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1、A Relialble Approach to Compute the Forward Kinematics of Robot with Uncertain G 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
2、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
3、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
4、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 t
5、he 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
6、 parameterIntroduction 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 f
7、rom 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
8、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 th
9、e 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
10、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
11、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
12、 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 obv
13、iate 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 e
14、xamples 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
15、 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 , , and as shown in Fig. 1. The homogenous transformation Ai is
16、given in Eq. (1) 34567 (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 i, di, i,
17、 ai have no fixed value but having the values falling in the intervals i, di, i, 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 in
18、terval 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 nu
19、mbers 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. 34567
