-
Notifications
You must be signed in to change notification settings - Fork 0
/
robot_mpc_obs_CBF_opt.py
544 lines (440 loc) · 21.4 KB
/
robot_mpc_obs_CBF_opt.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
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
import casadi as ca
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.patches import Rectangle, Circle
def shift(u, x_n):
u_end = np.concatenate((u[1:], u[-1:]))
x_n = np.concatenate((x_n[1:], x_n[-1:]))
return u_end, x_n
class Car:
def __init__(self):
# Vehicle parameters
self.L = 2.5 # Wheelbase length in meters
self.d = 0.2 # Safety margin in meters
# Control input constraints
self.min_a = -2.0 # Minimum acceleration (m/s^2)
self.max_a = 2.0 # Maximum acceleration (m/s^2)
self.min_omega = -np.radians(30) # Minimum steering rate (rad/s)
self.max_omega = np.radians(30) # Maximum steering rate (rad/s)
# State constraints
self.min_v = 0.0 # Minimum velocity (m/s)
self.max_v = 5.0 # Maximum velocity (m/s)
self.min_delta = -np.radians(30) # Minimum steering angle (rad)
self.max_delta = np.radians(30) # Maximum steering angle (rad)
class MPCControllerFixedGamma:
def __init__(self, car: Car, T=0.1, N=50, gamma=0.2, Q=np.diag([100.0, 100.0, 1.0, 1.0, 1.0]), R=np.diag([0.1, 0.1])):
self.car = car
self.T = T # time step
self.N = N # horizon length
self.gamma = gamma # Fixed gamma value
self.Q = Q
self.R = R
self.next_states = np.zeros((self.N + 1, 5))
self.u0 = np.zeros((self.N, 2))
def control_barrier_function(self, x, y, obs_x, obs_y, obs_r):
safety_margin = 0.1 # Additional safety margin
return (x - obs_x) ** 2 + (y - obs_y) ** 2 - (obs_r + self.car.d + safety_margin) ** 2
def setupController(self, obs, use_cbf=True):
self.opti = ca.Opti()
self.U = self.opti.variable(self.N, 2)
control_a = self.U[:, 0]
control_omega = self.U[:, 1]
self.X = self.opti.variable(self.N + 1, 5)
state_x = self.X[:, 0]
state_y = self.X[:, 1]
state_psi = self.X[:, 2]
state_v = self.X[:, 3]
state_delta = self.X[:, 4]
f = lambda x, u: ca.vertcat(*[
x[3] * ca.cos(x[2]),
x[3] * ca.sin(x[2]),
x[3] / self.car.L * ca.tan(x[4]),
u[0],
u[1]
])
self.x_ref = self.opti.parameter(2, 5)
self.u_ref = self.opti.parameter(self.N, 2)
num_obs = len(obs)
self.x_obs = self.opti.parameter(num_obs)
self.y_obs = self.opti.parameter(num_obs)
self.r_obs = self.opti.parameter(num_obs)
self.opti.subject_to(self.X[0, :] == self.x_ref[0, :])
for i in range(self.N):
x_next = self.X[i, :] + self.T * f(self.X[i, :], self.U[i, :]).T
self.opti.subject_to(self.X[i + 1, :] == x_next)
obj = 0
goal_state = self.x_ref[-1, :]
for i in range(self.N):
obj += ca.mtimes([(self.X[i, :] - goal_state), self.Q, (self.X[i, :] - goal_state).T]) + \
ca.mtimes([self.U[i, :], self.R, self.U[i, :].T])
self.opti.minimize(obj)
# Bounds on states and controls
self.opti.subject_to(self.opti.bounded(self.car.min_v, state_v, self.car.max_v))
self.opti.subject_to(self.opti.bounded(self.car.min_delta, state_delta, self.car.max_delta))
self.opti.subject_to(self.opti.bounded(self.car.min_a, control_a, self.car.max_a))
self.opti.subject_to(self.opti.bounded(self.car.min_omega, control_omega, self.car.max_omega))
# CBF constraints with fixed gamma
for i in range(self.N):
for j in range(num_obs):
h = self.control_barrier_function(state_x[i], state_y[i], self.x_obs[j], self.y_obs[j], self.r_obs[j])
h_next = self.control_barrier_function(state_x[i + 1], state_y[i + 1], self.x_obs[j], self.y_obs[j], self.r_obs[j])
if use_cbf:
self.opti.subject_to(h_next - h + self.gamma * h >= 0)
else:
self.opti.subject_to(h_next >= 0)
opts_setting = {'ipopt.max_iter': 5000,
'ipopt.print_level': 0,
'print_time': 0,
'ipopt.acceptable_tol': 1e-8,
'ipopt.acceptable_obj_change_tol': 1e-6}
self.opti.solver('ipopt', opts_setting)
def solve(self, x_ref, u_ref, obs):
self.opti.set_value(self.x_ref, x_ref)
self.opti.set_value(self.u_ref, u_ref)
self.opti.set_value(self.x_obs, obs[:, 0])
self.opti.set_value(self.y_obs, obs[:, 1])
self.opti.set_value(self.r_obs, obs[:, 2])
x0 = x_ref[0, :]
# Initial guesses
self.opti.set_initial(self.X, np.tile(x0, (self.N + 1, 1)))
self.opti.set_initial(self.U, np.zeros((self.N, 2)))
try:
sol = self.opti.solve()
u_opt = sol.value(self.U)
x_opt = sol.value(self.X)
except RuntimeError:
# If solver fails, return previous control and state
print("Solver failed, using previous control inputs.")
