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:
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]
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/
