A new method and algorithm of solution of multijoint manipulators with rotational pairs inverse kinematic problem was created. The method is based on the principally new approach-spinor representation of the spatial generalized rotations. One of the advantages of the method is that it allows elaborating formulation of technological tasks for manipulators in terms of Cartesian coordinates instead of traditional angles’ terms. Besides the method allows using only one (absolute) coordinate system and does not require a set of different (relative) coordinate systems. It provides easy, reliable and efficient way of solution of inverse kinematics problem of multijoints manipulators with rotational pairs.spinors, rotations, Euler’s angles, basic representations, kinematics inverse problem, orthogonal transformations

HÃ¼seyin Ã–nal

English

Research Papers in Economics

A Spinor Method of Solution of Manipulators' Inverse 
Kinematics Problem with Rotational Pairs
Hüseyin ÖNAL
Abstract:  A  new  method  and  algorithm  of  solution  of  multijoint 
manipulators with rotational pairs inverse kinematic problem was created. The 
method is based on the principally new approach-spinor representation of the 
spatial generalized rotations. One of the advantages of the method is that it allows 
elaborating  formulation  of  technological  tasks  for  manipulators  in  terms  of 
Cartesian coordinates instead of traditional angles' terms. Besides the method 
allows using only one (absolute) coordinate system and does not require a set of 
different (relative) coordinate systems. It provides easy, reliable and efficient way 
of  solution  of  inverse  kinematics  problem  of  multijoints  manipulators  with 
rotational pairs.
Keywords:  spinors,  rotations,  Euler's  angles,  basic  representations, 
kinematics inverse problem, orthogonal transformations
Introduction
 Within the wide complex of problems in kinematics and dynamic 
parameters of spatial mechanism and mechanism with rotational pairs 
motions  the  inverse  problem  of  kinematics  (Shahinpoor  M.,  (1988)-
Gelgamd I.M., Milnos R.A., Shapiro Z.A., (1972)) is the most important. 
The principal difficulty in its solution is the problem connected with three 
dimensional  rotations.  The  problem  for  mechanisms  with  rotating 
kinematics pairs leads to considerable difficulties in calculation that result 
in reducing the accuracy of estimation of these angles worsens process 
controlling  quality.  For  spatial  mechanisms  with  spherical  joints  and 
rotational pairs it apparently has no solution at all. The reason for the 
situation is that the applied method of presenting the three dimensional 
rotations by means of the third order orthogonal matrix of (so called basic 
presentation of the three dimensional rotation group), whose elements are 
trigonometric functions of Euler angles (Shahinpoor M., (1988)-Gelgamd 
Hüseyin ÖNAL is an associate professor in Faculty of Computer Technologies and Engineering at 
International Black Sea University, Georgia
170
IBSU Scientific Journal      2 (1), 2008I.M., Milnos R.A., Shapiro Z.A., (1972)). This method, firstly, is for 
description of separate concrete rotations with the zero center (placed at the 
beginning of coordinates), and, secondly, it doesn't allow to express Euler 
angles in a form of coordinate functions of the three points determining 
rotating: center, beginning and final. 
Hence, the purpose of the present work is to develop a new method 
of solution of the inverse kinematics problem for spatial mechanisms with 
rotational pairs based on the spinor presentation of three dimensional 
rotation groups.
Definition of the problem
In  the  works  (Milnikov A.A.,  Prangishvili A.I.,  Rodonaia  I.D. 
(2005) - Milnikov A., Onal H., Partskhaladze R., Rodonaia I., (2004)), on 
the basis of spinor model of three-dimensional Euclidian space the simple 
relationships between elements of three-dimensional orthogonal matrix of 
the basic presentation and Eulerian angles on the one side and coordinates 
of the starting and final points of rotation, on the other side have been 
obtained. The mentioned results allowed developing a new method and an 
algorithm  of  solutions  of  the  inverse  problem  for  manipulators  with 
rotational pairs.
Let us consider the mechanism (manipulator) with n rotational pairs, the 
principal diagram of which is given in fig.1. We apply one system of 
coordinates which may be chosen quite arbitrarily. 
Figure 1 principal diagram of multijoints manipulator with rotational pairs: y-goal point 
of manipulator's rotation
171
IBSU Scientific Journal      2 (1), 2008The new algorithm leads to the formulization of the inverse task 
1 different from the traditional one : 
Given: Cartesian coordinates of all n joints 
1 2 3 1 coordinates of the point y(y , y , y ), where the manipulator final point x (x ,  n n
2 3 x , x ) should be moved; n n
