I encountered similar issues while attempting to solve this for a Catmull-Rom spline without access to the source code in to fetch the coefficient derivatives.

My spline was based on time parameters of 0 - 1000, but this could easily be adapted for 0 - 1.

Since the position and time do not have 1:1 ratio, tracking the difference in position of the next desired step and then adjusting the next time step accordingly gave me the uniform values I was looking for.

My basic approach was as follows:

        Vector nextPosition = spline.GetPointAtTime(mSplineTime + 1);
        Vector positionDelta = (nextPosition - position);

        float distance = positionDelta.Magnitude();

        float ratio = 1 / distance;
        splineTime = splineTime + speed * ratio;

Where the speed is a some predefined constant.

I encountered similar issues while attempting to solve this for a Catmull-Rom spline without access to the source code in to fetch the coefficient derivatives.

My spline was based on time parameters of 0 - 1000, but this could easily be adapted for 0 - 1.

Since the position and time do not have 1:1 ratio, tracking the difference in position of the next desired step and then adjusting the next time step accordingly gave me the uniform values I was looking for.

My basic approach was as follows:

        Vector nextPosition = spline.GetPointAtTime(splineTime + 1);
        Vector positionDelta = (nextPosition - position);

        float distance = positionDelta.Magnitude();

        float ratio = 1 / distance;
        splineTime = splineTime + speed * ratio;

Where the speed is some predefined constant. The object position is then set per frame by getting the amended spline time in the last line of the code.