-
Notifications
You must be signed in to change notification settings - Fork 1
/
DampedNewton.py
437 lines (347 loc) · 14.5 KB
/
DampedNewton.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
import numpy as np
import scipy.sparse as sparse # Sparse matrices
from scipy.sparse.linalg import norm, lsqr
from scipy.linalg import eig
import matplotlib.pyplot as plt
newparams = {'figure.figsize': (6.0, 6.0), 'axes.grid': True,
'lines.markersize': 8, 'lines.linewidth': 2,
'font.size': 14}
plt.rcParams.update(newparams)
# function to get configuration space
def config_space(l):
r_max = np.sum(l)
l_max = np.max(l)
r_min = 2 * l_max - r_max
return r_min, r_max
def get_config_space_to_plot(r_max, r_min, p, N=1000):
x_conf = np.linspace(-r_max, r_max, N)
y_conf_up = np.sqrt(r_max ** 2 - x_conf ** 2)
y_conf_low = np.zeros(N)
min_dist = 0
if r_min > 0:
arg_low = np.argwhere(np.abs(x_conf) <= r_min).flatten()
y_conf_low[arg_low] = np.sqrt(r_min ** 2 - x_conf[arg_low] ** 2)
length_p = np.linalg.norm(p)
min_dist = r_min - length_p
print("Not possible to reach p.")
return x_conf, y_conf_up, y_conf_low, min_dist
def get_xy_to_plot(l, thetas):
n = len(l)
# start in the Origin
x = [0]
y = [0]
for i in range(n):
Stheta = np.sum(thetas[:i + 1])
x.append(x[i] + l[i] * np.cos(Stheta))
y.append(y[i] + l[i] * np.sin(Stheta))
x = np.asarray(x)
y = np.asarray(y)
return x, y
def get_string(l):
string = ""
for i in range(len(l)):
string += str(l[i])
if i != len(l) - 1:
string += ","
return string
def Plot(ax, l, thetas, p):
x, y = get_xy_to_plot(l, thetas)
r_min, r_max = config_space(l)
x_conf, y_conf_up, y_conf_low, min_dist = get_config_space_to_plot(r_max, r_min, p)
dist = np.linalg.norm(np.array([x[-1], y[-1]]) - p)
print("Minimal distance from Robot arm end to p:", min_dist)
print("Robot arm end to p:", dist)
epsilon = 2e-15
if min_dist - epsilon <= dist <= min_dist + epsilon:
print("Minimum is reached.")
ax.plot(x, y, "b-o", label="Robot")
ax.plot(x[-1], y[-1], "r.", label="End of robot arm")
ax.plot(p[0], p[1], "rx", label="$p$")
ax.fill_between(x_conf, y_conf_up, y_conf_low, label="Configuration space", color='g')
ax.fill_between(x_conf, - y_conf_up, - y_conf_low, color='g')
ax.set_xlim(-1.02 * r_max, 1.02 * r_max)
ax.set_ylim(-1.02 * r_max, 1.02 * r_max)
l_string = ""
ax.set_title("$\ell=(" + get_string(l) + ")$, $p=(" + get_string(p) + ")$,"
+ " $\|\|F(\\vartheta) - p \|\|_2=" + "{:.4f}".format(dist) + "$")
ax.set_xlabel("$x$")
ax.set_ylabel("$y$")
ax.grid(True)
return ax
def get_conv_to_plot(l ,thetas_list, p):
dist = []
for thetas in thetas_list:
x, y = get_xy_to_plot(l, thetas)
arm_end = np.array([x[-1], y[-1]])
dist.append(np.linalg.norm(arm_end - p))
dist = np.asarray(dist)
return dist
def Plot_convergence(ax, l, thetas_list, p):
dist = get_conv_to_plot(l, thetas_list, p)
ax.semilogy(dist, "b-*", label="$\|\|F(\\vartheta) - p \|\|_2$")
ax.set_title("$\ell=(" + get_string(l) + ")$, $p=(" + get_string(p) + ")$")
ax.set_xlabel("Iterations")
ax.set_ylabel("$\|\|F(\\vartheta) - p \|\|_2$")
ax.grid(True)
return ax
def r1(l, thetas, p1):
n = len(l)
return np.sum([l[i] * np.cos(np.sum(thetas[:i + 1])) for i in range(n)]) - p1
def del_r1(l, thetas, k):
# k, number of theta to differentiate on
n = len(l)
return - np.sum([l[i] * np.sin(np.sum(thetas[:i + 1])) for i in range(k, n)])
def del2_r1(l, thetas, k1, k2):
n = len(l)
s = max(k1, k2)
return - np.sum([l[i] * np.cos(np.sum(thetas[:i + 1])) for i in range(s, n)])
def r2(l, thetas, p2):
n = len(l)
