Analisis de Componentes Principales
4. Incidencia del fallo sobre el tema de tesis La CSJN establece una doctrina que aspira a impedir el ejercicio del derecho de huelga a
To illustrate our proposed method for computing reach-avoid sets, we present two numerical examples. The first shows the computational procedure in a simple optimal control scenario with a moving target and a moving obstacle, and the obtained reach-avoid set is validated against the analytic result. The second example presents a two-player reach-avoid game with moving target and constraint sets; our method is benchmarked against the space-time state augmentation approach (as proposed in [89]), reaching the same computed set (within one grid cell of accuracy) at drastically lower computational cost (by two orders of magnitude).
3.4.1
Example 1: Reachability Problem
We begin with a comparatively simple geometric problem that we can use to validate the accuracy of the numerical results against the analytic ground truth. Consider the optimal control problem of guiding a massless vehicle that can move in any direction at a bounded speed (simple motion dynamics) trying to reach a moving target set while avoiding a moving obstacle. The problem is defined on a finite time interval [0, T ] with T = 0.5. The state x(t) = (x(t), y(t)) represents the vehicle’s position on the plane, with dynamics
˙x = vu(t), u(t)∈ U, (3.29) where v = 0.5 is the maximum speed of the vehicle and U is the unit disc.
The target set is a square with side length 0.4 that moves downward (i.e. in the negative y direction) with velocity vT = 1.5. The center of the target set is initially located at (0, 0.75)
at t = 0, and reaches (0, 0) at t = 0.5.
T (t) = {(x, y) : max(|x|, |y − (0.75 − vTt)|) ≤ 0.2}. (3.30)
We represent this moving target set using a signed distance function, l(x, y, t) := sT(x, y, t). The failure set is a square obstacle with side length 0.2 that also moves downward, with velocity vF = 1. The center of the obstacle is at (0, 0) at t = 0, and (0,−0.5) at t = 0.5.
F(t) = {(x, y) : max(|x|, |y − (−vFt)|) < 0.1}. (3.31)
The constraint set K(t) = Fc(t) can then be encoded through the signed distance function
g(x, y, t) := sK(x, y, t).
Figure 3.1 shows the backward-time evolution of the reach-avoid set for the example problem described above. The lower boundary of the reach-avoid set for t = 0.45 consists of states from which the vehicle can meet the target set exactly at its final, lowermost position by constantly moving upward. This lower boundary progresses down in backward time (as the vehicle is given more time to arrive at this final position), but eventually gets “blocked” by the obstacle (t = 0.3), excluding any vehicle trajectories violating the constraint on their way to reaching the target. For earlier times (t = 0.1), the boundary is “pinched inwards”
t=0.45 -1 0 1 -1 0 1 t=0.10 -1 0 1 -1 0 1 t=0.30 -1 0 1 -1 0 1 t=0.00 -1 0 1 -1 0 1 Reach-avoid set Obstacle Target Set
Figure 3.1: Backward-time evolution of the reach-avoid set for an optimal control problem with a target (large square) moving downward at speed 1.5, and an obstacle (small square) moving downward at speed 1. The inside of the dashed boundary represents the set of states that can reach the target set while avoiding the obstacle.
again, including nearby states from which the vehicle can move around the obstacle to safely reach the target; yet, there remains a triangular region directly below the obstacle, as seen in the t = 0 plot, that is not part of the reach-avoid set, because starting from those states the vehicle is unable to avoid the obstacle that is moving down. The diagonal boundaries of the reach-avoid set at its upper region are formed by those states from which the vehicle can meet the target set between its initial and final positions. Lastly, the target set is always part of the reach-avoid set, since a vehicle starting inside the target has immediately succeeded in reaching it without constraint violations (note that the target and the obstacle never overlap); conversely, the failure set is always excluded from the reach-avoid set for the opposite reason.
Analytic Solution
The reach-avoid set boundary for this example problem can be computed analytically, and thus used as ground truth for the numerically obtained boundary. Because the problem is symmetric about the y axis, we will consider the reach-avoid set in the region x ≤ 0. We
now derive the analytic boundary by considering several different segments separately; the segments are labeled with numbers 1–7 in Figure 3.2.
t = 0 x -0.5 0 0.5 y -0.5 0 0.5 1
Numerical reach-avoid set Target set
Obstacle
Analytic reach-avoid set
1 2 3 4 5 6 7
Figure 3.2: Analytic and numerical reach-avoid set. The analytic boundary is only shown on the left half of the domain to facilitate visual comparison.
Based on the uniform motion of the target and obstacle and the simple motion dynamics of the vehicle, we can reason about each of these segments geometrically as follows.
1. Upper diagonal segment: this straight segment contains states from which the vehicle can reach the top left corner of the target at an intermediate position by optimally moving in a straight line, perpendicular to the segment, towards the interception point.
