties and sliding mode controller
decrease this kind of challenge.
2. THEOREM: DYNAMIC FORMULATION OF ROBOTIC MANIPULATOR,
COMPUTED TORQUE FORMULATION AND APPLIED TO ROBOT ARM
Dynamic of robot arm: The equation of an n-DOF robot manipulator governed by the following
equation [1, 4, 15]:
(1)
Where τ is actuation torque, M (q) is a symmetric and positive define inertia matrix, is the
vector of nonlinearity term. This robot manipulator dynamic equation can also be written in a
following form [1-29]:
(2)
Where B(q) is the matrix of coriolios torques, C(q) is the matrix of centrifugal torques, and G(q) is
the vector of gravity force. The dynamic terms in equation (2) are only manipulator position. This
is a decoupled system with simple second order linear differential dynamics. In other words, the
component influences, with a double integrator relationship, only the joint variable ,

independently of the motion of the other joints. Therefore, the angular acceleration is found as to
be [3, 15-62]:
(3)
This technique is very attractive from a control point of view.
Computed Torque Controller
The central idea of computed torque controller (CTC) is feedback linearization. Originally this
algorithm is called feedback linearization method. It has assumed that the desired motion
trajectory for the manipulator , as determined, by a path planner. Defines the tracking error
as:
(4)
Where e(t) is error of the plant, is desired input variable, that in our system is desired
displacement, is actual displacement. If an alternative linear state-space equation in the
form can be defined as
(5)
With and this is known as the Brunousky canonical form. By
equation (4) and (5) the Brunousky canonical form can be written in terms of the state
as [1]:
(6)
With
(7)
Then compute the required arm torques using inverse of equation (7), is;
(8)
This is a nonlinear feedback control law that guarantees tracking of desired trajectory. Selecting
proportional-plus-derivative (PD) feedback for U(t) results in the PD-computed torque controller
[6];
(9)
and the resulting linear error dynamics are
(10)
According to the linear system theory, convergence of the tracking error to zero is guaranteed [6].
Where and are the controller gains. The result schemes is shown in Figure 1, in which two
feedback loops, namely, inner loop and outer loop, which an inner loop is a compensate loop and
an outer loop is a tracking error loop.

FIGURE 1: Block diagram of PD-computed torque controller (PD-CTC)
The application of proportional-plus-derivative (PD) computed torque controller to control of
PUMA 560 robot manipulator introduced in this part. Suppose that in (9) the nonlinearity term
defined by the following term;
(11)
Therefore the equation of PD-CTC for control of PUMA 560 robot manipulator is written as the
equation of (12);
(12)
The controller based on a formulation (12) is related to robot dynamics therefore it has problems
in uncertain conditions.

3. METHODOLOGY: BASELINE ON-LINE TUNING FOR STABLE
COMPUTED TORQUE CONTROLLER
Computed torque controller has difficulty in handling unstructured model uncertainties. It is
possible to solve this problem by combining CTC and baseline tuning method which this method
can helps to improve the system’s tracking performance by online tuning method. In this research
the nonlinear equivalent dynamic (equivalent part) formulation problem in uncertain system is
solved by using on-line linear error-based tuning theorem. In this method linear theorem is
applied to CTC to adjust the coefficient. CTC has difficulty in handling unstructured model
uncertainties and this controller’s performance is sensitive to controller coefficient. It is possible to
solve above challenge by combining linear error-based tuning method and CTC. Based on above
discussion, compute the best value of controller coefficient has played important role to improve
system’s tracking performance especially when the system parameters are unknown or
uncertain. This problem is solved by tuning the controller coefficient of the CTC continuously in
real-time. In this methodology, the system’s performance is improved with respect to the pure
CTC. Figure 2 shows the baseline tuning CTC. Based on (23) and (27) to adjust the controller
coefficient we define as the baseline tuning.
FIGURE 2: Block diagram of a baseline computed torque controller
(13)
If minimum error ( ) is defined by;
(14)
Where is adjusted by an adaption law and this law is designed to minimize the error’s
parameters of adaption law in linear error-based tuning CTC is used to adjust the
controller coefficient. Linear error-based tuning part is a supervisory controller based on the
following formulation methodology. This controller has three inputs namely; error change of
error ( ) and the integral of error ( ) and an output namely; gain updating factor . As a
summary design a linear error-based tuning is based on the following formulation:
) (15)

Where is gain updating factor, ( ) is the integral of error, ( ) is change of error, is error
and K is a coefficient.
4. RESULTS
This part is focused on compare between PD computed torque controller (CTC) and baseline
error-based tuning computed torque controller (BCTC). These controllers were tested by step
responses. In this simulation, to control position of PUMA robot manipulator the first, second, and
third joints are moved from home to final position without and with external disturbance.
Tracking Performances: In baseline error-based tuning CTC the controller’s gain is adjusted
online depending on the last values of error , change of error ( ) and the integral of error ( )
by gain updating factor ( . Figure 3 shows tracking performance in BCTC and CTC without
disturbance for step trajectory.
FIGURE 3: BCTC and CTC for first, second and third link step trajectory performance without disturbance
Based on Figure 3 it is observed that, the overshoot in BCTC is 0% and in CTC’s is 7%, and the
rise time in BCTC’s is 0.5 seconds and in CTC’s is 0.4 second.
Disturbance Rejection
Figure 4 shows the power disturbance elimination inn BCTC and CTC with disturbance for step
trajectory. The disturbance rejection is used to test the robustness comparisons in these two
controllers for step trajectory. A band limited white noise with predefined of 40% the power of
input signal value is applied to the step trajectory.

FIGURE 4: BCTC and CTC for first, second and third link trajectory with 40%external disturbance
Based on Figure 4; by comparing step response trajectory with 40% disturbance of relative to the
input signal amplitude in BCTC and CTC, BCTC’s overshoot about (0%) is lower than CTC’s
(12%). CTC’s rise time (1 seconds) is lower than BCTC’s (1.4 second). Besides the Steady
State and RMS error in BCTC and CTC it is observed that, error performances in BCTC (Steady
State error =1.3e-5 and RMS error=1.8e-5) are about lower than CTC’s (Steady State
error=0.01 and RMS error=0.015). Based on Figure 4, CTC has moderately oscillation in
trajectory response with regard to 40% of the input signal amplitude disturbance but BCTC has
stability in trajectory responses in presence of uncertainty and external disturbance. Based on
Figure 4 in presence of 40% unstructured disturbance, BCTC’s is more robust than CTC because
BCTC can auto-tune the coefficient as the dynamic manipulator parameter’s change and in
presence of external disturbance whereas CTC cannot. The BCTC gives significant steady state
error performance when compared to CTC. When applied 40% disturbances in BCTC the RMS
error increased approximately 15.5% (percent of increase the BCTC RMS
error= ) and in CTC the RMS error increased
approximately 125% (percent of increase the PD-SMC RMS
error= ).
5. CONCLUSION
In this research, a baseline error-based tuning computed torque controller (BCTC) is design and
applied to robot manipulator. Pure CTC has difficulty in handling unstructured model
uncertainties. It is possible to solve this problem by combining CTC and baseline error-based
tuning. The controller gain is adjusted by baseline error-based tuning method. The gainupdating
factor ( ) of baseline error-based tuning part can be changed with the changes in error, change of
error and the integral (summation) of error. In pure CTC the controller gain is chosen by trial and
error, which means pure CTC had to have a prior knowledge of the system uncertainty. If the
knowledge is not available error performance is go up.
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