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
alexandr.klimchik@mines-nantes.fr, yier.wu@mines-nantes.fr, stephane.caro@irccyn.ec-nantes.fr,
anatol.pashkevich@mines-nantes.fr
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
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
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.
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

i


y     li0  li ·sin   q 0j  q j  
i 1 
j 1



n
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
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.
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
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)
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 Q i 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 )
 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 / Πi  0 for i  1, 2n with respect to
Π  {li , i , i  1, n} . Corresponding derivations yield
J
(i )
yq
 l1  cos 1(i )
... ln  sin  n( i ) 
1n
... l j  cos  
(i )
j
1n
J (xli )   cos 1( i )
... cos  n( i ) 
J (yil)  sin 1( i )
... sin  n( i ) 

1n
1n
Taking into account (6), the function (3) can be rewritten as


т

F    J i  П  Pi 
i 1
T
 Ji  П  Pi    min 
where Pi  Pi  P0i and expressions (4) are reduced to
т
 (J

i 1
i
T
 J i )·П   (J i  Pi ) 
T
1
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
i 1
So, the unknown parameters П , can be computed as

E  П    JTa  J а   JTa  Pа 

т
T
 where ε a  ε1 ε 2 ... ε m  .
As follows from (13), the latter expression produces
unbiased estimates
cov(П)  (J Ta  J а ) 1  J Ta  E  ε а ·εTa   J а  (J Ta  J а ) 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:
Then, taking into account that E  ε а ·εTa    2 ·I 2 n , where I 2n
Step 1. Carry out experiments and collect the input data
in the vectors of generalized coordinates Q i and endeffector position Pi ( xi , yi ) . Initialize П  0 .
Step 2. Compute end-effector position via direct
kinematic model (1) using initial generalized coordinates Q i
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 } .

T
IV.
1
Pi  P0i  J i  П  εi 
 m

cov(П)   2    J iT  J i   
 i 1

T
m
1
 Ji  .
i1
For the considered model (1), this sum can be expressed
as
 J
m

i1
T
i
A B 
 Ji   

 C D

where
A  l j lk ·c jk  ;

B  C  l j s jk  ;
m
m
i 1
i 1
D  c jk  ;
c jk   cos( j(i )   k(i ) ); s jk   sin( j(i )   k(i ) );  
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   JTa  J а   JTa   Pa  ε а  
T
i
j  1, n; k  1, n
errors. So, expression (11) for computing the vector of the
desired parameters П has to be rewritten as

J
defined by the matrix sum
J
m
i1
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
ACCURACY OF CALIBRATION EXPERIMENT
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

is the identity matrix of the size 2n  2n , the expression (15)
can be simplified to
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 ,
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.
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
J

 J i  that is similar to the “information
m
of the matrix
T
i
which leads to
i1
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
J
m
trace of the matrix
T
i
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
i1
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  0.7 o , q4  0.3o . All experiments
were carried out for 10 random experimental points, the
results are summarized in the Figure 1. They show that

 random plans give rather poor results both for D-optimality
det  C  max ;
det  S  min 
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)
Optimal plan
1

c jk  0; s jk  0;
j  1, n; k  1, n; j  k 

0.8
Random plan
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
m
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
Experimental points
OPTIMAL PLAN CONDITIONS FOR 2-, 3- AND 4-LINKS MANIPULATORS
c
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
m
s
0,
2i
i 1
3i
i 1
i 1
 0;
m
3i
 0;  c23i  0;
3i
 0;
i 1
m
s
i 1
m
s
i 1
c2i  cos q2i ; s2i  sin q2i ; i  1, m
m
s
i 1
Notation
24 i
m
c
i 1
 0;
 0;
4i
m
34 i
0
23i
i 1
m
s
 0;
 0;
4i
i 1
c
i 1
m
s
m
s
i 1
34 i
c3i  cos q3i ; s3i  sin q3i ;
c23i  cos(q2i  q3i ) ; s3i  sin(q2i  q3i ) ; i  1, m
c4i  cos q4i ; s4i  sin q4i ;
c24i  cos(q2i  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 ,



 l4  100 mm and their deviations are equal to 1.5 mm,  qi 
;
 Li 
;
i  1, n 
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  l , m  l2 , m  l3 , m  l4 , m, m, m, m)
 q1 

m  l1
2
TABLE III.
2
m  l1
;  q2 
 q1 
2
1

 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
Manipulator
2-links
manipulator
3-links
manipulator
4-links
manipulator
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  0.018deg
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
0.1
H (l1 ), mm
H (l2 ), mm
0.1
0.1
l1
0.08
l2
0.08
2
0.04
0.06
0.06
0.04
0.04
1
0.02
5
15
20
m
10
15
20
m
q2
0.025
0.015
0.02
2
0.02
5
1
10
15
20
m
20
m
2
0.06
q3
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
0.02
5
0.04
1
10
15
0.04
0.01
1
2
0.04
0.015
0.01
0.06
0.05
0.03
q1
l4
0.08
H (q3 ),deg
H (q2 ),deg
0.02
H (l4 ), mm
1
0.02
5
H (q1 ),deg
0.005
5
l3
2
1
10
0.1
0.08
2
0.06
H (l3 ), mm
0.02
1
1
10
15
20
m
0.01
5
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.
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.
[3]
[4]
[5]
[6]
[7]
[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.
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