Sensitivity analysis for parallel robots.

Ibraim, Cristina ; Chelaru, Elena Silvia ; Predincea, Nicolae 等

1. INTRODUCTION

Numerical methods that are known as Monte Carlo methods can be loosely described as statistical simulation methods, where statistical simulation is defined to be any method that utilizes sequences of random numbers to perform the simulation.

Monte Carlo is now used routinely in many diverse fields, from the simulation of complex physical phenomena such as radiation transport in the earth's atmosphere and the simulation of the esoteric sub nuclear processes in high energy physics experiments.

Statistical simulation methods may be contrasted to conventional numerical discretization methods (Germer et al., 2003), which typically are applied to ordinary or partial differential equations that describe some underlying physical or mathematical system. In many applications of Monte Carlo, the physical process is simulated directly, and there is no need to even write down the differential equations that describe the behaviour of the system. The essential characteristic of Monte Carlo is the use of random sampling techniques to arrive at a solution of the physical problem.

In contrast, a conventional numerical solution approach would start with the mathematical model of the physical system, discretizing the differential equations and then solving a set of algebraic equations for the unknown state of the system.

1.1. Gauss Distribution

The normal (or Gaussian) distribution is one which appears in an incredible variety of statistical applications.

The Monte Carlo simulation consists of many repetitions of the random experiment with a changing random number input list.

The basic idea is to perform the direct kinematical algorithm for a huge numbers of parameters variations. The choice of the right distribution for parameters variation depends on the given task; in this case the variation will be for each parameter a Gauss Distributions.

2. PROGRAM DESCRIPTION

2.1 Data representation

Information about all parameters is stored in KGraph matrix as:

First we will store the reference frame in data structure as following:

KGraph(1,2).u KGraph(7,6).u

KGraph(1,2).v > Reference frame < KGraph(7,6).v

KGraph(1,2).w KGraph(7,6).w

Information about joint i

KGraph(i,i).type--type of Joint >Revolute Traslation Universal Spherical

KGraph(i,i).driving--kind of Joint > 0--driven Joint; 1--driving Joint

KGraph(i,i).teta--initial value for angle theta

KGraph(i,i).a--KGraph(i,i).b--KGraph(i,i).c

KGraph(i,i).alfa--KGraph(i,i).beta--KGraph(i,i).gama

For the first direction

Information about position of Joint i: KGraph(i,i-1).poz this is a vector with 3 elements [ x y z ]

Information about Inlet orientation of joint i: KGraph(i,i+1).u KGraph(i,i+1).v KGraph(i,i+1).w

Information about outlet orientation of joint i: KGraph(i,i-1).x; KGraph(i,i-1).y; KGraph(i,i-1).z

For the second direction

Information about position of Joint i: KGraph(i,i+1) poz this is a vector with 3 elements [ x y z ]

Information about inlet orientation of joint i: KGraph(i,i-1).u; KGraph(i,i-1).v; KGraph(i,i-1).w

Information about outlet orientation of joint i: KGraph(i,i+1).x; KGraph(i,i+1).y; KGraph(i,i+1).z

Information about the Cut Joint

There are two different positions and two different orientations for the Cut Joint NRCJ--Number of cut joint:

* First position--KGraph(NRCJ+1,NRCJ).poz

* Second position--KGraph(NRCJ+1,NRCJ+1).poz

[FIGURE 1 OMITTED]

With this method, the local coordinate systems are automatically arranged in the correct order, and in this way, the efficient recursive coordinate transformations can be performed.

2.2 Functions description

The following functions are used in the program in the purpose of reducing the errors (Szatmari, 1999).

Read_Input_Data.m--this function is used to input the parameters

We need to enter the next variables:

* Number of joints

* Number of Cut Joint

For each joint we need to enter the next parameters:

* Type of Joint (R for rotation and T for translation)

* Kind of Joint (1 for driving Joint and 0 for driven Joint)

If we have to describe a Revolute Joint, we must enter the next values: parameter for joint transformation Theta, parameters for link transformation: A (Alfa), B (Betha), C (Gama)

If we have to describe a prismatic/translation Joint, we must enter only the distance.

Build Matrix Structure.m--this function create the complete structure of information based on the input parameters and transformation rules.

The next transformations will be computed:

First Direction

1. Link Transformation between Reference Frame and First Joint

2. Joint Transformation and Link Transformation until to the Cut Joint

Second Direction

1. Link Transformation between Reference Frame and Last Joint

2. Joint transformation and Link Transformation until to the Cut Joint

Calcerrors.m--this function compute the error between the two ccordinate systems of the cut Joint. The error is the distance between the two positions: Pxyz=KGraph(i,i-1).Poz; Puvw=KGraph(i,i+1).Poz; DP=sqrt((Pxyz(1)-Puvw(1))A2+(Pxyz(2)-Puvw(2))A2).

Minimisation.m--this function adjust the joint variables one by one until the objective function can not be further reduced.

Gauss_Distribution.m--this function generate random values for each parameter. We have to define here 3 variables: maximum variation for distance, maximum variation for angles--number of variation (the number of points).

[FIGURE 2 OMITTED]

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After that we have to create the order for combinations, Fig. 4 (method to create Monte Carlo Simulation) shows how this algorithm works.

Monte_Carlo_Simulation.m--based on the Gauss Distribution this function store the parameters in data structure. We have after Gauss distribution the vector A1 with normal distributed values and vector permutA1 with a random permutation.

For example:

for i:1:1000
KGraph (2,2).a=A1(permutA1(i));
...
end

CCD_for_Mesh_Analysis.m--this is similar with function CCDcomplete.m this time we need to store in matrix information about each point from workspace.

InverseKinematik.m--compute the inverse Kinematical solutions (Li-Chun & Cheng Chen 1993) q1 and q2 starting with the value P(x,y) from mesh surface.

MeshAnalysis.m--this function is described in the chapter 'Mesh analysis over the entire workspace'

3. CONCLUSIONS

With this program the users have a powerful tool to reduce the errors and to identify which parameter have a big influence on the absolute accuracy and witch can be neglected in calibration process (Conrad, 2000). To verify this method presented in this paper, it is applied to 'FUNFGELENK' robot. The simulation results have proved that for finding a realistic prediction of absolute accuracy.

This article is intended to determine the influence of geometric parameter variation on the absolute accuracy of parallel robots.

4. REFERENCES

Conrad, K. L.; Shiakolas, P.S.& Yih, T. C (2000). "Robotic calibration issues: Accuracy, Repeatability and Calibration", Mediterranean Conference on Control & Automation, Patras, Greece

Costa, M. & Smaby, N. (1997). "Calibration and precision manufacturing", ME 319 Robotics and Vision Lab

Germer, C.; Hansen, U.; Franke, H.-J. & Buttgenbach, S. (2003). "Development of a 3D-CAD add-in for tolerance analysis and synthesis in micro systems" Symposion on Design, Test, Integration and Packing of MEMS/MOMS, ISBN: 0-7803-7066-X

Li-Chun, W. & Cheng Chen, C. (1993). "On the Numerical Kinematic Analysis of general Parallel Robotic Manipulators Robotics and Automation, IEEE Transactions on, ISSN: 1042-296X

Szatmari, S. (1999). "Geometrical errors of parallel robots" Periodica Polytechnica Ser. Mech. Eng. Vol. 43, No. 2