-
Notifications
You must be signed in to change notification settings - Fork 1
/
Path_Optimizer.py
504 lines (463 loc) · 27 KB
/
Path_Optimizer.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
#!/usr/bin/env python3
import numpy as np
import scipy.optimize
import scipy.integrate
import sys
from math import sin, cos, pi, sqrt
class PathOptimizer:
def __init__(self):
self._xf = 0.0
self._yf = 0.0
self._tf = 0.0
######################################################
# PARAMETER OPTIMIZATION FOR POLYNOMIAL SPIRAL
######################################################
# Sets up the optimization problem to compute a spiral to a given
# goal point, (xf, yf, tf).
def optimize_spiral(self, xf, yf, tf):
"""Optimization function used for finding the optimization parameters.
Assumptions:
1. The first point in the spiral is in origin of the vehicle frame
2. Assumes that the curvature for the endpoints to be zero
(i.e. p0 and p3 are zero of the vector p = [p0, p1, p2, p3, sf])
args:
xf: Final x position (m) for the given goal state.
yf: Final y position (m) for the given goal state.
tf: Final yaw position (rad) for the given goal state.
returns:
spiral: The resulting optimized path that best fits the goal state.
The path is a list of points of the following format:
[x_points, y_points, t_points]:
x_points: List of x values (m) along the spiral
y_points: List of y values (m) along the spiral
t_points: List of yaw values (rad) along the spiral
"""
# Save the terminal x, y, and theta.
self._xf = xf
self._yf = yf
self._tf = tf
# The straight line distance serves as a lower bound on any path's
# arc length to the goal.
sf_0 = np.linalg.norm([xf, yf])
max = sys.maxsize
# The initial variables correspond to a straight line with arc length
# sf_0. Recall that p here is defined as:
# [p1, p2, sf]
#, where p1 and p2 are the curvatures at points p1 and p2
#, and sf is the final arc length for the spiral.
# Since we already set p0 and p4 (being the curvature of
# the initial and final points) to be zero.
p0 = [0.0, 0.0, sf_0]
# Here we will set the bounds [lower, upper] for each optimization
# variable.
# The first two variables correspond to the curvature 1/3rd of the
# way along the path and 2/3rds of the way along the path, respectively.
# As a result, their curvature needs to lie within [-0.5, 0.5].
# The third variable is the arc length, it has no upper limit, and it
# has a lower limit of the straight line arc length.
# ------------------------------------------------------------------
bounds = [[-0.5, 0.5], [-0.5, 0.5], [sf_0, max]]
# ------------------------------------------------------------------
# Here we will call scipy.optimize.minimize to optimize our spiral.
# The objective and gradient are given to you by self.objective, and
# self.objective_grad. The bounds are computed above, and the inital
# variables for the optimizer are set by p0. You should use the L-BFGS-B
# optimization methods.
# ------------------------------------------------------------------
res = scipy.optimize.minimize(fun=self.objective, x0=p0, method='L-BFGS-B', jac=self.objective_grad, bounds=bounds)
# ------------------------------------------------------------------
spiral = self.sample_spiral(res.x)
return spiral
######################################################
# COMPUTE LIST OF THETAS
######################################################
# This function computes the theta values for a given list of
# arc lengths, and spiral parameters a, b, c, d.
# Recall that the equation of a cubic spiral is
# kappa(s) = a + b*s + c*s^2 + d*s^3
# and since theta(s) is the integral of kappa(s) with respect to
# arc length, then theta(s) = a*s + b/2*s^2 + c/3*s^3 + d/4*s^4.
# Try to vectorize this function using numpy for speed, if you can.
# Inputs: a - the first term of kappa(s).
# b - the second term of kappa(s).
# c - the third term of kappa(s).
# d - the fourth term of kappa(s).
def thetaf(self, a, b, c, d, s):
pass
# ------------------------------------------------------------------
# Remember that a, b, c, d and s are lists
thetas = a*s + (b/2)*(s**2) + (c/3)*(s**3) + (d/4)*(s**4)
return thetas
# ------------------------------------------------------------------
######################################################
# SAMPLE THE SPIRAL PATH
######################################################
# This function samples the spiral along its arc length to generate
# a discrete set of x, y, and theta points for a path.
def sample_spiral(self, p):
"""Samples a set of points along the spiral given the optimization
parameters.
args:
p: The resulting optimization parameters that minimizes the
objective function given a goal state.
