Relación entre los “que” y los “como”

intersection point as a pseudo-tangent point, and the two intersecting parabolas as one super-parabola.

The pseudo-focus points effectively increase the eccentricity, e, of the PE from a standard circle (e = 0) to one similar to an ellipse (0 < e < 1, e =q1 − b2

a2). This

causes the two inverted S-Mi axis parabolas of a PE to intersect along the velocity line if there is nothing inside, as their corresponding circles are not tangentially aligned. When merging all four individual PE parabolic segments together to form the super-parabola, the intersection point between S-Mi axis parabolas are treated as if it were a tangent point. This prevents the algorithm drawing an incorrect Voronoi line along the velocity line, as the two parabolas are treated as a complete parabolic curve.

The velocity line problem could also be used as an advantage if non-convex shapes need to be described. Intersecting parabolic segments that describe the same shape would not trace Voronoi lines, and instead would be treated as one parabolic curve. Although not used in this thesis, this implementation could be used to describe other aspects of the environment, such as irregular objects.

4.2

Enhancing Accuracy

The PVPs of the PEs are guaranteed to be successfully calculated by the sweepline algorithm, provided the computational accuracy required can be achieved (Sec- tion 4.1, Item a1). However, as PEs are approximations of ellipses, contained within

the ellipse, Fig. 4.4a, discrepancies between PVPs and EVPs may occur. Additional EVPs may occur as a result of intersecting ellipses, Fig. 4.8, by creating more unique neighbours between ellipses than there were between intersecting PEs. By compar- ing the internal and external orientations of PVPs to its PE and associated ellipse, it can be determined if an additional EVP may be required.

4.2.1 Accuracy Required for the Considerate Path Planner

The important EVPs occur externally to the ellipses, as this is the safest area of the roadmap, allowing the CPP to generate a path that moves around potential collisions. All external EVPs can be found through this novel algorithm as the PE is enclosed within the ellipse, and so an external EVP will always begin with an associated external PVP. Fortunately the errors that do occur involve only internal VPs, and only arise over very small spaces, Fig. 4.8. Therefore, it does not affect the performance of the CPP in any significant way, however it does mean that a completely accurate VD may not be possible.

Measures are now introduced to find additional EVPs that cannot be found through direct convergence of PVPs. These additional EVPs should occur approx-

(a) _(b)

(c) (d)

Fig. 4.8 As the PEs are only approximations of ellipses they will always be smaller in area when they do not degenerate to a circle, Fig. 4.4. Therefore, over small areas (pink square in (a)), there may be more EVPs than PVPs, and so not all EVPs can be found. To increase the chance of finding all the EVPs additional points are calculated, Section 4.2, that should converge on EVPs not represented by PVPs. (a) Using four ellipses there are ellipse-ellipse intersections that are not represented with similar PE-PE intersections. The pink square highlights small area where there are more EVPs than PVPs. (b) PVPs that are found for the VD of PEs. (c) EVPs that are found for the VD of ellipses, when using only the PVPs as initial approximations. (d) EVPs that are found for the VD of ellipses, when using additional approximations to the initial PVP approximations, Section 4.2.1.1.

4.2 Enhancing Accuracy 117 imately along the first internal edge of the ellipses. This additional accuracy will therefore allow the VD to be approximated externally to the ellipses, as well as just within their internal perimeter. Any VD further inside the uncertainty ellipses is not beneficial, as the very nature of the ‘safest path’ that the VD provides would form paths that actually increase the collision likelihood.

4.2.1.1 Estimating Additional Ellipse Vertex Points

To establish if an additional EVP may occur, the PVP must be external to the PE and internal to its elliptical equivalent, or vice versa. This quick check will establish if neighbouring ellipses have intersected when their PE approximations have not. The bi-section method is used to converge on the EVP from an initial approximation.

Approximating the new EVP If determined that an additional EVP may exist, a normal intersection point, pp, is created between the PVP and the closest edge of the ellipse that the PVP is on the opposite side of. From pp, normal intersections are made between it and the other two ellipses that are represented by the other two PEs. The intersection point between the perpendicular bi-section of these connections then creates an additional PVP to be tested, this time on the correct side of the ellipse.

Converging onto the EVP As a PVP converges on an EVP root, it may cross an ellipse edge. This will be due to the orientation between three PEs differing to their equivalent ellipses. If this occurs, an additional start point is created, to test for alternative EVPs that occur due to additional ellipse-ellipse intersections.

Once the EVP is successfully converged upon, if it occurs on the opposite edge of an ellipse edge, then a new start point is created to test for an additional EVP. As with Approximating the new EVP, a normal intersection point, pp, is created between the PVP and the closest edge of the ellipse the PVP is on the opposite side of. This is repeated for the new EVP and creates another point, pe. The intersecting perpendicular bisectors between PVP - pp and EVP - pe generate the new start point.

When comparing the difference between enhancing the accuracy and not, Figs. 4.8c and 4.8d, the enhanced accuracy appears to allow additional EVPs to be discovered that cross the opposite edge of an ellipse. This level of accuracy is acceptable for the CPP, as ideally the CPP will never venture far into an ellipse when predicting a path. This will aide in avoiding paths that are likely to form collisions when executed. In order to generate a perfect VD of ellipses, the construction of the PE will have to be re-evaluated in order to be more similar in shape.

Fig. 4.9 Visualisation of re-creating the PE with additional circles. An additional circle is introduced, which is tangential to the ellipse in between the S-Ma and S-Mi axis limits. This will help to create a PE that is more similar in shape to the original ellipse, resulting in a VD of PEs that has a higher fidelity to the VD of ellipses. The addition of the extra circle will increase the processing time of the sweepline algorithm, however it will still operate with a logarithmic time complexity, Section 4.1.3. The introduction of a line segment must also be included at the S-Mi axis limit, as the additional circle cannot tangentially align with the axis and the new point of the ellipse. However, the line segment is tangential to the circle, and so the novel tangent function, Section 4.1.2, can still be applied.

4.2.2 The Potential for Complete Accuracy

When using just four circles to construct the PE, the accuracy is at worst 15% less than the area of the ellipse, Fig. 4.4b, regardless of the ratio between the ellipse axes. To create a better PE approximation, more tangential circles must be used. Fig. 4.9 provides an example of introducing an additional point along the ellipse that a circle can align tangentially with. The four circles of the PE used in this algorithm align tangentially with the ellipse axes, with parametric values: 0,1

2π, π, 3

2π. Intro-

ducing another tangent point half way between the ellipse axes, with parametric values: 1 4π, 3 4π, 5 4π, 7

4π, causes the accuracy of the PE to increase by becoming more

comparable in shape.

Regarding the performance costs, the time complexity still remains logarithmic, Section 4.1.3, however each new tangent point causes another event to be evaluated. This version of the PE must also connect the two circles either side of the S-Mi axis

4.3 Approximating the Voronoi Diagram of Ellipses 119