vzatsiorsky65

01-29-2004, 02:56 AM

Dear Richard and all:

I agree with what you wrote and want to add a simple 'geometric' explanation

of what is happening when power is computed in the ICS axes. To make the

understanding easier I replaced your vector a with F (force) and vector b

with V(velocity).

In the JCS system, the axes i and k are in the plane that is orthogonal to

axis j (a floating axis). Let's introduce in this plane an axis l that is

orthogonal to axis i and consider an orthogonal system of coordinates ijl.

In this reference system, power FV can be represented as a sum of three

terms associated with the individual force-velocity projections on the

coordinate axes:

FV=fi*vi+fj*vj+fl*vl

Because power is invariant [its magnitude does not depend on the selected

(inertial) system of coordinates] and the first two terms in the above

equation are equally valid for the JCS system, the following equality is

also valid:

fl*vl= fk*vk+(fk*vi+fi*vk) (i.k)

In this equation: fl*vl is a power term representing the power associated

with the force and velocity components along the orthogonal axis l; fk*vk

is a similar term representing the power associated with the non-orthogonal

axis k, (fk*vi+fi*vk) (i.k) is the difference (DELTA) between the above

power values. Because (i.k.) is simply a cosine of angle A formed by the

axes i and k, the DELTA = 0 when k is along axis l (i.e. the system ijk is

orthogonal) and DELTA is not equal to zero in all other cases. DELTA=1 when

axes i and k are along the same direction. In the latter case the

singularity occurs.

Sincerely,

Vladimir Zatsiorsky

----- Original Message -----

From: "Richard Baker"

To:

Sent: Wednesday, January 28, 2004 5:49 PM

Subject: Re: [BIOMCH-L] More angles and powers

> Dear Jonas and all,

>

> Ton and I have had a brief exchange of notes on a sub-topic of the current

> discussion - that of whether the decomposition of the dot product a.b

> =ax.bx+ay.by+az.bz works in non-orthogonal axis system. I resorted to an

> old textbook on mathematical physics which said "no" with some fairly

heady

> math to explain. Ton's come up with the same answer but in a much more

> insightful manner:

>

> Let a and b be vectors, and i,j,k be unit vectors along the coordinate

axes

> which may be non-orthogonal. Let ai, aj, ak be the scalar components

along

> each axis. Then:

>

> a = ai.i + aj.j + ak.k

> b = bi.i + bj.j + bk.k

>

> Hence:

>

> (a.b) = ai.bi.(i.i) + ai.bj.(i.j) + ai.bk.(i.k) +

> aj.bi.(j.i) + aj.bj.(j.j) + aj.bk.(k.k) +

> ak.bi.(k.i) + ak.bj.(k.j) + ak.bk.(k.k)

>

> = ai.bi + aj.bj + ak.bk +

> (aj.bi + bj.ai).(i.j) +

> (ak.bi + bk.ai).(i.k) +

> (ak.bj + bk.aj).(j.k)

>

> In the JCS, axes 1 and 2 are orthogonal, and axis 2 and 3 are orthogonal,

> so we lose the cross terms with (i.j) and (j.k). The (i.k) term

> remains :-(. So it seems that you need to add this term if you wanted

> to compute total power from JCS angular velocities and JCS moment

> components.

>

> We are both agreed that vector relationships must hold whatever the

> co-ordinate system used (whether orthogonal or not) but the way in which

> these are calculated from the basic components will depend on the

> characteristics of the co-ordinate system.

>

> Richard

>

>

> Richard Baker

>

> Gait Analysis Service Manager, Royal Children's Hospital

> Flemington Road, Parkville, Victoria 3052

> Tel: +613 9345 5354, Fax +613 9345 5447

>

> Adjunct Associate Professor, Physiotherapy, La Trobe University

> Honorary Senior Fellow, Mecahnical and Manufacturing Engineering,

Melbourne

> University

>

> -----------------------------------------------------------------

> 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

> -----------------------------------------------------------------

>

>

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I agree with what you wrote and want to add a simple 'geometric' explanation

of what is happening when power is computed in the ICS axes. To make the

understanding easier I replaced your vector a with F (force) and vector b

with V(velocity).

