A Weak Generalized Inverse Applied to Redundancy Solving of Serial Chain Robots

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A Weak Generalized Inverse Applied to Redundancy Solving of Serial Chain Robots

Bertrand Tondu, Philippe Souères

To cite this version:

Bertrand Tondu, Philippe Souères. A Weak Generalized Inverse Applied to Redundancy Solving of

Serial Chain Robots. International Conference on Intelligent Autonomous Systems, Jul 2014, Padova,

Italy. �hal-02004970�

A Weak Generalized Inverse Applied to Redundancy Solving of Serial Chain Robots

Bertrand Tondu^1,2 and Philippe Soueres²

1Institut National de Sciences Appliquées, University of Toulouse, 31077 Toulouse

2LAAS/CNRS, 7 avenue du colonel Roche, 31400 Toulouse

Abstract: A peculiar form of right inverse derived from the theory of rectangu- lar matrix determinants is considered instead of the classic Moore-Penrose psueodinverse with the aim to get compact symbolic expressions for the redun- dancy solving of serial chain robots. Such an approach, essentially based on the of the closed-form expressions of the mmminors of the mnrobot Jacobian (m<n), is proposed as a new way for fast computation in inverse kin- ematic control.

Keywords: generalized inverse, redundant robot limbs.

Introduction

Redundant robot-limbs are essentially controlled in their operational space by a nu- merical computation of the Moore-Penrose pseudoinverse of their Jacobian. In partic- ular, the possibility to arbitrarily choose the associated projection operator is a power- ful mean to derive benefit from the kinematic redundancy for avoiding singularities or obstacles [1], [3]. The increasing computing power of robot controllers has made possible the on-line implementation of such methods although mathematical forums sometimes emphasized the computing burden involved by the use of ‘pinv’ or

‘pinv2’-type algorithms [4]. The goal of this paper is to analyze with a fresh eye the interest of weaker generalized inverses for deriving relatively compact symbolic ex- pressions which could be used in some cases for a direct and simple on-line imple- mentation.

Our start point is the classical definition of a generalized inverse of any mnmatrix M as being the matrix X satisfying at least the first or the second of the four following equations:















) 4 . 1 ( )

(

) 3 . 1 ( )

(

(1.2)

(1.1)

XM XM

MX MX

M XMX

M MXM

T T

The Moore-Penrose pseudoinverse is the unique matrix satisfying the four equations.

Matrices satisfying only the first condition are called {1}-Inverses and they are con- sidered as being the simplest ones [5]. We will consider in this paper a right inverse of M – i.e. a matrix X verifying MX=I_m and as a consequence a {1,2,3}-Inverse – intro- duced in the 80’ by the Indian mathematician Joshi [6] in a work first devoted to de- terminant definition of non-rectangular matrices. In section 2, we introduce Joshi’s determinant that we relate to zonotope theory and then Joshi’s weak generalized in- verse for which we propose an associated symbolic expression of the projection oper- ator. In section 3, the approach is applied to the look for compact symbolic expres- sions of redundant robot inverse kinematic models including singularity avoidance and multiple tasks realization.

Joshi’s weak generalized inverse definition and its use in linear underdetermined systems solving

Matrix determinant is a fundamental algebra notion essentially associated to square matrices. Attempts have been made to extend this notion to non-square matrices. If no global theory exists and, may be, would be meaningless in the general case of a rec- tangular matrix, the problem can more easily be approached if it is limited to mn matrices with mn. In this case, the matrix M can be read as a sequence of n vectors (j₁, j₂,…, jn) belonging to a m-dimensional vector space. An intuitive way to define a determinant associated to this vector sequence consists to consider all mmminors by a sign, computed from their components. Radic [7] proposed at his time a first definition of the determinant of such rectangular matrices. We will however prefer a more recent definition proposed by Joshi [6] for which it is easy to show that it is equivalent to the simple form:

det( ) det( , ,..., )

2 2

1

1

...

