An Automated Parallel Parking Strategy Using Reach Control Theory

Numerical Values Used for Simulations in “An Automated Parallel Parking Strategy Using Reach Control Theory” Melkior Ornik∗

Miad Moarref∗

Mireille E. Broucke∗

March 14, 2017

Abstract This document contains raw numerical values omitted from [1] for lack of space. It gives the dimensions of the vehicle, coordinates of the obstacles, and the location of initial and desired goal sets used in the parallel parking strategy presented in that paper. Additionally, this document lists the coordinates of vertices, simplices and polytopes used in [1], as well as controls assigned at each vertex and each simplex of the state space. For details on the proposed parking maneuver, we invite the reader to consult [1].

1

Vehicle Size Dimensions of the car: wheelbase width distance of the front end from the front axle distance of the back end from the rear axle

2

2.468m, 1.952m, 0.911m, 0.819m.

Obstacle Positions

Unless otherwise stated, all values in the remainder of this file are given in metres. Position of the front parked car: rear left corner rear right corner front left corner front right corner

Position of the rear parked car:

(2, 0), (2, −1.952), (6.198, 0), (6.198, −1.952).

rear left corner rear right corner front left corner front right corner

(−10.198, 0), (−10.198, −1.952), (−6, 0), (−6, −1.952).

Position of the curb: {(x, y) ∈ R2 | y < −2.002}. ∗ Department of Electrical and Computer Engineering, University of Toronto, Toronto ON Canada M5S 3G4, {[email protected], [email protected], [email protected]}

1

3

Positions of the Initial and Ending Box Coordinates of the initial box I: top left corner bottom left corner top right corner bottom right corner

4

Coordinates of the ending box E:

(2.319, 1.976), (2.319, 1.726), (2.819, 1.976), (2.819, 1.726).

top left corner bottom left corner top right corner bottom right corner

(−5.181, −0.676), (−5.181, −0.976), (−1.379, −0.676), (−1.379, −0.976).

Polytope Construction

Coefficients ∆xk , ∆yk , wkx , and wky used in the construction of polytopes Pk , k ∈ {1, . . . , 8}, are as follows: k 1 2 3 4 5 6 7 8

∆xk -0.850 -0.834 -0.785 -0.707 -0.707 -0.785 -0.834 -0.850

∆yk -0.094 -0.286 -0.445 -0.592 -0.592 -0.445 -0.286 -0.094

wkx 0 0 0 0 0 0 0 0

wky 0.09375 0.12 0.12 0.12 0.12 0.12 0.12 0.09375

In total, polytopes Pk contain 40 vertices, labeled by vi , i ∈ {1, . . . , 40}. Vertices of polytopes Pk for k ∈ {1, 2, 3, 4} are v4(k−1)+1 , . . . , v4(k+1) , and vertices of polytopes Pk for k ∈ {5, 6, 7, 8} are v4k+1 , . . . , v4(k+2) . Vertices v4k+1 , . . . , v4(k+1) are vertices of exit facets Fkout for k ∈ {1, 2, 3, 4}, and vertices v4(k+1)+1 , . . . , v4(k+2) are vertices of exit facets Fkout for k ∈ {5, 6, 7, 8}. Vertices v17 , . . . , v20 and v21 , . . . , v24 , respectively, coincide, but are treated as separate in the remainder of the calculations. The coordinates (x, y, θ) of vertices vi are as follows: i 1 2 3 4 5 6 7 8 9 10