u_opt = self.u0
x_opt = self.next_states
self.u0, self.next_states = shift(u_opt, x_opt)
return u_opt[:, 0], u_opt[:, 1]
# def is_goal_reached(self, current_state, goal_state, tolerance=0.5, speed_threshold=0.1):
# distance = np.linalg.norm(current_state[:2] - goal_state[:2])
# speed = current_state[3]
# return distance < tolerance and speed < speed_threshold
def is_goal_reached(self, current_state, goal_state, tolerance=0.1):
distance = np.linalg.norm(current_state[:2] - goal_state[:2])
return distance < tolerance
def run_until_goal(self, x_ref, u_ref, obs, tolerance=0.5, use_cbf=True):
current_state = x_ref[0, :]
goal_state = x_ref[-1, :]
trajectory = [current_state]
control_inputs = []
plt.ion() # Turn on interactive mode
fig, ax = plt.subplots()
ax.set_xlim(-1, 13)
ax.set_ylim(-1, 13)
# Initialize plot elements
actual_line, = ax.plot([], [], 'b-', label='Actual Trajectory', linewidth=2)
predict_line, = ax.plot([], [], 'r--', label='Predicted Trajectory', linewidth=1)
rect = Rectangle((0, 0), 1, 1, angle=0, color='g', fill=True, label='Robot Pose', alpha=0.2)
ax.add_patch(rect)
obs_circles = [plt.Circle((obs[i, 0], obs[i, 1]), obs[i, 2], color='r', fill=True, alpha=0.2) for i in range(len(obs))]
for circle in obs_circles:
ax.add_artist(circle)
# Plot start and goal markers
ax.plot(x_ref[0, 0], x_ref[0, 1], 'g*', label='Start')
ax.plot(x_ref[-1, 0], x_ref[-1, 1], 'ro', label='Goal')
ax.legend()
def update_plot():
trajectory_np = np.array(trajectory)
actual_line.set_data(trajectory_np[:, 0], trajectory_np[:, 1])
# Update predicted trajectory
predicted_trajectory = self.next_states
predict_line.set_data(predicted_trajectory[:, 0], predicted_trajectory[:, 1])
# Update rectangle to represent the robot pose
x, y, psi = current_state[0], current_state[1], current_state[2]
robot_width, robot_length = 0.5, 1.0
rear_axle_x = robot_length / 4
rect.set_width(robot_length)
rect.set_height(robot_width)
rect.angle = np.degrees(psi)
center_x = x - robot_length / 2 * np.cos(psi) + robot_width / 2 * np.sin(psi)
center_y = y - robot_length / 2 * np.sin(psi) - robot_width / 2 * np.cos(psi)
rx = center_x + rear_axle_x * np.cos(psi)
ry = center_y + rear_axle_x * np.sin(psi)
rect.set_xy((rx, ry))
ax.relim()
ax.autoscale_view()
plt.draw()
plt.pause(0.001)
# Function to handle key press events
def on_key(event):
if event.key == 'escape':
plt.close(fig)
# Connect the key press event to the figure
fig.canvas.mpl_connect('key_press_event', on_key)
simulation_running = True
while simulation_running and not self.is_goal_reached(current_state, goal_state, tolerance):
self.setupController(obs, use_cbf=use_cbf)
u_a, u_omega = self.solve(x_ref, u_ref, obs)
current_state = self.next_states[1]
trajectory.append(current_state)
control_inputs.append([u_a[0], u_omega[0]]) # Store only the first control input
x_ref = np.concatenate(([current_state], x_ref[1:]))
u_ref = np.concatenate((self.u0, u_ref[self.N:]))
update_plot()
if not plt.get_fignums(): # Check if the figure has been closed
simulation_running = False
plt.ioff()
plt.close(fig)
return np.array(trajectory), np.array(control_inputs)
# The code for the Car class and shift function remains the same as in Method 1
class MPCControllerAdaptiveGamma:
def __init__(self, car: Car, T=0.1, N=50, gamma_desired=0.2, Q=np.diag([1000.0, 1000.0, 1.0, 1.0, 1.0]), R=np.diag([0.1, 0.1])):
self.gamma = None
self.car = car
self.T = T # time step
self.N = N # horizon length
self.gamma_desired = gamma_desired # Desired gamma value
self.Q = Q
self.R = R
self.next_states = np.zeros((self.N + 1, 5))
self.u0 = np.zeros((self.N, 2))
