我从事航天器的轨迹设计工作,我正在实现一种算法,用于查找周期性轨迹,该轨迹需要在轨迹越过特定平面时进行检测。我正在使用scipy.integrate
的solve_ivp
及其events
功能。
理想情况下,我想计算事件触发的时间,仅在触发一定数量的事件后才停止集成,但是由于我在solve_ivp's documentation中找不到任何类似的选项,因此我只是使用for循环链这些集成弧并计算事件的数量。但是,我的问题是,当初始条件触发事件和solve_ivp
时,event.terminal = True
卡住了。
您可以在下面找到示例代码片段:
import numpy as np
from scipy.integrate import solve_ivp
def eq_motion(t,Y):
Ydot = np.zeros((6,1))
r = (Y[0,0]**2 + Y[1,0]**2 + Y[2,0]**2)**0.5
Ydot[0] = Y[3,0]
Ydot[1] = Y[4,0]
Ydot[2] = Y[5,0]
Ydot[3] = 2*Y[4,0] + 3*Y[0,0] - Y[0,0]/r**3
Ydot[4] = -2*Y[3,0] - Y[1,0]/r**3
Ydot[5] = -Y[2,0] - Y[1,0]/r**3
return Ydot
def event_xz_plane(t,Y):
return Y[1]
event_xz_plane.terminal = True
event_xz_plane.direction = 0
# initial conditions
Y0 = np.array([-0.006489114696252, 0, 0.107462057974196 + 0.0006, 0, 4.165257687202607,0])
t_span = np.array([0, 1.892845635529040*2])
n_plane_intersections = 4
for i in np.arange(n_plane_intersections):
integrate = solve_ivp(fun=eq_motion, t_span=t_span, y0=Y0, method = 'LSODA',
vectorized = True, rtol = 1e-13, atol = 1e-14,
events = (event_xz_plane),dense_output=True)
print('Number of integrations before event: {}'.format(integrate.y.shape[1]-1))
print(integrate.y_events[0])
print('\n')
# attempting to chain the different integration arcs
Y0 = integrate.y[:,-1]
选择的初始条件接近周期轨道的初始条件,并产生以下trajectory,在其中您可以看到与xz平面有4个交点,包括该平面上的初始条件。
但是,以上脚本打印出以下内容:
Number of integrations before event: 1
[[-0.00648911 0. 0.10806206 0. 4.16525769 0. ]]
Number of integrations before event: 1
[[-0.00648911 0. 0.10806206 0. 4.16525769 0. ]]
Number of integrations before event: 1
[[-0.00648911 0. 0.10806206 0. 4.16525769 0. ]]
Number of integrations before event: 1
[[-0.00648911 0. 0.10806206 0. 4.16525769 0. ]]
您可以看到它基本上卡在了初始状态。当然,发生这种情况是因为初始条件位于终止条件,但是,似乎没有选择“通过”积分的第一次迭代。
我尝试将事件函数编写为
def event_xz_plane(t,Y):
return Y[1] + 1e3*(t==0)
因此它将忽略初始状态,但随后它仅会“卡住”在下一个平面相交处,而无论哪种方式,它似乎都不是一个好的解决方案。
我想到的唯一的解决方案是在事件函数上设置terminate=False
并检查这些实例处触发的事件数和状态向量。但是,这对我的实现而言并不理想,因为它取决于t_span
时间设置,这在算法中是未知的。也就是说,如果我需要获得n
个平面相交的数量,则必须迭代调整t_span直到达到该数量,而不是仅允许积分运行直到触发n
个平面相交。
反正有什么要克服的?
首先检查微分方程组。线性项如何解释?加速度的第三部分的中心力部分与第三坐标Y[2]
成比例。
如果您决定在矢量化版本中实现ODE功能,请完全符合要求。您不能假设输入向量仅包含一个状态,而使输出Ydot
与输入Y
具有相同的格式。
def eq_motion(t,Y): Ydot = np.zeros(Y.shape) r = (Y[0]**2 + Y[1]**2 + Y[2]**2)**0.5 Ydot[0] = Y[3] Ydot[1] = Y[4] Ydot[2] = Y[5] Ydot[3] = - Y[0]/r**3 + 2*Y[4] + 3*Y[0] Ydot[4] = - Y[1]/r**3 - 2*Y[3] Ydot[5] = - Y[2]/r**3 - Y[2] return Ydot
最好同时更新
t0
和Y0
,因此设置
t0, tfinal = 0, 10
[使用事件方向禁止求解器再次检测到最后一个事件,理想情况下,您将从
Y0
和eq_motion(t0,Y0)
中沿正向时间方向(此处为正向)检测事件的当前符号,并设置与该方向相反的方向] >event_xz_plane.direction = -1 n_plane_intersections = 4 for i in np.arange(n_plane_intersections): integrate = solve_ivp(fun=eq_motion, t_span=[t0,tfinal], y0=Y0, method = 'LSODA', vectorized = True, rtol = 1e-13, atol = 1e-14, events = (event_xz_plane),dense_output=True) print(integrate.message) print('Number of integrations before event: {}'.format(integrate.y.shape[1]-1)) print(integrate.t_events[0],integrate.y_events[0]) print('\n') # attempting to chain the different integration arcs Y0 = integrate.y[:,-1] t0 = integrate.t[-1] event_xz_plane.direction *= -1
这给出了输出
A termination event occurred. Number of integrations before event: 288 [0.96633212] [[ 3.54891748e-01 -6.67868538e-17 -8.77207053e-01 5.03973235e-02 -7.78052855e-01 -1.76219528e-02]] A termination event occurred. Number of integrations before event: 263 [2.03187275] [[ 7.62458860e-03 1.66901228e-15 1.66823579e-01 -2.71527186e-01 3.27865867e+00 1.07443749e-01]] A termination event occurred. Number of integrations before event: 300 [3.22281063] [[ 3.86773384e-01 -1.92554306e-16 -8.24376230e-01 -6.86117012e-02 -8.96669320e-01 2.07695448e-01]] A termination event occurred. Number of integrations before event: 258 [4.10715034] [[-2.07546473e-02 -9.56266316e-16 1.42957911e-01 9.37459643e-02 3.56344204e+00 7.24687825e-02]]
与之相比,Dormand-Prince 853方法要快得多(而Radau则要慢得多,并且得出相同的结果
A termination event occurred. Number of integrations before event: 67 [0.96633212] [[ 3.54891748e-01 -2.46764414e-16 -8.77207053e-01 5.03973235e-02 -7.78052855e-01 -1.76219528e-02]] A termination event occurred. Number of integrations before event: 52 [2.03187275] [[ 7.62458861e-03 -5.10008702e-16 1.66823579e-01 -2.71527186e-01 3.27865867e+00 1.07443749e-01]] A termination event occurred. Number of integrations before event: 58 [3.22281063] [[ 3.86773384e-01 1.21430643e-17 -8.24376230e-01 -6.86117012e-02 -8.96669320e-01 2.07695448e-01]] A termination event occurred. Number of integrations before event: 52 [4.10715034] [[-2.07546473e-02 9.19403442e-17 1.42957911e-01 9.37459642e-02 3.56344204e+00 7.24687825e-02]]