{(x, y) : −0.2 + 3−12(y− 0.95), y ∈ [0.375, 0.95]} (3.32)
2. Upper transition arc: this arc is formed by states from which the vehicle can move towards the point (−0.1, 0.2) and reach the top left corner of the moving target at exactly the final time.
{(x, y) : x = −0.2 −p0.252− (y − 0.2)2, y∈ [0.2, 0.375]} (3.33)
3. Side vertical segment: this straight segment contains all states from which the vehicle can reach the left side of the obstacle at its final position by traveling in a straight
horizontal line at maximum speed.
{(x, y) : x = −0.45, y ∈ [−0.2, 0.2]} (3.34) 4. Outer lower arc: this arc is formed by states from which the vehicle can move towards the point (−0.1, −0.2) and reach the bottom left corner of the moving target at exactly the final time.
{(x, y) : x = −0.2 −p0.252− (y + 0.2)2, y∈ [−0.2, −0.45]} (3.35)
5. Lower horizontal segment: this straight segment contains all states from which the vehicle can reach the lower side of the obstacle at its final position by traveling in a straight vertical line at maximum speed.
{(x, y) : x ∈ [−0.2, −0.1], y = −0.45} (3.36) 6. Lower arc below obstacle: this arc contains states from which the vehicle can move in a straight line to the point (xc, yc) = (−0.1, −0.31) by time tc= 0.28 (barely avoiding
the obstacle’s incoming lower left corner) and subsequently move vertically upwards, reaching the target at point (−0.1, −0.2) at the final time.
{(x, y) : x ∈ [−0.1, 0], y = −0.31 −p0.142− (x + 0.1)2} (3.37)
7. Obstacle’s “shadow”: this straight segment contains states from which the vehicle can barely avoid collision with the incoming obstacle by moving in a straight line to the future location of its lower left corner (analogously to Segment 1). A vehicle initially within the region enclosed by the segment and the obstacle cannot avoid a collision.
{(x, y) : x ∈ [−0.1, 0], y = −0.1 −√6.25 (x + 0.1)} (3.38) Numerical Convergence
Using the scheme described in Section 3.3, we numerically solved the double-obstacle Hamilton- Jacobi variational inequality (3.17) on a computation domain consisting of N×N grid points for N = 51, 101, 151, 201, 251, 301. We compared each of the numerical solutions to the de- rived analytic solution by the following procedure:
1. Construct the (non-negative) distance function to the zero level set of the numerically computed value function encoding the boundary of the computed reach-avoid set (for instance using [53]).
2. Evaluate the distance function at approximately 20 000 points distributed on the ana- lytically determined boundary of the reach-avoid set.
Number of grid points 50 100 150 200 250 300 350 Error 10-4 10-3 10-2 10-1 e avg emax dx
Figure 3.3: Convergence of the proposed numerical Hamilton-Jacobi solution scheme for Example 1 as the grid resolution is increased. Average error (distance) is consistently an order of magnitude smaller than the grid spacing, with the maximum error being roughly half of the grid size.
The resulting values measure the separation between points on the analytic (ground truth) reach-avoid set boundary and the numerically computed boundary. These values are used to construct error metrics for the numerical approximation.
Figure 3.3 shows in logarithmic scale the mean error and maximum error over all an- alytic points plotted against the number of grid points per dimension. An additional line is provided to give the scale of error in terms of the size of spatial discretization or grid spacing. Consistently across the different grid spacings, the mean error is roughly one tenth of the grid spacing, and the maximum error is approximately half of the grid spacing. The numerical scheme therefore exhibits desirable convergence both in terms of the mean error and the maximum error.
3.4.2
Example 2: Reach-Avoid Game
Figure 3.5 allows us to compare the reach-avoid sets computed by two alternative methods: 1. Space-time state augmentation method (4D): an augmented state representation (and numerical grid) is constructed with time xt ∈ [0, T ] as an auxiliary state, with trivial dy-
namics ˙xt= 1. All time-dependence is now reduced to state-dependence and therefore
the traditional Hamilton-Jacobi-Isaacs equation [88, 93] and numerical schemes [53] for time-invariant problems can be used on this higher-dimensional system.
2. Time-varying double-obstacle method (3D): the problem is solved in the original state space of the problem (with a numerical grid constructed accordingly), using the formu-
lation introduced in this chapter and the numerical scheme outlined in Algorithm 3.1 for time-varying problems.
The next example introduces additional complexity with a two-player zero-sum differ- ential game, and is used to illustrate the substantial computational savings introduced by our method relative to former approaches that required augmenting the state space with an auxiliary time variable (cf. [89]). Consider a reach-avoid game played on the finite time interval [0, T ], T = 1, during which the attacker moves freely in a square domain [−1, 1]2
while the defender moves on the vertical segment {0.05} × [−1, 1]. Let rA= (xA, yA) be the
position of the attacker, and yD be the vertical position of the defender, with the state of
the system x = (xA, yA, yD) governed by dynamics
˙pA= vAa(t), kak2 ≤ 1 ,
˙yD = vDb(t), b∈ [−1, 1] .
