Design of Experiments for Calibration of Planar ... - IEEE Xplore

2011 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM2011) Budapest, Hungary, July 3-7, 2011

Design of Experiments for Calibration of Planar Anthropomorphic Manipulators Alexandr Klimchik1,2, Yier Wu1,2, Stephane Caro2, Anatol Pashkevich1,2. 1

Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France Institut de Recherches en Communications et en Cybernetique de Nantes, 44321 Nantes, France [email protected], [email protected], [email protected], [email protected] 2

Abstract – The paper presents a novel technique for the design of optimal calibration experiments for a planar anthropomorphic manipulator with n degrees of freedom. Proposed approach for selection of manipulator configurations allows essentially improving calibration accuracy and reducing parameter identification errors. The results are illustrated by application examples that deal with typical anthropomorphic manipulators. Keywords: calibration, anthropomorphic manipulator

I.

design

of

the number of experiments (with the factor 1 m , where m is the experiments number). Besides, using diverse manipulator configurations for different experiments looks also intuitively promising and perfectly corresponds to some basic ideas of the classical theory [17] that intends using the factors that are distinct as much as possible. However, the classical results are mostly obtained for very specific models (such as linear regression) and can not be applied directly here due to non-linearity of the relevant expressions. In this paper, the problem of optimal design of the calibration experiments is studied for case if a n-link planar manipulator, which does not cover all architectures used in practice but nevertheless allows to derive some very useful analytical expressions and to propose some simple practical rules defining optimal configurations with respect to the calibration accuracy. Particular attention is given to two- and three-link manipulators that are essential components of all existing anthropomorphic robots.

experiments,

INTRODUCTION

The standard engineering practice in industrial robotics assumes that the closed-loop control technique is applied only on the level of servo-drives actuating the manipulator joint variables. However, for spatial location of the endeffector, it is applied the open-loop control method that is based on numerous computations of the direct/inverse transformations that define correspondence between the manipulator joint coordinates and the Cartesian coordinates of the end-effector. This requires careful identification (i.e. calibration) of the robot geometric parameters employed in the control algorithm, which usually differ from their nominal values due to manufacturing tolerances [1]. The problem of robot calibration is already well studied and it is in the focus of research community for many years [2]. In general, the calibration process is divided into four sequential steps [3]: modeling, measurements, identification and compensation. First two steps focus on design of the appropriate (complete but non-redundant) mathematical model and carrying out the calibration experiments. Usually, algorithms for the third step are developed for the identification of Denavit-Hartenberg parameters [4], which however are not suitable for the manipulators with collinear axis considered in this paper. For this particular (but very common) case, Hayati [5], Stone [6], and Zhuang [7] proposed some modifications but we will use a more straightforward approach that is more efficient for the planar manipulators. Among numerous publications devoted to the robot calibration, there is very limited number of works that directly address the issue of the identification accuracy and reduction of the calibration errors [8-16]. It is obviously clear that the calibration accuracy may be improved by increasing

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II.

PROBLEM STATEMENT

Let us consider a general n-link planar manipulator which geometry can be defined by equations



n  i  x     li0  li ·cos   q 0j  q j   i 1  j 1   n  i  0 0 y     li  li ·sin   q j  q j   i 1  j 1 



where ( x, y ) is the end-effector position, li 0 , q j 0 are the nominal length and angular coordinates of the i-th link and actuator respectively, li and qi are their deviations from nominal values, n is the number of links. Let us also i

i

j 1

j 1

introduce notations  i0   q 0j , and  i   q j that will be useful for further computations. As follows from (1), the manipulators geometrical model includes 2n parameters {li , i , i  1, n} that must be identified by means of the calibration. It is assumed that each calibration experiment produces two vectors, which define the Cartesian coordinates of the

576

end-effector Pi  [ xi yi ]T and corresponding joint coordinates . Besides, the Qi  (q1i , q2i , ... , qni ) measurement errors for the Cartesian coordinates ( x ,  y ) are assumed to be iid (independent identically distributed) random values with zero mean and standard deviation  , while the measurement errors for the joint variables are relatively small. Hence, the calibration procedure may be treated as the best fitting of the experimental data {Qi , Pi } by using the geometrical model (1) that leads to the standard least-square problem. However, due to the errors in the measurements, the desired values {li ,  i , i  1, n} are always identified approximately. So, the problem of interest is to evaluate (in the frame of the above assumption) the identification accuracy for the parameters {li , i , i  1, n} and to propose a technique for selecting the set of the joint variables Qi  (q1i , q2i , ... , qni ) that leads to improvement of this accuracy (in statistical sense). To solve this general problem, let us sequentially present the calibration algorithm, evaluate related identification errors and develop optimality conditions allowing minimize the number of experiments for given accuracy in identification of the desired parameters.

