View Full Version : Re: Estimation of angular velocity/acceleration from 3x3 matrices

11-12-2007, 09:44 AM
> > -----Original Message-----
> > I am interested in how to calculate angular
> > velocity/acceleration from a time serie of 3x3 rotation
> > matrices which I get from marker positions measured by a
> > Vicon motion analysis system.
> > Oliver Rettig
> > Stiftung Orthopdische Universittsklinik Heidelberg
> > Ganglabor Schlierbacher Landstr. 200a
> > 69118 Heidelberg
> > Germany

comments provided below

> > 1. First calculate time series of cardan angles with an
> > arbitray rotation order. Then you can estimate the
> > derivations of these three time series of double values by a)
> > simply calculating differences or by b) low order polynom
> > derivations (a kind of filtering and differentiating in one
> > step, different orders and techniques are possible). One
> > disadvantage of this method can be gimbal lock.

Cardan/Bryant/Euler angles are non-linear functions of
rotation matrices. Taking derivatives of non-linear
functions particularly near ill-conditioned values (gimbal
lock) will magnify noise.

> > 2. Use the formula [~omega]=3D[M'(t)][M(t)]^-1 to calculate the
> > Tensor which includes the components of the angular velocity
> > by time derivatives of each component of the rotation Matrix
> > M(t) multiplicated with the inverse of the rotations matrix.
> > For the estimation of the time derivative of the component
> > the same methods from 1. a),b) can be used. In Comparison
> > with 1. gimbal lock should be no problem.

This provides a first-order finite-difference first-derivative
estimate of angular velocity with no inherent filtering.

> > 3. You can simply estimate the velocity by calculating dot
> > products of the columns between two sequent matrices. The
> > implementation of Vicon PiG seems to do this by using
> > matrices of frames with a time distance of 0.05s by 120Hz frame rate.

This is essentially the same as method 2 above.

> > 4. Because my rotations matrices are typically based on cross
> > products of vectors between markers I can analytical
> > calculate formulars of the angular velocity as function from
> > the derivations of the marker positions.

Please see
Sommer, H. J. 1992.
Determination of First and Second Order Instant
Screw Parameters from Landmark Trajectories.
ASME J. Mechanical Design 114(2):274-282.

Similar methods have been discussed by Angeles
and Brodeur/Soutas-Little.

This method uses time derivatives of marker motion and
provides linear least-squares equations for angular velocity
and angular acceleration. Because you can use higher
order derivative estimates or splines applied directly to
raw data and not to ill-conditioned non-linear intermediate
functions, this method provides improved linear estimates.

We have recently extended the method to include
sensitivity equations, angular jerk and experimental
measurement of axode invariants.


************************************************** *****************
H.J. Sommer III, Ph.D., Professor of Mechanical Engineering
Department of Mechanical and Nuclear Engineering
The Pennsylvania State University
337 Leonhard Building, University Park, PA 16802
(814)863-8997 FAX (814)865-9693
hjs1@psu.edu http://www.mne.psu.edu/sommer