-
Notifications
You must be signed in to change notification settings - Fork 15
/
dynamic_extension_tracking.py
211 lines (174 loc) · 5.43 KB
/
dynamic_extension_tracking.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
"""
Example dynamic_extension_tracking.py
Author: Joshua A. Marshall <[email protected]>
GitHub: https://github.com/botprof/agv-examples
"""
# %%
# SIMULATION SETUP
import numpy as np
import matplotlib.pyplot as plt
from mobotpy.models import DiffDrive
from mobotpy.integration import rk_four
from scipy import signal
# Set the simulation time [s] and the sample period [s]
SIM_TIME = 15.0
T = 0.04
# Create an array of time values [s]
t = np.arange(0.0, SIM_TIME, T)
N = np.size(t)
# %%
# COMPUTE THE REFERENCE TRAJECTORY
# Radius of the circle [m]
R = 10
# Angular rate [rad/s] at which to traverse the circle
OMEGA = 0.1
# Pre-compute the desired trajectory
x_d = np.zeros((3, N))
u_d = np.zeros((2, N))
xi_d = np.zeros((4, N))
ddz_d = np.zeros((2, N))
for k in range(0, N):
x_d[0, k] = R * np.sin(OMEGA * t[k])
x_d[1, k] = R * (1 - np.cos(OMEGA * t[k]))
x_d[2, k] = OMEGA * t[k]
u_d[0, k] = R * OMEGA
u_d[1, k] = OMEGA
# Pre-compute the extended system reference trajectory
for k in range(0, N):
xi_d[0, k] = x_d[0, k]
xi_d[1, k] = x_d[1, k]
xi_d[2, k] = u_d[0, k] * np.cos(x_d[2, k])
xi_d[3, k] = u_d[0, k] * np.sin(x_d[2, k])
# Pre-compute the extended system reference acceleration
for k in range(0, N):
ddz_d[0, k] = -u_d[0, k] * u_d[1, k] * np.sin(x_d[2, k])
ddz_d[1, k] = u_d[0, k] * u_d[1, k] * np.cos(x_d[2, k])
# %%
# VEHICLE SETUP
# Set the track length of the vehicle [m]
ELL = 1.0
# Create a vehicle object of type DiffDrive
vehicle = DiffDrive(ELL)
# %%
# SIMULATE THE CLOSED-LOOP SYSTEM
# Initial conditions
x_init = np.zeros(3)
x_init[0] = 0.0
x_init[1] = 10.0
x_init[2] = 0.0
# Setup some arrays
x = np.zeros((3, N))
xi = np.zeros((4, N))
u = np.zeros((2, N))
x[:, 0] = x_init
# Set the initial speed [m/s] to be non-zero to avoid singularity
w = np.zeros(2)
u_unicycle = np.zeros(2)
u_unicycle[0] = u_d[0, 0]
# Initial extended state
xi[0, 0] = x_init[0]
xi[1, 0] = x_init[1]
xi[2, 0] = u_d[0, 0] * np.cos(x_init[2])
xi[3, 0] = u_d[0, 0] * np.sin(x_init[2])
# Defined feedback linearized state matrices
A = np.array([[0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]])
B = np.array([[0, 0], [0, 0], [1, 0], [0, 1]])
# Choose pole locations for closed-loop linear system
p = np.array([-1.0, -2.0, -2.5, -1.5])
K = signal.place_poles(A, B, p)
for k in range(1, N):
# Simulate the vehicle motion
x[:, k] = rk_four(vehicle.f, x[:, k - 1], u[:, k - 1], T)
# Update the extended system states
xi[0, k] = x[0, k]
xi[1, k] = x[1, k]
xi[2, k] = u_unicycle[0] * np.cos(x[2, k])
xi[3, k] = u_unicycle[0] * np.sin(x[2, k])
