Avoidance trajectory design for mobile robots.

Pozna, Claudiu ; Alexandru, Catalin

1. INTRODUCTION

After we have solved the trajectory definition and control problems (Pozna & Alexandru 2008) we are confronted with the avoiding obstacle problem. In (Brandt 2005) and (Hrich 2005) two solutions are presented: the first is based on the mathematical theory of differential games, and the second, which is called "the elastic bands", was developed by Quinlan and Khatib for mobile robots path planning. In (Brandt 2005) the two methods are compared and it is proved that the "elastic band" method is more suitable for path planning. This method is based on repulsive potential field generation. More precisely the road (the right and left kerb) and the obstacles are modeled by an elastic network which is composed by several nodes linked by springs. The nodes are disposed on the right and left kerbs. The spring's stiffness is related to the desired trajectory which is the equilibrium position of the network in the absence of the obstacle. The presence of the obstacle (a new point and new springs) will perturb the initial position of the network and will generate a new equilibrium position. Using this equilibrium position a smooth avoiding trajectory is designed.

From our point of view this method has the following inconvenient: first, because of the nonlinear stiffness definition, the equilibrium position computation is a time consuming process, and second, in practice, the shape and the dimensions of the obstacle are discovered during the avoiding maneuver.

2. MINIMUM POTENTIAL FIELD CONCEPT

The car is following a trajectory and in a certain moment, discovers an obstacle which makes the initial trajectory impossible to pursuit. The car control system must react and perform the following on line tasks: discover the shape and dimensions of the obstacle; compute the avoiding trajectory; control the robot on the trajectory and, in the end, return to the initial trajectory. Some comments are necessary: because the obstacle is discovered on line, during the avoiding manoeuvre, the avoiding trajectory will be composed by several parts; this means that, on line, the control system must perform a loop which is composed by: discovering the obstacle and the road (the universe), designing a part of the avoiding trajectory and controlling the robot (car) on this trajectory part.

Because this is an on line process we must avoid time consuming computation. For this reason we must link together, from mathematical point of view, the universe knowledge with the trajectory generation. This means that it is important to consider the sensors behavior when we design the avoiding trajectory. In fact, if we analyze the "elastic bands" solution we can see that the escaping trajectory is the static deformation of a hypothetically elastic network associated with the car, road and obstacle (Brock 1999; 2002).

It is also know that the equilibrium of these kinds of structures can be obtained by imposing a minimum potential energy condition. Our idea starts from this point: it is not necessary to construct such a complicated potential field and to compute the nonlinear solution of equilibrium; it is more suitable to construct a simple potential field which can be used very quickly.

The construction of the potential field is related on finding a mathematical function which has the following properties: in absence of the obstacle, the minimum of this function is the initial trajectory (see figure 1); the presence of the obstacle will generate a new minimum which will avoid the obstacle; the computation of minimum must be a non time consuming process; the function definition can be linked to the relative speed between the obstacle and the car; the function can be easily constructed, based on the data sensors.

The mathematical expression of this function (in the car referential system) is:

[z.sub.R] = 1/2([m.sub.2](sign(y - [e.sub.y]) + 1) - [m.sub.1](sign(y - [e.sub.y]) - 1)) x [absolute value of y - [e.sub.y]] (1)

[e.sub.y] = [X.sub.0] - e/cos([psi]) - x tan([psi]), [m.sub.1] = 1/L - e, [m.sub.2] = 1/e (2)

where e is the desired position of the car in the road referential frame, [X.sub.0] & [Y.sub.0]--the car position on the road referential frame, [psi]--the car orientation, L--the wide of the road.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Because we intend to use a laser scanner the obstacle(s) is a collection of scanned points. The idea is to generate a potential function for each point, so in the end the obstacle potential function will be a sum.

[z.sub.0] = [n.summation over (i=1)] exp(-[c.sub.1][(y - [y.sub.1]).sup.2] + [c.sub.2][(x - [x.sub.1]).sup.2]/2[[sigma].sup.2] (3)

where [c.sub.1], [c.sub.2] & [sigma] are parameters linked to the steering motor performance and the relative speed between the car and the obstacle, [x.sub.i] & [y.sub.i]--the position of the scanned point i = [bar.1, n].

To obtain the minimum points we must solve the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [x.sub.i] = 0:delta:Lr, Lr--the road length in the xoyz referential frame, I--the parameter which restrict the search area in order to avoid shaded minimum points.

3. AVOIDING TRAJECTORY--SIMULATION RESULTS

With equation (4) we have obtained a collection of minimum potential points. Using this result we must design a smooth C2 curve which will be the avoiding trajectory of the mobile robot. Because the obstacle is discovered during the avoiding maneuver the avoiding trajectory will be composed by several parts, for each part definition we imagine a strategy composed by two steps: first, recognize the environment obtain the minimum potential points, use the first k points and define the trajectory, and second, control the car (robot) on this trajectory.

In order to simulate this strategy we have designed in Matlab a program structure able to reproduce the car behavior during the avoiding maneuver. The road and the obstacles potential function is presented in figure 3. During the avoiding maneuver this function will change for each scanning step.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

From several simulations the avoiding trajectories, which are constructed step by step, are presented in the diagrams shown in figure 4.

4. CONCLUSIONS

Avoiding obstacle means first of all, to design a trajectory and after this to control the car on the trajectory. This paper focuses on the trajectory design problem, where a new concept: the potential field of the road is proposed. The designing of this trajectory is an "on-line process" because the obstacles forms and the obstacles dimensions are discovered during the avoiding maneuver.

Using the concept of the potential field we have integrated the sensors output in a mathematical function. The minimum of this function, which is the avoiding trajectory, can be easily computed.

Even the simulation results confirm our idea to obtain all the parameters of the function, needs experimental tries. In order to pursuit these try we must design the controller which allows following the designed trajectory.

5. REFERENCES

Pozna, C. & Alexandra, C. (2007). Mobile Robot Control by Learned Behaviour, Annals of DAAAM for 2007 & Proceedings of the 18 th International DAAAM Symposium: "Inteligent Manufacturing & Automation: Focus on Creativity, Responsability and Ethics of Engineers", Katalinic, B., pp. 605-606, ISBN 3-901509-58-5, October 2007, DAAAM International, Zadar

Brandt, T.; Sattel, T. & Wallaschek, J. (2005). On Automatic Collision Avoidance Systems, Proceedings of SP-1920 SAE World Congress, pp. 286-291, ISBN 978-0-7680-1565-2, April 2005, SAE International, Detroit

Hrich, K. (2005). Optimization of Emergency Trajectories of Autonomous Vehicles with Respect to Linear Vehicles Dynamic, Proceedings of IEEE/ASME 2005 International Conference on Advance Intelligent Mechatronic, pp. 135-144, ISBN 0-7803-9047-4, July 2005, IEEE Xplore, Monterey, CA

Brock, O. (1999). Generating Robot Motion the Integration of Planning and Execution, PhD thesis, Stanford University

Brock, O.; Khatib, O. & Viji, S. (2002). Task--Consistent Obstacle Avoidance and Motion Behaviour for Mobile Manipulation, Proceedings of the 2002 IEEE International Conference on Robotics & Automation--ICRA, pp. 388-393, ISBN 0-7803-7273-5, May 2002, IEEE Xplore, Washington DC