4.1. Numerical simulation

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

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

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 through the formulation are obtained in Table 2.

2. Location of the repulsion forces on the Bézier curve

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


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

Figure 3. (a) Control points and future predictions of Bézier trajectory and (b) deformation of eight concatenated Bézier curves.

In Figure 3(a), an example is shown, where a straight line represents the predicted optimal trajectory for a mobile robot obtained with the PFP algorithm. The control points needed to obtain the Bézier curves are displayed with red circles. The repulsive forces are placed in the proper positions of the predicted path. In this graphic example, there are eight points in the prediction horizon, and consequently, eight Bézier curves are concatenated in a straight line. The time devoted to perform trajectory is defined by the PFP prediction and has to be of 14 seconds. The time intervals corresponding to each curve, respectively, are [0,1.33], [1.33,3], [3,5], [5,7], [7,9], [9,11], [11,12.66], [12.66,14]. The representation of the resampling for the concatenation of eight Bézier curves is represented in Figure 3(b).