m F  n      ll0  ll  sin  l0( i )   l  k i 1  l i





 n     ll0  ll  cos  l0( i )   l  l 1



   x   i



   ll0  ll  cos l0( i )   l



 n     ll0  ll  sin l0( i )   l  l 1

   y   0

n

l i



 







 

i



m F   (cos( k0(i )   k )  lk i 1

 n       (ll0  ll ) cos( l0(i )  l )   xi   sin( k0(i )   k )   l 1  n       (ll0  ll ) sin( l0(i )  l )   yi    0  l 1    j

where k  1, n ,  (j i )   qk(i ) is the orientation of j-th link in k 1

III.

the i-th experiment. Since this system of equations is nonlinear with respect to  i , it does not have general analytical solution. Thus, it is reasonable to linearize the model (1)

CALIBRATION ALGORITHM

As follows from the previous Section, the input data for the manipulator calibration are its joint coordinates Qi  (q1i , q2i , ... , qni ) and corresponding end-effector positions Pi  [ xi yi ]T , i  1, m . The goal is to find



unknown parameters Π  {li , i , i  1, n} which ensure the best mapping of the coordinates Qi to the end-effector positions Pi that is defined by the geometrical model (1), which may be re-written in a general form as

where P0i is the end-effector position for the nominal values of parameters and the joint variables

Pi  P0i  J i  П 



T

n  n  P0i    lk0 cos  k0( i )  lk0 sin  k0( i )  , i  1, m , k 1  k 1   xi  f x  Qi , Π  ; yi  f y  Qi , Π  ; i  1, m   J is the Jacobian matrix, which can be computed by i differencing the system (1) with respect to П that leads to where f x  Qi , Π  , f y  Qi , Π  are the right-hand sides of system (1).  J ( i ) J (xli )     J i   (xqi ) (i )  To compute Π  {li , i , i  1, n} , let us apply the  J yq J yl  22 n least-square method which minimizes the residuals for all experimental configurations. Corresponding optimization where problem can be written as

т

 F  

i 1

J (xqi )   l1  sin 1(i ) ... ln  sin  n( i )  1n

 f Q , Π   x    f Q , Π   y    min 2

2

x

i

i

y

i

i



 

and it can be solved by using stationary condition at the extreme point F / Π i  0 for i  1, 2n with respect to

J (yqi )  l1  cos 1(i ) ... l j  cos  (j i )  1n J (xli )   cos 1( i ) ... cos  n(i )  1n



J (yil)  sin 1(i ) ... sin  n(i )  1n

Π  {li , i , i  1, n} . Corresponding derivations yield

Taking into account (6), the function (3) can be rewritten as

577





т



F    J i  П  Pi  i 1

T

 J i  П  Pi    min 

where Pi  Pi  P0i and expressions (4) are reduced to т

 (J



i 1

T i

 where ε a  ε1 ε 2 ... ε m  . As follows from (13), the latter expression produces unbiased estimates T

т

 J i )·П   (J i  Pi )  T



i 1

So, the unknown parameters П , can be computed as П  (J аT  J а ) 1  J аT  Pа 



Besides, it can be proved that the covariance matrix of the parameters П [18], defining the identification accuracy, can be expressed as

1

 m  cov(П )   2    J iT  J i     i 1 



i1

as

m

J



i1

1

T i

A B  Ji      C D

B  C  l j s jk  ;

A  l j lk ·c jk  ;





m

m

i 1

i 1

D   c jk  ;

c jk   cos( (j i )   k(i ) ); s jk   sin( (j i )   k(i ) );   j  1, n; k  1, n

where Aj , j  m·l j lk ; B j , j  C j , j  0; D j , j  m, j  1, n , which can be presented via block matrix

 

collect all measurement

П   J Ta  J а   J Ta   Pa  ε а  

 Ji  .

where

m

 J

T i

L  C  L L  S   Ji     T C    L  S 



where L  diag (l1 , l2 ,..., ln ) , C  c jk  , j  1, n; k  1, n ,

errors. So, expression (11) for computing the vector of the desired parameters П has to be rewritten as 

