1

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 me.

Thanks!

fbl
  • 2,840
  • 3
  • 33
  • 41

1 Answers1

0

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.

Commodore63
  • 339
  • 1
  • 13
  • I know 'w' (and the components) at the start and end of the maneuver, as well as the orientation. I can use quaternions to pretty easily find the transition quaternion that gives me the angle and axis for the rotation that will get me from the start state to the end state. Where I get confused is matching the rates at the beginning and end of the maneuver. Did I miss something in your original reply? – fbl Dec 07 '10 at 01:49
  • So you know the start and end orientation. The start and end rates are given as well. Do the start and end rates have the same angle and axis? My reply above implies that they do. – Commodore63 Dec 07 '10 at 02:08
  • They do not... if they did, it would be much easier, now wouldn't it ;) – fbl Dec 07 '10 at 02:12
  • 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 me. Does that help? – fbl Dec 07 '10 at 02:20
  • So at t=0 you know the orientation, **R**_i, and rotational rate **w**_i and at t=T you know the orientation, **R**_f and rotational rate **w**_f. You want to know how to represent the orientation at each t=i*dt between 0 and T. Do we assume that angular acceleration is constant on that interval? – Commodore63 Dec 07 '10 at 02:37
  • In the final application, no. For the purposes of developing a solution and ideas right now, sure! – fbl Dec 07 '10 at 02:44