已知齐次变换矩阵H= [0, 1, 0, 0; 0, 0, -1, 0; -1, 0, 0, 0; 0, 0, 0, 1]求出当Rot(f,θ)=H时,f和θ的值,并用MATLAB代码将其表示出来
时间: 2023-10-03 07:06:15 浏览: 196
根据齐次变换矩阵的定义,我们可以得到:
Rot(f,θ) = [cosθ+(1-cosθ)f1^2, (1-cosθ)f1f2-sinθf3, (1-cosθ)f1f3+sinθf2, 0;
(1-cosθ)f1f2+sinθf3, cosθ+(1-cosθ)f2^2, (1-cosθ)f2f3-sinθf1, 0;
(1-cosθ)f1f3-sinθf2, (1-cosθ)f2f3+sinθf1, cosθ+(1-cosθ)f3^2, 0;
0, 0, 0, 1]
将H代入上式,得到:
[cosθ, -sinθ, 0, 0;
sinθ, cosθ, 0, 0;
0, 0, 1, 0;
0, 0, 0, 1] = [0, 1, 0, 0;
0, 0, -1, 0;
-1, 0, 0, 0;
0, 0, 0, 1]
通过比较矩阵的对应元素,我们可以得到:
cosθ = 0
-sinθ = 1
1 = 0
0 = 0
由此可知,上式无解,即无法找到对应的f和θ。
MATLAB代码如下:
syms f1 f2 f3 theta
H = [0, 1, 0, 0; 0, 0, -1, 0; -1, 0, 0, 0; 0, 0, 0, 1];
R = [cos(theta)+(1-cos(theta))*f1^2, (1-cos(theta))*f1*f2-sin(theta)*f3, (1-cos(theta))*f1*f3+sin(theta)*f2, 0;
(1-cos(theta))*f1*f2+sin(theta)*f3, cos(theta)+(1-cos(theta))*f2^2, (1-cos(theta))*f2*f3-sin(theta)*f1, 0;
(1-cos(theta))*f1*f3-sin(theta)*f2, (1-cos(theta))*f2*f3+sin(theta)*f1, cos(theta)+(1-cos(theta))*f3^2, 0;
0, 0, 0, 1];
eqns = [R == H];
[Sf, Stheta] = vpasolve(eqns, [f1, f2, f3, theta]);
f1 = double(Sf.f1)
f2 = double(Sf.f2)
f3 = double(Sf.f3)
theta = double(Stheta)
阅读全文