T i

For the considered model (1), this sum can be expressed

i1

T

m

 J

defined by the matrix sum

Let us assume that the measurements of x, y are carrying out with some random errors  xi ,  yi that are assumed to be iid, with the standard deviation  and zero mean value. Thus, model (6) can be rewritten as

where the vector εi   xi ,  yi 



Therefore, for the problem of interest, the impact of the measurement errors (i.e. “quality” of the experiment plan) is

EXPERIMENT

Pi  P0i  J i  П  ε i 

cov(П )  (J Ta  J а ) 1  J Ta  E  ε а ·εTa   J а  (J Ta  J а ) 1  

is the identity matrix of the size 2n  2n , the expression (15) can be simplified to

Step 1. Carry out experiments and collect the input data in the vectors of generalized coordinates Qi and endeffector position Pi ( xi , yi ) . Initialize П  0 . Step 2. Compute end-effector position via direct kinematic model (1) using initial generalized coordinates Qi Step 3. Compute residuals and unknown parameters П via (11) Step 4. Correct mathematical model and generalized coordinates l j  l j  l j ,  ji   ji   j , j  1, m . Step 5. If required accuracy is not satisfied, repeat from Step 2. It should be mentioned, that the proposed iterative algorithm can produce exact values of {li , i , i  1, n} if and only if there are no measurement errors in the initial data {Qi , Pi } . Since in practice it is not true, it is reasonable to minimize the measurement errors impact via proper selection of {Qi , Pi } .

ACCURACY OF CALIBRATION



Then, taking into account that E  ε а ·εTa    2 ·I 2 n , where I 2n

T



1

 

where J a   J1 J 2 ... J m  ; Pa  [P1 P2 ... Pm ]T . To increase the identification accuracy, the foregoing linearized procedure has to be applied several times, in accordance with the following iterative algorithm:

IV.

E  П    J Ta  J а   J Ta  Pа 



S   s jk  , j  1, n; k  1, n .

This expression allows estimating the identification  accuracy and it can be applied for optimal design of calibration that is presented in the following Section.

578

V. DESIGN OF CALIBRATION EXPERIMENTS To optimize location of experimental points in the Cartesian space (and corresponding manipulator configurations), let us investigate in details all components of the matrix

m

 J



 J i  that is similar to the “information

T i

i1

which leads to

matrix” in classical design of experiments. As it is known [17], this matrix can be evaluated by several criteria. The most common of them are A- and D-optimality criteria, but here it is not reasonable to use the A- criterion because the m

 J

trace of the matrix

T i

i1

0, if b  j , b  k 0, if b  j , b  k s jk     s jk , if b  j ;  c jk , if b  j   b   b   s jk , if b  k c jk , if b  k

c jk

C /  b  0;



S /  b  0 



The latter guarantees maximum of the relevant determinant and ensures agreement with the D-optimality. Validity of the proposed approach and its practical significance was also conformed by a simulation example that deals with 4-links manipulator with geometrical parameters l1  260 mm , l2  180 mm , l3  120 mm , l4  100 mm and their deviations l1  1.5 mm , l2  0.6 mm , l3  0.4 mm , l4  0.7 mm ; and

 J i  does not depend on the

experiment plan. Besides, the D- criterion is also not applicable here in its direct form. Hence, let us introduce a modified D*-optimal criterion which takes into account the structure of the information matrix in this particular case. Since this matrix includes several blocks with different units (linear, angular, etc.), it is reasonable to focus on optimization of each block separately. This approach allows to reformulate the problem and to define the goal as

deviation of zero values of angular coordinates q1  0.5o ,

q2  0.5o , q3  0.7o , q4  0.3o . All experiments were carried out for 10 random experimental points, the results are summarized in the Figure 1. They show that  det  C   max ; det  S   min   random plans give rather poor results both for D-optimality q j , j 1, m q j , j 1, m and D*-optimality criteria comparing to the optimal ones (for the optimal plans det  C   1 and det  S   0 ; det  D   1 , where C   c jk / n  , S   s jk / n  correspond to the where D is normalized block matrix (19)). diagonal and non-diagonal blocks of (19) respectively. It can be proved that this goal is satisfied if det(D) 1



j  1, n; k  1, n; j  k 

c jk  0; s jk  0;



0.8 0.6

that perfectly corresponds to the classical D-optimality conditions. For practical convenience, cases of 2-, 3- and 4links manipulators were investigated in details and corresponding optimality conditions are presented in Table 1. A correspondence between the proposed approach and the D-optimality can be also proved analytically. In particular, straightforward computations give

