A model for the kinematical analysis of a three degrees of freedom mechanism.

Ciupan, Emilia ; Itul, Tiberiu ; Morar, Liviu 等

1. INTRODUCTION

The authors make a kinematical analysis of a three degrees of movement type BONEV mechanism using a neural model.

Figure 1 shows the schema of a mechanism of this type. This consists of two platforms: a rigid platform [B.sub.1][B.sub.2][B.sub.3] and a smaller, mobile one.

The movement of the mobile platform with the mobile reference system Oxy is done with the aid of three motor couplings [q.sub.1], [q.sub.2] and [q.sub.3]. These produce the modification of the position angles 01, 02 and 03 of the arms. The couplings [u.sub.1], [u.sub.2] and [u.sub.3] are passive because their position is determined implicitly.

It will be possible to move the mobile reference system Oxy tied to the centre of the mobile platform in the workspace and related under an angle [phi] depending on the position of the motor couplings [q.sub.1], [q.sub.2] and [q.sub.3] (Itul, 2000).

2. THE MATHEMATICAL MODEL

The kinematical analysis used in the field of industrial robots assumes establishing certain equations between the input values (done by the motor drive) and the output values (the position, speed and acceleration of the final effecter).

[FIGURE 1 OMITTED]

The kinematical analysis is comprised of two methods (Ispas & Blebea, 2003):

1. The direct kinematics: the movement laws of the motors are considered to be known and the movement law of the effecter is determined;

2. The inverse kinematics: the movement laws of the effecter are considered to be known and the movement law of the motors is determined.

The mathematical model for the analysis of the inverse kinematics is briefly described by equation (1)-(3) (Itul, 2000):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where

[[delta].sub.i] = 2[pi] / 3 x (i-1), i [member of] {1,2,3} (4)

and the angles [[theta].sub.i] are calculated by the following equation:

[[theta].sub.i] = [alpha] + [phi] + [[delta].sub.i]. (5)

The mathematical model for the analysis of the direct kinematics is described by the following system of equations (Itul, 2000):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The following constructive feature has been considered for the mechanism that is studied in the following part of the paper: the side of the fixed platform, meaning the side of the equilateral triangle [B.sub.1][B.sub.2][B.sub.3] is L=40 cm long.

3. THE NEURAL MODEL

A neural model was created for the inverse kinematical analysis of the mechanism described prior (Ciupan, 2008; Zilouchian & Jamshidi, 2001). The chosen neural network is three-layer type perceptron having architecture 3-50-3. The input layer has three neurons that correspond to coordinates X, Y and [phi] of the effectors' position. The three neurons belonging to the output layer correspond to motor couplings coordinates [q.sub.i], i=1, 2, 3. Good results for a hidden layer consisting of 50 neurons were obtained when the network was trained. The activation functions are sigmoid for the neurons in the intermediate layer and linear for the ones in the output layer.

The training was carried out with a training set having 121 input/output pairs. The training set was obtained using the mathematical model designed in MATHCAD. Back propagation was chosen training algorithm using the Levenberg-Marquardt method (Hagan & Menhaj, 1994). A mean square error of 3.5 x [10.sup.-5] was reached following training.

The model was tested conceiving some applications. The effecter's movement along certain trajectories in the plane describes by the positions (X, Y, [phi]) was taken into account. The coordinates of the motor couplings were calculated using the neural model. These were than used as input data for the mathematical model for the direct kinematical analysis and lead to the recalculation of the coordinates (XR, YR, [phi]R).

The approximation error of the robot's trajectory through the neural model results from the comparison of the initial positions (X,Y, [phi]) with the recalculated positions (XR, YR, [phi]R).

3.1 Application 1: a bunch of straight lines

The trajectories chosen for the testing of the model consist of a bunch of straight lines with different angles of the bow (30[degrees], 45[degrees] and 90[degrees]). The performance obtained when simulating fit between [10.sup.-5] and [10.sup.-3].

Figure 2 presents the approximation of the theoretical trajectories by the neural model. The approximation errors charts when the effecter moves on a straight trajectory having a 45[degrees] angle of bow are presented in figure 3.

3.2 Application 2: a circular trajectory

The robot's effecter movement was simulated on a circular trajectory in the workspace. The approximation error is around [10.sup.-4].

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The way in which the model approximates the movement of the robot on this trajectory is shown in figure 4. Figure 5 illustrates the charts of the approximation errors in this case.

4. CONCLUSIONS

Starting from the analysis of the above results as well as from the results of some research that is not presented in this paper, it is possible to affirm the following:

1. The neural model simulates the movement on different trajectories of the plane mechanism described at the beginning well enough;

2. Training the neural network on a small set of examples doesn't ensure a good performance in the generalization stage, even if the training performance is good;

3. Better result may be obtained through training on a larger set of examples that are equally distributed within the workspace;

4. A well built neural model can be used for the control of a mechanism the same type as the one presented in the paper successfully.

5. REFERENCES

Ciupan, E. (2008) Integrated Management of the Systems Using Open Control Platforms. Ph D Thesis, Technical University of Cluj-Napoca, pp. 115-128, 2008.

Hagan, M. T. & Menhaj, M. B. (1994) Training Feedforward Networks with the Marquardt Algorithm. IEEE Transactions on Neural Networks, vol. 5, no. 6, pp 989-993, November 1994.

Ispas, V. & Blebea, I. (2003) Robotics. Parallel Robots. Service Robots. UT Pres, Cluj-Napoca, 2003.

Itul, T. P. (2000) Roboti paraleli de tip B. Geometrie si cinematica (Type B Parallel Robots. Geometry and Kinematics). Todesco, Cluj-Napoca, 2000.

Zilouchian, A & Jamshidi, M. (2001) Intelligent Control Systems using Soft Computing Methodologies, CRC Press LLC, ISBN 0-8493-1875-0.