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Optimization-based subdivision algorithm for reachable sets

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  • Reachable sets for nonlinear control systems can be computed via the use of solvers for optimal control problems. The paper presents a new improved variant which applies adaptive concepts similar to the framework of known subdivision techniques by Dellnitz/Hohmann. Using set properties of the nearest point projection, the convergence and rigorousness of the algorithm can be proved without the assumption of diffeomorphism on a nonlinear mapping. The adaptive method is demonstrated by two nonlinear academic examples and for a more complex robot model with box constraints for four states, two controls and five boundary conditions. In these examples adaptive and non-adaptive techniques as well as various discretization methods and optimization solvers are compared.

    The method also offers interesting features, like zooming into details of the reachable set, self-determination of the needed bounding box, easy parallelization and the use of different grid geometries. With the calculation of a 3d funnel in one of the examples, it is shown that the algorithm can also be used to approximate higher dimensional reachable sets and the resulting box collection may serve as a starting point for more sophisticated visualizations or algorithms.

    Mathematics Subject Classification: Primary: 93B03, 49M37; Secondary: 49M25, 49J53, 93C10.


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  • Figure 1.  Reachable set of the bilinear problem for a full $ 33 \times 33 $ grid

    Figure 2.  Reachable set of the bilinear problem, generated with (right) and without (left) the subdivision algorithm

    Figure 3.  First step of the subdivision algorithm. Choose an initial bounding box (left), solve the optimization problems to approximate the map $ \mathcal{P} $ (middle) and select the boxes for the next step (right)

    Figure 4.  Selection of boxes for the next subdivision steps

    Figure 5.  Inner and outer approximation of the reachable set

    Figure 6.  Demonstration of reusability of the results within the subdivision algorithm

    Figure 7.  $ B = [-1,1] $ covered by boxes (left) or by balls (right)

    Figure 8.  $ B = [-1,1]^2 $ covered by boxes (left) or by balls (right)

    Figure 9.  Kenderov problem using different discretization methods for the ODE-constraints with 16 timesteps each: Explicit Euler (left), implicit Euler (middle), implicit trapezoidal rule (right)

    Figure 10.  Kenderov problem using explicit Euler (left), implicit Euler (middle) and the implicit trapezoidal rule (right) with 261 timesteps each

    Figure 11.  Illustration of the robot from Ex. 4.3

    Figure 12.  Rough approximation of the reachable set of the robot problem

    Figure 13.  Finer approximation of the reachable set of the robot problem

    Figure 14.  Reachable set of the robot without the transformation in the objective function (left) and the transformed set using the same results and colorcode (right)

    Figure 15.  Objective function of the non-transformed robot problem (left) and the transformed version (right)

    Figure 16.  Approximated transformed reachable set of the robot with a less elaborate strategy for initial guesses

    Figure 17.  Reachable set of the robot problem, generated with Ipopt (left) and WORHP (right)

    Figure 18.  Robot model using a circular grid

    Figure 19.  Graphical (left) and computational (right) zoom on the example of the Kenderov Problem

    Figure 20.  Self-finding of the bounding box

    Figure 21.  Piecewise construction of the whole reachable set, by choosing new bounding boxes at locations containing target points from previous runs or cut off looking edges

    Figure 22.  Solution funnel of the bilinear problem for $ T \in [0, 1] $. The plot only shows the slices at $ t_i = \frac{1}{8} \cdot i,\; i\in \{0, 1, \dots, 8\} $ to make it easier to read (cpu-time 103 534s)

    Figure 23.  Final box collections of the solution funnel of the bilinear problem for $ T \in [0, 1] $, rendered with POV-Ray

    Table 1.  Comparison of the computational effort of the algorithm with (S) and without (N) using the subdivision algorithm for a few different grids

    Number of optimization problems
    Grid N S S/N
    3 × 3 9 9 1
    5 × 5 25 18 0.72
    9 × 9 81 41 0.506
    17 × 17 289 92 0.318
    33 × 33 1 089 231 0.212
    65 × 65 4 225 635 0.150
    129 × 129 16 641 1948 0.119
    257 × 257 66 049 6822 0.103
    513 × 513 263 169 25477 0.097
    1025 × 1025 1 050 625 98040 0.093
    CPU time
    Grid N S S/N
    3 × 3 0.13s 0.13s 1
    5 × 5 0.35s 0.25s 0.714
    9 × 9 1.2s 0.57s 0.475
    17 × 17 4.3s 1.24s 0.288
    33 × 33 16.12s 3.12s 0.194
    65 × 65 62.74s 8.55s 0.136
    129 × 129 253.53s 26.29s 0.104
    257 × 257 1128.09s 92.63s 0.082
    513 × 513 8234.45s 355.29s 0.043
    1025 × 1025 94221.4s 1741.58s 0.018
     | Show Table
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    Table 2.  Computational times for generating parts of the reachable set with the shown boxes

    Box Times
    (with subdivision)
    (without subdivision)
    [-1.1, 0.4]×[-0.5, 0.5] 7.49 s 18.21 s
    [ 0.4, 1.9]×[-0.5, 0.5] 7.00 s 17.58 s
    [-1.1, 0.4]×[ 0.5, 1.5] 9.35 s 18.12 s
    [ 0.4, 1.9]×[ 0.5, 1.5] 10.55 s 18.69 s
     | Show Table
    DownLoad: CSV
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