How to correctly obtain the signs of the measurement function h(x) from the kinematics of non holonomic filter?

In the script "NHConstrainedIMUGPSFuserBase" from the Sensor Fusion and Tracking Toolbox, there is the function named "measurementFcnKinematics."

 function h = measurementFcnKinematics(~, x)
            %MEASUREMENTFCNKINEMATICS Measurement function h(x) for state vector x
            %   2 measurements from kinematic constraints
            %   [velY, velZ];
            
            q0 = x(1);
            q1 = x(2);
            q2 = x(3);
            q3 = x(4);
            gbX = x(5); %#ok<NASGU>
            gbY = x(6); %#ok<NASGU>
            gbZ = x(7); %#ok<NASGU>
            pn = x(8); %#ok<NASGU>
            pe = x(9); %#ok<NASGU>
            pd = x(10); %#ok<NASGU>
            vn = x(11);
            ve = x(12);
            vd = x(13);
            abX = x(14); %#ok<NASGU>
            abY = x(15); %#ok<NASGU>
            abZ = x(16); %#ok<NASGU>
            
            h = ...
                [
                q0*(q1*vd + q0*ve - q3*vn) + q1*(q0*vd - q1*ve + q2*vn) + q2*(q3*vd + q2*ve + q1*vn) - q3*(q3*ve - q2*vd + q0*vn);
                q0*(q0*vd - q1*ve + q2*vn) - q1*(q1*vd + q0*ve - q3*vn) + q2*(q3*ve - q2*vd + q0*vn) + q3*(q3*vd + q2*ve + q1*vn);
                ];
    end

The equations of h are described on the referenced bibliography of the insfilternonholonomic [1] Munguía, R. "A GPS-Aided Inertial Navigation System in Direct Configuration." Journal of applied research and technology. Vol. 12, Number 4, 2014, pp. 803 – 814. as follows:

Where the rotation matrix is:

I have been trying to understand the signs of the equations in MATLAB, because when I calculate them by hand, the result I obtain is as follows:

    h = ...
    [
    q0*(-q1*vd + q0*ve + q3*vn) + q1*(-q0*vd - q1*ve + q2*vn) + q2*(q3*vd + q2*ve + q1*vn) + q3*(-q3*ve + q2*vd + q0*vn);
    q0*(q0*vd + q1*ve - q2*vn) - q1*(q1*vd - q0*ve - q3*vn) + q2*(q3*ve - q2*vd - q0*vn) + q3*(q3*vd + q2*ve + q1*vn);
    ];

So, in summary, I would like to know if anyone else has noticed this difference or if they are able to obtain the same result as in MATLAB using the equations from the mentioned paper.

Thank you very much in advance.

Answer 1

[Brian Fanous]

The quaternion to rotation matrix conversion equation you've posted is for an active rotation (what the quatenrion class calls a "point" rotation).

The convention in these insfilters is to use passive (or "frame") rotations, regardless of what the original paper does. So the conversion is different.