Kinematics Analysis of Parallel Mechanism Based on Force Feedback Device

International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 3 Issue 11 ǁ November. 2015 ǁ PP.55-61
www.ijres.org 55 | Page
Kinematics Analysis of Parallel Mechanism Based on Force
Feedback Device
LU Jiu-ru, HANG Lu-bin, QIN wei
(College of Mechanical Engineering, Shanghai University of Engineering Science, Shanghai 201620)
Abstract: Kinematic analysis of mechanism is the fundamental work of force feedback device research.The
composition of Delta mechanism based on Omega.7 force feedback device was illustrated in this paper.The
kinematic loop equations of Delta mechanism was established according to its geometric relationship,also the
inverse kinematics solution of Delta mechanism were obtained. And the numerical forward kinematics were
calculated by Newton iteration algorithm.Finally,The analysis of velocity and acceleration was carried out
through matrix operations.Kinematic analysis of Delta mechanism provides a theoretical basis for following
study.
Keywords：Force Feedback Device; Delta Mechanism; Forward Kinematics; Inverse Kinematics
I. Introduction
The force feedback device, which based on the high performance data processing computer and information
communication technology, can realize the operator’s action input by adding force feedback on the base of the
traditional human-computer interaction[1]
, and the force feedback device can use the haptic rendering to enhance
the man-machine communication, which is widely used in many fields,such as aviation, medical, military,
manufacturing, entertainment.[2~4]
The interaction mechanism of the force feedback device is a human-computer interaction interface, which
has a higher demand for the configuration of the mechanism. Because of the high system stiffness, small
cumulative error, high positioning accuracy, large working space, and all the joints are rotation joints, the
moving linkages can be made of light material. The drive motor is installed on the static platform,so the
system’s inertia is low.These greatly satisfy the requirements of force feedback device. The excellent
characteristic of Delta mechanism make it widely used in the field of force feedback.[5]
The force feedback device collects a series of motion data through the joint actuated encoder, but these data
can not directly reflect the user's operation, the kinematic analysis of the interaction mechanism should be
carried on, and the kinematic relationship between the end of the moving platform and the joint actuated should
be established, so that the coordinate parameters of the real-time position of the operator can be calculated.
The kinematic analysis of mechanism is to study the correlation and regularity of displacement, velocity
and acceleration between the input link and the output link without considering the movement generated by
force and torque, and kinematic analysis is the basis of some succeeding researches such as dynamic analysis
and workspace analysis.[6~7]
In this paper, the Delta mechanism of the force feedback device Omega.7 which is manufactured by Force
Dimension is taken as the research object, kinematic equations are established to solve the forward and inverse
kinematics according to the geometrical-restriction relation, and this provides a theoretical basis for the
succeeding analysis of dynamics and workspace .
II. The Delta Mechanism Based on Omega.7 Force Feedback Device
2.1 Omega.7 Force Feedback Device
Omega.7 is a kind of haptic feedback device manufactured by Force Dimension which located in
Switzerland[8]
, and it is shown in Figure.1. Omega.7 mainly made of aluminum alloy materials, and the output
power component is moving-coil machine Maxon. The system inertia decreased significantly, and the response
speed is improved, because of the usage of wire rope traction drive. And the usage of hybrid of parallel
mechanism and serial mechanism enhances the reliability and stability of the system.

Kinematics Analysis of Parallel Mechanism Based on Force Feedback Device
Fig.1 The graph of Omega.7
The number of DOF of the interaction mechanism is 7, it can be considered as a mechanism series
connected of Delta parallel mechanism, wrist joint mechanism and holding jaw mechanism, and the use of the
Delta parallel mechanism is to provide three degrees translation and obtain the operating arm’s moving position.
Fig.2 The Delta mechanism of Omega.7
Fig.3 Wrist joint mechanism and holding jaw mechanism
2.2 Structure analysis of the Delta mechanism
The Delta mechanism is composed of static platform, moving platform and the three links which connected
the static platform and the moving platform[9]
, and the mechanism sketch of Delta mechanism is shown in
Figure.4, as is shown in it, the structure of each branched chain are same, and it series connected of active arm
and driven arm.The active arm is a planar linkage with rotation joint on the two sides, the driven arm is a
parallelogram mechanism, and the two parallel links in the parallelogram mechanism of the driven arm connect
the active arm and the moving platform through the rotation joints. The three parallelogram mechanisms in the
Delta mechanism restrict all the rotations, so the motion of moving platform is a pure translation in the space.
Fig.4 The mechanism sketch of Delta mechanism

