A Relialble Approach to Compute the Forward Kinematics of Robot with Uncertain
Geometric Parameters
注意：本论文已在《Asian
simulation conference: The fifth international conference on system simulation
and scientific computing (Shanghai) (ICSC'2002) 》November, 36, 2002.
Shanghai, China.发表
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ABSTRACT:Uncertainties
widely exist in engineering structural analysis and mechanical equipment
designs, and they cannot always be neglected. The probabilistic method,
the fuzzy method and the interval method are the three major approaches
to model uncertainties at present. By representing all the uncertain length
and the uncertain twist of the link parameters, and the uncertain distance
and the uncertain angle between the links as interval numbers, the static
pose (position and orientation) of the robot end effector in space was
obtained accurately by evaluating interval functions. Overestimation is
a major drawback in interval computation. A reliable computation approach
is proposed to overcome it. The presented approach is based on the inclusion
monotone property of interval mathematics and the physical/real means
expressed by the interval function. The interval function was evaluated
by solving the corresponding optimization problems to determine the endpoints
/ bounds of every interval element of the solution. Moreover, an intelligent
algorithm named as realcode genetic algorithm was used to locate the
global optima of these optimization problems. Before using the present
approach to determine the response interval of uncertain robot system,
some mathematical examples were used to examine its efficiency also.
Key words: robot kinematics; interval analysis; global optimization;
uncertain geometry parameter
Introduction
When computing the robot forward kinematic, the
nominal values for the link and joint parameters provided in the user manuals
are used. Due to the manufacturing tolerance, the assembling error and part
wear, the actual values for the kinematic parameters are always different from
the given one. So the actual working envelop is different from the one reading
from the robot controller computing with the nominal parameters. The Monte Carlo
method is applied in a statistic way, but the computation is timeconsuming to
emulate all states [1].
The
probabilistic method, the fuzzy method and the interval method are the three
major approaches to model uncertainties at present [3]. Probabilistic approaches
are not deliver reliable results at the required precision without sufficient
experimental data to validate the assumptions made regarding the joint
probability densities of the random variables or functions involved [4]. When
the fuzzysetbased approach is used, sufficient experimental data are needed to
determine the subject function also. As to obtain these sufficient experimental
data is so difficult and expensive in some engineering cases, analyzers or
designers have to select the probability density function or the subject
function subjectively. In this situation, the reliability of the given results
is doubtable. A realistic or natural way of representing uncertainty in
engineering problems might be to consider the values of unknown variables within
intervals that possess known bounds [2]. This approach is so called interval
method (or interval analysis).
In the last 20 years, both of
the algorithmic components of interval arithmetic and their relation on
computers were further developed. However, overestimation of an interval
function is still a major drawback in interval analysis.
By representing uncertain
geometric parameters as interval numbers, this paper presents a novel approach
to compute the forward kinematics of robot by solving a series of interval
functions. And a reliable approach to evaluate the interval functions’ values
was proposed also to obviate overestimation, the major drawback in interval
computation. In this approach, these interval functions were estimated by
solving a series of global optimization problems. An intellective algorithm
named as realcode genetic algorithm was used to solve the optimization problems
also. Numerical examples were given to illustrate the feasibility and the
efficiency.
the interval
computational model to compute
the forward kinematics of robot with uncertain geometric parameters
(1)
Determinate computational model of robot
Fig. 1 DH convention for robot link coordinate system
The
robot kinematic model is based on the DenavitHartenberg (DH) convention. The
relative translation and rotation between link coordinate frame i1
and i can be described by a homogenous
transformation matrix, which is a function of four kinematic parameters
,
,
and
as shown in Fig. 1.
The homogenous
transformation A_{i} is given
in Eq. (1)
(1)
Using the homogenous transformation matrix the relationship of the endeffector
frame with respect to the robot base frame can be represented as in Eq. (2):
(2)
(2) The robot
kinematic model using parameters with interval uncertainty
When
the kinematic parameters θ_{i},
d_{i}, α_{i}, a_{i}
have no fixed value but having the values falling in the intervals [θ_{i}],
[d_{i}], [α_{i}],
[a_{i}] randomly, expanding
the Eq. (2) with the intervals, we get,
(3)
with
solution
of the interval computational model of robot with uncertain geometric parameters
(1) Brief review of some definitions and properties in interval
mathematics [78]
For
two interval number
and
, (
,
is the set of real compact intervals), the interval arithmetic was defined as
follows.
,
,
and
(for
).
If
, then the interval
degenerates to a real number a, i.e.
. In this way, interval mathematics can be considered as a generation of real
numbers mathematics. However, only some of the algebraic laws, valid for real
numbers, remain valid for intervals. The other laws hold only in a weaker form.
For example, a nondegenerate interval
has no inversion with respect to
addition or multiplication. Even the distributive law has to be replaced by the
socalled subdistributivity
(4)
Let
be given by a mathematical
expression
, which is composed by finitely many elementary operations
and standard functions
. The following inclusion monotone holds.
for
(5)
where,f([x])
is an interval also and which
stands for an interval arithmetic evaluation of
f
over
.As
x∈[x]
, the relation (6) can be obtained.
(6)
whence
(7)
Where
R(f,[x])
denotes the range of
f
over
.
(2) A new approach to evaluate interval functions
Overestimation
is a major drawback in interval computation. Based on the inclusion monotone
relation (7) and the physical/real means expressed by the interval function, a
new approach to evaluate interval functions was proposed in this work.
Relation (7) is the fundamental
property on which nearly all applications of interval arithmetic are based. It
shows that it is possible to compute lower and upper bounds for the range over
an interval by using only the bounds of the given interval without any further
assumption.
Obviously,
the true value of
is existing and unique.
means the range of
over
. One of the original idea to introduce the interval function
is to evaluate the range of the
value of the function
when the variable
changes in the range of
in a statement of interval way.
However, because only some of the algebraic laws which valid for real numbers
hold only in a weaker form for interval numbers, the computational results of
depends on the calculating order
severely, and they are often larger than the value of
. A number of literatures took efforts on finding the skills to obtain the
better results of
. And some valuable rules were found. For example, (1) If each variable
,
, occurs at most once in
, then
; (2) To make the most of the subdistributivity, i.e., to execute the addition
and subtraction operation first, to execute multiplication and division
operations then. For instance, the better result of the polynomial
can be obtained through computing
its reformed form
. However, the similar results to improve the results of a rational function
have not been found.
In
fact, the best result of
can be obtained through calculate
directly. The bounds of
can be obtained through solving the following two optimization problems.
(8)
(9)
It
is clear that the optima indicated in optimization problem (8) and optimization
problem (9) refer to the global optima of
in
. When
is monotone in
, there are only one local maximum and one local minimum of
, they are the global maximum and the global minimum of
in
respectively. Many optimization
algorithms (for example, Newton algorithm, NowtonRaphson algorithm, Gauss
Newton algorithm, etc) that only can locate the local optima of the problems
could be used. However, when the expression
is not monotonic in
, or the monotone property is unknown, the global optimization method that has
good capability to locate the global optima of
in
is needed.
In this work, a realcode genetic algorithm is used
to locate the global optima of optimization problem (8) and (9). It can be
briefly described as: (1) The fitness function of individuals is defined by
and
for minimum and maximum problems
respectively.
and
respectively are the maximum and
the minimum of
in he generations up to now. The
Goldberg’s linear scaling formulation
(
) is used also for fitness scaling [9]. (2) The proportional selection model was
used. (3) The arithmetic crossover operator [10] was used. (4) Both of the
nonuniform mutation operator [10] and the boundary mutation operator [10] were
used in this study. (5) The elitist strategy was used to add the best individual
in the previous population to the next generation, in place of its worst
individual. (6) A maximum number of generations
is specified for stopping the
evaluation.
(3) Mathematic examples to examine the presented approach
Example 1. Consider the polynomial function
in the interval
=[5,5].
The figure
of
was showed in Fig.2. It is to see
from Fig.2 that
is not monotonic in [5, 5]. The
global minimum and the global maximum value of
in
are
and
respectively. Whence
. The accurate result
could be easily obtained by using
the present method.
However,
the solution
was obtained by directly using the
interval arithmetic operations. A better result,
, was obtained by using the reformed form
.
Numerical examples
The
nominal parameters for MOTOMAN SV3 industrial robot were shown in Table 1.
Table1 Design parameters of MOTOMAN SV3 robot
Link coordinate system

