Dear Tomislav and Ton,

> If your result is correct, this brings us back to the original

> question of joint power.

Let me assure you one more time. The angular velocity can be computed

from the first time-derivatives of the Cardan angles or from the

transformation matrix as I demonstrated before. They yield identical

outcomes.

> The two ways to compute joint power are 1.

> Sum of moment-angular velocity products in Cardanic (joint coordinate

> system) axis system 2. Dot product of moment and angular velocity in a

> cartesian reference frame.

Be careful in using the first method. As I tried to emphasize all along,

Pj mi.wi + mj.wj + mk.wk, [1]

where i, j, and k are the three non-orthogonal Cardanic axes, m = moment

components, and w = ang vel components. This relationship only holds

when the system is orthogonal. Rather, joint power must be computed as

Pj = M*W

= M*(Wi + Wj + Wk)

= M*Wi + M*Wj + M*Wk, [2]

where * = dot product operator, and Wi, Wj, and Wk = three angular

velocity vectors decomposed along the three Cardanic axes. Or

Wi = wi.I = da/dt.I

Wj = wj.J = db/dt.J

Wk = wk.K = dc/dt.K, [3]

where I, J, and K = unit vectors of the Cardan axes, and a, b, and c =

three Cardan angles. All these vectors (M, Wi, Wj, and Wk) must be

described in a unified orthogonal reference frame, such as the global

inertial frame or a particular local segmental reference frame. Each of

the three terms in the above equation shows how the moment vector is

associated with the three angular velocity vectors. One can easily see

that

M*Wi + M*Wj + M*Wk = mx.wx + my.wy + mz.wz, [4]

where x, y, and z = Cartesian axes. It is just a matter of how to break

down the power into meaningful terms and Eq. 4 shows two different ways.

Tomislav brought up an interesting issue: JCS. To me, a coordinate

system just means how to describe a vector into components: rectangular,

polar, cylindrical, hyperbolic, etc. What's more important in describing

motions of the body parts is the reference frame or the perspective of

description. The only entities that can carry perspectives to me are the

segments, not the joints. As the orientation of a segment changes, the

reference frame fixed to it also moves with the segment. However, the

relative relationship between the segment and the local frame does not

change. On the other hand, joint motion is simply the relative motion of

the distal segment to its linked proximal segment. The relative

orientation of the Cardanic axes (JCS) changes as the joint motion

progresses. The JCS may be used to decompose the angular velocity vector

into meaningful sub vectors, but JCS shouldn't be treated as a reference

frame (or perspective). The confusion shown in Eq. 1 originates from the

concept of JCS. In Eq. 2, the vectors can be described in any reference

frame (global or segmental). However, the perspective used in the

computation does not affect the three power terms to be obtained from

Eq. 2.

Young-Hoo

------------------------------------------------------

- Young-Hoo Kwon, Ph.D.

- Biomechanics Lab, Texas Woman's University

- kwon3d@kwon3d.com

- http://kwon3d.com

------------------------------------------------------

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> If your result is correct, this brings us back to the original

> question of joint power.

Let me assure you one more time. The angular velocity can be computed

from the first time-derivatives of the Cardan angles or from the

transformation matrix as I demonstrated before. They yield identical

outcomes.

> The two ways to compute joint power are 1.

> Sum of moment-angular velocity products in Cardanic (joint coordinate

> system) axis system 2. Dot product of moment and angular velocity in a

> cartesian reference frame.

Be careful in using the first method. As I tried to emphasize all along,

Pj mi.wi + mj.wj + mk.wk, [1]

where i, j, and k are the three non-orthogonal Cardanic axes, m = moment

components, and w = ang vel components. This relationship only holds

when the system is orthogonal. Rather, joint power must be computed as

Pj = M*W

= M*(Wi + Wj + Wk)

= M*Wi + M*Wj + M*Wk, [2]

where * = dot product operator, and Wi, Wj, and Wk = three angular

velocity vectors decomposed along the three Cardanic axes. Or

Wi = wi.I = da/dt.I

Wj = wj.J = db/dt.J

Wk = wk.K = dc/dt.K, [3]

where I, J, and K = unit vectors of the Cardan axes, and a, b, and c =

three Cardan angles. All these vectors (M, Wi, Wj, and Wk) must be

described in a unified orthogonal reference frame, such as the global

inertial frame or a particular local segmental reference frame. Each of

the three terms in the above equation shows how the moment vector is

associated with the three angular velocity vectors. One can easily see

that

M*Wi + M*Wj + M*Wk = mx.wx + my.wy + mz.wz, [4]

where x, y, and z = Cartesian axes. It is just a matter of how to break

down the power into meaningful terms and Eq. 4 shows two different ways.

Tomislav brought up an interesting issue: JCS. To me, a coordinate

system just means how to describe a vector into components: rectangular,

polar, cylindrical, hyperbolic, etc. What's more important in describing

motions of the body parts is the reference frame or the perspective of

description. The only entities that can carry perspectives to me are the

segments, not the joints. As the orientation of a segment changes, the

reference frame fixed to it also moves with the segment. However, the

relative relationship between the segment and the local frame does not

change. On the other hand, joint motion is simply the relative motion of

the distal segment to its linked proximal segment. The relative

orientation of the Cardanic axes (JCS) changes as the joint motion

progresses. The JCS may be used to decompose the angular velocity vector

into meaningful sub vectors, but JCS shouldn't be treated as a reference

frame (or perspective). The confusion shown in Eq. 1 originates from the

concept of JCS. In Eq. 2, the vectors can be described in any reference

frame (global or segmental). However, the perspective used in the

computation does not affect the three power terms to be obtained from

Eq. 2.

Young-Hoo

------------------------------------------------------

- Young-Hoo Kwon, Ph.D.

- Biomechanics Lab, Texas Woman's University

- kwon3d@kwon3d.com

- http://kwon3d.com

------------------------------------------------------

-----------------------------------------------------------------

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

-----------------------------------------------------------------