“Average” of multiple quaternions?

Question

I\'m trying to make the switch from matrices to quaternions for skeletal animation in my OpenGL program, but I\'ve encountered a problem:

Given a number of unit quat

Answer 1

Since there are different approaches here, I wrote a Matlab script to compare them. These results seem to suggest that simply averaging and normalizing the quaternions (the approach from the unity wiki, called simple_average here) might be enough for cases where the quaternions are sufficiently similar and small deviations are acceptable.

Here's the output:

everything okay, max angle offset == 9.5843
qinit to average: 0.47053 degrees
qinit to simple_average: 0.47059 degrees
average to simple_average: 0.00046228 degrees
loop implementation to matrix implementation: 3.4151e-06 degrees

And here's the code:

%% Generate random unity quaternion
rng(42); % set arbitrary seed for random number generator
M = 100;
qinit=rand(1,4) - 0.5;
qinit=qinit/norm(qinit);
Qinit=repmat(qinit,M,1);

%% apply small perturbation to the quaternions 
perturb=0.05; % 0.05 => +- 10 degrees of rotation (see angles_deg)
Q = Qinit + 2*(rand(size(Qinit)) - 0.5)*perturb;
Q = Q ./ vecnorm(Q, 2, 2); % Normalize perturbed quaternions
Q_inv = Q * diag([1 -1 -1 -1]); % calculated inverse perturbed rotations

%% Test if everything worked as expected: assert(Q2 * Q2_inv = unity)
unity = quatmultiply(Q, Q_inv);
Q_diffs = quatmultiply(Qinit, Q_inv);
angles = 2*acos(Q_diffs(:,1));
angles_deg = wrapTo180(rad2deg(angles));
if sum(sum(abs(unity - repmat([1 0 0 0], M, 1)))) > 0.0001
    disp('error, quaternion inversion failed for some reason');
else
    disp(['everything okay, max angle offset == ' num2str(max(angles_deg))])
end

%% Calculate average using matrix implementation of eigenvalues algorithm
[average,~] = eigs(transpose(Q) * Q, 1);
average = transpose(average);
diff = quatmultiply(qinit, average * diag([1 -1 -1 -1]));
diff_angle = 2*acos(diff(1));

%% Calculate average using algorithm from https://stackoverflow.com/a/29315869/1221661
average2 = quatWAvgMarkley(Q, ones(M,1));
diff2 = quatmultiply(average, average2 * diag([1 -1 -1 -1]));
diff2_angle = 2*acos(diff2(1));

%% Simply add coefficients and normalize the result
simple_average = sum(Q) / norm(sum(Q));
simple_diff = quatmultiply(qinit, simple_average * diag([1 -1 -1 -1]));
simple_diff_angle = 2*acos(simple_diff(1));
simple_to_complex = quatmultiply(simple_average, average * diag([1 -1 -1 -1]));
simple_to_complex_angle = 2*acos(simple_to_complex(1));

%% Compare results
disp(['qinit to average: ' num2str(wrapTo180(rad2deg(diff_angle))) ' degrees']);
disp(['qinit to simple_average: ' num2str(wrapTo180(rad2deg(simple_diff_angle))) ' degrees']);
disp(['average to simple_average: ' num2str(wrapTo180(rad2deg(simple_to_complex_angle))) ' degrees']);
disp(['loop implementation to matrix implementation: ' num2str(wrapTo180(rad2deg(diff2_angle))) ' degrees']);