4. Simulation results

2 cos <sup>∅</sup> <sup>þ</sup> <sup>μ</sup> � �

vkð Þ <sup>k</sup> <sup>þ</sup> <sup>2</sup>

vki obsð Þ k þ 2

∗i

≤ r ∗i

2 4

3.4. Consensus with Q-CAC–based avoidance

<sup>u</sup> ¼ �<sup>η</sup> log <sup>10</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup><sup>t</sup>

<sup>u</sup> ¼ �<sup>η</sup> log <sup>10</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup><sup>t</sup>

logarithmic term log10(k + 1) and the term <sup>Δ</sup><sup>t</sup>

unbounded u using Eqs. (4) or (5), then for each u<sup>i</sup> > <sup>r</sup>∗<sup>i</sup>

And for the leaderless architecture,

found in Ref. [14].

Once a safe attitude vector vi

20 Advanced Path Planning for Mobile Entities

size within the interval 0 < u<sup>i</sup>

region at any single time step.

vi

computed as a point a distance r

vki obsð Þ k þ 2

5 M

(k) is normalized to keep the computed control bounded. Whether there are intersections of the safety regions or not, one can guarantee the safety of the algorithm by bounding the control

Another important consideration is the size of control computed at each time using Laplacian matrices, which is directly proportional to the algebraic connectivity of the communication graph, and inversely proportional to the magnitude of the current time k. This means that, while the early values of u are large and therefore unsafe for collision avoidance (and must be bounded), the latter values of u are very small and therefore slow down the rate of convergence. One can observe that collisions are less likely to occur in the latter times when the vehicles are closer to their goal positions; consequently, convergence is slower at that time. Therefore, there is need to obtain constantly bounded control u which can guarantee both collision avoidance and a high speed of convergence. The following modifications to Eq. (4) and Eq. (5) were proposed in our previous works [5, 9, 14]. For the leader-follower architecture,

<sup>2</sup>λ2ð Þ <sup>L</sup> <sup>Γ</sup> <sup>x</sup> � xoff � � � <sup>β</sup> log <sup>10</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup><sup>t</sup>

<sup>2</sup>λ2ð Þ <sup>L</sup> <sup>Γ</sup> <sup>x</sup> � xoff � � � <sup>β</sup> log <sup>10</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup><sup>t</sup>

where λ2(L) is the second smallest eigenvalue of the Laplacian L. The parameter η is for scaling the consensus term and β is for scaling the proportional term in Eqs. (32) and (33). The

increase kuk when k is large. The choices of parameters η and β should depend on the radius of S and safety region ε for each vehicle. Alternatively, one may choose to compute an

The step-by-step procedure for implementing the algorithm including a flowchart can be

2 4

3

vkð Þ <sup>k</sup> <sup>þ</sup> <sup>2</sup>

3 5 T

(k) is computed at time k for any i, the next position x<sup>i</sup>

/2 from the current position, along the vector vi

/2. This means that a vehicle never steps beyond its safety

≥ 0: (31)

<sup>2</sup>λ2ð Þ <sup>L</sup> <sup>K</sup> <sup>x</sup> � xoff � �: (32)

<sup>2</sup>λ2ð Þ <sup>L</sup> <sup>K</sup>ð Þ <sup>x</sup> � xd , (33)

and set ui <sup>¼</sup> <sup>r</sup>∗<sup>i</sup>

<sup>2</sup> ui .

<sup>2</sup>λ2ð Þ <sup>L</sup> are used to reduce <sup>k</sup>u<sup>k</sup> when <sup>k</sup> is small and

<sup>2</sup> , normalize ui

(k + 1) is

(k). Note that

To demonstrate the solutions developed in this chapter, we revisit the experiment presented in Figure 1. The robots are homogeneous, and S for each robot is 85 mm, ε = 90 mm, whereas the dimensions of the soccer pitch are 6050 mm x 4050 mm. In Figure 14 (a), Eq. (5) was applied with the cyclic communication topology with one leader (Figure 2). In Figure 14 (b), Eq. (33) was applied with a full communication topology (i.e., every vehicle can communicate with each other). The simulation was done with MATLAB R2009a on an Intel® Core(TM)2 Duo P8600 @ 2.40 GHz with 2 GB RAM, running Windows 7. For Figure 14(a), the multipath planning problem took 244 time-steps to solve, resulting in a total computation time of 7.343 s, in which 203 avoidance attempts were made, and there were no collisions. For Figure 14(b), using a full communication topology, the computational time was 0.0131 s, and there were no collisions.

In [14], more simulations and analyses are presented, together with the limitations of this approach, which remains to be explored for future development.

Figure 14. Collision-free reconfiguration: (a) using topology with Eq. (5) and (b) using fully connected graph with Eq. (33).