return np.sum([l[i] * np.sin(np.sum(thetas[:i + 1])) for i in range(n)]) - p2
def del_r2(l, thetas, k):
# k, number of theta to differentiate on
n = len(l)
return np.sum([l[i] * np.cos(np.sum(thetas[:i + 1])) for i in range(k, n)])
def del2_r2(l, thetas, k1, k2):
n = len(l)
s = max(k1, k2)
return - np.sum([l[i] * np.sin(np.sum(thetas[:i + 1])) for i in range(s, n)])
def Hessian_ri(l, thetas, del2_ri):
n = len(l)
ri = sparse.dok_matrix((n, n))
for k1 in range(n):
for k2 in range(n):
ri[k1, k2] = del2_ri(l, thetas, k1, k2)
return ri
def r(l, thetas, p):
r = sparse.dok_matrix((2, 1))
r[0] = r1(l, thetas, p[0])
r[1] = r2(l, thetas, p[1])
return r
def Jacobi(l, thetas):
n = len(l)
J = sparse.dok_matrix((2, n))
for k in range(n):
J[0, k] = del_r1(l, thetas, k)
J[1, k] = del_r2(l, thetas, k)
return J
def thetas_0_2pi(thetas):
for i in range(len(thetas)):
while thetas[i] >= 2 * np.pi or thetas[i] < 0:
if thetas[i] >= 2 * np.pi:
thetas[i] -= 2 * np.pi
if thetas[i] < 0:
thetas[i] += 2 * np.pi
return thetas
def wolfeLineSearch(l, thetas_k, p, fk, gk, pk, c1, c2, rho, ak=1.0, nmaxls=100):
pkgk = pk @ gk
# Increase the step length until the Armijo rule is (almost) not satisfied
while 0.5 * norm(r(l, thetas_k + rho * ak * pk, p)) < fk + c1 * rho * ak * pkgk[0]:
ak *= rho
# Use bisection to find the optimal step length
aU = ak # upper step length limit
aL = 0 # lower step length limit
for i in range(nmaxls):
# Find the midpoint of aU and aL
ak = 0.5 * (aU + aL)
if 0.5 * norm(r(l, thetas_k + ak * pk, p)) > fk + c1 * ak * pkgk:
# Armijo condition is not satisfied, decrease the upper limit
aU = ak
continue
gk_ak_pk = Jacobi(l, thetas_k + ak * pk).T @ r(l, thetas_k + ak * pk, p)
if pk @ gk_ak_pk > -c2 * pkgk:
# Upper Wolfe condition is not satisfied, decrease the upper limit
aU = ak
continue
if pk @ gk_ak_pk < c2 * pkgk:
# Lower Wolfe condition is not satisfied, increase the lower limit
aL = ak
continue
# Otherwise, all conditions are satisfied, stop the search
break
return ak
def searchDirection(gk, Hk, epsilon=1e-8):
pk = - lsqr(Hk.tocsr(), gk.toarray().ravel())[0] # compute the search direction
if - pk @ gk <= epsilon * norm(gk) * np.linalg.norm(pk):
gk = gk.toarray() # needed here to not get fail later
return - gk.ravel() # ensure that the directional derivative is negative in this direction
return pk
def DampedNewton(l, p, thetas0, tol=1e-10, nmax=1000, nmaxls=100,
c1=1e-4, c2=0.9, rho=2, epsilon=1e-8):
thetas_k_list = [thetas0] # Start condition
ak = 1 # current step length
count = 0
n = len(l)
for i in range(nmax):
thetas_k = thetas_0_2pi(thetas_k_list[-1]) # thetas between 0 and 2*pi
fk = 0.5 * norm(r(l, thetas_k, p)) # current function value
Jk = Jacobi(l, thetas_k) # current Jacobi
rk = r(l, thetas_k, p)
gk = Jk.T @ rk # current gradient
Hk = Jk.T @ Jk + rk[0, 0] * Hessian_ri(l, thetas_k, del2_r1) \
+ rk[1, 0] * Hessian_ri(l, thetas_k, del2_r2) # current Hessian
# Compute the search direction
pk = searchDirection(gk, Hk, epsilon)
# Perform line search
ak = wolfeLineSearch(l, thetas_k, p, fk, gk, pk, c1, c2, rho, ak=1.0, nmaxls=nmaxls)
# Perform the step, add the step to the list, and compute the f and the gradient of f for the next step
thetas_k_list.append(thetas_k + ak * pk)
# count number of iterations
count += 1
if norm(gk) < tol:
if len(l) <= 5: # the problem is small enough, run test
eigval= eig(Hk.toarray())[0]
min_eigval = np.min(eigval)
max_eigval = np.max(eigval)
if min_eigval < 0 < max_eigval: # converged to a saddle point
print("Converged to å saddle point, trying with new initial values.")
