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Dear Richard and all,
>... but think it is instructive to look at what happens at the gimbal
lock point. At this point (regardless of "true" angular velocity or
moment) a and c are not defined and thus neither are da/dt or dc/dt.
Neither can any component of M which is perpendicular to the plane in
which the three (co-planar) axes lie be represented within the
equations. The second equation thus breaks down at this point.
I agree on the fact that when the second rotation angle (b) is 90 deg,
gimbal lock occurs and you will not be able to determine a and c
separately from the transformation matrix. However, there are ways to
determine a and c at gimbal lock in the real situations. For example, in
a continuous motion you can treat the gimbal lock frames as missing and
generate interpolated angles for a and c later as long as you keep an
eye on the continuity of the orientation angles.
>Gimbal lock doesn't affect moments and angular velocities represented
conventionally about orthogonal axis system so the first equation is
still valid. If one expression is undefined at a point at which another
is valid then the two expressions cannot be equivalent.
It is a bit too much. We cannot compute a and c directly at gimbal lock
but it does not mean that the true a and c do not exist. There are the
true a and c from the motion but we simply don't know how to compute
them from the transformation matrix alone. When you find another way to
separate a from c (perhaps, the interpolation approach), it is not a
problem any more. The bottom line is that the relationship is still
correct regrdless of whether it is in the gimbal lock or not.
Young-Hoo
------------------------------------------------------
- Young-Hoo Kwon, Ph.D.
- Biomechanics Lab, Texas Woman's University
- kwon3d@kwon3d.com
- http://kwon3d.com
------------------------------------------------------
-----Original Message-----
From: Biomechanics and Movement Science listserver
[mailto:BIOMCH-L@NIC.SURFNET.NL] On Behalf Of Richard Baker
Sent: Thursday, February 05, 2004 3:40 PM
To: BIOMCH-L@NIC.SURFNET.NL
Subject: Re: [BIOMCH-L] Angular velocity vector
Ton, Young-Hoo and everyone else
At 10:37 AM 04/02/2004 -0600, you wrote:
>I still think intuitively that total power should be the same, whether
>we use dot product of the w and M vectors:
>
> P = w.M
>
>or sum the three "motor-equivalent" powers from a joint coordinate
>system:
> P = da/dt*Mi + db/dt*Mj + dc/dt*Mk.
I haven't worked through all of Young-Hoo's equations looking at this
but think it is instructive to look at what happens at the gimbal lock
point. At this point (regardless of "true" angular velocity or moment) a
and c are not defined and thus neither are da/dt or dc/dt. Neither can
any component of M which is perpendicular to the plane in which the
three (co-planar) axes lie be represented within the equations. The
second equation thus breaks down at this point.
Gimbal lock doesn't affect moments and angular velocities represented
conventionally about orthogonal axis system so the first equation is
still valid.
If one expression is undefined at a point at which another is valid then
the two expressions cannot be equivalent.
Richard
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>... but think it is instructive to look at what happens at the gimbal
lock point. At this point (regardless of "true" angular velocity or
moment) a and c are not defined and thus neither are da/dt or dc/dt.
Neither can any component of M which is perpendicular to the plane in
which the three (co-planar) axes lie be represented within the
equations. The second equation thus breaks down at this point.
I agree on the fact that when the second rotation angle (b) is 90 deg,
gimbal lock occurs and you will not be able to determine a and c
separately from the transformation matrix. However, there are ways to
determine a and c at gimbal lock in the real situations. For example, in
a continuous motion you can treat the gimbal lock frames as missing and
generate interpolated angles for a and c later as long as you keep an
eye on the continuity of the orientation angles.
>Gimbal lock doesn't affect moments and angular velocities represented
conventionally about orthogonal axis system so the first equation is
still valid. If one expression is undefined at a point at which another
is valid then the two expressions cannot be equivalent.
It is a bit too much. We cannot compute a and c directly at gimbal lock
but it does not mean that the true a and c do not exist. There are the
true a and c from the motion but we simply don't know how to compute
them from the transformation matrix alone. When you find another way to
separate a from c (perhaps, the interpolation approach), it is not a
problem any more. The bottom line is that the relationship is still
correct regrdless of whether it is in the gimbal lock or not.
Young-Hoo
------------------------------------------------------
- Young-Hoo Kwon, Ph.D.
- Biomechanics Lab, Texas Woman's University
- kwon3d@kwon3d.com
- http://kwon3d.com
------------------------------------------------------
-----Original Message-----
From: Biomechanics and Movement Science listserver
[mailto:BIOMCH-L@NIC.SURFNET.NL] On Behalf Of Richard Baker
Sent: Thursday, February 05, 2004 3:40 PM
To: BIOMCH-L@NIC.SURFNET.NL
Subject: Re: [BIOMCH-L] Angular velocity vector
Ton, Young-Hoo and everyone else
At 10:37 AM 04/02/2004 -0600, you wrote:
>I still think intuitively that total power should be the same, whether
>we use dot product of the w and M vectors:
>
> P = w.M
>
>or sum the three "motor-equivalent" powers from a joint coordinate
>system:
> P = da/dt*Mi + db/dt*Mj + dc/dt*Mk.
I haven't worked through all of Young-Hoo's equations looking at this
but think it is instructive to look at what happens at the gimbal lock
point. At this point (regardless of "true" angular velocity or moment) a
and c are not defined and thus neither are da/dt or dc/dt. Neither can
any component of M which is perpendicular to the plane in which the
three (co-planar) axes lie be represented within the equations. The
second equation thus breaks down at this point.
Gimbal lock doesn't affect moments and angular velocities represented
conventionally about orthogonal axis system so the first equation is
still valid.
If one expression is undefined at a point at which another is valid then
the two expressions cannot be equivalent.
Richard
-----------------------------------------------------------------
To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l
Please consider posting your message to the Biomch-L Web-based
Discussion Forum: http://movement-analysis.com/biomch_l
-----------------------------------------------------------------