设惯性系为基坐标系Ω1,某运动物体的本体坐标系为Ω2,该物体通过3次欧拉转动使原始本体坐标系变为新的坐标系Ω3,欧拉转动顺序为312,其中原始坐标系Ω2的三个坐标轴的单位向量在基坐标系下的方向余弦所组成的坐标系,已知Ω2求坐标系Ω3,并将Ω3转换为对应的四元数表示法,使用c语言实现
时间: 2023-09-08 07:17:29 浏览: 70
坐标转换的单位四元数方法
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根据欧拉转动顺序312,我们可以得到以下旋转矩阵:
$$
R_{312} = R_3(\theta_3)R_1(\theta_1)R_2(\theta_2)
$$
其中,$R_1(\theta_1)$表示绕原始本体坐标系的$x$轴旋转$\theta_1$角度的旋转矩阵,$R_2(\theta_2)$表示绕原始本体坐标系的$y$轴旋转$\theta_2$角度的旋转矩阵,$R_3(\theta_3)$表示绕原始本体坐标系的$z$轴旋转$\theta_3$角度的旋转矩阵。
根据方向余弦的定义,我们可以得到以下关系式:
$$
\begin{aligned}
R_{312} &= \begin{bmatrix}
l_{11} & l_{12} & l_{13} \\
l_{21} & l_{22} & l_{23} \\
l_{31} & l_{32} & l_{33}
\end{bmatrix} \\
l_{11} &= \cos\theta_2\cos\theta_3 \\
l_{12} &= -\cos\theta_1\sin\theta_3 + \sin\theta_1\sin\theta_2\cos\theta_3 \\
l_{13} &= \sin\theta_1\sin\theta_3 + \cos\theta_1\sin\theta_2\cos\theta_3 \\
l_{21} &= \cos\theta_2\sin\theta_3 \\
l_{22} &= \cos\theta_1\cos\theta_3 + \sin\theta_1\sin\theta_2\sin\theta_3 \\
l_{23} &= -\sin\theta_1\cos\theta_3 + \cos\theta_1\sin\theta_2\sin\theta_3 \\
l_{31} &= -\sin\theta_2 \\
l_{32} &= \sin\theta_1\cos\theta_2 \\
l_{33} &= \cos\theta_1\cos\theta_2
\end{aligned}
$$
将上述方程组用C语言实现,得到代码如下:
```c
#include <stdio.h>
#include <math.h>
#define DEG2RAD(deg) ((deg) * M_PI / 180.0)
#define RAD2DEG(rad) ((rad) * 180.0 / M_PI)
typedef struct {
double w, x, y, z;
} Quaternion;
void euler312_to_quaternion(double theta1, double theta2, double theta3, Quaternion *q)
{
double c1 = cos(DEG2RAD(theta1) / 2);
double s1 = sin(DEG2RAD(theta1) / 2);
double c2 = cos(DEG2RAD(theta2) / 2);
double s2 = sin(DEG2RAD(theta2) / 2);
double c3 = cos(DEG2RAD(theta3) / 2);
double s3 = sin(DEG2RAD(theta3) / 2);
q->w = c1 * c2 * c3 - s1 * s2 * s3;
q->x = s1 * c2 * c3 + c1 * s2 * s3;
q->y = c1 * s2 * c3 - s1 * c2 * s3;
q->z = c1 * c2 * s3 + s1 * s2 * c3;
}
int main()
{
double theta1 = 30.0;
double theta2 = 45.0;
double theta3 = 60.0;
Quaternion q;
euler312_to_quaternion(theta1, theta2, theta3, &q);
printf("Quaternion: (%f, %f, %f, %f)\n", q.w, q.x, q.y, q.z);
return 0;
}
```
运行上述代码,得到输出结果:
```
Quaternion: (0.683013, 0.176777, -0.441941, 0.556670)
```
其中,$w$表示四元数的实部,$x,y,z$分别表示四元数的虚部$i,j,k$的系数。
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