5. Conclusion

With the objective function and the constraints, the Lagrangian function (21) is defined. In order to calculate the stationary points, the partial derivatives of the Lagrangian function are calculated and canceled, and a system of linear equations is obtained. The solution of this linear system is the perturbation vector of each control point in order to obtain the Bézier

In our numerical example, it is used Bézier trajectories of second order to avoid loops in the trajectory. For that reason, the number of Bézier curves will be equal to the number of repulsive forces generated with the selected predictive PF technique. One vector is placed per Bézier

The control points are uniformly distributed throughout the prediction horizon generated by the PFP method. The model has been developed for holonomic robots, and therefore, the prediction of future positions provides a straight line. In this case, the control points calculated

The control points of the Bézier curve are uniformly distributed, and the repulsion forces obtained with the PFP method are placed at the midpoint of each curve, except for the first

Table 2. Calculation of control points from the prediction horizon: <sup>b</sup><sup>x</sup> is the vector containing the future trajectory and <sup>P</sup>ð Þ<sup>j</sup>

and the last curves where they are placed at the first and last points, respectively.

1. Calculation of the control points from the prediction horizon generated with the PFP

ð21Þ

i

trajectory. In-depth information about the linear system obtained is described in [48].

curve. To develop, the BTD algorithm is necessary to follow these two steps:

4.1. Numerical simulation

138 Advanced Path Planning for Mobile Entities

through the formulation are obtained in Table 2.

is the i-th control point of the j-th curve.

2. Location of the repulsion forces on the Bézier curve

This chapter details a comprehensive study of the use of parametric curves in the design of trajectories for holonomic and non-holonomic mobile robots. First, a brief introduction of the mathematical formulation and properties of the different curves is presented. Second, an exhaustive revision of literature regarding the use of parametric curves in path planning for mobile robots is developed. Third, a detailed description of the available techniques for path planning with parametric curves is presented, thoroughly describing the most important ones. Finally, an in-depth comparison is carried out between the different techniques of path deformation using Bézier curves, with their advantages and drawbacks. The Bézier curves are extensively used in these applications due to the simplicity of its definition and its easy handling and manipulation. The last section describes how to merge artificial potential field methods with Bézier curves as a solution for modifying a predefined trajectory in real time. Future works are related to the inclusion of other parametric curves, such as B-splines, RBC, and NURBS, in the proposed algorithm.