x 2.319 2.819 2.819 2.319 1.469 1.969 1.969 1.469 0.635 1.135

y 1.976 1.976 1.726 1.726 1.976 1.976 1.539 1.539 1.820 1.820

θ (in rad) 0.000 0.000 0.000 0.000 0.196 0.196 0.196 0.196 0.393 0.393

i 11 12 13 14 15 16 17 18 19 20

2

x 1.135 0.635 -0.150 0.350 0.350 -0.150 -0.857 -0.357 -0.357 -0.857

y 1.123 1.123 1.505 1.505 0.547 0.547 1.043 1.043 -0.175 -0.175

θ (in rad) 0.393 0.393 0.589 0.589 0.589 0.589 0.785 0.785 0.785 0.785

i 21 22 23 24 25 26 27 28 29 30

5

x -0.857 -0.357 -0.357 -0.857 -1.563 -1.063 -1.063 -1.563 -2.349 -1.849

y 1.043 1.043 -0.175 -0.175 0.580 0.580 -0.897 -0.897 0.265 0.265

θ (in rad) 0.785 0.785 0.785 0.785 0.589 0.589 0.589 0.589 0.393 0.393

i 31 32 33 34 35 36 37 38 39 40

x -1.849 -2.349 -3.182 -2.682 -2.682 -3.182 -4.032 -3.532 -3.532 -4.032

y -1.472 -1.472 0.109 0.109 -1.888 -1.888 0.109 0.109 -2.076 -2.076

θ (in rad) 0.393 0.393 0.196 0.196 0.196 0.196 0.000 0.000 0.000 0.000

Polytope Triangulation

In addition to vertices v1 , . . . , v40 , triangulations of polytopes Pk , k ∈ {1, . . . , 8}, make use of 8 additional points denoted by vi , i ∈ {41, . . . , 48}. Their coordinates are given as follows: i 41 42 43 44 45 46 47 48

x 2.144 1.302 0.493 -0.253 -0.960 -1.706 -2.516 -3.357

y 1.804 1.614 1.249 0.730 0.138 -0.381 -0.747 -0.936

θ (in rad) 0.098 0.295 0.491 0.687 0.687 0.491 0.295 0.098

Each polytope Pk , k ∈ {1, . . . , 8}, is triangulated into 12 simplices which are denoted by S12(k−1)+1 , . . . , S12k . Each simplex Sj , j ∈ {1, . . . , 96}, is a convex hull of four vertices vj1 , . . . , vj4 , with indices j1 , . . . , j4 ∈ {1, . . . , 48}. The vertices of all simplices are given as follows: j 1 2 3 4 5 6 7 8 9 10 11 12

j1 41 41 41 41 4 4 4 4 41 41 4 4

j2 6 3 1 6 3 41 8 41 7 8 41 1

j3 1 6 6 3 41 8 41 1 6 7 3 41

j4 5 7 2 2 7 7 5 5 5 5 2 2

j 13 14 15 16 17 18 19 20 21 22 23 24

j1 42 42 42 42 42 42 42 42 42 42 42 42

j2 10 5 8 12 5 10 7 8 11 12 7 7

3

i3 5 8 12 8 10 7 10 7 10 11 5 8

j4 9 9 9 11 6 6 11 11 9 9 6 5

j 25 26 27 28 29 30 31 32 33 34 35 36

j1 43 43 43 43 43 43 43 43 43 43 43 43

j2 14 9 14 9 11 12 12 16 11 11 15 16

i3 11 14 9 12 14 11 16 12 9 12 14 15

j4 10 10 13 13 15 15 13 15 10 9 13 13

j 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

6

j1 44 44 44 44 44 44 44 44 44 44 44 44 45 45 45 45 45 45 45 45

j2 16 15 16 20 13 18 13 18 20 19 16 15 26 23 21 24 26 23 28 24

i3 15 18 20 16 18 15 16 13 19 18 13 16 21 26 26 21 23 24 24 28

j4 19 19 17 19 14 14 17 17 17 17 14 14 22 22 25 25 27 27 25 27

j 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

j1 45 45 45 45 30 30 28 28 28 27 27 27 30 30 27 27 34 34 32 32

j2 27 26 24 21 46 25 32 46 25 28 46 30 32 46 46 46 47 29 36 47

i3 28 27 23 23 25 46 46 32 46 46 30 46 46 32 28 25 29 47 47 36

j4 25 25 21 22 29 26 31 29 29 31 31 26 29 31 25 26 33 30 35 33

j 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

j1 32 31 31 31 34 34 31 31 38 38 38 38 36 36 36 36 38 38 36 36

j2 29 32 47 34 36 47 47 47 48 33 48 35 48 33 40 48 40 48 48 35

i3 47 47 34 47 47 36 32 29 33 48 35 48 40 48 48 35 48 40 33 48

j4 33 35 35 30 33 35 29 30 37 34 34 39 37 37 39 39 37 39 34 34

Proposed Control Law

Speeds and front wheel angles assigned at each simplex vertex vi , i ∈ {1, . . . , 48}, are given as follows: i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

