Positioning a multi-robot system formation using potential field method.

Milic, Vladimir ; Kasac, Josip ; Situm, Zeljko 等

1. INTRODUCTION

In practical applications of mobile robots, autonomous motion in an unknown environment and robots interaction are most often required.

Mathematical modelling of robots and robot control is considered in the reference (de Wit et al., 1997). For this work the most important concepts are treated in detail in the third part, where is presented the general formalism for the modelling and control of wheeled mobile robots.

Reference (Krick et al., 2008) deals with the control of multi-robot systems. Different variants of the application of PFMs developed for planning the movement of multiple robots are discussed. Doctoral thesis (Ogren, 2003), represents a set of papers that refer to navigate a multi-robotic system, avoiding obstacles in the formation, implementation of the Lyapunov theory for the control of mobile robots and collective robotics.

In this paper, the control problem of multi-robot system to form a prescribed geometric arrangement is considerd. The usual approach to the control law synthesis requires solving the inverse kinematic problem. In our approach the control law is derived using an analytic fuzzy approach based on the kinematics of rigid body which removes numerical problems of classical approach. The desired trajectory of motion is generated by using PFM. Method of potential fields in the last few decades, is very popular in the control of mobile robots due to its mathematical simplicity.

2. PROBLEM FORMULATION

The Figure 1. shows the problem of positioning a multi-robotic systems where robots must achieve the desired formation. It is assumed that the three robots must achieve formation of the equilateral triangle, while the fourth robot is in focus of this triangle.

It is known that the radius of the circumscribed circle of this triangle is R = D[square root]3/3 where D is the length of the side of the equilateral triangle. Furthermore, from the analytic geometry we know that the distance between any two points in the plane can be calculated from the expression

d([T.sub.1], [T.sub.2]) = [square root of [([x.sub.2] - [x.sub.1]).sup.2] + [([y.sub.2] - [y.sub.1]).sup.2], (1)

[FIGURE 1 OMITTED]

where ([x.sub.1], [y.sub.1]) and ([x.sub.2], [y.sub.2]) are the coordinates of the points [T.sub.1] and [T.sub.2], respectively.

We assume that all robots are modelled as wheeled mobile robots of the unicycle type (Belkhouche, F. & Belkhouche, B., 2005; de Wit et al., 1997; Kasac et al., 2002)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

for i = 1,2,..,4 where ([x.sub.i], [y.sub.i]) are the coordinates of the reference point of i-th robot in the Cartesian frame of reference. [[theta].sub.i] is its orientation angle with respect to the positive x-axis. [v.sub.i](t) and [[omega].sub.i](t) are the linear and agular velocities, respectively.

This model applies to a large class of mobile robots with differential drives. Although the control inputs are at the velocity level, this is not restrictive for real mobile robot control because the modelling can be easily extended to include system dynamic. The main difficulties in dealing with the system (2) are getting from the fact that it is essentially underactuated, having less independent inputs then motion planning variables (de Wit et al., 1997).

3. CONTROL LAW SYNTHESIS

3.1 Potential Field Based Approach

Let's now define the vector x = [[[x.sub.1] ... [x.sub.4]].sup.T] and vector y = [[[y.sub.1] ... [y.sub.4]].sup.T]. The potential function is given as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where q = [[x y].sup.T] is the configuration vector the robots, [d.sub.12], [d.sub.23], [d.sub.31], [r.sub.42], [r.sub.34] are second power of the distances between robots defined by (1), [x.sub.r] and [y.sub.r] are coordinates of desired position, a and b are the gain factors that specifies the strength of the attractive potential.

The desired configuration vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be obtained using gradient descent scheme (Kasac et al., 2002)

[[??].sub.d] = -[[nabla].sub.q] V(q) (4)

3.2 Kinematics control

In this work control law will be performed by applying the basic principles of kinematics. First, we define the following vectors:

* position of i-th mobile robot: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

* desired trajectory: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

* distance between i-th robot and trajectory: [[??].sub.t] = [[??].sub.d] - [[??].sub.m,i],

* orientation of i-th robot: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Based on the previously defined vectors, control law for ith mobile robot has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [k.sub.1], [k.sub.2] and [k.sub.3] are the constant gains. The control law (5) represents the analytic formulation of the following fuzzy rules: a) if the robot direction [[??].sub.e,i] is on the right/left side from the vector [[??].sub.i] then angular velocity [[omega].sub.i] is positive/negative; b) the linear velocity [v.sub.i] is proportional to the distance [parallel][[??].sub.i][parallel]; c) the linear velocity [v.sub.i] has small value for large values of angular velocity [[omega].sub.i].

4. SIMULATION RESULTS

The selected values for the initial positions and orientations of robots are shown in Table 1. The gain factors that specifies the strength of the attractive potential from expression (3) are a=10; b=5, while the length of the side of the equilateral triangle is D=2 m.

In Figure 2 are shown trajectories of the robots from their initial positions towards the desired position with coordinates [x.sub.r]=5 m and [y.sub.r]=5 m.

The convergence rate of the desired formation error is illustrated in Figure 3. Figure 4 shows the convergence rate of the desired position error. It is obvious from the figure that the formation error and positioning error tend asymptotically to zero.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. CONCLUSION

In this paper, we have presented a new approach to control law synthesis with analytic fuzzy rules of the multi-robot system based on basic principles of kinematics. Potential field method was used to generate the robots reference trajectories. The control law strategy is illustrated in simulation example of positioning a formation of robots. A natural extension of this work is to consider problem of obstacle avoidance in formation. Future work also includes the implementation of this method on real multi-robot system including complete robot and actuator dynamics.

6. REFERENCES

Belkhouche, F. & Belkhouche, B. (2005). Modeling and Controlling a Robotic Convoy Using Guidance Laws Strategies. IEEE Transactions On Systems, Man., And Cybernetics--Part B: Cybernetics, Vol. 35, No. 4, August 2005, pp. 813-825, ISSN: 1083-4419

Kasac, J.; Brezak, D.; Majetic, D. & Novakovic, B. (2002). Mobile Robot Path Planing Using Gauss Potential Functions and Neural Network, In: DAAAM International Scientific Book 2002, Katalinic, B., (Ed.), pp. 287-298, DAAAM International Vienna, ISBN: 3-901509-30-5, Vienna

Krick, L.; Broucke, M. & Francis, B. (2008). Getting Mobile Autonomous Robots to Form a Prescribed Geometric Arrangement, In: Recent Advances in Learning and Control, Blondel, V. D.; Boyd, S. P. & Kimura, H. (Eds.), pp. 149-159, Springer-Verlag, ISBN: 978-1-84800-154-1, Berlin

Ogren, P. (2003). Formations and Obstacle Avoidance in Mobile Robot Control. Doctoral thesis, Department of Mathematics, Royal Institute of Technology Stockholm, ISBN: 91-7283-521-4, Stockholm

de Wit, C. C.; Siciliano, B. & Bastin, G. (1997). Theory of Robot Control, Springer-Verlag, ISBN: 3-540-76054-7, London

Tab. 1. Initial positions and orientations of robots

 robot 1 robot 2 robot 3 robot 4

x, m 6 3 1.5 1.5
y, m 1.5 7 4 1.5
[theta], rad [pi] -[pi]/2 -[pi]/2 [pi]/2