Format: [p1, p2, sf], Unit: [1/m, 1/m, m]
, where p1 and p2 are the curvatures at points p1 and p2
and sf is the final arc length for the spiral.
returns:
[x_points, y_points, t_points]:
x_points: List of x values (m) along the spiral
y_points: List of y values (m) along the spiral
t_points: List of yaw values (rad) along the spiral
"""
# These equations map from the optimization parameter space
# to the spiral parameter space.
p = [0.0, p[0], p[1], 0.0, p[2]] # recall p0 and p3 are set to 0
# and p4 is the final arc length
a = p[0]
b = -(11.0*p[0]/2.0 - 9.0*p[1] + 9.0*p[2]/2.0 - p[3])/p[4]
c = (9.0*p[0] - 45.0*p[1]/2.0 + 18.0*p[2] - 9.0*p[3]/2.0)/p[4]**2
d = -(9.0*p[0]/2.0 - 27.0*p[1]/2.0 + 27.0*p[2]/2.0 - 9.0*p[3]/2.0)/p[4]**3
# Set the s_points (list of s values along the spiral) to be from 0.0
# to p[4] (final arc length)
s_points = np.linspace(0.0, p[4])
# Compute the theta, x, and y points from the uniformly sampled
# arc length points s_points (p[4] is the spiral arc length).
# Use self.thetaf() to compute the theta values from the s values.
# Recall that x = integral cos(theta(s)) ds and
# y = integral sin(theta(s)) ds.
# You will find the scipy.integrate.cumtrapz() function useful.
# Try to vectorize the code using numpy functions for speed if you can.
# Try to vectorize the code using numpy functions for speed if you can.
# ------------------------------------------------------------------
t_points = self.thetaf(a, b, c, d, s_points)
x_points = scipy.integrate.cumtrapz(np.cos(t_points), s_points, initial=None)
y_points = scipy.integrate.cumtrapz(np.sin(t_points), s_points, initial=None)
return [x_points, y_points, t_points]
# ------------------------------------------------------------------
######################################################
######################################################
# BELOW ARE THE FUNCTIONS USED FOR THE OPTIMIZER.
######################################################
######################################################
def objective(self, p):
"""
The optimizer can freely move 3 of the spiral parameter variables.
The other two are fixed due to boundary conditions.
"""
p = [0.0, p[0], p[1], 0.0, p[2]]
return self.fbe(p) + 25*(self.fxf(p) + self.fyf(p)) + 30*self.ftf(p)
def objective_grad(self, p):
"""
The optimizer can freely move 3 of the spiral parameter variables.
The other two are fixed due to boundary conditions.
"""
p = [0.0, p[0], p[1], 0.0, p[2]]
return np.add(np.add(np.add(self.fbe_grad(p), np.multiply(25, self.fxf_grad(p))), \
np.multiply(25, self.fyf_grad(p))), np.multiply(30, self.ftf_grad(p)))