a
/mm

α
/℃

d
/mm

θ_{i}
/ °

1

150

90

0

170~
170

2

260

0

0

45~
150

3

60

90

0

70~
190

4

0

90

260

180~
180

5

0

90

0

135~
135

6

0

0

90

350~
350

Based on the nominal (design)
kinematic parameters those were shown in Table 1, the endeffector working
envelope can be calculated as follows by the presented method in this paper.
Due
to the tolerance and manufacturing error, 0.1% of the design value is taken for
every kinematic parameter as the parameter deviation from the nominal one, that
is, the value is fall in the interval [10.05%,1+0.05%] after normalization. The
actual working envelop for the robot endeffector could be obtained as following
shows by using the presented method.
Conclusion and remarks
By
representing all uncertain geometric parameters, a new approach to determine the
static pose (position and orientation) of the robot end effector in space was
proposed through evaluating interval functions. A reliable computation strategy
to is proposed also to overcome overestimation, the major drawback in
conventional interval computation.
Parameters
with interval uncertainties instead of fixed values are used to compute the
forward kinematic. In this way, the actual robot endeffector envelop can be
determined, which is essential for the robot offline programming, obstacle
autonomous avoidance, etc.
In
most cases, the error distribution should be identified, that is, with known
endeffector position error to determine the robot kinematic parameter
deviation. It is important for the robot calibration and robot production. Using
the interval theory to solve this problem inversely is ongoing.
ACKNOWLEDGEMENTS
The
research reported in this paper was supported by China Postdoctoral Science
Foundation and the National Nature Science Foundation of China (10072014).
Reference
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Gong C H. Nongeometric error identification and
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Wang D G. Nonlinear inversion algorithms and their
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4
Chen S H, Yang X W. Interval finite element method for beam structures.
Finite elements in analysis and design, 2000, 34:7588.
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Zhang J Y, Shen S F. Interval analysis method for
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Alefeld G, Herzberger J. Introduction to interval
computations. Academic Press, New York. 1983.
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Alefeld G, Claudio D. The basic properties of
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Goldberg D E. Genetic algorithms in search,
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