2.3 The system of coordinates
According to the geometrical characteristics of Delta mechanism, the space coordinates ' ' ' 'O x y z 、
O xyz , which is shown in Figure 5, are both established at the geometrical center of the Delta mechanism’s
moving platform and static platform. The positive direction of y axis is along the vertical direction of the origin
O to the point 1A , the positive direction of x axis is horizontal to the right, and the direction of z axis can be
defined according to the right-hand rule. The positive direction of 'y axis is along the vertical direction of the
origin O to the point 1A , the positive direction of x axis is horizontal to the right, and the direction of z
axis can be defined according to the right-hand rule.
Fig .5 The space coordinates of Delta mechanism
The radius of the center circle of the base shaft is denoted by 1r ,and is the radius of the center circle of the
moving platform is denoted by 2r , 1l and 2l respectively represent the length of the active arm and the driven
arm.
1iOA r ; 2' iO C r ; 1i iA B l ; 2i iB C l ; (1)
The angle between the active arm i iA B and plane xoy is denoted by i ,  , ,x y z is the coordinate of
the center point of the moving platform in the coordinates O xyz .
III. The position analysis of Delta mechanism
3.1 The inverse kinematics analysis of Delta mechanism
The analysis of inverse kinematics is to solve the input angles  1 2 3, ,   of the active arm, as the center
position coordinates  , ,x y z of the moving platform are provided.[10~11]
The projection relations of static platform in planar xoy and the projections relations of moving platform
in planar ' ' 'x o y are shown in Figure.6 and Figure.7 respectively. Let i be the angle between iOA and the
positive direction of x axis, and the angle between iO A  and the positive direction of x axis equals i ,
1,2,3i  .
Fig.6 The projection relation of static platform

Fig.7 The projection relation of moving platform
It can be derived from the geometric relation:
2
3 6
i i
 
   (2)
The position vector of point iA in the static platform:
1
cos
sin
0
i
i iOA r


 
   
  

(3)
The position vector of point iB in the static platform:
1 1
cos cos cos
sin cos sin
0 sin
i i i
i i i i i i i
i
OB OA A B r l
  
  

   
         
      
  
(4)
The position vector of point iC in the static platform:
2
cos
' ' sin
0
i
i i i
x
OC OO O C y r
z


   
         
      
  
(5)
The vector loop relationship can be established by the geometric relation:
i i i iOC OB BC 
  
(6)
The equation can be established according to the constraint of rod length 2i iB C l :
   
 
2 2
2 1 1 2 1 1
2 2
1 2
cos cos cos cos sin sin cos sin
sin
i i i i i i i i
i
x r r l y r r l
z l l
       

      
  
(7)
Expand the formula 7 and obtain:
2 2 2 2 2 2
2 1 1 2 2 1
1 2 1 1
( ) 2( )( cos sin )
2 ( cos sin )cos 2 sin 0
i i
i i i i
x y z r r l l r r x y
l r r x y zl
 
   
        
     
(8)
By using the trigonometrical transform:
2
2 2 2
2tan 1 tan tan
2 2 2sin ;cos ;tan ;
1 tan 1 tan 1 tan
2 2 2
a a a
a a a
a a a

  
  
(9)
Transform the formula 8 into the equation for it ( tan
2
i
it

 ):
2 2 2 2 2
2 1 2 1 1 2 1
2 2 2 2 2 2
2 1 2 1 1 2 1
1
[ ( )] 2[ ( )]( cos sin )
{ [ ( )] 2[ ( )]( cos sin )}tan
2
4 tan 0
2
i i
i
i i
i
x y z l l r r l r r x y
x y z l l r r l r r x y
zl
 

 

          
         
 
(10)
Transform the formula 10 into the quadratic equation for it :
2
0i i i i ia t bt c   ; (11)

2 2 2 2
2
2 1
2
1 1
1
2
1 1
;
cos sin ;
;
( ) 2 ( );
4 ;
( ) 2 ( );
i i i
i i
i
i i
u x y z l
v x y
r r r
a u l r v l r
b zl
c u l r v l r
 
    

 
  

      
  

       
(12)
According to the formula of root of quadratic equation:
2
4
2
i i i i
i
i
b b a c
t
a
  
 ；
When the center position coordinate  , ,x y z of the moving platform of the Delta mechanism are provided,
according to 2arctani it  , the inverse positional problem of two positions can be calculated, then the unique
solution can be determined by the initial position and the continuous change rule of active arm’s movement.
3.2 The forward kinematics analysis of Delta mechanism
The forward kinematics analysis of Delta mechanism is to solve the coordinate  , ,x y z of the center
position of moving platform, in the case of the active arm’s input angle  1 2 3, ,   are provided.Because of the
complexity of the parallel mechanism, it is difficult to get the analytic solution, so the question of solving
forward kinematics is generally converted to the question of obtaining the numerical solution of nonlinear
equations. In this paper, the forward kinematics solution is solved by the Newton iteration method.[12]
Equation 7 is the constraint equation established based on the Delta mechanism’s vector loop and the three
elements nonlinear equations of variables x, y, z can be got by equation 7:
2 2 2 2
1 1 1 2
2 2 2 2
2 2 2 2
2 2 2 2
3 3 3 2
( ) ( ) ( ) 0;
( ) ( ) ( ) 0;
( ) ( ) ( ) 0;
x A y B z C l
x A y B z C l
x A y B z C l
       

