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AtsushiSakaiAtsushiSakai
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enhance cubic spline path doc (AtsushiSakai#698)
* enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cublic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc * enhance cubic spline path doc
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‎PathPlanning/CubicSpline/cubic_spline_planner.py‎

Lines changed: 213 additions & 63 deletions
Original file line numberDiff line numberDiff line change
@@ -9,87 +9,133 @@
99
import bisect
1010

1111

12-
class Spline:
12+
class CubicSpline1D:
1313
"""
14-
Cubic Spline class
14+
1D Cubic Spline class
15+
16+
Parameters
17+
----------
18+
x : list
19+
x coordinates for data points. This x coordinates must be
20+
sorted
21+
in ascending order.
22+
y : list
23+
y coordinates for data points
24+
25+
Examples
26+
--------
27+
You can interpolate 1D data points.
28+
29+
>>> import numpy as np
30+
>>> import matplotlib.pyplot as plt
31+
>>> x = np.arange(5)
32+
>>> y = [1.7, -6, 5, 6.5, 0.0]
33+
>>> sp = CubicSpline1D(x, y)
34+
>>> xi = np.linspace(0.0, 5.0)
35+
>>> yi = [sp.calc_position(x) for x in xi]
36+
>>> plt.plot(x, y, "xb", label="Data points")
37+
>>> plt.plot(xi, yi , "r", label="Cubic spline interpolation")
38+
>>> plt.grid(True)
39+
>>> plt.legend()
40+
>>> plt.show()
41+
42+
.. image:: cubic_spline_1d.png
43+
1544
"""
1645

1746
def __init__(self, x, y):
18-
self.b, self.c, self.d, self.w = [], [], [], []
1947

48+
h = np.diff(x)
49+
if np.any(h < 0):
50+
raise ValueError("x coordinates must be sorted in ascending order")
51+
52+
self.a, self.b, self.c, self.d = [], [], [], []
2053
self.x = x
2154
self.y = y
22-
2355
self.nx = len(x) # dimension of x
24-
h = np.diff(x)
2556

26-
# calc coefficient c
57+
# calc coefficient a
2758
self.a = [iy for iy in y]
2859

2960
# calc coefficient c
3061
A = self.__calc_A(h)
31-
B = self.__calc_B(h)
62+
B = self.__calc_B(h, self.a)
3263
self.c = np.linalg.solve(A, B)
33-
# print(self.c1)
3464

3565
# calc spline coefficient b and d
3666
for i in range(self.nx - 1):
37-
self.d.append((self.c[i + 1] - self.c[i]) / (3.0 * h[i]))
38-
tb = (self.a[i + 1] - self.a[i]) / h[i] - h[i] * \
39-
(self.c[i + 1] + 2.0 * self.c[i]) / 3.0
40-
self.b.append(tb)
67+
d = (self.c[i + 1] - self.c[i]) / (3.0 * h[i])
68+
b = 1.0 / h[i] * (self.a[i + 1] - self.a[i]) \
69+
- h[i] / 3.0 * (2.0 * self.c[i] + self.c[i + 1])
70+
self.d.append(d)
71+
self.b.append(b)
4172

42-
def calc(self, t):
73+
def calc_position(self, x):
4374
"""
44-
Calc position
75+
Calc `y` position for given `x`.
4576
46-
if t is outside of the input x, return None
77+
if `x` is outside the data point's `x` range, return None.
4778
79+
Returns
80+
-------
81+
y : float
82+
y position for given x.
4883
"""
49-
50-
if t < self.x[0]:
84+
if x < self.x[0]:
5185
return None
52-
elif t > self.x[-1]:
86+
elif x > self.x[-1]:
5387
return None
5488

55-
i = self.__search_index(t)
56-
dx = t - self.x[i]
57-
result = self.a[i] + self.b[i] * dx + \
89+
i = self.__search_index(x)
90+
dx = x - self.x[i]
91+
position = self.a[i] + self.b[i] * dx + \
5892
self.c[i] * dx ** 2.0 + self.d[i] * dx ** 3.0
5993

60-
return result
94+
return position
6195

62-
def calcd(self, t):
96+
def calc_first_derivative(self, x):
6397
"""
64-
Calc first derivative
98+
Calc first derivative at given x.
99+
100+
if x is outside the input x, return None
65101
66-
if t is outside of the input x, return None
102+
Returns
103+
-------
104+
dy : float
105+
first derivative for given x.
67106
"""
68107

69-
if t < self.x[0]:
108+
if x < self.x[0]:
70109
return None
71-
elif t > self.x[-1]:
110+
elif x > self.x[-1]:
72111
return None
73112