In the JCS system, the axes i and k are in the plane that is orthogonal to

axis j (a floating axis). Let's introduce in this plane an axis l that is

orthogonal to axis i and consider an orthogonal system of coordinates ijl.

In this reference system, power FV can be represented as a sum of three

terms associated with the individual force-velocity projections on the

coordinate axes:

FV=fi*vi+fj*vj+fl*vl

Because power is invariant [its magnitude does not depend on the selected

(inertial) system of coordinates] and the first two terms in the above

equation are equally valid for the JCS system, the following equality is

also valid:

fl*vl= fk*vk+(fk*vi+fi*vk) (i.k)

In this equation: fl*vl is a power term representing the power associated

with the force and velocity components along the orthogonal axis l; fk*vk

is a similar term representing the power associated with the non-orthogonal

axis k, (fk*vi+fi*vk) (i.k) is the difference (DELTA) between the above

power values. Because (i.k.) is simply a cosine of angle A formed by the

axes i and k, the DELTA = 0 when k is along axis l (i.e. the system ijk is

orthogonal) and DELTA is not equal to zero in all other cases. DELTA=1 when

axes i and k are along the same direction. In the latter case the

singularity occurs.

Sincerely,

Vladimir Zatsiorsky

----- Original Message -----

From: "Richard Baker"

To:

Sent: Wednesday, January 28, 2004 5:49 PM

Subject: Re: [BIOMCH-L] More angles and powers

> Dear Jonas and all,

>

> Ton and I have had a brief exchange of notes on a sub-topic of the current

> discussion - that of whether the decomposition of the dot product a.b

> =ax.bx+ay.by+az.bz works in non-orthogonal axis system. I resorted to an

> old textbook on mathematical physics which said "no" with some fairly

heady

> math to explain. Ton's come up with the same answer but in a much more

> insightful manner:

>

> Let a and b be vectors, and i,j,k be unit vectors along the coordinate

axes

> which may be non-orthogonal. Let ai, aj, ak be the scalar components

along

> each axis. Then:

>

> a = ai.i + aj.j + ak.k

> b = bi.i + bj.j + bk.k

>

> Hence:

>

> (a.b) = ai.bi.(i.i) + ai.bj.(i.j) + ai.bk.(i.k) +

> aj.bi.(j.i) + aj.bj.(j.j) + aj.bk.(k.k) +

> ak.bi.(k.i) + ak.bj.(k.j) + ak.bk.(k.k)

>

> = ai.bi + aj.bj + ak.bk +

> (aj.bi + bj.ai).(i.j) +

> (ak.bi + bk.ai).(i.k) +

> (ak.bj + bk.aj).(j.k)

>

> In the JCS, axes 1 and 2 are orthogonal, and axis 2 and 3 are orthogonal,

> so we lose the cross terms with (i.j) and (j.k). The (i.k) term

> remains :-(. So it seems that you need to add this term if you wanted

> to compute total power from JCS angular velocities and JCS moment

> components.

>

> We are both agreed that vector relationships must hold whatever the

> co-ordinate system used (whether orthogonal or not) but the way in which

> these are calculated from the basic components will depend on the

> characteristics of the co-ordinate system.

>

> Richard

>

>

> Richard Baker

>

> Gait Analysis Service Manager, Royal Children's Hospital

> Flemington Road, Parkville, Victoria 3052

> Tel: +613 9345 5354, Fax +613 9345 5447

>

> Adjunct Associate Professor, Physiotherapy, La Trobe University

> Honorary Senior Fellow, Mecahnical and Manufacturing Engineering,

Melbourne

> University

>

> -----------------------------------------------------------------

> 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

> -----------------------------------------------------------------

>

>

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

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

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