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# A Relialble Approach to Compute

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ABSTRACT:Uncertainties widely exist in engineering structural analysis and mechanical equipment designs, and they cannot always be neglected. 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?XML:NAMESPACE PREFIX = O /

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 .

The probabilistic method, the fuzzy method and the interval method are the three major approaches to model uncertainties at present . 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 . 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 . 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. , 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 functionsˇŻ 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

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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, is a function of four kinematic parameters,,andas shown in Fig. 1.

The homogenous transformation Ai is given in Eq. (1)



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