74-
i = self.__search_index(t)
75-
dx = t - self.x[i]
76-
result = self.b[i] + 2.0 * self.c[i] * dx + 3.0 * self.d[i] * dx ** 2.0
77-
return result
113+
i = self.__search_index(x)
114+
dx = x - self.x[i]
115+
dy = self.b[i] + 2.0 * self.c[i] * dx + 3.0 * self.d[i] * dx ** 2.0
116+
return dy
78117

79-
def calcdd(self, t):
118+
def calc_second_derivative(self, x):
80119
"""
81-
Calc second derivative
120+
Calc second derivative at given x.
121+
122+
if x is outside the input x, return None
123+
124+
Returns
125+
-------
126+
ddy : float
127+
second derivative for given x.
82128
"""
83129

84-
if t < self.x[0]:
130+
if x < self.x[0]:
85131
return None
86-
elif t > self.x[-1]:
132+
elif x > self.x[-1]:
87133
return None
88134

89-
i = self.__search_index(t)
90-
dx = t - self.x[i]
91-
result = 2.0 * self.c[i] + 6.0 * self.d[i] * dx
92-
return result
135+
i = self.__search_index(x)
136+
dx = x - self.x[i]
137+
ddy = 2.0 * self.c[i] + 6.0 * self.d[i] * dx
138+
return ddy
93139

94140
def __search_index(self, x):
95141
"""
@@ -112,30 +158,82 @@ def __calc_A(self, h):
112158
A[0, 1] = 0.0
113159
A[self.nx - 1, self.nx - 2] = 0.0
114160
A[self.nx - 1, self.nx - 1] = 1.0
115-
# print(A)
116161
return A
117162

118-
def __calc_B(self, h):
163+
def __calc_B(self, h, a):
119164
"""
120165
calc matrix B for spline coefficient c
121166
"""
122167
B = np.zeros(self.nx)
123168
for i in range(self.nx - 2):
124-
B[i + 1] = 3.0 * (self.a[i + 2] - self.a[i + 1]) / \
125-
h[i + 1] - 3.0 * (self.a[i + 1] - self.a[i]) / h[i]
169+
B[i + 1] = 3.0 * (a[i + 2] - a[i + 1]) / h[i + 1]\
170+
- 3.0 * (a[i + 1] - a[i]) / h[i]
126171
return B
127172

128173

129-
class Spline2D:
174+
class CubicSpline2D:
130175
"""
131-
2D Cubic Spline class
132-
176+
Cubic CubicSpline2D class
177+
178+
Parameters
179+
----------
180+
x : list
181+
x coordinates for data points.
182+
y : list
183+
y coordinates for data points.
184+
185+
Examples
186+
--------
187+
You can interpolate a 2D data points.
188+
189+
>>> import matplotlib.pyplot as plt
190+
>>> x = [-2.5, 0.0, 2.5, 5.0, 7.5, 3.0, -1.0]
191+
>>> y = [0.7, -6, 5, 6.5, 0.0, 5.0, -2.0]
192+
>>> ds = 0.1 # [m] distance of each interpolated points
193+
>>> sp = CubicSpline2D(x, y)
194+
>>> s = np.arange(0, sp.s[-1], ds)
195+
>>> rx, ry, ryaw, rk = [], [], [], []
196+
>>> for i_s in s:
197+
... ix, iy = sp.calc_position(i_s)
198+
... rx.append(ix)
199+
... ry.append(iy)
200+
... ryaw.append(sp.calc_yaw(i_s))
201+
... rk.append(sp.calc_curvature(i_s))
202+
>>> plt.subplots(1)
203+
>>> plt.plot(x, y, "xb", label="Data points")
204+
>>> plt.plot(rx, ry, "-r", label="Cubic spline path")
205+
>>> plt.grid(True)
206+
>>> plt.axis("equal")
207+
>>> plt.xlabel("x[m]")
208+
>>> plt.ylabel("y[m]")
209+
>>> plt.legend()
210+
>>> plt.show()
211+
212+
.. image:: cubic_spline_2d_path.png
213+
214+
>>> plt.subplots(1)
215+
>>> plt.plot(s, [np.rad2deg(iyaw) for iyaw in ryaw], "-r", label="yaw")
216+
>>> plt.grid(True)
217+
>>> plt.legend()
218+
>>> plt.xlabel("line length[m]")
219+
>>> plt.ylabel("yaw angle[deg]")
220+
221+
.. image:: cubic_spline_2d_yaw.png
222+
223+
>>> plt.subplots(1)
224+
>>> plt.plot(s, rk, "-r", label="curvature")
225+
>>> plt.grid(True)
226+
>>> plt.legend()
227+
>>> plt.xlabel("line length[m]")
228+
>>> plt.ylabel("curvature [1/m]")
229+
230+
.. image:: cubic_spline_2d_curvature.png
133231
"""
134232

135233
def __init__(self, x, y):
136234
self.s = self.__calc_s(x, y)
137-
self.sx = Spline(self.s, x)
138-
self.sy = Spline(self.s, y)
235+
self.sx = CubicSpline1D(self.s, x)
236+
self.sy = CubicSpline1D(self.s, y)
139237