1

m m

i i n i i i

i j j

j

M













 (2)

i.e. the sum of the 







mn mmminors of the ordered sequence of the matrix n column vectors. We will adopt this definition first due to its simplicity and also because we can geometrically justify it by means of the zonotope theory [8]. It is indeed well known that the determinant of m vectors in a m-dimensional space can be interpreted as the oriented volume of the m-parallelepiped that the m vectors span. In some ex- tent, we propose to interpret the determinant of n vectors in a m-dimensional space, as defined in Equ. (2), as the sum of the oriented volumes of the 



 



 m

n m-parallelepipeds resulting from what is called a “cubical” dissection of the zonotope whose

) ,..., ,

(j₁ j₂ j_n are the generators. This idea is illustrated in Fig.1: let us consider the three vectors (j₁,j₂,j₃)of the 2-dimensional vector space; the three 22 minors

) ,

det(j₁ j₂ , det(j₁,j₃)and det(j₂,j₃)determine three oriented areas filling the zonotope











  







i n

i i i 1 3 2

1, , ) ,0 1

(j j j  j 

Z . From the famous Shepard-McMullen

theorem [9] the volume of this zonotope is equal to











i i i n

i i i

m

m

...

1 _{1 2}

2 1, ,..., ) det(j j j . Because however minors are signed, according to the order in the vector sequence, the determinant can be equal to the zonotope volume (Fig. 1.a) or not (Fig. 1.b).

Figure 1. Geometric interpretation of the determinant definition as a sum of signed parallelepi- ped volumes (here parallelogram areas) resulting from a “cubical” dissection of the zonotope whose matrix column vectors j1 , j2 , j3 are the generators , (a) Case of det(j1, j2 , j3): all involved minors are positive and the determinant is equal to the zonotope area, (b) Case of det(j₁ , j₃ , j₂):

two minors are positive, one is negative and the determinant is not equal to the zonotope area.

The volumic interpretation of the considered non-square matrix determinant also em- phasizes a major property shared with classic determinant theory: it is independent on the base in which matrix vectors are expressed. This will be an interesting property for application to robotics. From a similar determinant definition, Joshi proposes a very appealing definition of the inverse of a rectangularmnmatrix with mn, we will note M^Z in reference to zonotope theory. We put:

T

Z M C

M (1/det( )) (3) where C is a cofactor matrix-like, whose elements c_ij,1im, 1 jnare defined as follows: let E, F, G, H be respectively the submatrices of M of the order

j

j

j

det(j ,j )

det(j ,j )

det(j ,j )

 





j

j

j





(a) (b)

) 1 ( ) 1

(i  j , (i1)(n j), (mi)(n j), (mi)(j1)such that



















G H

F E M





 

aij where a_ij is the i^th row and j^th column element of M. We have there- fore:

)

det( 



 

 EH GF

cij (4) Let us illustrate this computation mode in the case of the Jacobian matrix J_3R of a classic 3R-planar robot made of three links whose lengths are respectively noted l₁, l₂, l₃ and associated rotation variables ₁, ₂, ₃ ; in frame R₀ shown in Fig. 2.a we get – Si and Ci respectively denote sin(i), cos(i), Sij and Cij , sin(i+j) and cos(i+j) :















     



123 3 123 3 12 2 123 3 12 2 1 1

123 3 123 3 12 2 123 3 12 2 1

3 l1C l C l C l C l C l C

S l S l S l S

l S l S

J _R l (5)

from which we derive the three 22 minors involving the three column vectors )

, ,

(j₁ j₂ j₃ that we will note D₁₂, D₁₃, D₂₃ as follows:









  

  



)

, det(

) (

) , det(

) (

) , det(

3 3 2 3 2 23

23 1 3 2 3 3 1 13

23 3 2 2 1 2 1 12

S l l D

S l S l l D

S l S l l D

j j

j j

j j

(6) and therefore the following expression of the generalized inverse results:























    



123 3 12 2 1 1 123 3 12 2 1 1

12 2 1 1 12

2 1 1

123 3 12 2 123

3 12 2 3

3

2 2

2 2

2 2

det 1

S l S l S l C l C l C l

S l S l C

l C l

S l S l C

l C l J J

R Z

R (7)

where: detJ_3R D₁₂D₁₃D₂₃l₁l₂S₂2l₂l₃S₃2l₁l₃S₂₃.

It is particularly interesting to note that the matrix of “cofactors” has a complexity in terms of algebraic or sinus/cosinus operations similar to this of the Jacobian matrix from which it results, with an associated relatively compact determinant expression.

Although, in the case of this example, (J_3RJ₃^T_R) is a 22matrix, the components of the closed-form of the Moore-Penrose pseudoinverse J₃^_R J₃^T_R(J_3RJ₃^T_R)^1are much more complex to express, even after all trigonometric simplifications were made. The symbolic compactness of Joshi’s inverse of any mnmatrix

) ,..., ,

(j₁ j₂ j_n

M  however has a major drawback: while the pseudoinverse becomes

singular if and only if det ( , ,..., ) 0

...