def fxf(self, p):
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = self._xf-p[4]*(cos(p[0]*p[4]-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*t14*(1.0/2.0))+cos(p[0]*p[4]*(1.0/2.0)-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)+p[4]*t14*(1.0/8.0))*2.0+cos(p[0]*p[4]*(3.0/4.0)-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*t14*(9.0/3.2E1))*2.0+cos(p[0]*p[4]*(1.0/4.0)-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)+p[4]*t14*(1.0/3.2E1))*2.0+cos(p[0]*p[4]*(3.0/8.0)-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)+p[4]*t14*(9.0/1.28E2))*4.0+cos(p[0]*p[4]*(1.0/8.0)-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4+p[4]*t14*(1.0/1.28E2))*4.0+cos(p[0]*p[4]*(5.0/8.0)-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*t14*(2.5E1/1.28E2))*4.0+cos(p[0]*p[4]*(7.0/8.0)-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*t14*(4.9E1/1.28E2))*4.0+1.0)*(1.0/2.4E1)
t0 = t15*t15
return t0
def fxf_grad(self, p):
grad = [0.0, 0.0, 0.0]
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[0]*p[4]*(3.0/4.0)
t18 = p[0]*p[4]*(1.0/4.0)
t19 = p[0]*p[4]*(3.0/8.0)
t20 = p[0]*p[4]*(1.0/8.0)
t21 = p[0]*p[4]*(5.0/8.0)
t22 = p[0]*p[4]*(7.0/8.0)
t0 = p[4]*(self._xf-p[4]*(cos(t15-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*t14*(1.0/2.0))+cos(t16-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)+p[4]*t14*(1.0/8.0))*2.0+cos(t17-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*t14*(9.0/3.2E1))*2.0+cos(t18-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)+p[4]*t14*(1.0/3.2E1))*2.0+cos(t19-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)+p[4]*t14*(9.0/1.28E2))*4.0+cos(t20-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4+p[4]*t14*(1.0/1.28E2))*4.0+cos(t21-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*t14*(2.5E1/1.28E2))*4.0+cos(t22-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*t14*(4.9E1/1.28E2))*4.0+1.0)*(1.0/2.4E1))*(p[4]*sin(t15-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*(p[3]-t2+t3-t4)*(1.0/2.0))*(3.0/8.0)+p[4]*sin(t16-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)+p[4]*(p[3]-t2+t3-t4)*(1.0/8.0))*(5.1E1/6.4E1)+p[4]*sin(t17-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*(p[3]-t2+t3-t4)*(9.0/3.2E1))*8.701171875E-1+p[4]*sin(t18-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)+p[4]*(p[3]-t2+t3-t4)*(1.0/3.2E1))*3.544921875E-1+p[4]*sin(t19-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)+p[4]*(p[3]-t2+t3-t4)*(9.0/1.28E2))*1.2161865234375+p[4]*sin(t20-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4+p[4]*(p[3]-t2+t3-t4)*(1.0/1.28E2))*2.259521484375E-1+p[4]*sin(t21-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*(p[3]-t2+t3-t4)*(2.5E1/1.28E2))*1.7669677734375+p[4]*sin(t22-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*(p[3]-t2+t3-t4)*(4.9E1/1.28E2))*1.5970458984375)*(1.0/1.2E1)