(3.39)
In this reach-avoid game, the attacker wishes to reach a target set that is moving upwards at speed vT = 1.5, while the defender tries to prevent the attacker from succeeding by
intercepting or delaying its advance. The attacker is additionally required to avoid a growing obstacle whose lower edge is expanding downwards at a rate vF = 0.5. The players have
maximum speeds vA = 2 and vD = 3. Interception is defined as the two players coming
within a distance of 0.1 of each other. Figure 3.5 shows the initial configuration of the moving target and the moving obstacle, as well the interception set centered at four possible defender positions.
The reach-avoid set that we seek to compute comprises the set of joint player configu- rations from which the attacker is guaranteed the ability to reach the target while avoiding both interception by the defender and collision with the obstacle. In game-theoretic terms, it is the attacker’s victory domain.
To compute the reach-avoid set, we solve (3.17) with the Hamiltonian H(x,∇xV, t) = min
kak2≤1
max
b∈[−1,1]∇rAV · vAa(t) +∇yDV · vDb(t) . (3.40)
It is straightforward to see that the decoupling between the motion of both players makes the order of optimizations irrelevant, that is, the minimax and the maximin are identical to each other. Therefore, Isaacs’ condition holds and the upper and lower values coincide—we can simply speak of the value of the game. This implies that neither player needs (nor can benefit from) instantaneous information on the others’ control input. The optimal Hamiltonian of the game can be directly expressed as
H(x,∇xV, t) =−vAk∇rAVk2+ vD|∇yDV|. (3.41)
Since the state space of the reach-avoid game is three-dimensional, we visualize two- dimensional cross-sections of the three-dimensional reach-avoid set at t = 1, taken at various initial defender positions.
t=0.92 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 t=0.72 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 t=0.25 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 t=0.00 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Reach-avoid set (3D) Interception set Target set Obstacle
Figure 3.4: Backward-time evolution of the reach-avoid set for a reach-avoid differential game. As t decreases, the attacker has more time to reach the target, so the reach-avoid set grows. The growth of the reach-avoid set is inhibited by the defender’s interception set and the obstacle.
Figure 3.4 shows multiple stages of the backward-time evolution of the reach-avoid set, sliced at the same defender position. At t = 0.92, there is a relatively small region in the state space from which the attacker can reach the target by the end of the game (T = 1). At earlier times t, the attacker has more time to reach the target and thus the reach-avoid set becomes larger; however, its growth is inhibited by the presence of the obstacle and the presence of the defender, who will actively try to intercept the attacker if it comes within range.
The shape of the reach-avoid set across the different initial defender positions is presented in Figure 3.5. If the defender starts the game near the bottom of the domain (top left plot), it will be able to block the attacker from traversing the narrowing gap below the obstacle. Thus we see that the reach-avoid set boundary does not extend into the left half quadrant of the domain. However, in this case, the attacker is free to cross the gap above the top edge of the obstacle, which leads to a large area of the top left quadrant being inside the reach-avoid set.
As the defender’s starting position gradually becomes higher, we see a shift towards the opposite reach-avoid set layout with the bottom gap now becoming less well protected. However, the reach-avoid set extends into the bottom left quadrant to a lesser extent than in the top left quadrant. This is due to the fact that the passage under the obstacle is closing and the target is moving away from it: therefore an attacker not starting close enough to the opening will either get blocked out or not be able to make it through in time to reach the target.
Computational Efficiency
Computations for both methods were run in MATLAB using [53] on a computer with a Core i7-2640M processor. As can be appreciated in the figure, the reach-avoid set boundaries computed by the two methods are extremely similar (in fact, the computed reach-avoid boundaries are well within a grid cell of each other throughout the state space); this similarity contrasts with the very different amounts of computation required to obtain the two solutions. The space-time state augmentation method took approximately 1 hour and 50 minutes on a 454 state grid. The time-varying double-obstacle method took approximately 3 minutes on
a 513 state grid.
Given that the dynamic programming computation is roughly linear in the number of state grid cells, it is expected that having about 30 times fewer grid cells would lead to a comparable speedup in computation, and indeed computation was roughly 36 times faster. This scaling property is important, since the grid used for this toy problem is of moderate size. As we will see in Chapter 4, many engineering problems will require much larger grids and longer time horizons (possibly with hundreds or thousands of time steps), in which case the difference in computational cost between the two methods can be of multiple orders of magnitude.
t=0 -1 0 1 -1 -0.5 0 0.5 1 t=0 -1 0 1 -1 -0.5 0 0.5 1 t=0 -1 0 1 -1 -0.5 0 0.5 1 Reach-avoid set (4D) Reach-avoid set (3D) Interception set Target set Obstacle t=0 -1 0 1 -1 -0.5 0 0.5 1
Figure 3.5: Reach-avoid set computed through the space-time state augmentation method (4D) and the time-varying double-obstacle method (3D). 2D cross-sections of the set are shown at the initial time for four different defender positions.