% MATRIX2QUATERNION - Homogeneous matrix to quaternion % % Converts 4x4 homogeneous rotation matrix to quaternion % % Usage: Q = matrix2quaternion(T) % % Argument: T - 4x4 Homogeneous transformation matrix % Returns: Q - a quaternion in the form [w, xi, yj, zk] % % See Also QUATERNION2MATRIX % Copyright (c) 2008 Peter Kovesi % School of Computer Science & Software Engineering % The University of Western Australia % pk at csse uwa edu au % http://www.csse.uwa.edu.au/ % % Permission is hereby granted, free of charge, to any person obtaining a copy % of this software and associated documentation files (the "Software"), to deal % in the Software without restriction, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % The Software is provided "as is", without warranty of any kind. function Q = matrix2quaternion(T) % This code follows the implementation suggested by Hartley and Zisserman R = T(1:3, 1:3); % Extract rotation part of T % Find rotation axis as the eigenvector having unit eigenvalue % Solve (R-I)v = 0; [v,d] = eig(R-eye(3)); % The following code assumes the eigenvalues returned are not necessarily % sorted by size. This may be overcautious on my part. d = diag(abs(d)); % Extract eigenvalues [s, ind] = sort(d); % Find index of smallest one if d(ind(1)) > 0.001 % Hopefully it is close to 0 warning('Rotation matrix is dubious'); end axis = v(:,ind(1)); % Extract appropriate eigenvector if abs(norm(axis) - 1) > .0001 % Debug warning('non unit rotation axis'); end % Now determine the rotation angle twocostheta = trace(R)-1; twosinthetav = [R(3,2)-R(2,3), R(1,3)-R(3,1), R(2,1)-R(1,2)]'; twosintheta = axis'*twosinthetav; theta = atan2(twosintheta, twocostheta); Q = [cos(theta/2); axis*sin(theta/2)];