grad[0] = t0
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[4]*t14*(1.0/8.0)
t18 = t16+t17-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)
t19 = p[0]*p[4]*(3.0/4.0)
t20 = p[0]*p[4]*(1.0/4.0)
t21 = p[4]*t14*(1.0/3.2E1)
t22 = t20+t21-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)
t23 = p[0]*p[4]*(3.0/8.0)
t24 = p[4]*t14*(9.0/1.28E2)
t25 = t23+t24-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)
t26 = p[0]*p[4]*(1.0/8.0)
t27 = p[4]*t14*(1.0/1.28E2)
t28 = t26+t27-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4
t29 = p[0]*p[4]*(5.0/8.0)
t30 = p[0]*p[4]*(7.0/8.0)
t0 = p[4]*(self._xf-p[4]*(cos(t15-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*t14*(1.0/2.0))+cos(t19-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*t14*(9.0/3.2E1))*2.0+cos(t29-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*t14*(2.5E1/1.28E2))*4.0+cos(t30-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*t14*(4.9E1/1.28E2))*4.0+cos(t18)*2.0+cos(t22)*2.0+cos(t25)*4.0+cos(t28)*4.0+1.0)*(1.0/2.4E1))*(p[4]*sin(t15-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*(p[3]-t2+t3-t4)*(1.0/2.0))*(3.0/8.0)+p[4]*sin(t19-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*(p[3]-t2+t3-t4)*(9.0/3.2E1))*3.955078125E-1+p[4]*sin(t29-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*(p[3]-t2+t3-t4)*(2.5E1/1.28E2))*2.838134765625E-1+p[4]*sin(t30-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*(p[3]-t2+t3-t4)*(4.9E1/1.28E2))*1.2740478515625-p[4]*sin(t18)*(3.0/6.4E1)-p[4]*sin(t22)*1.201171875E-1-p[4]*sin(t25)*2.669677734375E-1-p[4]*sin(t28)*9.70458984375E-2)*(1.0/1.2E1)
grad[1] = t0
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[0]*p[4]*(3.0/4.0)
t18 = p[0]*p[4]*(1.0/4.0)
t19 = p[0]*p[4]*(3.0/8.0)
t20 = p[0]*p[4]*(1.0/8.0)
t21 = p[0]*p[4]*(5.0/8.0)
t22 = p[0]*p[4]*(7.0/8.0)
t23 = p[4]*(p[3]-t2+t3-t4)*(1.0/2.0)
t39 = p[4]*t9*(1.0/4.0)
t40 = p[4]*t13*(1.0/3.0)
t24 = t15+t23-t39-t40
t25 = p[4]*(p[3]-t2+t3-t4)*(1.0/8.0)
t41 = p[4]*t9*(1.0/6.4E1)
t42 = p[4]*t13*(1.0/2.4E1)
t26 = t16+t25-t41-t42
t27 = p[4]*(p[3]-t2+t3-t4)*(1.0/3.2E1)
t45 = p[4]*t9*9.765625E-4
t46 = p[4]*t13*(1.0/1.92E2)
t28 = t18+t27-t45-t46
t29 = p[4]*(p[3]-t2+t3-t4)*(9.0/3.2E1)
t43 = p[4]*t9*7.91015625E-2
t44 = p[4]*t13*(9.0/6.4E1)
t30 = t17+t29-t43-t44