1 2 3 Required to determine: new positions x(x , x , x ) (i=1,…,n-1) ;  i i i i
of  all intermediate joints and Eulerian angles  θ, ψ and φ (i=1,…,n) of all  i i i
manipulator segments, providing rotation of rotational joints of the final 
1 2 3 1 2 3 1 2 3 one x (x , x , x ) into y(y , y , y ) and intermediate ones x(x , x , x ) into  n n n n i i i i
1΄ 2΄ 3΄ x΄(x , x , x ) (i=1,…,n-1). i i i i
To solve the formulized problem an algorithm consisting of two 
stages has been developed.
The following important circumstances should be pointed out.  The 
nature of spinor method enables to work with Cartesian coordinates and use 
the unique system of coordinate which presents indubitable advantage 
compared with the classical method requiring to formulate the tasks for 
manipulators in terms of rotation angle.
Principal Part
Calculation of Euler angles for a Three-Member Manipulator 
with Rotational Pairs
The flow diagram of such a manipulator is shown in Figure 3.1. The 
initial positions of the manipulator joints are given as follows: x (0,0,0);   0
x (20,0,0); x (20,0,-10); x (20,30,-10).The joint lengths are equal to l =20,   1 2 3 1
l =10 and  l =20. It is required to move the terminal point of the manipulator  2 3
to the position y=(27.765,30.095,5.829). 
It is not difficult to see that by giving the initial positions of the 
manipulator joints in this manner we have succeeded in obtaining the 
following relative positions of the joints: the first joint (x  - x ) is positioned  0 1
1 1 along the Ox -axis, the second joint (x -  x ) is positioned in the plane x  = 20  1 2
2 3 (parallel to the  Ox x -plane), and the third joint (x  - x ) is orthogonal to the  2 3
1  second joint and also in the plane x = 20.
1 2 3 x(x , x , x ) (i=1,…,n) ;  i i i i
1 Before introducing its formulization it is necessary to make a remark concerning notations:  hereafter 
1 2 3 Cartesian coordinates of joints are presented by three-dimensional vector x(x , x , x ) , where the lower index  i i i i
is the joint's number  and the upper index is the number of coordinates; the length of the segment (link) 
between i-1 and i joints is designated by l.  Despite that the below mentioned considering is in  i
orthonormalized spaces, we preserved for vectors' coordinates upper index that are traditional for 
contravariant vectors.
172
IBSU Scientific Journal      2 (1), 2008Moreover, since the pairs are rotational, the first joint can rotate 
3 only in the plane x  = 0, the second joint – in the plane orthogonal to the 
3 plane x  = 0, and the third joint – in the plane orthogonal to the second plane. 
The solution of this problem by means of the spinor method is carried out 
by  the  same  algorithm  as  was  used  for  the  solution  of  problems  of 
manipulators with spherical pairs[A. Milnikov, H. Onal, C. Erguven, I. 
Rodonaia., (2006)]. The only difference consists in that in this case the 
plane       should be replaced by the equation of the plane     w h e r e  
rotation is to be performed. If the plane          of rotation centers was 
previously used only for the purpose of solving the nonlinear system (3.1), 
now the equation of the rotation plane         is the necessary restriction 
(constraint) imposed on joint motion.
In the context of the problem under consideration, the succession of 
calculations is as follows. 
We obviously see that joint 2 cannot serve as the center of rotation  
x  " y and therefore we should define its such position (if it exists) that  3
satisfies the above conditions. To this end we solve the system 
   ;
        ;
        , (1)
2 i i Where  А  = 1 А  = А  =0;  l  = 10; l  = 30; x  and y are the coordinates of the  1 2 3 2 3 1
initial position of the first joint and the target point. 
Table 3.1 presents the results of the solution of system (3.1). Note 
that since, as different from the problem with spherical joints, in this case 
the points x1 and x2 lie in the plane, the solution procedure becomes a little 
simpler. 
j jx x ¢ X
1 + Q
i ix x
j jx x ¢ X
1 + Q
i ix x
0 1 = +
¢
- D x A
i
n i
å
=
¢ = -
3
1
2
3
2
2) (
i
i i l x y
å
=
¢ = -
3
1
2
2
2
2 1 ) (
i
i i l x x
2 Such a choice of coefficients is due to the fact that the rotation of the point x2 can take place only in the 
1 plane x =20. 
173
IBSU Scientific Journal      2 (1), 2008Table 1 Succession of Calculations for the Solution of System (1)
Notation of the 
variable 
Geometrical description of the 
variable   Calculated value 
= D  
Free term on the equation of the 
plane 
2 x 1 x Θ  
-20 
2
3
2
1
A
A
A
A
 
Coefficients of the plane 
equation and the norm of the 
respective vector  
1
0
0
1
 
= -
= -
3 3
1 1
z x
z x
 
Distances from the points x   1
and  x4  to the plane 
y x4 Ξ   7.765
0
 
-
=
=
=
=
=
=
3
3
2
3
1
3
3
1
2
1
1
1
z
z
z
z
z
z
 
Coordinates of the points z1 and 
z3 which the orthogonal 
projections of the points x1 and 
x3 onto the plane 
2 x 1 x Θ  
829 . 5
095 . 30
20
0
0
20
 
2 = r
 
Square of the distance between 
the points z   2
and z4. 
 