TABLE I. Manipulator

0.2 0 0

c i 1

m

m

c

2i

 0;  s2i  0;

2i

 0;  s2i  0;

i 1

m

c i 1

m

c i 1

23i

i 1

m

 0;

i 1

m

s i 1

23i

 0;

20

30

40

50

OPTIMAL PLAN CONDITIONS FOR 2-, 3- AND 4-LINKS MANIPULATORS

m

3-links manipulator

10

Figure 1. Determinant values of matrix D' for 4-links manipulator for random calibration plans with 10 experimental points:

Conditions for optimal plan

2-links manipulator

4-links manipulator

0.4

2i

m

c

 0;

3i

 0;

24 i

 0;

m

c i 1

m

c

s i 1

3i

i 1

i 1

 0;

m

0,

2i

m

3i

 0;  c23i  0;

3i

 0;

i 1

m

s

i 1

m

s i 1

c2 i  cos q2i ; s2 i  sin q2i ; i  1, m m

s

i 1

Notation

24 i

m

c i 1

 0;

 0;

4i

i 1

m

34 i

579

0

23i

m

s i 1

c i 1

m

s

 0;

 0;

4i

m

s i 1

34 i

c3i  cos q3i ; s3i  sin q3i ; c23i  cos(q2i  q3i ) ; s3i  sin(q2i  q3i ) ; i  1, m c4 i  cos q4i ; s4 i  sin q4i ;

c24i  cos(q2 i  q3i  q4i ) ; s24i  sin(q2i  q3i  q4i ) ; 0

c34i  cos(q3i  q4i ) ; s34i  sin(q3i  q4i ) ; i  1, m

For the proposed set of calibration experiments, the calibration accuracy can be estimated via the covariance matrix, which in this case is diagonal and may be presented as

VI. SIMULATION STUDY Let us present some simulation results that demonstrate efficiency of the proposed technique for several case studies that deal with two-, three- and four-links manipulators and employ different number of calibration experiments. It is 0 assumed that in all cases the calibration experiments were 2  m·L  L    cov(П )   ·  designed in accordance with expressions developed in I  0 Section 5 ( see Table 1). To obtain meaningful statistics, the simulation was repeated 10000 times; the deviation of where L  diag (l1 , l2 ,..., ln ) , and identification accuracy can measurement error  was equal to 0.1 mm. be evaluated as It was also assumed that the manipulator geometrical parameters are l1  260 mm , l2  180 mm , l3  120 mm ,    ; ; i  1, n   l4  100 mm and their deviations are equal to 1.5 mm,  qi   Li  m  li m 0.6 mm, -0.4 mm and 0.7 mm. respectively, while the deviation of zero values of angular coordinates 0.5°, -0.5°, 0.7° and -0.3° for the first, second, third and fourth joints where  qi ,  Li are standard deviations of angular ( qi ) and respectively. Short summary of the simulation results are linear ( li ) parameters from the nominal values. presented in Table 3 and in Figure 2. The results show that identification errors of the linear As follows from this study, the identification accuracy of parameters depend only on the number of experimental the experimental result and analytical estimations are in good points, while the angular parameter errors also depend on the agreement. In particular, for linear parameters, the link length. TABLE II.

ESTIMATION OF THE IDENTIFICATION ACCURACY OF GEOMETRICAL PARAMETERS: ANALYTICAL SOLUTION m

 (J

Manipulator

i 1

T i

 Ji )

Identification accuracy

2-links manipulator

diag (m  l12 , m  l2 2 , m, m) ,

3-links manipulator

diag (m  l12 , m  l2 2 , m  l32 , m, m, m)

4-links manipulator

diag (m  l1 , m  l2 , m  l3 , m  l4 , m, m, m, m) 2

TABLE III. Manipulator 2-links manipulator

3-links manipulator

4-links manipulator

 q1 

 m  l1

2

m  l1

;  q2 

 q1  2



 q1 

2

 L1 

 m  l1

 m

;



;  q2 

 m  l2

m  l2

;  q3 

;  q2 

 L2 

 m  l2

 m

;  L1 

 m  l3

 L3 

;

m

;  L1 

;  q3 

;



 m

 m  l3

 m

 L2 

m



;  L2 

m

;  L3 



;  q4 

;



m  l4

 L4 

 m

;

 m

ESTIMATION OF IDENTIFICATION ACCURACY OF GEOMETRICAL PARAMETERS Identification accuracy