v (in m/s) -1.654 -1.391 -2.225 -1.747 -1.773 -2.319 -2.447 -1.711 -1.815 -2.348 -2.289 -1.674 -1.818 -2.358 -2.227 -1.667

i 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

ϕ (in rad) -0.484 -0.141 -0.273 -0.499 -0.505 -0.095 -0.509 -0.533 -0.506 -0.077 -0.488 -0.518 -0.504 -0.071 -0.506 -0.516

4

v (in m/s) -1.822 -2.198 -2.159 -1.760 -1.854 -2.264 -2.186 -1.767 -1.818 -2.358 -2.227 -1.667 -1.815 -2.348 -2.289 -1.674

ϕ (in rad) -0.459 0.244 -0.504 -0.493 0.466 0.136 0.511 0.493 0.504 0.071 0.506 0.516 0.506 0.077 0.488 0.518

i 33 34 35 36 37 38 39 40

v (in m/s) -1.773 -2.319 -2.447 -1.711 -1.643 -1.406 -2.198 -1.734

ϕ (in rad) 0.505 0.095 0.509 0.533 0.495 -0.195 0.166 0.502

i 41 42 43 44 45 46 47 48

v (in m/s) -1.909 -2.047 -2.025 -2.001 -2.018 -2.025 -2.047 -1.904

ϕ (in rad) -0.380 -0.404 -0.398 -0.351 0.400 0.398 0.404 0.326

If Sj is a simplex with vertices vj1 , . . . , vj4 , the feedback control uj (x) = Kj x + gj is obtained by affinely extending the control values assigned to vj1 , . . . , vj4 above (with x1 corresponding to speed v, and x2 corresponding to front wheel angle φ). This produces the following feedback controls at each simplex Sj : j 

-1.092 0.820



-1.510 0.605



0.527 0.685



-2.081 0.684



-0.957 0.451



-1.473 0.048



-0.621 0.500



-0.525 0.499



-1.092 0.820



-1.473 0.048



-0.957 0.451



0.527 0.685

1 2 3 4 5 6 7 8 9 10 11 12

uj (x) = Kj x + gj   -0.456 -5.332 x+ 0.526 3.442   0.293 -7.388 x+ 0.946 2.324   -0.456 -2.446 x+ 0.526 3.202   3.338 -13.736 x+ 0.528 3.194   1.097 -4.228 x+ 0.721 1.440   1.097 -5.146 x+ 0.721 0.721   -0.142 -2.643 x+ 0.063 2.052   0.369 -2.879 x+ 0.059 2.053   0.293 -6.643 x+ 0.946 2.707   -0.142 -8.498 x+ 0.063 -1.059   3.338 -6.011 x+ 0.528 1.593   0.369 -1.003 x+ 0.059 2.385