def control_barrier_function(self, x, y, obs_x, obs_y, obs_r):
safety_margin = 0.1 # Additional safety margin
return (x - obs_x) ** 2 + (y - obs_y) ** 2 - (obs_r + self.car.d + safety_margin) ** 2
def setupController(self, obs, use_cbf=True, gamma_desired=0.2):
self.opti = ca.Opti()
self.U = self.opti.variable(self.N, 2)
control_a = self.U[:, 0]
control_omega = self.U[:, 1]
self.X = self.opti.variable(self.N + 1, 5)
state_x = self.X[:, 0]
state_y = self.X[:, 1]
state_psi = self.X[:, 2]
state_v = self.X[:, 3]
state_delta = self.X[:, 4]
# Declare gamma as an optimization variable
self.gamma = self.opti.variable(self.N)
f = lambda x, u: ca.vertcat(*[
x[3] * ca.cos(x[2]),
x[3] * ca.sin(x[2]),
x[3] / self.car.L * ca.tan(x[4]),
u[0],
u[1]
])
self.x_ref = self.opti.parameter(2, 5)
self.u_ref = self.opti.parameter(self.N, 2)
num_obs = len(obs)
self.x_obs = self.opti.parameter(num_obs)
self.y_obs = self.opti.parameter(num_obs)
self.r_obs = self.opti.parameter(num_obs)
self.opti.subject_to(self.X[0, :] == self.x_ref[0, :])
for i in range(self.N):
x_next = self.X[i, :] + self.T * f(self.X[i, :], self.U[i, :]).T
self.opti.subject_to(self.X[i + 1, :] == x_next)
obj = 0
goal_state = self.x_ref[-1, :]
for i in range(self.N):
# Objective function includes penalty on gamma deviation
obj += ca.mtimes([(self.X[i, :] - goal_state), self.Q, (self.X[i, :] - goal_state).T]) + \
ca.mtimes([self.U[i, :], self.R, self.U[i, :].T]) + \
1 * (self.gamma[i] - gamma_desired) ** 2 # Weight for gamma deviation
self.opti.minimize(obj)
# Bounds on states and controls
self.opti.subject_to(self.opti.bounded(self.car.min_v, state_v, self.car.max_v))
self.opti.subject_to(self.opti.bounded(self.car.min_delta, state_delta, self.car.max_delta))
self.opti.subject_to(self.opti.bounded(self.car.min_a, control_a, self.car.max_a))
self.opti.subject_to(self.opti.bounded(self.car.min_omega, control_omega, self.car.max_omega))
# Bounds on gamma
gamma_min = 0.01
gamma_max = 0.2
self.opti.subject_to(self.opti.bounded(gamma_min, self.gamma, gamma_max))
# add delta_gamma constraints
for i in range(self.N - 1):
self.opti.subject_to(self.gamma[i + 1] - self.gamma[i] <= 0.1)
self.opti.subject_to(self.gamma[i + 1] - self.gamma[i] >= -0.1)
# CBF constraints with adaptive gamma
for i in range(self.N):
for j in range(num_obs):
h = self.control_barrier_function(state_x[i], state_y[i], self.x_obs[j], self.y_obs[j], self.r_obs[j])
h_next = self.control_barrier_function(state_x[i + 1], state_y[i + 1], self.x_obs[j], self.y_obs[j], self.r_obs[j])
if use_cbf:
self.opti.subject_to(h_next - h + self.gamma[i] * h >= 0)
# h(x) is small, gamma should be small, and vice versa
else:
self.opti.subject_to(h_next >= 0)
opts_setting = {'ipopt.max_iter': 3000,
'ipopt.print_level': 0,
'print_time': 0,
'ipopt.acceptable_tol': 1e-6,
'ipopt.acceptable_obj_change_tol': 1e-6}
self.opti.solver('ipopt', opts_setting)
def solve(self, x_ref, u_ref, obs):
self.opti.set_value(self.x_ref, x_ref)
self.opti.set_value(self.u_ref, u_ref)
self.opti.set_value(self.x_obs, obs[:, 0])
self.opti.set_value(self.y_obs, obs[:, 1])
self.opti.set_value(self.r_obs, obs[:, 2])
x0 = x_ref[0, :]
# Initial guesses
self.opti.set_initial(self.X, np.tile(x0, (self.N + 1, 1)))
self.opti.set_initial(self.U, np.zeros((self.N, 2)))
self.opti.set_initial(self.gamma, np.full(self.N, self.gamma_desired))
try:
sol = self.opti.solve()
u_opt = sol.value(self.U)
x_opt = sol.value(self.X)
gamma_opt = sol.value(self.gamma)
except RuntimeError:
# If solver fails, return previous control and state
print("Solver failed, using previous control inputs.")