t31 = p[4]*(p[3]-t2+t3-t4)*(1.0/1.28E2)
t49 = p[4]*t9*6.103515625E-5
t50 = p[4]*t13*6.510416666666667E-4
t32 = t20+t31-t49-t50
t33 = p[4]*(p[3]-t2+t3-t4)*(9.0/1.28E2)
t47 = p[4]*t9*4.94384765625E-3
t48 = p[4]*t13*(9.0/5.12E2)
t34 = t19+t33-t47-t48
t35 = p[4]*(p[3]-t2+t3-t4)*(2.5E1/1.28E2)
t51 = p[4]*t9*3.814697265625E-2
t52 = p[4]*t13*8.138020833333333E-2
t36 = t21+t35-t51-t52
t37 = p[4]*(p[3]-t2+t3-t4)*(4.9E1/1.28E2)
t53 = p[4]*t9*1.4654541015625E-1
t54 = p[4]*t13*2.233072916666667E-1
t38 = t22+t37-t53-t54
t0 = (self._xf-p[4]*(cos(t15-t39-t40+p[4]*t14*(1.0/2.0))+cos(t16-t41-t42+p[4]*t14*(1.0/8.0))*2.0+cos(t18-t45-t46+p[4]*t14*(1.0/3.2E1))*2.0+cos(t17-t43-t44+p[4]*t14*(9.0/3.2E1))*2.0+cos(t20-t49-t50+p[4]*t14*(1.0/1.28E2))*4.0+cos(t19-t47-t48+p[4]*t14*(9.0/1.28E2))*4.0+cos(t21-t51-t52+p[4]*t14*(2.5E1/1.28E2))*4.0+cos(t22-t53-t54+p[4]*t14*(4.9E1/1.28E2))*4.0+1.0)*(1.0/2.4E1))*(cos(t24)*(1.0/2.4E1)+cos(t26)*(1.0/1.2E1)+cos(t28)*(1.0/1.2E1)+cos(t30)*(1.0/1.2E1)+cos(t32)*(1.0/6.0)+cos(t34)*(1.0/6.0)+cos(t36)*(1.0/6.0)+cos(t38)*(1.0/6.0)-p[4]*(sin(t24)*(p[0]*(1.0/8.0)+p[1]*(3.0/8.0)+p[2]*(3.0/8.0)+p[3]*(1.0/8.0))+sin(t26)*(p[0]*(1.5E1/1.28E2)+p[1]*(5.1E1/1.28E2)-p[2]*(3.0/1.28E2)+p[3]*(1.0/1.28E2))*2.0+sin(t28)*(p[0]*1.2060546875E-1+p[1]*1.7724609375E-1-p[2]*6.005859375E-2+p[3]*1.220703125E-2)*2.0+sin(t30)*(p[0]*1.1279296875E-1+p[1]*4.3505859375E-1+p[2]*1.9775390625E-1+p[3]*4.39453125E-3)*2.0+sin(t32)*(p[0]*8.7615966796875E-2+p[1]*5.6488037109375E-2-p[2]*2.4261474609375E-2+p[3]*5.157470703125E-3)*4.0+sin(t34)*(p[0]*1.24237060546875E-1+p[1]*3.04046630859375E-1-p[2]*6.6741943359375E-2+p[3]*1.3458251953125E-2)*4.0+sin(t36)*(p[0]*1.11541748046875E-1+p[1]*4.41741943359375E-1+p[2]*7.0953369140625E-2+p[3]*7.62939453125E-4)*4.0+sin(t38)*(p[0]*1.19842529296875E-1+p[1]*3.99261474609375E-1+p[2]*3.18511962890625E-1+p[3]*3.7384033203125E-2)*4.0)*(1.0/2.4E1)+1.0/2.4E1)*-2.0
grad[2] = t0
return grad
def fyf(self, p):
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = self._yf-p[4]*(sin(p[0]*p[4]-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)+p[4]*t14*(1.0/2.0))+sin(p[0]*p[4]*(1.0/2.0)-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)+p[4]*t14*(1.0/8.0))*2.0+sin(p[0]*p[4]*(3.0/4.0)-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)+p[4]*t14*(9.0/3.2E1))*2.0+sin(p[0]*p[4]*(1.0/4.0)-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)+p[4]*t14*(1.0/3.2E1))*2.0+sin(p[0]*p[4]*(3.0/8.0)-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)+p[4]*t14*(9.0/1.28E2))*4.0+sin(p[0]*p[4]*(1.0/8.0)-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4+p[4]*t14*(1.0/1.28E2))*4.0+sin(p[0]*p[4]*(5.0/8.0)-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2+p[4]*t14*(2.5E1/1.28E2))*4.0+sin(p[0]*p[4]*(7.0/8.0)-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1+p[4]*t14*(4.9E1/1.28E2))*4.0)*(1.0/2.4E1)