1

2

2 1

2

1 





i i i n

i i i

m

jm

j

j , Joshi’s inverse becomes

singular if and only if det( , ,..., ) 0

...

1 _{1 2}

2

1 





i i i n

i i i

m

jm

j

j . In other words, by comparison

with Moore-Penrose pseudoinverse, Joshi’s inverse introduces the algorithmic singu- larities corresponding to a sum of considered minors equal to zero without any single minor is equal to zero. We will see however in next paragraph how to avoid any sin- gularity.

Let us consider now the underdetermined linear equation v=Mu where v is a m- dimensional vector and u a n-dimensional vector (m < n). Because Joshi’s inverse is a {1,2,3}-Inverse of M, the general solution of the linear equation is given, if

0 )

det(M  , by:

z M M I v M

u ^Z ( _n ^Z ) (8) where z is an arbitrary n1matrix. According to our look for compact symbolic ex- pressions the question is: what is the compactness of the projection operator (InM^ZM)? We first want to emphasize a point shared by any right inverse and there- fore also by the Moore-Penrose inverse: (I_nInverse(M)M)is independent on the base in which column vectors of M are expressed. Let us consider M=M₁M₂ where M₂ is a mnmatrix (m < n) and M₁ is a square mmmatrix, uM^Zv(I_nM₂^ZM₂)z is a general solution of v=Mu since: MuMM^Zv(MM₁M₂M₂^ZM₂)zv. Let us consider a reference 0-base in which are expressed the column vectors of the matrix

0M; it follows that the general solution ⁰v = ⁰Mu is given, ifdet(M)0, by:

z M M I v M

u^{0 Z0} ( _n^{k Zk} ) (9) where z is an arbitrary n1matrix and ‘k’ designates an arbitrary base of the consid- ered m-dimensional space. In what follows, the base in which the projector is ex- pressed will be omitted. This result suggests that (I_nM^ZM)could be expressed only with minors since these last ones are base-independent. In the case where m=2 and any n, we get, for M=[a_ij] with i=1 to 2 and j=1 to n, and C its cofactor matrix:



















































       









n n n

n

n n

n n

T

a a a

a a a a a

a a

a a

a a

a a

a a

a a

a a

a a

2 22 21

1 12 11 1 1 12

11 1 2 22

21

1 13

11 2

23 21

1 13

12 2

23 22

...

...

...

...

...

...

...

...

 M 

C (10)

whose all resulting terms clearly correspond to a sum of minors D_kj=(a_1ka_2ja_1ja_2k), k<j, by a sign. A similar result can also be simply deduced in the case where m=3 and any n. This suggests the following conjecture we however not tried to demonstrate in the general case to privilege the robotic point of view:

 , _{1 ,1 j n} ,...,

, ...

1

...

...

3 2

3 2

3 2 3

) 2

det(







  





 

















n j i

i k k k

n k k k

k k jk k k ik Z

m m

m mD M 

M

M (11)

where

km

k ik_{2 3}...

 is equal to 1 if i < k₂ and 1 if not, and )

,..., ,

det( 2

3

2k...k_m j k k_m

Djk  j j j . In the case of our planar 3R-robot, we get:













 

 



 



















   



12 12 12

13 13 13

23 23 23 23 13 12 3

3

23 13 12 12

13 23 12 13

23 23

13 12 3

3 3

) 1 (

)

det(

D D D

D D D

D D D D D D

D D D D

D D

D D

D D

D D

R Z R

R Z R R

J J I

J J J

3

(12)

The claimed result of Equ. (11) opens the way to the development of original algo- rithms for solving underdetermined problems based on the computation of the matrix

mmminors in some alternative way to the classic singular value decomposition applied to the Moore-Penrose inverse. Moreover, it is easy to show that the number of minors involved in the computation of each term of the matrix det(M)(I_nM^ZM)

is equal to 





 









1 1 mn

mn on the diagonal and to 





  1 2 m

n outside. Consequently, very compact symbolic expressions can be hoped for a low degree r of redundancy defined as being r = (nm), especially for r = 1 or r = 2, which concerns most of redundant industrial robots; in particular, if r=1, each term of the projection operator matrix is equal to one single minor.