u_opt = self.u0
x_opt = self.next_states
gamma_opt = np.full(self.N, self.gamma_desired)
self.u0, self.next_states = shift(u_opt, x_opt)
return u_opt[:, 0], u_opt[:, 1], gamma_opt
# def is_goal_reached(self, current_state, goal_state, tolerance=0.5, speed_threshold=0.1):
# distance = np.linalg.norm(current_state[:2] - goal_state[:2])
# speed = current_state[3]
# return distance < tolerance and speed < speed_threshold
def is_goal_reached(self, current_state, goal_state, tolerance=0.1):
distance = np.linalg.norm(current_state[:2] - goal_state[:2])
return distance < tolerance
def run_until_goal(self, x_ref, u_ref, obs, tolerance=0.5, use_cbf=True):
current_state = x_ref[0, :]
goal_state = x_ref[-1, :]
trajectory = [current_state]
control_inputs = []
gamma_values = []
plt.ion() # Turn on interactive mode
fig, ax = plt.subplots()
ax.set_xlim(-1, 13)
ax.set_ylim(-1, 13)
# Initialize plot elements
actual_line, = ax.plot([], [], 'b-', label='Actual Trajectory', linewidth=2)
predict_line, = ax.plot([], [], 'r--', label='Predicted Trajectory', linewidth=1)
rect = Rectangle((0, 0), 1, 1, angle=0, color='g', fill=True, label='Robot Pose', alpha=0.2)
ax.add_patch(rect)
obs_circles = [plt.Circle((obs[i, 0], obs[i, 1]), obs[i, 2], color='r', fill=True, alpha=0.2) for i in range(len(obs))]
for circle in obs_circles:
ax.add_artist(circle)
# Plot start and goal markers
ax.plot(x_ref[0, 0], x_ref[0, 1], 'g*', label='Start')
ax.plot(x_ref[-1, 0], x_ref[-1, 1], 'ro', label='Goal')
ax.legend()
def update_plot():
trajectory_np = np.array(trajectory)
actual_line.set_data(trajectory_np[:, 0], trajectory_np[:, 1])
# Update predicted trajectory
predicted_trajectory = self.next_states
predict_line.set_data(predicted_trajectory[:, 0], predicted_trajectory[:, 1])
# Update rectangle to represent the robot pose
x, y, psi = current_state[0], current_state[1], current_state[2]
robot_width, robot_length = 0.5, 1.0
rear_axle_x = robot_length / 4
rect.set_width(robot_length)
rect.set_height(robot_width)
rect.angle = np.degrees(psi)
center_x = x - robot_length / 2 * np.cos(psi) + robot_width / 2 * np.sin(psi)
center_y = y - robot_length / 2 * np.sin(psi) - robot_width / 2 * np.cos(psi)
rx = center_x + rear_axle_x * np.cos(psi)
ry = center_y + rear_axle_x * np.sin(psi)
rect.set_xy((rx, ry))
ax.relim()
ax.autoscale_view()
plt.draw()
plt.pause(0.1)
# Function to handle key press events
def on_key(event):
if event.key == 'escape':
plt.close(fig)
# Connect the key press event to the figure
fig.canvas.mpl_connect('key_press_event', on_key)
simulation_running = True
safety_margins = []
while simulation_running and not self.is_goal_reached(current_state, goal_state, tolerance):
gamma_ref = self.gamma_desired
# decrease gamma if the robot is close to the obstacle
# dis_obs = np.linalg.norm(current_state[:2] - obs[0, :2])
# # linearly decrease gamma from 0.5 to 0.01 as the robot gets closer to the obstacle
# if dis_obs < 5.1:
# # 5 is the distance threshold to start decreasing gamma
# # dis = 5 --> gamma = 0.2, dis = 1 --> gamma = 0.01
# # more small distance to obs, more small gamma
# gamma_ref = 0.2 - (0.2 - 0.01) * (dis_obs - 1) / 4 # dis = 5, gamma = 0.2, dis = 1, gamma = 0.01
#
self.setupController(obs, use_cbf=use_cbf, gamma_desired = gamma_ref)