t0 = t15*t15
return t0
def fyf_grad(self, p):
grad = [0.0, 0.0, 0.0]
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[0]*p[4]*(3.0/4.0)
t18 = p[0]*p[4]*(1.0/4.0)
t19 = p[0]*p[4]*(3.0/8.0)
t20 = p[0]*p[4]*(1.0/8.0)
t21 = p[0]*p[4]*(5.0/8.0)
t22 = p[0]*p[4]*(7.0/8.0)
t23 = p[4]*t14*(1.0/2.0)
t24 = t15+t23-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)
t25 = p[4]*t14*(1.0/8.0)
t26 = t16+t25-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)
t27 = p[4]*t14*(9.0/3.2E1)
t28 = t17+t27-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)
t29 = p[4]*t14*(1.0/3.2E1)
t30 = t18+t29-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)
t31 = p[4]*t14*(9.0/1.28E2)
t32 = t19+t31-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)
t33 = p[4]*t14*(1.0/1.28E2)
t34 = t20+t33-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4
t35 = p[4]*t14*(2.5E1/1.28E2)
t36 = t21+t35-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2
t37 = p[4]*t14*(4.9E1/1.28E2)
t38 = t22+t37-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1
t0 = p[4]*(self._yf-p[4]*(sin(t24)+sin(t26)*2.0+sin(t28)*2.0+sin(t30)*2.0+sin(t32)*4.0+sin(t34)*4.0+sin(t36)*4.0+sin(t38)*4.0)*(1.0/2.4E1))*(p[4]*cos(t24)*(3.0/8.0)+p[4]*cos(t26)*(5.1E1/6.4E1)+p[4]*cos(t28)*8.701171875E-1+p[4]*cos(t30)*3.544921875E-1+p[4]*cos(t32)*1.2161865234375+p[4]*cos(t34)*2.259521484375E-1+p[4]*cos(t36)*1.7669677734375+p[4]*cos(t38)*1.5970458984375)*(-1.0/1.2E1)
grad[0] = t0
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[4]*t14*(1.0/8.0)
t18 = t16+t17-p[4]*t9*(1.0/6.4E1)-p[4]*t13*(1.0/2.4E1)
t19 = p[0]*p[4]*(3.0/4.0)
t20 = p[0]*p[4]*(1.0/4.0)
t21 = p[4]*t14*(1.0/3.2E1)
t22 = t20+t21-p[4]*t9*9.765625E-4-p[4]*t13*(1.0/1.92E2)
t23 = p[0]*p[4]*(3.0/8.0)
t24 = p[4]*t14*(9.0/1.28E2)
t25 = t23+t24-p[4]*t9*4.94384765625E-3-p[4]*t13*(9.0/5.12E2)
t26 = p[0]*p[4]*(1.0/8.0)
t27 = p[4]*t14*(1.0/1.28E2)
t28 = t26+t27-p[4]*t9*6.103515625E-5-p[4]*t13*6.510416666666667E-4
t29 = p[0]*p[4]*(5.0/8.0)
t30 = p[0]*p[4]*(7.0/8.0)
t31 = p[4]*t14*(1.0/2.0)
t32 = t15+t31-p[4]*t9*(1.0/4.0)-p[4]*t13*(1.0/3.0)
t33 = p[4]*t14*(9.0/3.2E1)
t34 = t19+t33-p[4]*t9*7.91015625E-2-p[4]*t13*(9.0/6.4E1)
t35 = p[4]*t14*(2.5E1/1.28E2)
t36 = t29+t35-p[4]*t9*3.814697265625E-2-p[4]*t13*8.138020833333333E-2
t37 = p[4]*t14*(4.9E1/1.28E2)
