Elliptical obstacles

In this section we consider m elliptical obstacles inR2. For i= 1. . . m, letAi ∈ M2×2 be

symmetric and positive definite matrices, and let µi_min >0 be the minimum eigenvalue of matrixAi and define the following obstacle functions

βi(x) = (x−ci)>Ai(x−ci)−µiminr2i. (3.41)

where ci ∈ X is the center of the i-th ellipse and ri > 0 is the length of its largest axis. Each obstacle is then defined as

Oi =

x∈ X

βi(x)<0 . (3.42)

In these experiments we place the center of each ellipsoid in a different orthant. In particular, each center is set to be in the position L(±1,±1) and then we add a random variation drawn uniformly from [−∆,∆]2, where 0<∆< L. The maximum axis of the ellipse –ri– is drawn uniformly from [r0/10, r0/5] and the matricesAi fori= 1. . . m are such that they

are orthogonal and their eigenvalues are random and uniformly selected from the interval [1,2]. We verify that the obstacles resulting of the previous process do not intersect. If they do, we re draw all previous parameters. For the objective function we consider a quadratic cost given by f0(x) = (x−x∗)>Q(x−x∗), where x∗ is drawn uniformly over [−r0/2, r0/2]2

and we verify that it is in the free space. The matrix Q ∈ M2×2 _{is a random positive}

definite symmetric matrix whose eigenvalues are selected as follows. For each obstacle we compute the maximum condition number that Q could have in order to satisfy condition (2.18). Let Ncond be the maximum among these admissible condition numbers. Then, the eigenvalues of Q are selected randomly from [1, Ncond−1], hence ensuring that (2.18)

is satisfied. For the estimates of the objective function, its gradient, the distance to the obstacles, the direction defined by the position of the agent and its projection onto the obstacles and their curvature we consider independent gaussian additive noise with mean zero and standard deviationσq. The step size selected for the update (3.10) is of the form ηt=η0/(1 +ζt) and the initial position is selected randomly over [−r0, r0]2.

For this experiment we set the parameters to be c0 = 0, r0 = 20, L = 6, ∆ = 1,

σf0 =σ∇f0 = 1 and σdi =σRi =σni =di(x)/10. The selection of a variance that depends

on the the distance is done so to ensure that the closer the agent is to the boundary of the free space the better the estimation of the obstacle is. In particular, at the boundary we have that σdi = σRi = σni = 0. We set the constant at which the agent is able to

(a) Trajectories withk= 7

(b) Trajectories withk= 12.

Figure 3.1: The trajectories resulting of the navigation function approach – solid line– and its stochastic approximation given in (3.10) –stars–succeed in driving the agent to the goal configuration for five different initial positions as expected in virtue of Theorem 6. We observe that for the same world (cf., Figures 3.1(a) and 3.1(b)) the larger the order parameter kis, the closer the trajectory

Figure 3.2: The trajectories resulting of the navigation function approach withk= 15 – solid line– and its stochastic approximation given in (3.10) –stars–succeed in driving the agent to the goal configuration for five different initial positions as expected in virtue of Theorem 6.

(a) Local estimation of the obstacle with perfect measures.

(b) Stochastic estimation of the obstacle.

Figure 3.3: Estimation of the obstacles by the hallucinated osculating circle for a particular position 58

0 1000 2000 3000 4000 5000 6000 7000 0 5 10 15 20 25

Figure 3.4: Evolution of the distance to the goal in a world with elliptical obstacles. We set the order parameter of the navigation function tok= 12, and the step size to satisfy Assumption 5 with the following parametersη0= 1×10−7,ζ= 5×10−5.

measure an obstacle [cf., (A.46)] to be c = 7. Finally, the parameters of the step size are

η0 = 5×10−2 and ζ = 5×10−3 and we run each simulation 100 steps with a normalized

estimate.

In Figure 3.1 we observe the behavior of the system that follows the local and stochastic update (3.10) – marked with stars – and that of the system following the gradient dynamical system ˙x=−∇ϕk(x)– solid lines – for five different initial conditions. In Figure 3.1(a) the

order parameter is set to be k= 7 while in 3.1(b) it is set to be 12. In both cases it can be observed that the local and stochastic update succeeds in generating a sequence that remains in the free space and that converges to the minimum of the objective function. It is also observed that the direction in which the agent moves while following the local update differs from that of the agent following the gradient of the navigation function. This result is not surprising in virtue of the fact that as discussed in Section 2.2 the model selected results in a biased estimate of the gradient of the navigation function. However notice that by increasing k the two trajectories become closer to each other. This effect can be observed by comparing the trajectories depicted in figures 3.1(a) and 3.1(b) where the order parameter k is set to be 7 and 12 respectively. This result is expected because the norm of the bias decreases with 1/k. This is an Assumption in Section 3.2.1 but in Appendix A.2.1 we show that it is indeed the case for circular estimates of the obstacles. In particular by selectingk large enough the bias could be reduced arbitrarily. Another effect of having larger k is that of diminishing the relative weight of the ∇β(x) as compared to ∇f0(x) in

the gradient of the navigation function. Hence in a sense having large k is equivalent to follow only the direction −∇f0(x) and neglect the obstacles. Thus yielding shorter paths.

Since in the stochastic approximation we only consider nearby obstacles a similar effect is expected. This is what we observed in Figure 3.1(b).

The effect of the standard deviations of the noise in the estimation of the obstacles is illustrated in Figure 3.3. In particular, for the initial position of one of the trajectories depicted in Figure 3.1(a) we observe the estimation of the closest obstacle to that position in the noiseless case 3.3(a) and the estimate with noise 3.3(b). The fact that even for noiseless cases the estimation is not perfect is what yields a biased estimate.

In Figure 3.4 we consider the evolution of the distance between the agent and the destination withk= 12 for the same five initial conditions than in figures 3.1(a) and 3.1(b). For this simulation, we do not consider a normalized gradient and we take smaller step sizes. In particular we setη0 = 1×10−7,ζ = 5×10−5. Observe that the speed at which the

agent advances differs considerably depending on its position. The main reason for this to happen is that the number of obstacles considered for the estimate is not constant and in depends on the position of the agent. This results in a piece-wise continuous scaling α(x) with large differences of its value at the points of discontinuity (cf., (3.1)).