[Robotics: Aerial Robotics][quiz]2.1

Question1：

旋转矩阵是酉矩阵，满足 R*R^(T) = I

r = [0.212 0.7743 0.5963; 0.212 -0.6321 0.7454;0.9540 -0.0316 -0.2981];

r*r' =

1.0001 -0.0000 0.0000
-0.0000 1.0001 0.0000
0.0000 0.0000 1.0000

所以第一个是对的

r = [2^(0.5)/2, 0, 2^(0.5)/2;0,1,0;-2^(0.5)/2, 0, 2^(0.5)/2];

r*r'
ans =

1.0000 0 0
0 1.0000 0
0 0 1.0000

第二个也是对的

第三个是二维的旋转矩阵，满足要求。

第四个元素中出现绝对值大于1的元素不符合要求。

Question2

创建函数：

function [ mat ] = getRzyz( thetaZ1, thetaY, thetaZ2 )
%UNTITLED Summary of this function goes here
%   Detailed explanation goes here
mat1 = [cos(thetaZ1), -sin(thetaZ1), 0;
    sin(thetaZ1), cos(thetaZ1), 0;
    0, 0, 1];

mat2 = [cos(thetaY), 0, -sin(thetaY);
    0,1,0;
    sin(thetaY), 0, cos(thetaY)];
mat3 = [cos(thetaZ2), -sin(thetaZ2), 0;
    sin(thetaZ2), cos(thetaZ2), 0;
    0, 0, 1];
mat = mat1*mat2*mat3;
end

getRzyz(0.2,0.1,0.6)符合

Question3

根据：

计算

omega_b = [0,-1,0.9689;1,0,-0.2474;-0.9689,0.2474,0]

r = [0.675,-0.1724,0.7174;0.2474,0.9689,0;-0.6951,0.1775,0.6967]

r_dot = r*omega_b

omega_s = r_dot*r'

0 -0.6967 1.0000
0.6967 0 -0.7174
-1.0000 0.7174 0.0000

Question4

角度固定在了[0,pi]之间，那么除了特例的情况下，是有解的

r = [0.2919,0.643,-0.7081;-0.643, -0.4161, -0.643;-0.7081, 0.643, 0.2919]

trace(r) = 0.1677;

acos((trace(r)-1)*0.5)

2.0

0.5/sin(2)*(r-r')

ans =
0 0.7071 0
-0.7071 0 -0.7071
0 0.7071 0

u = [0.7071, 0, -0.7071]

Question5

trace(r) = -1;

cos(theta) = (-1-1)/2 = -1

theta = pi;

转轴为u或是-u