Computing Euler Angles from MoCap Markers 3D Data

I have obtained 3D (x,y,z) data from a Motion Capture (MoCap) system using markers placed on the subject's head and chin. The data is structured in the following format. I need to compute the Euler angles (yaw, pitch, and roll). Any assistanse in this matter would be greatly appreciated.

headFL = [data.HeadFL_X, data.HeadFL_Y, data.HeadFL_Z];

headFR = [data.HeadFR_X, data.HeadFR_Y, data.HeadFR_Z];

headBL = [data.HeadBL_X, data.HeadBL_Y, data.HeadBL_Z];

headBR = [data.HeadBR_X, data.HeadBR_Y, data.HeadBR_Z];

chin = [data.Chin_X, data.Chin_Y, data.Chin_Z];

headFL = [-97.507,-105.897,1594.74]

headFR = [-186.455,-9.66,1609.556]

headBL = [20.236,-15.579,1544.697]

headBR = [-86.745, 98.811,1557.386]

Chin = [-169.193,-74.004,1431.651]

Answer 1

[Simar]

Hi Allah,

I understand that you have collected 3D Spatial data from a MoCap system and want to compute the Euler Angles (yaw, pitch, and roll)

Following is a simplified example for calculating Euler angles. The data originates from markers positioned on specified locations around the head: front-left (FL), front-right (FR), back-left (BL), back-right (BR), and chin. Objective is to determine orientation of head using these angles. Given marker positions:

• headFL = ([-97.507, -105.897, 1594.74])
• headFR = ([-186.455, -9.66, 1609.556])
• headBL = ([20.236, -15.579, 1544.697])
• headBR = ([-86.745, 98.811, 1557.386])
• Chin = ([-169.193, -74.004, 1431.651])

Step 1: Define Vectors

First, compute vectors that define the head's orientation:

• Forward Vector ((F)): Points from the back to the front of the head.
• Upward Vector ((U)): Points from the chin upwards.
• Rightward Vector ((R)): Orthogonal to both (F) and (U), pointing from the head's left to right.

Step 2: Calculate Basis Vectors

Using the given marker positions, calculate the basis vectors:

% Define marker positions 
headFL = [-97.507, -105.897, 1594.74]; 
headFR = [-186.455, -9.66, 1609.556]; 
headBL = [20.236, -15.579, 1544.697]; 
headBR = [-86.745, 98.811, 1557.386]; 
chin = [-169.193, -74.004, 1431.651]; 
% Calculate midpoints for simplicity 
headFront = (headFL + headFR) / 2; 
headBack = (headBL + headBR) / 2; 
headTop = (headFL + headFR + headBL + headBR) / 4; 
% Forward vector (F) 
F = headFront - headBack; 
F = F / norm(F); % Normalize 
% Upward vector (U) 
U = headTop - chin; 
U = U / norm(U); % Normalize 
% Rightward vector (R) - orthogonal to F and U 
R = cross(U, F); % Cross product 
R = R / norm(R); % Normalize 
% Adjust U for orthogonality 
U = cross(F, R); 
U = U / norm(U); 

Step 3: Compute Euler Angles

With basis vectors defined, calculate Euler angles. Assuming ZYX rotation order (yaw around Z, pitch around Y, roll around X):

% Yaw (around Z) 
yaw = atan2(R(2), R(1)); 
% Pitch (around Y) 
pitch = atan2(-F(3), sqrt(F(1)^2 + F(2)^2)); 
% Roll (around X) 
roll = atan2(U(3), sqrt(U(1)^2 + U(2)^2)); 

Step 4: Convert to Degrees

Finally, convert the angles from radians to degrees for easier interpretation:

yaw_deg = rad2deg(yaw); 
pitch_deg = rad2deg(pitch); 
roll_deg = rad2deg(roll); 
% Display the results 
fprintf('Yaw: %f degrees\n', yaw_deg); 

Yaw: -48.267532 degrees

fprintf('Pitch: %f degrees\n', pitch_deg); 

Pitch: -19.133081 degrees

fprintf('Roll: %f degrees\n', roll_deg); 

Roll: 70.752575 degrees

The example demonstrates calculating Euler angles representing orientation of a head model based on MoCap data. The angles provide insight into how the head is oriented in space, which can be crucial for various applications, from animation to biomechanical analyses.

Please refer to the following documentation links-

• rad2deg- https://www.mathworks.com/help/matlab/ref/rad2deg.html?searchHighlight=rad2deg&s_tid=srchtitle_support_results_1_rad2deg
• atan2- https://www.mathworks.com/help/matlab/ref/atan2.html?searchHighlight=atan2&s_tid=srchtitle_support_results_1_atan2
• cross- https://www.mathworks.com/help/matlab/ref/cross.html?searchHighlight=cross&s_tid=srchtitle_support_results_1_cross

Hope it helps!

Best Regards,

Simar