# Compute the extended linear system input control signals
eta = K.gain_matrix @ (xi_d[:, k - 1] - xi[:, k - 1]) + ddz_d[:, k - 1]
# Compute the new (unicycle) vehicle inputs
B_inv = np.array(
[
[np.cos(x[2, k - 1]), np.sin(x[2, k - 1])],
[-np.sin(x[2, k - 1]) / u_unicycle[0], np.cos(x[2, k - 1]) / u_unicycle[0]],
]
)
w = B_inv @ eta
u_unicycle[0] = u_unicycle[0] + T * w[0]
u_unicycle[1] = w[1]
# Convert unicycle inputs to differential drive wheel speeds
u[:, k] = vehicle.uni2diff(u_unicycle)
# %%
# MAKE PLOTS
# Change some plot settings (optional)
plt.rc("text", usetex=True)
plt.rc("text.latex", preamble=r"\usepackage{cmbright,amsmath,bm}")
plt.rc("savefig", format="pdf")
plt.rc("savefig", bbox="tight")
# Plot the states as a function of time
fig1 = plt.figure(1)
fig1.set_figheight(6.4)
ax1a = plt.subplot(411)
plt.plot(t, x_d[0, :], "C1--")
plt.plot(t, x[0, :], "C0")
plt.grid(color="0.95")
plt.ylabel(r"$x$ [m]")
plt.setp(ax1a, xticklabels=[])
plt.legend(["Desired", "Actual"])
ax1b = plt.subplot(412)
plt.plot(t, x_d[1, :], "C1--")
plt.plot(t, x[1, :], "C0")
plt.grid(color="0.95")
plt.ylabel(r"$y$ [m]")
plt.setp(ax1b, xticklabels=[])
ax1c = plt.subplot(413)
plt.plot(t, x_d[2, :] * 180.0 / np.pi, "C1--")
plt.plot(t, x[2, :] * 180.0 / np.pi, "C0")
plt.grid(color="0.95")
plt.ylabel(r"$\theta$ [deg]")
plt.setp(ax1c, xticklabels=[])
ax1d = plt.subplot(414)
plt.step(t, u[0, :], "C2", where="post", label="$v_L$")
plt.step(t, u[1, :], "C3", where="post", label="$v_R$")
plt.grid(color="0.95")
plt.ylabel(r"$\bm{u}$ [m/s]")
plt.xlabel(r"$t$ [s]")
plt.legend()
# Save the plot
plt.savefig("../agv-book/figs/ch4/dynamic_extension_tracking_fig1.pdf")
# Plot the position of the vehicle in the plane
fig2 = plt.figure(2)
plt.plot(x_d[0, :], x_d[1, :], "C1--", label="Desired")
plt.plot(x[0, :], x[1, :], "C0", label="Actual")
plt.axis("equal")
X_L, Y_L, X_R, Y_R, X_B, Y_B, X_C, Y_C = vehicle.draw(x[0, 0], x[1, 0], x[2, 0])
plt.fill(X_L, Y_L, "k")
plt.fill(X_R, Y_R, "k")
plt.fill(X_C, Y_C, "k")
plt.fill(X_B, Y_B, "C2", alpha=0.5, label="Start")
X_L, Y_L, X_R, Y_R, X_B, Y_B, X_C, Y_C = vehicle.draw(
x[0, N - 1], x[1, N - 1], x[2, N - 1]
)
plt.fill(X_L, Y_L, "k")
plt.fill(X_R, Y_R, "k")
plt.fill(X_C, Y_C, "k")
plt.fill(X_B, Y_B, "C3", alpha=0.5, label="End")
plt.xlabel(r"$x$ [m]")
plt.ylabel(r"$y$ [m]")
plt.legend()
# Save the plot
plt.savefig("../agv-book/figs/ch4/dynamic_extension_tracking_fig2.pdf")
# Show all the plots to the screen
plt.show()
# %%
# MAKE AN ANIMATION
# Create and save the animation
ani = vehicle.animate_trajectory(
x, x_d, T, True, "../agv-book/gifs/ch4/dynamic_extension_tracking.gif"
)
# Show the movie to the screen
plt.show()
# # Show animation in HTML output if you are using IPython or Jupyter notebooks
# plt.rc('animation', html='jshtml')
# display(ani)
# plt.close()