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Rodrigues Parameters from Rotation Matrix
In this example we derive Rodrigues parameters/Gibbs Vector from the rotation matrix elements. The Rodrigues parameters have a singularity at 180 deg and use is limited for principal rotations which are less than 180 deg.
clc; clear all; % Rotation matrix Rm = [0.36 0.48 -0.8; -0.8 0.6 0; 0.48 0.64 0.6 ]; g1 = (Rm(2,3)- Rm(3,2))/(1 + trace(Rm)); g2 = (Rm(3,1)- Rm(1,3))/(1 + trace(Rm)); g3 = (Rm(1,2)- Rm(2,1))/(1 + trace(Rm)); G = [g1 g2 g3]'
G =
-0.2500
0.5000
0.5000
Euler rotation example, Rotation matrix, Quaternion, Euler Axis and Principal Angle
A classical Euler rotation involves first a rotation about e3 axis, then one about the e1 axis and finally a rotation about the e3 axis.
% Compute the elements of rotation matrix R
clc;
psi = pi/4;
theta = pi/4;
fi = pi/6;
R3_psi = [cos(psi) sin(psi) 0;
-sin(psi) cos(psi) 0;
0 0 1]
R1_theta = [1 0 0;
0 cos(theta) sin(theta);
0 -sin(theta) cos(theta)]
R3_fi = [cos(fi) sin(fi) 0;
-sin(fi) cos(fi) 0;
0 0 1]
R3_psi =
0.7071 0.7071 0
-0.7071 0.7071 0
0 0 1.0000
R1_theta =
1.0000 0 0
0 0.7071 0.7071
0 -0.7071 0.7071
R3_fi =
0.8660 0.5000 0
-0.5000 0.8660 0
0 0 1.0000
Multiplying each rotation matrix
R = R3_fi*R1_theta*R3_psi
R =
0.3624 0.8624 0.3536
-0.7866 0.0795 0.6124
0.5000 -0.5000 0.7071
From Rotation Matrix Compute Quaternion
q4 = 0.5*(1 + R(1,1)+ R(2,2) + R(3,3))^0.5; q1 = (R(2,3) - R(3,2))/(4*q4); q2 = (R(3,1) - R(1,3))/(4*q4); q3 = (R(1,2) - R(2,1))/(4*q4); q = [q1 q2 q3 q4] norm_q = norm(q) % Checking that the norm of q = 1 qv = [q1 q2 q3];
q =
0.3794 0.0500 0.5624 0.7330
norm_q =
1.0000
Compute Euler axis and its components along the axis E1,E2,E3 unit vector. Compute the Euler principle angle.
Euler_angle = 2*acos(q4)*180/pi % [deg] Euler_axis = qv/sind(Euler_angle/2) % [E1,E2,E3] norm_E = norm(Euler_axis)
Euler_angle =
85.7293
Euler_axis =
0.5577 0.0734 0.8268
norm_E =
1.0000
Small Satellites 