t38 = t30+t37-p[4]*t9*1.4654541015625E-1-p[4]*t13*2.233072916666667E-1
t0 = p[4]*(self._yf-p[4]*(sin(t18)*2.0+sin(t22)*2.0+sin(t25)*4.0+sin(t28)*4.0+sin(t32)+sin(t34)*2.0+sin(t36)*4.0+sin(t38)*4.0)*(1.0/2.4E1))*(p[4]*cos(t18)*(3.0/6.4E1)+p[4]*cos(t22)*1.201171875E-1+p[4]*cos(t25)*2.669677734375E-1+p[4]*cos(t28)*9.70458984375E-2-p[4]*cos(t32)*(3.0/8.0)-p[4]*cos(t34)*3.955078125E-1-p[4]*cos(t36)*2.838134765625E-1-p[4]*cos(t38)*1.2740478515625)*(1.0/1.2E1)
grad[1] = t0
t2 = p[0]*(1.1E1/2.0)
t3 = p[1]*9.0
t4 = p[2]*(9.0/2.0)
t5 = p[0]*(9.0/2.0)
t6 = p[1]*(2.7E1/2.0)
t7 = p[2]*(2.7E1/2.0)
t8 = p[3]*(9.0/2.0)
t9 = t5-t6+t7-t8
t10 = p[0]*9.0
t11 = p[1]*(4.5E1/2.0)
t12 = p[2]*1.8E1
t13 = t8-t10+t11-t12
t14 = p[3]-t2+t3-t4
t15 = p[0]*p[4]
t16 = p[0]*p[4]*(1.0/2.0)
t17 = p[0]*p[4]*(3.0/4.0)
t18 = p[0]*p[4]*(1.0/4.0)
t19 = p[0]*p[4]*(3.0/8.0)
t20 = p[0]*p[4]*(1.0/8.0)
t21 = p[0]*p[4]*(5.0/8.0)
t22 = p[0]*p[4]*(7.0/8.0)
t23 = p[4]*(p[3]-t2+t3-t4)*(1.0/2.0)
t39 = p[4]*t9*(1.0/4.0)
t40 = p[4]*t13*(1.0/3.0)
t24 = t15+t23-t39-t40
t25 = p[4]*(p[3]-t2+t3-t4)*(1.0/8.0)
t41 = p[4]*t9*(1.0/6.4E1)
t42 = p[4]*t13*(1.0/2.4E1)
t26 = t16+t25-t41-t42
t27 = p[4]*(p[3]-t2+t3-t4)*(1.0/3.2E1)
t45 = p[4]*t9*9.765625E-4
t46 = p[4]*t13*(1.0/1.92E2)
t28 = t18+t27-t45-t46
t29 = p[4]*(p[3]-t2+t3-t4)*(9.0/3.2E1)
t43 = p[4]*t9*7.91015625E-2
t44 = p[4]*t13*(9.0/6.4E1)
t30 = t17+t29-t43-t44
t31 = p[4]*(p[3]-t2+t3-t4)*(1.0/1.28E2)
t49 = p[4]*t9*6.103515625E-5
t50 = p[4]*t13*6.510416666666667E-4
t32 = t20+t31-t49-t50
t33 = p[4]*(p[3]-t2+t3-t4)*(9.0/1.28E2)
t47 = p[4]*t9*4.94384765625E-3
t48 = p[4]*t13*(9.0/5.12E2)
t34 = t19+t33-t47-t48
t35 = p[4]*(p[3]-t2+t3-t4)*(2.5E1/1.28E2)
t51 = p[4]*t9*3.814697265625E-2
t52 = p[4]*t13*8.138020833333333E-2
t36 = t21+t35-t51-t52
t37 = p[4]*(p[3]-t2+t3-t4)*(4.9E1/1.28E2)
t53 = p[4]*t9*1.4654541015625E-1
t54 = p[4]*t13*2.233072916666667E-1
t38 = t22+t37-t53-t54
t0 = (self._yf-p[4]*(sin(t15-t39-t40+p[4]*t14*(1.0/2.0))+sin(t16-t41-t42+p[4]*t14*(1.0/8.0))*2.0+sin(t18-t45-t46+p[4]*t14*(1.0/3.2E1))*2.0+sin(t17-t43-t44+p[4]*t14*(9.0/3.2E1))*2.0+sin(t20-t49-t50+p[4]*t14*(1.0/1.28E2))*4.0+sin(t19-t47-t48+p[4]*t14*(9.0/1.28E2))*4.0+sin(t21-t51-t52+p[4]*t14*(2.5E1/1.28E2))*4.0+sin(t22-t53-t54+p[4]*t14*(4.9E1/1.28E2))*4.0)*(1.0/2.4E1))*(sin(t24)*(1.0/2.4E1)+sin(t26)*(1.0/1.2E1)+sin(t28)*(1.0/1.2E1)+sin(t30)*(1.0/1.2E1)+sin(t32)*(1.0/6.0)+sin(t34)*(1.0/6.0)+sin(t36)*(1.0/6.0)+sin(t38)*(1.0/6.0)+p[4]*(cos(t24)*(p[0]*(1.0/8.0)+p[1]*(3.0/8.0)+p[2]*(3.0/8.0)+p[3]*(1.0/8.0))+cos(t26)*(p[0]*(1.5E1/1.28E2)+p[1]*(5.1E1/1.28E2)-p[2]*(3.0/1.28E2)+p[3]*(1.0/1.28E2))*2.0+cos(t28