039.686 
=
=
2
2
2
1
r
r
 
Squares of the radii  
 
705 . 839
100
 
=
=
=
z
y
x
 
Vectors of the new coordinate 
system constructed on the plane 
y x4 Ξ
   ) 0 ; 0 20 (
) 82 . 5 ; 09 . 30 , 0 (
90 . 601 , 58 . 116 , 0 (
- -
-
 
=
=
=
ZN
YN
XN
 
The orthonormalized new basis 
which makes up the orthogonal 
transformation matrix М
  
(0;-0.19;0.982) 
(0;-0.98;-0.19) 
(1;0;0) 
 
=
=
Y
X
 
Coordinates of the point of 
intersection of the 
circumferences on the plane 
y x4 Ξ
45  . . 9
26 . 3
-
-
 
=
=
=
'
'
'
/
/
/
3
2
2
2
1
2
x
x
x
 
Coordinates of the point  of 
intersection of the 
circumferences in the initial 
basis – the sought new position 
of the point  66 x   3
. 8
5
20
 
-
= - '
= - '
= - '
1 2
2
3 2
x x
y x
x x
 
Check of the distances from the 
new point x3 '  to the points x4, y 
and  
 
x2 
10
30
30
 
 
;
174
IBSU Scientific Journal      2 (1), 2008As seen from Table 3.1, the found point x΄ (20,5,-8.66) lies in the  2
1 plane x  = 20, which is what we wanted to establish.
Now we are to calculate the Euler angles for two rotations
x "x΄   and  x΄ " y. 2     2 3  
2. Rotation x "x΄ . 2     2  
a. We choose arbitrary values for the parameter α  and calculate the 
values of the parameter β (Milnikov A.A., Prangishvili A.I., Rodonaia I.D. 
(2005)) 
b. we define the orthogonal matrix of the basic representation 
(Milnikov A.A., Prangishvili A.I., Rodonaia I.D. (2005))
from which, comparing the elements of the obtained matrix with 
Euler transformation matrix, we easily calculate the Euler angles 
It is not difficult to verify whether our calculations are correct. For 
this, it is sufficient to substitute the calculated values of the Euler angles 
into transformation matrix and verify the fulfillment of the equality  x΄     2
=Ax . 2  
3. Rotation  x΄ " y. 3  
c. We define the displacement of the point x  occurring during the  3
rotation 
x "x΄ :   . 2     2  
718 . 10 ; 268 . 0 ; 1 ; 40 2 1 2 1 - = - = = = b b a a
 