Application to redundancy solving of serial chain redundant robots

The matrix M is now the mnJacobian matrix denoted J of a serial chain robot with a number n of degrees of freedom greater than necessary to perform a given task in a m-dimensional operational space. The joint vector will be noted q and its components

n i

q_i,1  while the operational vector will be noted x. Since historical works by Nakamura [1] redundancy solving, in a kinematic point of view, can take two forms:

the optimization of a criterion on the one hand, the realization of multiple tasks on the other hand.

3.1 Criterion optimization

In the case of the pseudoinverse, the arbitrarily vector z associated to the projection operator (I_nJ^ZJ) can be used for maximizing a criterion thanks to the fact that

(I_nJ^ZJ)is a nonnegative definite matrix. That is not the case for a {1,2,3}-Inverse but this difficulty can be simply overcome if we define z as follows:

T n T

n Z

c q q q

k 



 















 

 Crit( )

) ...

( Crit ) ( ) Crit (

2 1

q q

J q J I

z (13)

where Crit(q) is a criterion to be maximized by the robot and k_c a positive real. By comparison with Moore-Penrose pseudoinverse computation, the supplementary term (I_nJ^ZJ)^Thas a negative effect on the look for a final compact symbolic expression of

q

but we think that this effect is limited by the symbolic form (I_nJ^ZJ) which can be derived from Equ. (11) especially for low redundancy degree, as we are going to illus- trate it. Moreover, in the practical important case of the look for singularity avoid- ance, and because our definition of the matrix determinant includes the robot singular- ities, we can consider, instead of the manipulability criterion, the following simpler one:

T n i i

T n

i i q

q _ _



 







 



 







 



1 1

2 det( )

) det(

) 2 ( Crit ) ( det ) (

Crit J

q J J

q (14)

where the terms det( )) (

qi



 J

are easy to deduce in closed form from the knowledge of minors expressions. Let consider for example the case of a 3R planar robot whose Jacobian inverse matrix J_3R^Z is given in Equ. (7). From the symbolic expression of its first minors D₁₂, D₁₃ and D₂₃ given in Equ. (6) and from Equ. (11), after all simplifica- tions were made, we finally get the final expression of including the satisfaction of q singularity avoidance by means of the positive real k_c – in fact it would be 6k_c from Equ. (13):



























12 13 23 23

3 2 2 1 13 12 23 1 3 2 3

3 (2 ( ) ( 2 ))

M M M C

l C l l M M C l C l l k J^Z_Rx _c

q  (15)

where C₂,C₃ and C₂₃ respectively denote cos(₂), cos(₃) and cos(₂+₃). We applied this equation to the example proposed by Yoshikawa in his book [10] (page 255): the planar 3R-robot with link lengths equal to l₁=l₂=1, l₃=0.3 must perform a straight-line path from the initial configuration q0 = [180°, 170°, 10°]^T, corresponding to the initial operational end-position [Px0 , Py0]^T[0.28, 0.17]^T with respect to frame (O0, X₀, Y₀), to the final one [P_x0 , 0.1]^T according to the desired trajectory





)

(t  P_x0 P_y0  t t² P_y0 ^T t

x (see Fig. 2.a). We give in Fig. 2.b

and 2.d the comparison between our approach and the classic pseudoinverse approach with maximization of the robot manipulability as defined by Yoshikawa

)

det(J_3RJ₃^T_R . It clearly appears that our approach can lead to a result very close to

this given by the pseudoinverse with very close changes for the manipulability crite- rion (see Fig. 2.c).

(a)

-1 -0.5 0 0.5

-0.5 0 0.5 1

X-position

Y-position

(b)

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Tim e (s)

Length squared dimension

| det(J3R)|

(c)

-1 -0.5 0 0.5

-0.5 0 0.5 1

X-position

Y-position

(d)

Figure 2. Simulation of a 3R-planar robot whose redundancy is used to avoid singularity in the Cartesian plane while its end-point must perform a straight-line path with some imposed veloc- ity profile, (a) Pseudoinverse without criterion, (b) Pseudoinverse with criterion, (c) Manipula- bility comparison between pseudoinverse with criterion (continuous line) and proposed method

with criterion (dotted line), (d) Proposed method with criterion (k_c=35).