)*(p[0]*1.2060546875E-1+p[1]*1.7724609375E-1-p[2]*6.005859375E-2+p[3]*1.220703125E-2)*2.0+cos(t30)*(p[0]*1.1279296875E-1+p[1]*4.3505859375E-1+p[2]*1.9775390625E-1+p[3]*4.39453125E-3)*2.0+cos(t32)*(p[0]*8.7615966796875E-2+p[1]*5.6488037109375E-2-p[2]*2.4261474609375E-2+p[3]*5.157470703125E-3)*4.0+cos(t34)*(p[0]*1.24237060546875E-1+p[1]*3.04046630859375E-1-p[2]*6.6741943359375E-2+p[3]*1.3458251953125E-2)*4.0+cos(t36)*(p[0]*1.11541748046875E-1+p[1]*4.41741943359375E-1+p[2]*7.0953369140625E-2+p[3]*7.62939453125E-4)*4.0+cos(t38)*(p[0]*1.19842529296875E-1+p[1]*3.99261474609375E-1+p[2]*3.18511962890625E-1+p[3]*3.7384033203125E-2)*4.0)*(1.0/2.4E1))*-2.0
grad[2] = t0
return grad
def ftf(self, p):
t2 = self._tf-p[0]*p[4]+p[4]*(p[0]*(1.1E1/2.0)-p[1]*9.0+p[2]*(9.0/2.0)-p[3])*(1.0/2.0)+p[4]*(p[0]*(9.0/2.0)-p[1]*(2.7E1/2.0)+p[2]*(2.7E1/2.0)-p[3]*(9.0/2.0))*(1.0/4.0)-p[4]*(p[0]*9.0-p[1]*(4.5E1/2.0)+p[2]*1.8E1-p[3]*(9.0/2.0))*(1.0/3.0)
t0 = t2*t2
return t0
def ftf_grad(self, p):
grad = [0.0, 0.0, 0.0]
t0 = p[4]*(self._tf-p[0]*p[4]+p[4]*(p[0]*(1.1E1/2.0)-p[1]*9.0+p[2]*(9.0/2.0)-p[3])*(1.0/2.0)+p[4]*(p[0]*(9.0/2.0)-p[1]*(2.7E1/2.0)+p[2]*(2.7E1/2.0)-p[3]*(9.0/2.0))*(1.0/4.0)-p[4]*(p[0]*9.0-p[1]*(4.5E1/2.0)+p[2]*1.8E1-p[3]*(9.0/2.0))*(1.0/3.0))*(-3.0/4.0)
grad[0] = t0
t0 = p[4]*(self._tf-p[0]*p[4]+p[4]*(p[0]*(1.1E1/2.0)-p[1]*9.0+p[2]*(9.0/2.0)-p[3])*(1.0/2.0)+p[4]*(p[0]*(9.0/2.0)-p[1]*(2.7E1/2.0)+p[2]*(2.7E1/2.0)-p[3]*(9.0/2.0))*(1.0/4.0)-p[4]*(p[0]*9.0-p[1]*(4.5E1/2.0)+p[2]*1.8E1-p[3]*(9.0/2.0))*(1.0/3.0))*(-3.0/4.0)
grad[1] = t0
t0 = (p[0]*(1.0/8.0)+p[1]*(3.0/8.0)+p[2]*(3.0/8.0)+p[3]*(1.0/8.0))*(self._tf-p[0]*p[4]+p[4]*(p[0]*(1.1E1/2.0)-p[1]*9.0+p[2]*(9.0/2.0)-p[3])*(1.0/2.0)+p[4]*(p[0]*(9.0/2.0)-p[1]*(2.7E1/2.0)+p[2]*(2.7E1/2.0)-p[3]*(9.0/2.0))*(1.0/4.0)-p[4]*(p[0]*9.0-p[1]*(4.5E1/2.0)+p[2]*1.8E1-p[3]*(9.0/2.0))*(1.0/3.0))*-2.0
grad[2] = t0
return grad
def fbe(self, p):
t0 = p[4]*(p[0]*p[1]*9.9E1-p[0]*p[2]*3.6E1+p[0]*p[3]*1.9E1-p[1]*p[2]*8.1E1-p[1]*p[3]*3.6E1+p[2]*p[3]*9.9E1+(p[0]*p[0])*6.4E1+(p[1]*p[1])*3.24E2+(p[2]*p[2])*3.24E2+(p[3]*p[3])*6.4E1)*(1.0/8.4E2)
return t0
def fbe_grad(self, p):
grad = [0.0, 0.0, 0.0]
t0 = p[4]*(p[0]*9.9E1+p[1]*6.48E2-p[2]*8.1E1-p[3]*3.6E1)*(1.0/8.4E2)
grad[0] = t0
t0 = p[4]*(p[0]*3.6E1+p[1]*8.1E1-p[2]*6.48E2-p[3]*9.9E1)*(-1.0/8.4E2)
grad[1] = t0
t0 = p[0]*p[1]*(3.3E1/2.8E2)-p[0]*p[2]*(3.0/7.0E1)+p[0]*p[3]*(1.9E1/8.4E2)-p[1]*p[2]*(2.7E1/2.8E2)-p[1]*p[3]*(3.0/7.0E1)+p[2]*p[3]*(3.3E1/2.8E2)+(p[0]*p[0])*(8.0/1.05E2)+(p[1]*p[1])*(2.7E1/7.0E1)+(p[2]*p[2])*(2.7E1/7.0E1)+(p[3]*p[3])*(8.0/1.05E2)
grad[2] = t0
return grad