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 real-code 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 time-consuming 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 fuzzy-set-based 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 real-code 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 D-H convention for robot link coordinate system
The robot kinematic model is based on the Denavit-Hartenberg (DH) convention. The relative translation and rotation between link coordinate frame i-1 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 Ai is given in Eq. (1)
(1)
Using the homogenous transformation matrix the relationship of the end-effector 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 θi, di, αi, ai have no fixed value but having the values falling in the intervals [θi], [di], [αi], [ai] 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 [7-8]
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 non-degenerate interval 
has no inversion with respect to addition or multiplication. Even the distributive law has to be replaced by the so-called 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]) 
.As
(6)
when ce
(7)
Where 
.
(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, Nowton-Raphson 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 real-code 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 non-uniform 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].
Fig. 2 the graph of f(x)
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 
.