5

1.779 -3.425



1.526 -3.612



-1.976 -3.112



-2.120 -3.112



-1.419 -2.788



-0.224 -1.853



-0.061 -1.767



-1.165 -1.759



0.557 -4.110



2.339 -0.491



-5.287 -2.456



-3.605 -2.190



j 

-1.065 0.858



-1.173 0.478



-1.189 0.466



-1.231 0.060



-1.092 0.820



-1.478 0.197



-1.380 0.290



-1.473 0.048



-1.065 0.858



-1.231 0.060



-1.092 0.820



-1.473 0.048



-1.175 0.323



-1.065 0.858



-1.081 0.867



-1.168 0.455



-1.162 0.358



-1.231 0.060

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

uj (x) = Kj x + gj   -0.047 -4.775 x+ 0.397 3.952   -0.142 -5.307 x+ 0.063 2.077   -0.203 -5.287 x+ 0.017 2.092   -0.165 -5.388 x+ 0.375 1.124   -0.047 -4.821 x+ 0.397 3.889   0.293 -6.190 x+ 0.946 1.680   -0.084 -5.233 x+ 0.589 2.586   -0.165 -5.798 x+ 0.375 1.103   -0.084 -4.699 x+ 0.589 3.550   -0.203 -5.576 x+ 0.017 -0.670   0.293 -3.568 x+ 0.946 5.910   -0.142 -5.816 x+ 0.063 1.345   -0.084 -4.888 x+ 0.589 2.272   -0.150 -4.555 x+ 0.266 3.891   -0.150 -4.578 x+ 0.266 3.905   -0.203 -5.010 x+ 0.017 1.857   -0.137 -4.728 x+ 0.455 2.672   -0.179 -5.132 x+ 0.275 0.952

6

0.823 -3.325



1.272 -1.740



1.385 -1.654



1.410 -1.418



0.871 -3.258



1.228 -2.682



1.427 -2.494



1.845 -1.396



0.859 -3.517



1.526 -0.311



-0.047 -4.739



1.813 -0.963



1.058 -2.408



0.924 -3.063



0.943 -3.074



1.263 -1.555



1.040 -2.454



1.325 -1.238



j 

-1.156 0.454



-1.119 0.021



-1.065 0.858



-1.231 0.060



-1.081 0.867



-1.119 0.021



-1.119 0.021



-0.838 0.720



-1.040 0.554



-0.798 -0.021



-1.081 0.867



-0.811 0.762



-1.067 0.550



-0.753 1.406



-0.798 -0.021



-0.753 1.406



-1.081 0.867



-1.119 0.021

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

uj (x) = Kj x + gj   -0.157 -5.051 x+ 0.013 1.860   -0.179 -4.969 x+ 0.275 0.895   -0.084 -4.168 x+ 0.589 5.771   -0.203 -5.102 x+ 0.017 1.283   -0.137 -4.612 x+ 0.455 3.411   -0.157 -4.809 x+ 0.013 -0.975   -0.162 -4.278 x+ 0.292 1.162   -0.032 -2.789 x+ 0.614 4.863   -0.051 -4.401 x+ 0.028 2.215   -0.162 -3.940 x+ 0.292 1.117   -0.013 -3.108 x+ 0.406 5.678   -0.137 -2.427 x+ 0.455 5.413   -0.157 -4.233 x+ 0.013 2.238   -0.013 -2.763 x+ 0.406 6.245   -0.051 -2.917 x+ 0.028 -1.320   -0.032 -2.700 x+ 0.614 5.584   -0.157 -4.247 x+ 0.013 2.572   -0.137 -4.324 x+ 0.455 0.859

7

1.221 -1.551



1.190 -1.191



0.651 -4.389



1.340 -1.078



0.943 -3.068



1.084 0.054



0.774 -1.357



-0.273 -3.958



0.797 -1.753



0.623 -1.338



-0.130 -4.330



-0.439 -4.210



0.752 -1.760



-0.284 -4.583



-0.162 0.531



-0.313 -4.281



0.758 -1.909



0.787 -1.268



j 

-0.818 -0.660



-1.001 -0.359



-1.081 -0.867



-0.920 -0.399



-0.995 -0.362



-0.838 0.035



-0.928 -0.395



-1.119 -0.021



-1.119 -0.021



-1.081 -0.867



-0.838 0.035



-0.818 -0.660



-1.065 -0.858



-1.081 -0.867



-1.231 -0.060



-1.154 -0.454



-1.160 -0.454



-1.119 -0.021

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

uj (x) = Kj x + gj   -0.132 3.737 x+ -0.196 3.170   -0.064 4.236 x+ -0.308 2.348   -0.132 4.014 x+ -0.196 3.388   -0.072 3.293 x+ -0.022 1.293   -0.089 4.116 x+ -0.295 2.411   -0.031 3.338 x+ -0.147 0.438   -0.102 3.206 x+ -0.008 1.333   -0.031 3.634 x+ -0.147 0.497   -0.102 4.383 x+ -0.008 -0.966   -0.089 4.207 x+ -0.295 2.943   -0.072 3.207 x+ -0.022 0.836   -0.064 3.111 x+ -0.308 4.200   -0.072 4.366 x+ -0.134 3.635   -0.072 4.388 x+ -0.134 3.648   -0.057 5.126 x+ -0.129 0.611   -0.081 4.889 x+ -0.007 1.828   -0.102 4.789 x+ -0.008 1.820   -0.057 4.964 x+ -0.129 0.555

