Dear all,

I post this to Biomch-L since there may be some information

of general interest in this reply.

Dong CHEN wrote:

> In my experiment, the global coordinations of four markers sticking on the

> back of the human body were sampled and stored in a computer while the

> object was walking on a treadmill. My task is to calculate the change of

> the Euler Angles in this procedure according to the global coordinations of

> the four markers.

>

> It is a over-determined problem. Because the error of the rigid body

> assumption on the human body and error in the data collection, this

> over-determined problem has no solutions !

The overdetermined data will allow you to find a least-squares

solution. On the ISB web site, you can find several algorithms

to do this either in Matlab or Fortran. Look under

"Movement Analysis" in http://www.lri.ccf.org/isb/software/

Least squares solutions are better because they are less sensitive

to errors in a single marker. And the more markers the better.

Input is marker coordinates in a segment-fixed coordinate system

and coordinates of the same markers in a laboratory coordinate

system. Output is a rotation matrix and translation vector

relating the two coordinate systems. Then you can extract Euler

angles from the rotation matrix. First you must define the

sequence of rotations and symbolically derive the equations for

all matrix elements in terms of the Euler angles you have chosen.

This is tedious, and below I have appended a Mathematica script

to automate this for the XYZ sequence, but the script is easily

modified for other sequences. After doing this,

you can derive simple expressions to compute Euler angles from

the matrix elements. Matlab routines for this can be found in

http://www.lri.ccf.org/isb/software/kinemat/ but before using those

take some time to make sure that you understand which Euler system

you are using.

Let R be defined as the rotation matrix relating the segment

coordinates to the laboratory coordinates of markers on a rigid

body:

poslab = R*posseg + translation

Then an Euler decomposition

R = Rx*Ry*Rz

means that the rotation is decomposed into rotations about the

Z-axis of the segment reference frame, the X-axis of the laboratory

reference frame, and a third ("Y") rotation about the mutual perpendicular axis.

This interpretation may help you choose a suitable Euler sequence.

To extract Euler angles from matrix elements, use the ATAN2 function

(Matlab or Fortran) as much as possible, because this gives you

a range of -180 to +180 degrees, as opposed to -90 to +90 for

ASIN or ACOS. This was not done in the Kinemat package, so this

is something you can improve upon.

For whole body orientation relative to the laboratory reference

frame, there is an ISB standardization proposal by M.R. Yeadon:

http://www.lri.ccf.org/isb/standards/yeadon1.html

This defines the Euler angles "somersault", "tilt", and "twist",

which is consistent with existing terminology used in sport.

I suggest that you use this Euler system for describing upper

body rotations.

Regards,

Ton van den Bogert

Dept. of Biomedical Engineering

Cleveland Clinic Foundation

bogert@bme.ri.ccf.org

Appendix: Mathematica script for generating symbolic rotation matrices

from Euler angles.

================================================== ===========================

SetOptions[$Output, PageWidth -> Infinity]

(* Print[ Options[$Output] ] *)

Rx = { { 1, 0, 0},

{ 0, Cos[a], -Sin[a]},

{ 0, Sin[a], Cos[a]}

}

Ry = { { Cos[b], 0, Sin[b] },

{ 0, 1, 0},

{ -Sin[b], 0, Cos[b]}

}

Rz = { { Cos[c], -Sin[c], 0},

{ Sin[c], Cos[c], 0},

{ 0, 0, 1}

}

Rxyz = Rx . Ry . Rz

Print["\n"]

Print[ MatrixForm[Rxyz] ]

Print[ TeXForm[Rxyz] ]

Print["\n"]

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