3.1. Multiple tasks

In the case of the redundancy use for performing multiple tasks, the following well- known formula was established by Nakamura [1]:

z J J I J J I x J J x J x J

q ~ ~ )

)(

(

~ )

~ (

2 2 1

1 1

1 2 2 2 1

1         

    _{n n}

 (16)

-1 -0.5 0 0.5

-0.5 0 0.5 1

X-p osition

Y-position

Start

Goa l End -point trajectory

X0

Y0

where x₁J₁q defines the first task with J₁ a mnmatrix (m < n), x₂J₂q de- fines the second task with J₂ a pnmatrix (p < m), ~ ( )

1 1 2

2 J I J J

J  _n ^ and z an arbitrary n1vector. The same approach applied to our {1,2,3}-Inverse leads to:

z J J I J J I x J J x J J J I x J

q ~ ~ )

)(

(

~ )

~ ( )

( _{1 1 2 2 2 1 1 1 1 2 2}

1

1 Z

Z n Z n

Z n Z

Z      

   

 (16)

where ~ ( )

1 1 2

2 J I J J

J  _n ^{Z and z} an arbitrary n1vector. But, by comparison with the {1,2,3,4}-Inverse, we can not write ‘ _{n 1}^Z ₁ ~₂^Z ~₂^Z

)

(I J J J J ’. However the ex- pected advantage of Equ. (17) by comparison with Equ. (16) is to derive benefit from compact symbolic expressions of J₁^Z, ~₂^Z

J , (I_nJ₁^ZJ₁)^{as from} ~ ~ ) (I_nJ₂^ZJ₂ in order to get a final symbolic expression of qexpression able to favour a fast computa- tion of the redundancy solving. Let us illustrate this approach in the case of a 4R- planar robot with a degree of redundancy equal to 2 i.e. n=4 and m=2: the first task consists in positioning the robot end-point in the Cartesian plane. From an obvious extension of J_3R-matrix expression given in Equ. (5) for defining J_4R, we easily derive the following expression of J₄^Z_R– as previously the successive link-lengths of the robot are denoted l_i and its joint variables i, for i=1 to 4 :































            



1234 4 123 3 12 2 1 1 1234 4 123 3 12 2 1 1

1234 4 123 3 12 2 1 1 1234 4 123 3 12 2 1 1

1234 4 12 2 1 1 1234

4 12 2 1 1

1234 4 123 3 12 2 1234

4 123 3 12 2 4

4

3 3

2 3

3 2

2 2

2 2

3

3 2

3 2

) det(

1

S l S l S l S l C l C l C l C l

S l S l S l S l C

l C l C l C l

S l S l S l C

l C l C l

S l S l S l C

l C l C l

R ZR

J J

(18)

The determinant of J_4R is the sum of the six minors det( , )

2 1 2

1i i i

Di  j j for

4

1i₁i₂ where j_k is the k^th column-vector of J_4R for k=1 to 4; we get:











    

      



);

(

) (

);

(

) (

) (

);

(

4 4 3 34 34

2 4 3 4 24

34 4 3 3 2 23 234 1 34 2 4 3 4 14

234 1 34 2 4 23 1 3 2 3 13 234 4 23 3 2 2 1 12

S l l D S

l S l l D

S l S l l D S l S l S l l D

S l S l l S l S l l D S l S l S l l D

(19)

and therefore:

234 4 1 34 4 2 23 3 1 4 4 3 3 3 2 2 2 1

1 2 2 2 4 3

detJ ll S  l l S  l l S  ll S  l l S  ll S (20) The second task consists in using the two degrees of freedom brought by the redun- dancy to avoid the obstacle shown in Fig. 3.a by imposing adapted joint 3 and joint 4 trajectories denoted respectively ^_3dand^_4d. We get, as a consequence:





 00 00 0110

J2 (21)

We will now note J_4R=J₁ . From Equ. (11) we derive:





























      

      

     





23 13 12 23 13 23

12 13

12

24 14 24

14 12 24 12 14

12

34 14 34

13 34

14 13 14 13

34 24 34

23 24

23 34

24 23

1 1 4

1)( )

det(

D D D D D D

D D

D

D D D

D D D D D

D

D D D

D D

D D D D

D D D

D D

D D

D D

ZJ J I J

(22)

from which we can easily get symbolic expressions of ~₂

J and then _{4 1}^Z ₁ ~₂^Z ) (I J J J . Finally we check that ₃ ^_3d,  ₄ ^_4d and we get:



 





 

 



 



 

 

 





 





d d y

Z x lower Z

upper a a

a a P

P a

a a a

4 3 22 21

12 11 2 _

22 1 21

12 11 2 _

1 1 2

1

~ ) det(

) 1

~ ) det(

( 1 ) det(

1







 











J J J J

J

(20) where J₁^Z_{_upper} , J₁^Z_{_lower}are the 22upper and lower sub-matrices of det(J₁)J₁^Z, as expressed in Equ. 18 and:



























   

   

   



23 14 24 13 34 1 12 2

24 13 34 12 34 1 14 22

23 14 34 12 24 1 13 21

23 14 34 12 13 1 24 12

24 13 34 12 14 1 23 11

) ) (det(

~ ) det(

) ) (det(

) ) (det(

) ) (det(

) ) (det(

D D D D D D

D D D D D D

a

D D D D D D

a

D D D D D D

a

D D D D D D

a

J J

J J J J

(23)

Moreover it is easy to show that _{1 1} ~₂~₂) ₄ )(

(I_nJ^J I_nJ^J O which means that all the redundancy capability is here used for avoiding the obstacle. The symbolic ex- pressions involved in the algorithm are now more complex than in the case of the previous 3R-planar robot example but they are still acceptable for an on-line compu- tation.

We show in Fig. 3 the simulation of the algorithm on the following example: the 4R- planar robot whose all links have a length equal to 0.25 is initially set in a start con- figuration q₀=[45°,10°,20°,30°]^T from which it must move in a straight-line accord- ing to a trapezoidal speed profile – constant cruising speed is equal to 1 (link length dimension)/s and constant initial/final accelerations are equal to 2 (link length dimen- sion)/s²) – to a final position with same X-value and a Y-value equal to zero. In order to avoid the obstacle, the trajectories of 3 and 4 are imposed also according to a given trapezoidal speed profile from their following initial positions to the final one (_3f, _4f) = (25°, 45°). If this second task is not specified, the robot end-point fol- lows the straight-line imposed by the first task but comes against the obstacle (Fig.

3.b). Thanks to the second task, the obstacle is avoided (Fig. 3.c) with joint trajecto- ries in 1 and 2 deprived of excessive slopes as shown in Fig. 3.d.

(a)

(b)

(c) (d)

Figure 3. Simulation of a 4R-planar robot whose redundancy is used for following a straight- line path (first task) while avoiding an obstacle thanks to predefined joint trajectories in 3 and

₄ (second task), (a) Initial configuration and location of the obstacle, (b) Redundancy solving without the second task, (c) Redundancy solving with the second task, (d) Corresponding joint

variables to the performance of both first and second tasks.

Conclusion

We proposed a symbolic approach for a fast computation of redundancy solving in- cluding optimization criterion and multiple tasks realization. Our approach is founded on the determinant definition of a mnrectangular matrix (m < n) as the sum of its



 



mn mmordered minors, and the right inverse which can be associated to it. We highlighted the fact that each term of the projection operator associated to this inverse can be written as a limited sum of these minors, leading to a general redundancy solv- ing algorithm based on the symbolic expressions of the robot Jacobian minors. The

0 0.5 1

-0.2 0 0.2 0.4 0.6

X-p osition

Y-position

X₀ Y₀

Start

Goa l

End-point tra jectory

Velocity profile

obsta c le

0 0.5 1

-0.2 0 0.2 0.4 0.6

X-p osition

Y-position

0 0.5 1

-0.2 0 0.2 0.4 0.6

X-p osition

Y-position

0 0.5 1 1.5 2

-60 -40 -20 0 20 40 60 80

Time (s)

Joint positions (deg) ₁

4

3

2

proposed method however suffers the combinatorial increasing of the number of con- sidered minors, especially when the robot is considered in the 6-dimensional opera- tional space with a high degree of redundancy. We think however that the method can be particularly efficient in some important cases: on the one hand, in the plane, for nR snake-like robots, even with a high n-number of degrees of freedom, due to the fact we can hope to get recursive equations for the minor expressions as this is suggested by the obtained expressions in 3R and 4R-planar robot cases; on the other hand, for any spatial robot whose degree of redundancy is equal to 1, as it is the case for the classic 7R-arm, involving therefore seven66 minors and a projection operator ma- trix det(J)(I_nJ^ZJ)whose each term is equal to only one minor.

References

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