Quaternion Interpolation w/ Rate Matching

I have an object w/ and orientation and the rotational rates about each of the body axis. I need to find a smooth transition from this state to a second state with a different set of rates. Additionally, I have constraints on how fast I can rotate/accelerate about each of the axis.

I have explored Quaternion slerp's, and while I can use them to smoothly interpolate between the states, I don't see an easy way to get the rate matching into it.

This feels like an exercise in differential equations and path planning, but I'm not sure exactly how to formulate the problem so that the algorithms that are out there can work on it.

Any suggestions for algorithms that can help solve this and/or tips on formulating the problem to work with those algorithms would be greatly appreciated.

[Edit - here is an example of the type of problem I'm working on]

Think of a gunner on a helicopter that needs to track a target as the helicopter is flying. For the sake of argument, he needs to be on the target from the time it rises over the horizon to the time it is no longer in view. The relative rate of this target is not constant, but I assume that through the aggregation of several 'rate matching' maneuvers I can approximate this tracking fairly well. I can calculate the gun orientation and tracking rates required at any point, it's just generating a profile from some discrete orientations and rates that is stumping 开发者_StackOverflow社区me.

Thanks!

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First of all your rotational rates about each axis should compose into a rotational rate vector (i.e. w = [w_x w_y w_z]^T). Then you can separate the magnitude of the rotation from the axis of the rotation. The magnitude is w_mag = w/|w|. Then the axis is the unit vector u = w/w_mag. You can then update your gross rotation by composing an incremental rotation using your favorite representation (i.e. rotation matrices, quaternions). If your starting rotation is R_0 and your incrementatl rotation is defined by R_inc(w_mag*dt, u) then you follow the following composition rules:

R_1 = R_0 * R_inc

R_k+1 = R_k * R_inc

enjoy.