# A Relialble Approach to Compute the Forward Kinematics of Ro

**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

The homogenous transformation *A*_{i} is given in Eq. (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) The robot kinematic model using parameters with interval uncertainty

When the kinematic parameters *θ _{i}*,

*d*,

_{i}*α*,

_{i}*a*have no fixed value but having the values falling in the intervals [

_{i}*θ*], [

_{i}*d*], [

_{i}*α*], [

_{i}*a*] randomly, expanding the Eq. (2) with the intervals, we get,

_{i}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

If

Let

where,*f([x]) *is an interval also and which stands for an interval arithmetic evaluation of

*f*over

**∈**[x] , the relation (6) can be obtained.

whence

Where 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

In fact, the best result of

It is clear that the optima indicated in optimization problem (8) and optimization problem (9) refer to the global optima of

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

(3) Mathematic examples to examine the presented approach

**Example 1. **Consider the polynomial function [-5,5].

Fig. 2 the graph of |

The figure of

However, the solution

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 |
| α /℃ |
| θ / ° |

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 end-effector 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 [1-0.05%,1+0.05%] after normalization. The actual working envelop for the robot end-effector 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 end-effector envelop can be determined, which is essential for the robot off-line programming, obstacle autonomous avoidance, etc.

In most cases, the error distribution should be identified, that is, with known end-effector 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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