# try new thetas0
thetas0 = thetas_0_2pi(thetas0 + np.random.rand(len(thetas0))) # Get new initial guess away form old
print("New thetas0: ", thetas0 / np.pi, " * pi")
thetas_k_list2, count2 = DampedNewton(l, p, thetas0, nmax=nmax-count)
for i in range(len(thetas_k_list2)):
thetas_k_list.append(thetas_k_list2[i])
count += count2
break
thetas_k_list = np.asarray(thetas_k_list)
return thetas_k_list, count
def print_thetas(thetas):
print("Thetas:", thetas)
print("Thetas:", thetas / np.pi, "* pi")
def run_test_cases(save=False):
l_dict = {0: [3, 2, 2], 1: [1, 4, 1], 2: [3, 2, 1, 1], 3: [3, 2, 1, 1]}
p_dict = {0: [3, 2], 1: [1, 1], 2: [3, 2], 3: [0, 0]}
thetas0_dict = {0: np.array([3/2, 1/2, 1/2])*np.pi, 1: np.array([4/5, 4/5, 4/5]) * np.pi,
2: np.array([0.80089218, 0.64076372, 0.61571798, 0.50032875]) * np.pi,
3: np.array([3/2, 1/2, 1/2, 1/2]) * np.pi}
# Number after "2:" are choosen by running the code once and taking the last thetas0 for New thetas0.
# 0.80089218 0.64076372 0.61571798 0.50032875
# 1.80234842, 1.2842212, 0.65702182, 1.95688468
# to save the thetas_lists
thetas_list_dict = {}
# Case plots
fig1 = plt.figure(num="caseplot", figsize=(18.5, 6), dpi=100)
fig1.suptitle("", fontsize=20)
ax11 = fig1.add_subplot(2, 2, 1)
ax12 = fig1.add_subplot(2, 2, 2)
ax13 = fig1.add_subplot(2, 2, 3)
ax14 = fig1.add_subplot(2, 2, 4)
axs1 = np.array([ax11, ax12, ax13, ax14])
for i in range(4):
l = l_dict[i]
p = p_dict[i]
thetas0 = thetas0_dict[i]
print("---------------------------------------")
print("Case: ", i + 1)
print("l:", l)
print("p:", p)
thetas_list, count = DampedNewton(l, p, thetas0)
print("Converged in", count, "iterations")
print_thetas(thetas_list[-1])
axs1[i] = Plot(axs1[i], l, thetas_list[-1], p)
thetas_list_dict[i] = thetas_list
print("---------------------------------------")
# legend
# asking matplotlib for the plotted objects and their labels
lines, labels = ax14.get_legend_handles_labels()
ax13.legend(lines, labels, loc=9, bbox_to_anchor=(1.15, -0.11), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "caseplot" + ".pdf", bbox_inches='tight')
# convergence plots
fig2 = plt.figure(num="convplot", figsize=(13.5, 6), dpi=100)
fig2.suptitle("", fontsize=20)
ax21 = fig2.add_subplot(2, 2, 1)
ax22 = fig2.add_subplot(2, 2, 2)
ax23 = fig2.add_subplot(2, 2, 3)
ax24 = fig2.add_subplot(2, 2, 4)
axs2 = np.array([ax21, ax22, ax23, ax24])
for i in range(4):
l = l_dict[i]
p = p_dict[i]
thetas_list = thetas_list_dict[i]
axs2[i] = Plot_convergence(axs2[i], l, thetas_list, p)
# legend
# asking matplotlib for the plotted objects and their labels
lines, labels = ax23.get_legend_handles_labels()
ax23.legend(lines, labels, loc=9, bbox_to_anchor=(1.15, -0.11), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "convplot" + ".pdf", bbox_inches='tight')
