Python编程实现欧拉角、方向余弦阵、四元数及等效旋转矢量之间的相互转换

欧拉角、方向余弦阵、四元数和等效旋转矢量都是描述旋转的方式。它们之间的相互转换可以通过数学计算来实现。下面是Python代码实现这些转换的示例：

1. 将欧拉角转换为方向余弦阵

import numpy as np

# 定义欧拉角
roll = np.radians(30)
pitch = np.radians(45)
yaw = np.radians(60)

# 计算方向余弦阵
R_x = np.array([[1, 0, 0],
                [0, np.cos(roll), -np.sin(roll)],
                [0, np.sin(roll), np.cos(roll)]])

R_y = np.array([[np.cos(pitch), 0, np.sin(pitch)],
                [0, 1, 0],
                [-np.sin(pitch), 0, np.cos(pitch)]])

R_z = np.array([[np.cos(yaw), -np.sin(yaw), 0],
                [np.sin(yaw), np.cos(yaw), 0],
                [0, 0, 1]])

R_xyz = np.dot(R_z, np.dot(R_y, R_x))

2. 将方向余弦阵转换为欧拉角

import math

# 定义方向余弦阵
R_xyz = np.array([[0.866, -0.354, 0.354],
                  [0.354, 0.866, -0.354],
                  [0.354, 0.354, 0.866]])

# 计算欧拉角
pitch = -math.asin(R_xyz[2, 0])
roll = math.atan2(R_xyz[2, 1], R_xyz[2, 2])
yaw = math.atan2(R_xyz[1, 0], R_xyz[0, 0])

pitch_deg = math.degrees(pitch)
roll_deg = math.degrees(roll)
yaw_deg = math.degrees(yaw)

3. 将方向余弦阵转换为四元数

# 定义方向余弦阵
R_xyz = np.array([[0.866, -0.354, 0.354],
                  [0.354, 0.866, -0.354],
                  [0.354, 0.354, 0.866]])

# 计算四元数
q_w = math.sqrt(1 + R_xyz[0, 0] + R_xyz[1, 1] + R_xyz[2, 2]) / 2
q_x = (R_xyz[2, 1] - R_xyz[1, 2]) / (4 * q_w)
q_y = (R_xyz[0, 2] - R_xyz[2, 0]) / (4 * q_w)
q_z = (R_xyz[1, 0] - R_xyz[0, 1]) / (4 * q_w)

4. 将四元数转换为方向余弦阵

# 定义四元数
q_w = 0.866
q_x = -0.354
q_y = 0.354
q_z = 0.354

# 计算方向余弦阵
R_xyz = np.array([[1 - 2 * (q_y ** 2 + q_z ** 2), 2 * (q_x * q_y - q_w * q_z), 2 * (q_x * q_z + q_w * q_y)],
                  [2 * (q_x * q_y + q_w * q_z), 1 - 2 * (q_x ** 2 + q_z ** 2), 2 * (q_y * q_z - q_w * q_x)],
                  [2 * (q_x * q_z - q_w * q_y), 2 * (q_y * q_z + q_w * q_x), 1 - 2 * (q_x ** 2 + q_y ** 2)]])

5. 将四元数转换为等效旋转矢量

# 定义四元数
q_w = 0.866
q_x = -0.354
q_y = 0.354
q_z = 0.354

# 计算等效旋转矢量
theta = 2 * math.acos(q_w)
sin_theta = math.sin(theta / 2)
v_x = q_x / sin_theta
v_y = q_y / sin_theta
v_z = q_z / sin_theta