I hypothesize that the mathematics of solid body collisions could be translated to N-dimensional. Another question is that of player input. In a 3-DOF game, the player requires 3 axes or 6 discrete inputs to apply force or rotation in 3-dimensions. Adding more controls to be able to simultaneously apply force and torque in all directions would be overwhelming beyond perhaps 4 dimensions. It may be best to give the player 3-DOF but allow him/her to quickly remap which directions are controlled which which control axes.

Rigid Body Kinetics

From my old quadrotor docs:

$\dot{\mathbf{q}} = \mathbf{A}_3 \boldsymbol\omega$ $\ddot{\mathbf{q}} = \dot{\mathbf{A}}_3 \boldsymbol\omega + \mathbf{A}_3 \dot{\boldsymbol\omega}$ $\ddot{\mathbf{q}} = \dot{\mathbf{A}}_3 \boldsymbol\omega + \mathbf{A}_3 \mathbf{I}^{-1} \left( \boldsymbol\tau - \boldsymbol\omega \times \left( \mathbf{I} \boldsymbol\omega \right) \right)$

Player Input

Another question is that of player input. In a 3-DOF game, the player requires 3 axes or 6 discrete inputs to apply force or rotation in 3-dimensions. Adding more controls to be able to simultaneously apply force and torque in all directions would be overwhelming beyond perhaps 4 dimensions. It may be best to give the player 3-DOF but allow him/her to quickly remap which directions are controlled which which control axes.

Charles Rymal

My Blog

Rigid Body Kinetics

Player Input