# A special case discussed in the theory
# There is a saddle point which the robot arm gets stuck in
def run_special_case(save=False):
l = [3, 2, 2]
p = [1, 0.5] # Somewhere along the first l
thetas0 = np.array([2728, 0, np.pi]) # [x, 0, pi], for x in |R \ 2728
print("---------------------------------------")
print("Case: 5")
print("l:", l)
print("p:", p)
thetas_list, count = DampedNewton(l, p, thetas0)
print("Converged in", count, "iterations")
print_thetas(thetas_list[-1])
fig3 = plt.figure(num="case5plot", figsize=(12, 6), dpi=100)
fig3.suptitle("", fontsize=20)
ax3 = fig3.add_subplot(1, 1, 1)
ax3 = Plot(ax3, l, thetas_list[-1], p)
lines, labels = ax3.get_legend_handles_labels()
ax3.legend(lines, labels, loc=9, bbox_to_anchor=(0.5, -0.1), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "case5plot" + ".pdf", bbox_inches='tight')
fig4 = plt.figure(num="case5convplot", figsize=(12, 3), dpi=100)
fig4.suptitle("", fontsize=20)
ax4 = fig4.add_subplot(1, 1, 1)
ax4 = Plot_convergence(ax4, l, thetas_list, p)
lines, labels = ax4.get_legend_handles_labels()
ax4.legend(lines, labels, loc=9, bbox_to_anchor=(0.85, 1.22), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
print("---------------------------------------")
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "case5convplot" + ".pdf", bbox_inches='tight')
# A special case discussed in the theory
# There is a saddle point which the robot arm gets stuck in
def run_big_case(size, nmax=1000, save=False):
l = np.random.randint(low=1, high=5, size=size)
p = np.random.randint(low=0, high=7, size=2)
thetas0 = 2 * np.pi * np.random.rand(size) # [x, 0, pi], for x in |R \ 2728
print("---------------------------------------")
print("Case: 6")
print("l:", l)
print("p:", p)
thetas_list, count = DampedNewton(l, p, thetas0, nmax=nmax)
print("Converged in", count, "iterations")
print_thetas(thetas_list[-1])
fig3 = plt.figure(num="case6plot", figsize=(12, 6), dpi=100)
fig3.suptitle("", fontsize=20)
ax3 = fig3.add_subplot(1, 1, 1)
ax3 = Plot(ax3, l, thetas_list[-1], p)
lines, labels = ax3.get_legend_handles_labels()
ax3.legend(lines, labels, loc=9, bbox_to_anchor=(0.5, -0.1), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "case6plot" + ".pdf", bbox_inches='tight')
fig4 = plt.figure(num="case6convplot", figsize=(12, 3), dpi=100)
fig4.suptitle("", fontsize=20)
ax4 = fig4.add_subplot(1, 1, 1)
ax4 = Plot_convergence(ax4, l, thetas_list, p)
lines, labels = ax4.get_legend_handles_labels()
ax4.legend(lines, labels, loc=9, bbox_to_anchor=(0.85, 1.22), ncol=4)
plt.subplots_adjust(hspace=0.45, wspace=0.3)
print("---------------------------------------")
# Set to "save" to True to save the plot
if save:
plt.savefig("Robot" + "case6convplot" + ".pdf", bbox_inches='tight')
"run code"
# save = True to save the plots
save = False
run_test_cases(save=save)
run_special_case(save=save)
run_big_case(5, nmax=1000)
plt.show()