u_a, u_omega, gamma_opt = self.solve(x_ref, u_ref, obs)
current_state = self.next_states[1]
trajectory.append(current_state)
control_inputs.append([u_a[0], u_omega[0]]) # Store only the first control input
gamma_values.append(gamma_opt[0]) # Store the first gamma value
x_ref = np.concatenate(([current_state], x_ref[1:]))
u_ref = np.concatenate((self.u0, u_ref[self.N:]))
h_values = [self.control_barrier_function(current_state[0], current_state[1], ob[0], ob[1], ob[2]) for ob in
obs]
min_h = min(h_values)
safety_margins.append(min_h)
update_plot()
if not plt.get_fignums(): # Check if the figure has been closed
simulation_running = False
plt.ioff()
plt.close(fig)
# After simulation
plt.figure()
time = np.arange(len(safety_margins)) * self.T
plt.plot(time, safety_margins, 'b-', label='Safety Margin (h)')
plt.xlabel('Time (s)')
plt.ylabel('Safety Margin')
plt.title('Safety Margin Over Time')
plt.legend()
plt.grid(True)
plt.show()
return np.array(trajectory), np.array(control_inputs), np.array(gamma_values)
def test_run():
car = Car()
# Simulation parameters
T = 0.1
N = 20
gamma_fixed = 0.1 # Fixed gamma value for Method 1
gamma_desired = 0.1 # Desired gamma value for Method 2
# Reference trajectory and initial control inputs
x_ref = np.array([[0, 0, 0, 0, 0], [10, 10, 0, 0, 0]])
u_ref = np.zeros((N, 2))
obs = np.array([[5, 5, 2]])
# Method 1: MPC with fixed gamma
mpc_controller_no = MPCControllerFixedGamma(car, T=T, N=N, gamma=gamma_fixed)
trajectory_no, control_inputs_no = mpc_controller_no.run_until_goal(
x_ref, u_ref, obs, tolerance=0.5, use_cbf=False)
# Method 1: MPC with fixed gamma
mpc_controller_fixed = MPCControllerFixedGamma(car, T=T, N=N, gamma=gamma_fixed)
trajectory_fixed, control_inputs_fixed = mpc_controller_fixed.run_until_goal(
x_ref, u_ref, obs, tolerance=0.5, use_cbf=True)
# Method 2: MPC with adaptive gamma
mpc_controller_adaptive = MPCControllerAdaptiveGamma(car, T=T, N=N, gamma_desired=gamma_desired)
trajectory_adaptive, control_inputs_adaptive, gamma_values_adaptive = mpc_controller_adaptive.run_until_goal(
x_ref, u_ref, obs, tolerance=0.5, use_cbf=True)
# Plotting the trajectories
plt.figure(figsize=(12, 8))
plt.plot(trajectory_no[:, 0], trajectory_no[:, 1], 'k:', label='No CBF')
plt.plot(trajectory_fixed[:, 0], trajectory_fixed[:, 1], 'b-', label='Fixed Gamma')
plt.plot(trajectory_adaptive[:, 0], trajectory_adaptive[:, 1], 'r--', label='Adaptive Gamma')
plt.plot(x_ref[0, 0], x_ref[0, 1], 'g*', markersize=10, label='Start')
plt.plot(x_ref[-1, 0], x_ref[-1, 1], 'ro', markersize=10, label='Goal')
# Plot obstacle
for ob in obs:
circle = plt.Circle((ob[0], ob[1]), ob[2], color='gray', fill=True, alpha=0.3)
plt.gca().add_artist(circle)
plt.xlabel('X Position (m)')
plt.ylabel('Y Position (m)')
plt.title('Comparison of Trajectories: Fixed Gamma vs Adaptive Gamma')
plt.legend()
plt.grid(True)
plt.axis('equal')
plt.show()
# Plot gamma values for adaptive method
time = np.arange(len(gamma_values_adaptive)) * T
plt.figure()
plt.plot(time, gamma_values_adaptive, 'k-', label='Adaptive Gamma')
plt.axhline(gamma_desired, color='r', linestyle='--', label='Desired Gamma')
plt.xlabel('Time (s)')
plt.ylabel('Gamma')
plt.title('Gamma Values over Time (Adaptive Gamma)')
plt.legend()
plt.grid(True)
plt.show()
if __name__ == '__main__':
test_run()