- #1
yahastu
- 79
- 7
In 1-dimensions, the final orientation after undergoing constant angular acceleration is:
[tex]\theta_f = \theta_0 + \omega_0 t + 0.5 \alpha t^2[/tex]
What is an analogous equation in the 3D case?
In other words, if I know the initial orientation (say, as a quaternion), angular velocity (as a 3-vector, representing axis and speed), and constant angular acceleration (as a 3-vector, about a different axis), what is the final orientation of an object (as a quaternion)?
Note: if there is a more elegant way to represent the problem (eg, not using quaternions) then that is fine. I'm really just looking for the most elegant/direct way to compute the final orientation with a closed form expression.
EDIT: I'm thinking the below might be a solution, though I haven't verified it yet...
[tex]Q_f = Q_0 + \frac{t}{2} Q_{\omega} Q_0 + \frac{t^2}{4} Q_{\alpha} Q_0[/tex]
where [tex]Q_0[/tex] is the initial orientation expressed as a quaternion, [tex]Q_{\omega} = (0, \omega)[/tex] is a non-normalized quaternion representation of the angular velocity [tex]\omega[/tex] expressed as axis multiplied by rate of spin, and [tex]Q_{\alpha} = (0, \Delta \omega)[/tex] is similarly defined as a quaternion representation of angular acceleration.
Can anyone confirm or deny this?
[tex]\theta_f = \theta_0 + \omega_0 t + 0.5 \alpha t^2[/tex]
What is an analogous equation in the 3D case?
In other words, if I know the initial orientation (say, as a quaternion), angular velocity (as a 3-vector, representing axis and speed), and constant angular acceleration (as a 3-vector, about a different axis), what is the final orientation of an object (as a quaternion)?
Note: if there is a more elegant way to represent the problem (eg, not using quaternions) then that is fine. I'm really just looking for the most elegant/direct way to compute the final orientation with a closed form expression.
EDIT: I'm thinking the below might be a solution, though I haven't verified it yet...
[tex]Q_f = Q_0 + \frac{t}{2} Q_{\omega} Q_0 + \frac{t^2}{4} Q_{\alpha} Q_0[/tex]
where [tex]Q_0[/tex] is the initial orientation expressed as a quaternion, [tex]Q_{\omega} = (0, \omega)[/tex] is a non-normalized quaternion representation of the angular velocity [tex]\omega[/tex] expressed as axis multiplied by rate of spin, and [tex]Q_{\alpha} = (0, \Delta \omega)[/tex] is similarly defined as a quaternion representation of angular acceleration.
Can anyone confirm or deny this?
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