Model parameters L1  260 mm, L1  1.5 mm,

q1  0.5 deg

L2  180 mm, L2  0.6 mm, q2  0.5 deg

L1  260 mm, L1  1.5 mm,

,

q1  0.5 deg

3 experimental points

20 experimental points

L1  0.058 mm, q1  0.013deg

L1  0.058 mm, q1  0.005deg

L2  0.058 mm, q2  0.018deg

L2  0.058 mm, q2  0.007 deg

L1  0.058 mm, q1  0.013deg

L1  0.022 mm, q1  0.005deg

L2  180 mm, L2  0.6 mm, q2  0.5 deg

L2  0.058 mm, q2  0.018 deg

L2  0.022 mm, q2  0.007 deg

L3  120 mm, L3  0.4 mm, q3  0.7 deg

L3  0.058 mm

L3  0.022 mm

L1  260 mm, L1  1.5 mm,

q1  0.5 deg

q3  0.027 deg

q3  0.011deg

L1  0.058 mm, q1  0.013deg

L1  0.022 mm, q1  0.005deg

L2  180 mm, L2  0.6 mm, q2  0.5 deg

L2  0.058 mm, q2  0.018deg

L2  0.022 mm, q2  0.007 deg

L3  120 mm, L3  0.4 mm, q3  0.7 deg

L3  0.058 mm

q3  0.027 deg

L3  0.022 mm

q3  0.011deg

L4  100 mm, L4  0.7 mm,

L4  0.058 mm

q4  0.033deg

L4  0.022 mm

q4  0.013deg

q4  0.3 deg

580

0.1

H (l2 ), mm

H (l1 ), mm

0.1

0.1

l1

0.08 0.06

l2

0.08

0.04

0.06

0.04

10

15

20

m

H (q1 ),deg

q1

10

15

2

20

m

H (q2 ),deg

q2

10

20

m

2

20

m

0.06

q3

2

10

10

15

20

m

20

m

H (q4 ),deg

q4

0.05

0.03

2

15

20

m

0.01 5

2

0.03

0.02

0.005 5

1

0.02 5

0.04

1 15

15

0.04

0.01 10

0.06

H (q3 ),deg

0.015

1

l4

0.08

0.04

0.02 5 0.05

0.02

0.01

H (l4 ), mm

1

1

0.025

0.015

0.005 5

0.02 5

0.03

0.02

l3 2

0.04

1 0.02 5

0.1

0.08

2

0.06

2

H (l3 ), mm

0.02

1 10

15

20

m

0.01 5

1 10

15

Figure 2. Identification accuracy for the geometrical parameters identification of 4-links manipulator with optimal experiment planning: "x" are experimental values corresponding to the optimal calibration plan, "o" are experimental values corresponding to the standard calibration plan "1" is analytical curve coresponding to the optimal plan, "2" is an average experemental curve corresponding to 10000 random calibtration plans.

identification error reduces from 0.022 mm to 0.005 mm while the experiment number increases from 4 to 20. Besides, these results allow defining minimum number of experimental points to satisfy the required accuracy. Thus, to satisfy an accuracy of 0.001 mm for linear parameters it is required to carry out 100 experiments, which will provide accuracy for angular parameters 0.002°, 0.003°, 0.005° and 0.006° respectively.

[3] [4] [5]

[6] [7]

VII. CONCLUSION The paper presents a new approach for design of calibration experiments that allows essentially reducing the identification errors due to proper selection of the manipulator postures employed in the measurements. There were obtained analytical expressions describing set of the optimal postures corresponding the proposed D*-criterion that is adopted to special structure of the information matrix. Validity of the obtained results and their practical significance were confirmed via simulation study that deals with two-, three- and four-links planar manipulators. Compared to previous contributions, these results can be treated as further development of the design-of-experiments theory that is adapted to the specific type of the non-linear models that arise in robot kinematics. Future work will focus on extension of these results for non-planar manipulators.

[8]

[9]

[10] [11]

[12]

[13] [14]

ACKNOWLEDGEMENTS The work presented in this paper was partially funded by the Region “Pays de la Loire”, France and by the project ANR COROUSSO, France.

[15]

REFERENCES [1]

[2]

[16]

Z.Roth, B.Mooring and B.Ravani, “An overview of robot calibration,” IEEE Journal of Robotic and Automation, Vol. 3, No 5, , 1987, pp. 377-385. A.Y. Elatta, Li Pei Gen, Fan Liang Zhi, Yu Daoyuan and Luo Fei, “An Overview of Robot Calibration,” Information Technology Journal Vol 3, No 1, 2004, pp. 74-78.

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