A potential field method approach to robotic convoy obstacle avoidance.

Milic, Vladimir ; Kasac, Josip ; Essert, Mario 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.

The problem of robotic convoy control has received a lot of attention in recent years. This problem is addressed from different points of view, e.g., artificial vision, nonlinear control, fuzzy control, etc. In (Belkhouche, F. & Belkhouche, B., 2005), the authors propose an approach based on guidance laws strategies, where the robotic convoy is modelled in terms of relative velocities of each lead robot with respect to its following robot.

In this paper, the control problem of robotic convoy obstacle avoidance 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 robotic convoy obstacle avoidance in the Cartesian frame. The aim is to design a control law for four unconnected wheeled autonomous robots in order to follow the lead robot while keeping a constant distance from each other. We assume that the robots move in the horizontal plane and initially all the robots are not in position of convoy.

[FIGURE 1 OMITTED]

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)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

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 modeling can be easily extended to include system dynamic. The main difficulties in dealing with the system (1) 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

For the purpose of reference trajectory generation for obstacle avoidance, the most commonly used form of attractive potential function is

[U.sub.a] ([x.sub.r], [y.sub.r], [x.sub.c], [y.sub.c]) = 1/2 a[[([x.sub.r]--[x.sub.c]).sup.2] + [([y.sub.r] [y.sub.c].sup.2])] (2)

where [x.sub.c] and [y.sub.c] are the coordinates of the goal position, [x.sub.r] and yr are the coordinates of reference trajectory, a is the gain factor that specifies the strength of the attractive potential. The repulsive potential has the following form (Kasac et al., 2002)

[U.sub.r] ([x.sub.r], [y.sub.r], [x.sub.p,i], [y.sub.p,i]) = 1/2 exp (-b[([x.sub.r]--[x.sub.p,i]).sup.2] + [([y.sub.r] - [y.sub.p,i]).sup.2]), (3)

for i = 1,2,...,n, where n is the number of the obstacles, [x.sub.pi] and [y.sub.p,i] are the obstacles coordinates, b is the gain factor that specifies the strength of the repulsive potential which can be adjusted to satisfy appropriate conditions like passage between closely spaced obstacles. The coordinates of reference trajectory can be obtain by solving the following diferential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [c.sub.1], [c.sub.2], [c.sub.3] are the constant gains.

Let's now define the vectors x = [[x.sub.1] ... [x.sub.4].sup.T], y = [[y.sub.1] ... [y.sub.4]].sup.T] and q = [[x y].sup.T]. To keep a constant distance between mobile robots we introduce a potential function as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [d.sub.12], [d.sub.23], [d.sub.34], [d.sub.13], [d.sub.14] are second power of the distances between robots, [a.sub.f] and [b.sub.f] are the gain factors that specifies the strength of the potential function. The desired guidance law can be obtained using gradient descent scheme (Kasac et al., 2002)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

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: [[bar.r].sub.r] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

* distance between i-th robot and trajectory: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

* 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] (7)

where [k.sub.1], [k.sub.2] and [k.sub.3] are the constant gains. The control law (7) 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 mi 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

This section presents the results of simulation verification of proposed control strategy of the robotic convoy obstacle avoidance.

The trajectories of robtic convoy in environment with obstacles from its initial positions towards the desired target are shown in Figure 2. Figure 3. shows the distances between robots obtained from simulation. It is obvious from the figures that the desired position, obstacle avoidance and distances between robots are achieved.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

5. CONCLUSION

In this paper, we have presented a new approach to control law synthesis with analytic fuzzy rules of the robotic convoy 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 robotic convoy obstacle avoidance. A natural extension of this work includes the implementation of this method on real robotic convoy 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