0
$\begingroup$

Lets say I have the a set of points $P = \{p_1, p_2, ...\}, p_i \in R^3$ that change position with time. These points are part of a rigid body and I record these positions in order to estimate its properties. Also, I know the linear velocity and acceleration of these points.

I was able to estimate the transformation matrix $T$ between two consequent time instances $t_1, t_2$, that best maps $P_i(t_2) = T P_i(t_1), \forall i$, meaning I know the orientation of the body. Now I need to estimate the angular velocity and angular acceleration with respect to some point $O$.

What are my options? I want to avoid numerical differentiation if possible.

$\endgroup$
0
$\begingroup$

If you know that your transform was caused by a rotation, then what you're looking at is a class Aerospace engineering problem called as Wahba's Problem which comes up in ADCS (attitude determination and control systems)

Wahba's problem asks for an orthogonal matrix $T$ (orthogonal because they represent rotations in 3D), that best minises the error function

$$ \delta \theta = \sum_{i=1}^{n}|P_i(t + 1) - w_i T P_i(t)| $$

where $P_i(t)$ are unit reference vectors, $w_i$ are their weights that reflect our "belief" in the correctness of the $P_i(t)$

The "best" algorithm that I am aware of is QUEST (QUaternion ESTimation). Here is a link to an explanation and historical overview of the algorithm by the creator of the algorithm himself.

Note that the problem cannot have a "perfect" solution by definition, since the system is forced to be either over or under damped.

This is because to specify a rotation, we require 3 parameters (commonly thought of as Euler angles)

However, since the $P(t)$ vectors are unit vectors, they are determined by just 2 components (say $x$ and $y$. The third component is derived from $z = \sqrt{x^2 + y^2}$. Hence, one vector provides 2 data points, while 2 vectors provide 4. We can never have 3 data points (which is what the problem essentially asks for).

$\endgroup$
  • $\begingroup$ Thank you for your answer. Actually, I have solved the problem of estimating the rotation matrix through SVD. My problem now is how to estimate the angular velocity and acceleration. $\endgroup$ – crow May 7 '16 at 12:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.