how to do coordinate transformation around a fixed axis using robotics toolbox or spatial math toolbox?

Dear friends,

If the initial frame {s} rotate around a screw axis S which is defined by rotaion unit axis omega=(0.7,0.0,0.7) , a vector q=(1,1,0.5) and a screw pitch h=1. the initial frame {s} rotate around the screw axis S, what the new frame looks like?

Your help would be highly appreciated.

Answer 1

[Askic V]

Hello Daniel,

I suggest you to look in the book "Modern Robotics" by K. Lynch and F. Park, which is available online for free.

However, I have use anothe online source which I find very good to understand:

https://www.mecharithm.com/screws-a-geometric-description-of-twists-in-robotics/

In my opinion, you need to find the homogeneous transformation matrix T. This is a transformation matrix between two frames.

I have written the following Matlab code:

% Source: 
% https://www.mecharithm.com/screws-a-geometric-description-of-twists-in-robotics/
w = [sqrt(2)/2, 0, sqrt(2)/2]';
q = [1, 1, 0.5]';
h = 1;

% Screw S = [omega; velocity]
Vs = [w; cross(-w,q) + h*w];

% Skew matrix omega
sk_w = [0 -w(3) w(2); w(3) 0 -w(1); -w(2) w(1) 0];
% norm of w is 1
theta = norm(w);
% using the Rodrigues’ formula 
% we can find the rotational part of the transformation matrix 
R = eye(3) + sin(theta)*sk_w + (1-cos(theta))*sk_w*sk_w;

% velocity = last three elements in Screw vector
v = Vs(4:6);
% Rodrigues formula for G
G = (eye(3)*theta + (1-cos(theta))*sk_w + (theta-sin(theta))*sk_w^2)*v;
T = [R G; 0, 0, 0, 1]

T = 4×4

0.7702 -0.5950 0.2298 1.4170 0.5950 0.5403 -0.5950 0.1622 0.2298 0.5950 0.7702 -0.0028 0 0 0 1.0000

So, when you have this transformation matrix T, you can now express the configuration (position and orientation) of a frame relative to a fixed (reference) frame.

https://www.mecharithm.com/homogenous-transformation-matrices-configurations-in-robotics/

[Daniel Niu]

thank you for your answer and information!

[Daniel Niu]

Dear Askic,

how to include the rotate angle, for example 265 degree in the code?

I see the theta is 1 in your code. And I don't quite understand how to get the Skew matrix omega, for

I didn't find the formula in Lynch's book

thank you!