8

-5.353 -2.385



-5.881 -1.515



-5.796 -2.733



-5.154 -0.868



-5.789 -1.564



-5.112 0.154



-5.098 -0.894



-5.585 0.059



-6.090 1.045



-5.935 -2.414



-5.016 -0.137



-4.933 -3.078



-6.013 -2.900



-6.051 -2.923



-6.664 -0.053



-6.425 -1.276



-6.393 -1.273



-6.393 0.041



j 

-1.043 -0.365



-1.028 -0.350



-1.065 -0.858



-1.231 -0.060



-1.119 -0.021



-1.081 -0.867



-1.092 -0.820



-1.065 -0.858



-1.473 -0.048



-1.303 -0.434



-1.316 -0.432



-1.231 -0.060



-1.046 -0.396



-1.020 -0.388



-1.092 -0.820



-1.473 -0.048



-1.231 -0.060



-1.065 -0.858

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

uj (x) = Kj x + gj   -0.034 4.588 x+ -0.236 2.245   -0.089 4.204 x+ -0.295 1.838   -0.081 4.307 x+ -0.007 4.471   -0.034 4.862 x+ -0.236 1.801   -0.102 4.731 x+ -0.008 1.190   -0.089 4.549 x+ -0.295 5.223   -0.021 4.440 x+ -0.111 3.574   -0.021 4.395 x+ -0.111 3.638   0.014 6.412 x+ -0.117 0.374   -0.031 5.787 x+ -0.014 1.796   -0.081 5.438 x+ -0.007 1.845   0.014 6.002 x+ -0.117 0.395   0.064 5.107 x+ -0.207 2.011   -0.034 4.212 x+ -0.236 1.743   -0.031 4.357 x+ -0.014 4.418   0.064 5.833 x+ -0.207 1.419   -0.081 5.294 x+ -0.007 1.212   -0.034 4.520 x+ -0.236 4.934

9

-6.069 -1.416



-5.876 -1.213



-5.987 -3.263



-6.525 -0.678



-6.295 -0.225



-6.136 -3.757



-6.118 -2.794



-6.038 -2.908



-7.629 0.087



-7.052 -1.227



-7.020 -1.232



-6.902 0.050



-6.134 -1.338



-5.879 -1.262



-6.101 -2.971



-7.422 -0.288



-6.765 -0.109



-6.084 -3.383



j 

0.473 -1.378



-1.092 -0.820



-0.104 -0.638



-0.159 -0.646



-0.901 -0.750



-0.914 -0.752



-0.927 -0.673



-1.473 -0.048



0.473 -1.378



-0.927 -0.673



-1.092 -0.820



-1.473 -0.048

85 86 87 88 89 90 91 92 93 94 95 96

uj (x) = Kj x + gj   0.301 -2.710 x+ -0.163 6.021   0.301 0.079 x+ -0.163 5.026   0.064 -4.200 x+ -0.207 4.237   0.362 -0.926 x+ -0.165 4.700   0.042 3.976 x+ -0.004 3.404   -0.031 3.294 x+ -0.014 3.310   0.048 4.086 x+ -0.022 3.090   0.048 5.058 x+ -0.022 1.974   0.042 -5.465 x+ -0.004 7.724   0.362 0.442 x+ -0.165 4.749   -0.031 3.611 x+ -0.014 3.431   0.064 5.212 x+ -0.207 0.179

0.230 -5.045



-5.297 -3.074



-1.779 -2.425



-2.009 -2.457



-5.279 -2.529



-5.326 -2.536



-5.373 -2.258



-7.300 -0.048



0.259 -5.063



-4.722 -2.555



-5.954 -2.777



-7.300 -0.045



References [1] M. Ornik, M. Moarref, M. E. Broucke, An automated parallel parking strategy using reach control theory, accepted to 20th IFAC World Congress, 2017.

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