a
0.999
0.043 -
0.025 -
0.05
0.865
0.499
0
0.5 -
0.866
æ
ç
ç
è
ö
÷
÷
ø
=
0 864 . 2 30
025 . 0 sin sin
0 sin sin
866 . 0 cos
/ /
0 0
/ = = =
- =
=
=
¢ ¢ ¢
x x x x x x x x x x x x y j q
q y
q j
q
) 66 . 8 ; 35 ; 20 ( 2 / 3 - = x
175
IBSU Scientific Journal      2 (1), 2008d. We move the origin to the rotation center x΄2
Next we choose arbitrary values for the parameter α and calculate 
the values of the parameter β (Milnikov A.A., Prangishvili A.I., Rodonaia 
I.D. (2005)).
e. We define the orthogonal matrix of the basic representation 
(Milnikov A.A., Prangishvili A.I., Rodonaia I.D. (2005))
from which, comparing the elements of the obtained matrix with 
transformation matrix, we easily obtain the Euler angles 
In this case, too, we can verify whether our calculations are correct 
through verifying the fulfillment of the equality y= αx΄ . 3
Thus we have defined the new positions of the joints x (0,0,0);    0
x (20,0,0); x (20,0,-10); x (20,30,-10); y=(27.765,30.095,5.829) and the  1 2 3
corresponding Euler angles, by means of which we can perform rotations of 
the joints from the initial positions to the new ones.
0
30
0
3
2
3
2 3
3
3
2
2
2
2 3
2
3
1
2
1
2 3
1
3
= - =
= - =
= - =
¢ ¢
¢ ¢
¢ ¢
x x
x x
x x
x
x
x
489 . 14
095 . 25
765 . 7
3
2
3 3
2
2
2 2
1
2
1 1
= - =
= - =
= - =
¢ ¢
¢ ¢
¢ ¢
x y y
x y y
x y y
005 . 13
634 . 17
1
4
2
1
2
1
- =
- =
=
=
b
b
a
a
0 0 0 024 . 55 16 . 52 41 . 57
665 . 0 sin sin
483 . 0 sin sin
539 . 0 cos
- = = =
=
=
=
y j q
q y
q j
q
176
IBSU Scientific Journal      2 (1), 2008Figure 2.  The flow diagram of a manipulator with Rotational Pairs:
a) ___ Initial position; b) …… Terminal position.
We conclude this subsection by making two remarks. 
Remark1: The use of the above approach to the solution of the 
inverse kinematic problem for a manipulator with rotational pairs requires 
that the rotation planes of manipulator joints be given explicitly. In our 
example the plane x = 20 is the given one, though, certainly, it can be chosen 
arbitrarily. 
Remark2: It has more than once been noted that the nature of the 
spinor method allows us to work directly with Cartesian coordinates and 
use, in that case, a single coordinate system. The latter fact is an undoubted 
advantage as compared with the classical method which requires that data 
for manipulators be given in terms of rotation angles. The considered 
problem can be reformulated in terms of rotation angles as follows: 
о 1 1. Rotate by 30  the second joint (x  _  x ) in the plane x  = 20; 1 2
о 2. Rotate by 15  the third joint (x  _ x ) in the plane orthogonal to the  2 3
second joint. 
It is not difficult to solve the problem in such a formulation, but it is 
rather  difficult  to  obtain  the  formulation  in  terms  of  rotation  angles 
177
IBSU Scientific Journal      2 (1), 2008proceeding from the formulation in terms of Cartesian coordinates. The 
latter is a separate problem which demands the development of special 
methods. 
Conclusion
Earlier obtained results of spinor representation for generalized 
three-dimensional    rotations to allowed to transform the statement of 
inverse task of kinematics: the spinor     method's nature allows to work 
with Cartesian coordinates and use the only coordinate system which is the 
doubtless advantage comparing with the classical method, requiring to 
formulate tasks for manipulators in terms of angle rotations. 
Such an approach allowed to develop the new method and two-
stage algorithm of solution of the inverse kinematics task for multilink 
spatial mechanisms with rotational pairs.
At the first stage the coordinates of new positions of all joints are 
being determined, at the second stage the orthogonal matrix of the basic 
representation  and  the  Euler  angles  are  being  calculated. A  concrete 
example of Euler angle calculation for the multilink mechanism with 
rotational pairs is given. 
References
1. Gelgamd I.M., Milnos R.A., Shapiro Z.A., (1972) Representations of 
Rotations Group and Lorentz Group, -425 p.  Nauka, Moscow
2. Milnikov A., H. Onal, C. Erguven, I. Rodonaia., (2006) A Spinor Method 
Solution of Manipulators' inverse Kinematics of spatial rotations and Euler's Angles: 
Problems of Mechanics International scientific Journal, No 1(22), p. 41-48, Tbilisi
3. Milnikov A., Onal H., Partskhaladze R., Rodonaia I., (2004) Spinor 
evaluation of Euler's angles: Jornal of applied mechanics, No2, p.p. 48-53.  Tbilisi 
4. Milnikov A.A., Prangishvili A.I., Rodonaia I.D. (2005) Spinor Model of 
Generalized Three-dimensional Rotations: Automation and remote Control, Vol.66, No 
6, p.p. 876-872, Moscow
5. Shahinpoor M., (1988) A Robot Engineering Textbook,.-527 p. Harper & 
Row publishers, New York 
178
IBSU Scientific Journal      2 (1), 2008

A Robot Engineering Textbook,.-527 p. Harper & Row publishers,

A Spinor Method Solution of Manipulators' inverse Kinematics of spatial rotations and Euler's Angles:

Spinor evaluation of Euler's angles:

Spinor Model of Generalized Three-dimensional Rotations: Automation and remote Control,

A Spinor Method of Solution of Manipulators’ Inverse Kinematics Problem with Rotational Pairs

http://journal.ibsu.edu.ge/index.php/ibsusj/article/download/38/35

A Spinor Method of Solution of Manipulators’ Inverse Kinematics Problem with Rotational Pairs

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