      

      
(13)
Some these are replaced by 2 1 1( cos )cosi i iA r r l     ; 2 1 1( cos )sini i iB r r l     ; 1 sini iC l   ;
Let 2 2 2 2
2( , , ) ( ) ( ) ( )i i i if x y z x A y B z C l       ,  1 2 3
T
f f fF ; the nonlinear equations 13 can
be expressed as follows in further steps:
( ) 0F x (14)
Let ( )F x be the Jacobian Matrix of the vector function ( )F x (namely the derivative of ( )F x ), 1
( )
F x
is the inversion of the Jacobian Matrix of ( )F x .
1 1 1
1 1 1
2 2 2
2 2 2
3 3 3
3 3 3
( ) ( ) ( )
2( ) 2( ) 2( )
( ) ( ) ( )
( ) 2( ) 2( ) 2( )
2( ) 2( ) 2( )
( ) ( ) ( )
   
 
       
                        
 
   
f f f
x y z
x A y B z C
f f f
x A y B z C
x y z
x A y B z C
f f f
x y z
x x x
x x x
F x
x x x
(15)
Apply the Newton method of the single equation on equation 14, then we can obtain the Newton iterative
method for nonlinear equations:
( 1) ( ) ( ) 1 ( )
( ) ( ), 0,1, , 
   k k k k
kx x F x F x (16)
Suppose ( 1) ( ) ( )
  k k k
x x x , the linear equation ( ) ( ) ( )
( ) ( )   k k k
F x x F x need to be solved to find out
the vector of ( )
 k
x , then let ( 1) ( ) ( )
 k k k
x x x , each step includes the calculations of vector ( )
( )k
F x and the
matrix ( )
( ) k
F x .
By setting the calculation accuracy, the numerical forward solution of position, which can meet the
requirement, can be obtained by the finite numbers of iterations.

IV. Velocity and acceleration analysis of Delta mechanism
4.1 Velocity analysis of Delta mechanism
After derivation calculus to formula 8, we can get[13]
:
2 1 1 2 1 1 1
2 1 1
[ ( cos )cos ] [ ( cos )sin ] ( sin )
[( cos sin )sin cos ]
    
    
        
     
  

i i i i i
i i i i i
x r r l x y r r l y z l z
x y r r z l
(17)
Suppose 2 1 1[( cos sin )sin cos ]        i i i i iD x y r r z l , then transform formula 16 as a matrix:
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
0 0
0 0
0 0



        
                
               



x A y B z C x D
x A y B z C y D
x A y B z C z D
(18)
Denoted by:
1 1 1
2 2 2
3 3 3
   
     
    
x
x A y B z C
J x A y B z C
x A y B z C
、
1
2
3
0 0
0 0
0 0
 
   
  
q
D
J D
D
；     
T
x y zx ；
1 2 3     
  
T
q ；
Formula 17 can be expressed as:
 x qJ Jx q (19)
Then forward solution of velocity of Delta mechanism can be expressed as:
1
  x qJ Jx q (20)
The inverse solution of velocity of Delta mechanism can be expressed as:
1
 q xJ Jq x (21)
4.2 Acceleration analysis of Delta mechanism
After two-order derivation calculus to formula 19, we can get:
      x x q qJ J J Jx x q q (22)
Then forward solution of acceleration of Delta mechanism can be expressed as:
1
( )
     x q q xJ J J Jx q q x (23)
The inverse solution of acceleration of Delta mechanism can be expressed as:
1
( )
     q x x qJ J J Jq x x q (24)
Of which:
1 1 1
2 2 2
3 3 3
     
       
      
  
   
  
x
x A y B z C
J x A y B z C
x A y B z C
，
1
2
3
0 0
0 0
0 0
 
   
  

q
D
J D
D
,
1
1
1
1
cos sin ;
sin sin ;
cos ;
{( cos sin )sin [( ) ]cos }
  
  
 
      
   

   

   

          



   
i i i i
i i i i
i i i
i i i i i i i i
A l
B l
C l
D x y z r z l
,

V. Conclusion
(1) The composition of Delta mechanism based on Omega.7 force feedback device was illustrated in this paper.
The kinematics loop equations of Delta mechanism was established and the inverse kinematics were
obtained.
(2) The numerical forward kinematics were calculated by Newton iteration algorithm.
(3) The analysis of velocity and acceleration are carried out through matrix operations.
(4) Kinematic analysis of Delta mechanism provides a theoretical basis for following study on the analysis of
dynamics and workspace.
VI. Acknowledgement
The paper is financially supported by Shanghai University of Engineering Science.
References
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