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 c语言实现

假设原始本体坐标系Ω2相对于基坐标系Ω1的欧拉转动顺序为yaw-pitch-roll，即绕z轴旋转yaw角度，绕y轴旋转pitch角度，绕x轴旋转roll角度，则变换矩阵为：

R1 = Rz(yaw) * Ry(pitch) * Rx(roll)

其中

Rz(yaw) = [ cos(yaw)  -sin(yaw)   0 ]
          [ sin(yaw)   cos(yaw)   0 ]
          [    0         0       1 ]

Ry(pitch) = [  cos(pitch)   0   sin(pitch) ]
            [      0        1        0     ]
            [ -sin(pitch)   0   cos(pitch) ]

Rx(roll) = [ 1     0           0      ]
           [ 0  cos(roll)  -sin(roll) ]
           [ 0  sin(roll)   cos(roll) ]

然后，假设新的本体坐标系Ω3相对于原始本体坐标系Ω2的欧拉转动顺序为yaw'-pitch'-roll'，变换矩阵为：

R2 = Rz(yaw') * Ry(pitch') * Rx(roll')

将变换矩阵相乘即可得到从基坐标系Ω1到新的本体坐标系Ω3的变换矩阵：

R = R2 * R1

C语言代码实现如下：

#include <math.h>

typedef struct {
    double x, y, z;
} Vector3;

typedef struct {
    double m[3][3];
} Matrix3x3;

// 绕x轴旋转roll角度的旋转矩阵
Matrix3x3 Rx(double roll) {
    Matrix3x3 R;
    double cos_a = cos(roll);
    double sin_a = sin(roll);
    R.m[0][0] = 1.0;
    R.m[0][1] = 0.0;
    R.m[0][2] = 0.0;
    R.m[1][0] = 0.0;
    R.m[1][1] = cos_a;
    R.m[1][2] = -sin_a;
    R.m[2][0] = 0.0;
    R.m[2][1] = sin_a;
    R.m[2][2] = cos_a;
    return R;
}

// 绕y轴旋转pitch角度的旋转矩阵
Matrix3x3 Ry(double pitch) {
    Matrix3x3 R;
    double cos_a = cos(pitch);
    double sin_a = sin(pitch);
    R.m[0][0] = cos_a;
    R.m[0][1] = 0.0;
    R.m[0][2] = sin_a;
    R.m[1][0] = 0.0;
    R.m[1][1] = 1.0;
    R.m[1][2] = 0.0;
    R.m[2][0] = -sin_a;
    R.m[2][1] = 0.0;
    R.m[2][2] = cos_a;
    return R;
}

// 绕z轴旋转yaw角度的旋转矩阵
Matrix3x3 Rz(double yaw) {
    Matrix3x3 R;
    double cos_a = cos(yaw);
    double sin_a = sin(yaw);
    R.m[0][0] = cos_a;
    R.m[0][1] = -sin_a;
    R.m[0][2] = 0.0;
    R.m[1][0] = sin_a;
    R.m[1][1] = cos_a;
    R.m[1][2] = 0.0;
    R.m[2][0] = 0.0;
    R.m[2][1] = 0.0;
    R.m[2][2] = 1.0;
    return R;
}

// 从基坐标系到新的本体坐标系的变换矩阵
Matrix3x3 transform(Vector3 angles) {
    double roll = angles.x;
    double pitch = angles.y;
    double yaw = angles.z;
    Matrix3x3 R1 = Rz(yaw) * Ry(pitch) * Rx(roll);

    double roll_prime = 0.0;    // 新的本体坐标系绕x轴旋转的角度
    double pitch_prime = 0.0;   // 新的本体坐标系绕y轴旋转的角度
    double yaw_prime = 0.0;     // 新的本体坐标系绕z轴旋转的角度
    Matrix3x3 R2 = Rz(yaw_prime) * Ry(pitch_prime) * Rx(roll_prime);

    Matrix3x3 R = R2 * R1;      // 从基坐标系到新的本体坐标系的变换矩阵
    return R;
}

其中，`Vector3`表示三维向量，`Matrix3x3`表示3x3的矩阵。`Rx`、`Ry`、`Rz`分别表示绕x、y、z轴旋转的旋转矩阵，`transform`函数即为从基坐标系到新的本体坐标系的变换矩阵计算函数。