Author details

Example 2. This example illustrates more the satisfaction of the objectives. In this example the safety circle radius is put as p ¼ 3. Here we choose the following

> , x<sup>1</sup> <sup>f</sup> ; y<sup>1</sup> <sup>f</sup> ; ϕ<sup>1</sup> f

, x<sup>3</sup> <sup>f</sup> ; y<sup>3</sup> <sup>f</sup> ; ϕ<sup>3</sup> f 

After applying our approach, you can see the resulting optimized motions in Figure 6. In Figure 6, errors in x- and y-coordinates and orientation of each robot are shown with respect to time. It is clear that errors of zero are achieved. In Figure 7 for each robot, constraint evaluations, i.e., safety clearance, are displayed for the other two robots throughout time. You can see that robots come close to each other sometimes but without violating the safety distance. This result is attained maybe because of special structure of initial and final positions and orien-

The paper investigated the time-energy minimization onto the multi-robot case. A global objective function is formulated as the sum of individual robot objectives in time and energy. Constraints are divided into two sets, namely, robot-individual constraints and robots' interaction constraints. The problem is decomposed into L subproblems with L being the number of robot systems. The subproblems are coupled with each other by the collision avoidance information. Applying a distributed algorithm solved the problem iteratively. The overall output gives optimized motions for all robots in time and energy while adhering and not colliding with each other. We applied our approach to the case of three wheeled mobile robots: we generated in parallel for each robot an optimized control input trajectory.

An extension to this study is to generate optimized motion trajectories and apply them experimentally. A possible area for experimentation is full-scale autonomous vehicles. Issues related to communication and distributing information during the parallel algorithm will need to be incorporated and investigated. Also, aspects of state estimation and localization of the robot system will come into the place which were not considered in this work. A possible other investigation is to distribute the problem further onto the time variable k; this will lead the problem to the domain of distributed model predictive control. This will, possibly, pave the way to faster

The authors would like to thank King Fahd University of Petroleum and Min-

<sup>f</sup> ; y<sup>2</sup> <sup>f</sup> ; ϕ<sup>2</sup> f  ¼ 8; 0;

¼ ð Þ 0; �8; 0

¼ �8; <sup>0</sup>; � <sup>π</sup>

π 2 

2

π 2 

2 

initial and final positions and orientations for the three robots:

Path Planning for Autonomous Vehicles - Ensuring Reliable Driverless Navigation…

tations. That could have given flexibility for the algorithm.

x1 <sup>0</sup>; y<sup>1</sup> <sup>0</sup>; ϕ<sup>1</sup> 0 ¼ �8; <sup>0</sup>;

x2 <sup>0</sup>; y<sup>2</sup> <sup>0</sup>; ϕ<sup>2</sup> 0 <sup>¼</sup> ð Þ <sup>0</sup>; <sup>8</sup>; <sup>0</sup> , x<sup>2</sup>

x3 <sup>0</sup>; y<sup>3</sup> <sup>0</sup>; ϕ<sup>3</sup> 0 <sup>¼</sup> <sup>8</sup>; <sup>0</sup>; � <sup>π</sup>

deployment into autonomous vehicles.

Acknowledgements

132

erals supporting this work.

5. Conclusion

Mohamad T. Shahab<sup>1</sup> \* and Moustafa Elshafei<sup>2</sup>

1 Department of Electrical and Computer Engineering, University of Waterloo, ON, Canada

2 Zewail City of Science and Technology, October Gardens, 6th of October City, Giza, Egypt

\*Address all correspondence to: m4shahab@uwaterloo.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