140238
def __calc_s(self, x, y):
141239
dx = np.diff(x)
@@ -148,35 +246,70 @@ def __calc_s(self, x, y):
148246
def calc_position(self, s):
149247
"""
150248
calc position
249+
250+
Parameters
251+
----------
252+
s : float
253+
distance from the start point. if `s` is outside the data point's
254+
range, return None.
255+
256+
Returns
257+
-------
258+
x : float
259+
x position for given s.
260+
y : float
261+
y position for given s.
151262
"""
152-
x = self.sx.calc(s)
153-
y = self.sy.calc(s)
263+
x = self.sx.calc_position(s)
264+
y = self.sy.calc_position(s)
154265

155266
return x, y
156267

157268
def calc_curvature(self, s):
158269
"""
159270
calc curvature
271+
272+
Parameters
273+
----------
274+
s : float
275+
distance from the start point. if `s` is outside the data point's
276+
range, return None.
277+
278+
Returns
279+
-------
280+
k : float
281+
curvature for given s.
160282
"""
161-
dx = self.sx.calcd(s)
162-
ddx = self.sx.calcdd(s)
163-
dy = self.sy.calcd(s)
164-
ddy = self.sy.calcdd(s)
283+
dx = self.sx.calc_first_derivative(s)
284+
ddx = self.sx.calc_second_derivative(s)
285+
dy = self.sy.calc_first_derivative(s)
286+
ddy = self.sy.calc_second_derivative(s)
165287
k = (ddy * dx - ddx * dy) / ((dx ** 2 + dy ** 2)**(3 / 2))
166288
return k
167289

168290
def calc_yaw(self, s):
169291
"""
170292
calc yaw
293+
294+
Parameters
295+
----------
296+
s : float
297+
distance from the start point. if `s` is outside the data point's
298+
range, return None.
299+
300+
Returns
301+
-------
302+
yaw : float
303+
yaw angle (tangent vector) for given s.
171304
"""
172-
dx = self.sx.calcd(s)
173-
dy = self.sy.calcd(s)
305+
dx = self.sx.calc_first_derivative(s)
306+
dy = self.sy.calc_first_derivative(s)
174307
yaw = math.atan2(dy, dx)
175308
return yaw
176309

177310

178311
def calc_spline_course(x, y, ds=0.1):
179-
sp = Spline2D(x, y)
312+
sp = CubicSpline2D(x, y)
180313
s = list(np.arange(0, sp.s[-1], ds))
181314

182315
rx, ry, ryaw, rk = [], [], [], []
@@ -190,14 +323,30 @@ def calc_spline_course(x, y, ds=0.1):
190323
return rx, ry, ryaw, rk, s
191324

192325

193-
def main(): # pragma: no cover
194-
print("Spline 2D test")
326+
def main_1d():
327+
print("CubicSpline1D test")
328+
import matplotlib.pyplot as plt
329+
x = np.arange(5)
330+
y = [1.7, -6, 5, 6.5, 0.0]
331+
sp = CubicSpline1D(x, y)
332+
xi = np.linspace(0.0, 5.0)
333+
334+
plt.plot(x, y, "xb", label="Data points")
335+
plt.plot(xi, [sp.calc_position(x) for x in xi], "r",
336+
label="Cubic spline interpolation")
337+
plt.grid(True)
338+
plt.legend()
339+
plt.show()
340+
341+
342+
def main_2d(): # pragma: no cover
343+
print("CubicSpline1D 2D test")
195344
import matplotlib.pyplot as plt
196345
x = [-2.5, 0.0, 2.5, 5.0, 7.5, 3.0, -1.0]
197346
y = [0.7, -6, 5, 6.5, 0.0, 5.0, -2.0]
198347
ds = 0.1 # [m] distance of each interpolated points
199348

200-
sp = Spline2D(x, y)
349+
sp = CubicSpline2D(x, y)
201350
s = np.arange(0, sp.s[-1], ds)
202351

203352
rx, ry, ryaw, rk = [], [], [], []
@@ -209,8 +358,8 @@ def main(): # pragma: no cover
209358
rk.append(sp.calc_curvature(i_s))
210359

211360
plt.subplots(1)
212-
plt.plot(x, y, "xb", label="input")
213-
plt.plot(rx, ry, "-r", label="spline")
361+
plt.plot(x, y, "xb", label="Data points")
362+
plt.plot(rx, ry, "-r", label="Cubic spline path")
214363
plt.grid(True)
215364
plt.axis("equal")
216365
plt.xlabel("x[m]")
@@ -235,4 +384,5 @@ def main(): # pragma: no cover
235384

236385

237386
if __name__ == '__main__':
238-
main()
387+
# main_1d()
388+
main_2d()

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