Consideraciones metodológicas aplicadas al SROI

Example Path Example Control Points

Adjusted Control Points Adjusted Spline

z1

z2

z3

Fig. 5.14 Definition of a spline with many segments using few parameters

5.7

N Control Point Splines (NCPS)

5.7.1

Method

In order to give a higher degree of control over the spline, without requiring an excessive number of parameters to optimise over, a further method was investigated. An example path was generated manually and subdivided into n spline segments, requiring n + 1 control points to define it. The control points were equally spaced along the example path. Perpendicular lines to the example path were defined at each of the control points, and the optimisation variable for each control point was the distance along these perpendiculars to the example spline away from the initial position, zi, ranging from −3 m to 3 m. The spline was required

to pass through all the control points as shown in Figure 5.14. The method to enforce this is described below. In this way, only one optimisation parameter was required to define the position of each control point, and none were needed for the start and end point, so a spline with n segments was defined by n − 1 parameters. This compares very favourably to the method used in section 5.6.2 which required 4 (n − 1) + 3 parameters. For a ten segment spline, this corresponds to nine parameters instead of 39. Using as many as ten spline segments is likely to cause over-fitting, or oscillations in the spline. However, it allows increased flexibility for the critical, tightly-constrained vehicles.

This type of spline is known as a Catmull-Rom spline [203]. A single spline segment passing through four control points, C1−C4, can be defined by

140 Manoeuvrability Modelling: Algorithm Selection

Fig. 5.15 Variation of the algorithm score for each manoeuvre with the number of control points used to define the spline

Q(t) = TC (5.17)

where Q is the coordinate for that particular spline segment, T is defined as above, and C is the coefficients matrix that encodes the control points. Following the method used above to find C from the boundary conditions yields

C = 1 2      −1 3 −3 1 2 −5 4 −1 −1 0 1 0 0 2 0 0           x₁ y₁ x₂ y₂ x₃ y₃ x₄ y₄      (5.18)

The outer two control points, C1and C4, are used to define the gradients at points C2and

C₃, thus the interpolation is only valid between C2and C3. In other words, at C2, t = 0 and at

C₃, t = 1. Continuity between the segments can be ensured by sharing control points.

5.7.2

Results

5.7.2.1 Investigation of the effect of number of control points

In order to assess the effect of the number of control points on the algorithm, success boundaries were plotted for a range of values of n. Figure 5.15 shows the effect of the number of control points on the score for the manoeuvre.

5.7 N Control Point Splines (NCPS) 141

The plots for both manoeuvres show a significant drop in performance at seven control points, compared to six or eight. Analysis of the example manoeuvres shows that, for these particular manoeuvres, an odd number of control points leads to one of the control points being directly in the centre of the manoeuvre, at the critical point. When the number of control points is even, the critical centre of the manoeuvre is not explicitly defined by a control point, but instead by the spline joining the control points on either side, which allows greater freedom. At higher numbers of control points, this effect is less pronounced, since the distances between the control points are lower, and therefore the positioning of the control points along the spline is less crucial.

Figure 5.15 suggests that more control points gives better performance, with peaks at 20 control points for the first manoeuvre and 17 control points for the second. Tests with more than 20 control points were slow to solve because of the increased number of optimisation variables. In addition, a spline with more than 20 control points can exhibit significant oscillation. This firstly increases the solution time, and secondly may yield a very unrealistic path. For this reason, it was decided to generate the results using 20 control points.

5.7.2.2 Results for 20 control points

Figure 5.16 shows the success boundaries for the two representative manoeuvres. The percentage of successful vehicles discovered was 96% for the first manoeuvre and 89% for the second manoeuvre. The algorithm was successful at locating the maximum wheelbase for both manoeuvres. This suggests that the algorithm is capable of planning paths which either counter-steer or overshoot the corner, unlike some of the other algorithms. The algorithm performs relatively poorly as the rear overhang increases. This is likely to be an effect of the oscillations previously described, as vehicles with very long rear overhangs will exhibit significant tailswing in the transient region of manoeuvres, which for a path with oscillations is a significant proportion of the path.

One adaptation of the algorithm was considered, where the freedom of movement of each control point varied. For example control points near the start and end of the manoeuvre were allowed only a small amount of movement, while control points near the centre could move much further from their starting position. However it was decided that this restricted the algorithm on manoeuvres where the most crucial part was not at the centre.

142 Manoeuvrability Modelling: Algorithm Selection

(a) 90° corner

(b) Chicane

Fig. 5.16 Success boundaries as found by Multiple Control Point Spline algorithm with 20 control points

Unrealistic Vehicles

True Performance Envelope NCPS Performance Envelope Unsuccessful Vehicles

5.8 Algorithm Selection 143