Rotation Matrix in estimateMo​noCameraPa​rameter.m

Hi, I need to find the distance of a vehicle from ego-vehicle using monocular camera. I can accomplish it using Visual Perception Using Monocular Matlab given. But I need to know the math behind it, on of them is the rotation matrix to align camera coordinate with vehicle coordinate in estimaMonoCameraParameter.m, In there I found

% R = [cos(a)*cos(c)-sin(a)*cos(b)*sin(c), -cos(a)*sin(c)-sin(a)*cos(b)*cos(c), sin(a)*sin(b);

% sin(a)*cos(c)+cos(a)*cos(b)*sin(c), -sin(a)*sin(c)+cos(a)*cos(b)*cos(c), -cos(a)*sin(b);

% sin(b)*sin(c), sin(b)*cos(c), cos(b)]

to be the rotation matrix. But I can decompose the rotation matrix to elementary matrix to find the order of rotation. Please help me. Thank you so much.

Answer 1

[Altaïr]

Hey @Mochammad Habibi,

The example on Matrix Rotations and Transformations illustrates how rotation matrices can be used to rotate a vector with respect to a frame. The example can be accessed through the following link.

https://www.mathworks.com/help/symbolic/rotation-matrix-and-transformation-matrix.html

From the example, Rx, Ry, and Rz represent the rotation matrices to rotate a vector by an angle t about the x, y, and z axis respectively.

Rx = [1 0 0; 0 cos(t) -sin(t); 0 sin(t) cos(t)];
Ry = [cos(t) 0 sin(t); 0 1 0; -sin(t) 0 cos(t)];
Rz = [cos(t) -sin(t) 0; sin(t) cos(t) 0; 0 0 1];

By experimenting with different combinations and orders of these matrices, the order and sense of each rotation can be determined. This process can be simplified using symbolic variables. Here’s a snippet that creates a rotation matrix as described:

syms a b c

% Rotation matrices for rotating vector about the frame
Rz1 = [cos(a) -sin(a) 0; sin(a) cos(a) 0; 0 0 1];
Rx = [1 0 0; 0 cos(b) -sin(b); 0 sin(b) cos(b)];
Rz2 = [cos(c) -sin(c) 0; sin(c) cos(c) 0; 0 0 1];

R = Rz1*Rx*Rz2

R = 

This implies that the rotation order is as follows:

1. Rotate the vector about z axis by an angle a
2. Rotate the vector about the new x axis by an angle b
3. Rotate the vector about the new z axis by an angle c

Starting from MATLAB R2022b, the rigidtform3d function is available to store an overall transformation matrix with three rotations and three translations. More information about this function can be found here:

https://www.mathworks.com/help/images/ref/rigidtform3d.html