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kwon3d
10-17-2005, 04:26 AM
Dear Tomislav and all,

It was interesting to read the summary regarding why the horizontal
components of the ground reaction torque SHOULD be 0. I was giving some
thoughts to why Tomislav asked the following question in the first place:

> Last question is concerned with cases where Ty and Tx do
> normally exists and particular platform construction. For
> example Kistler platform normally outputs data from eight
> channels which are originally provided by four force sensors.
> Data from those either channels are the combined to
> ultimately give ground reaction force in all three
> directions, cop and torque in only vertical direction.
> However, if I am not mistaken, based on particular derivation
> of equations, it seems to be necessary to assume Tx=Ty=0. In
> other words, in cases where we cannot assume Tx=Ty=0 usual
> eight channel data would not allow us to compute Tx and Ty as
> well. Under those circumstances are there any additional
> force (torque) sensors needed then on platform or what?

And Frank Borg's contributions may provide the answer to my curiosity.

Let me present the symbols and equations I used again:

Symbols:
Forces measured by the sensors: F1, F2, F3, & F4
Fi = (Fix, Fiy, Fiz)
Relative positions of the sensors to the center of the surface (plate frame
origin): r1, r2, r3, & r4
Position of the CP relative to the plate frame origin: R = ( X, Y, 0 )
Torques produced by the forces about the origin: M1, M2, M3, & M4
Ground reaction torque: T
Relative positions of the sensors sto the CP: R1, R2, R3, & R4
Ri = ( xi, yi, zi )
Ground reaction force (total force): F = F1 + F2 + F3 + F4
Resultant torque about the plate origin: M = M1 + M2 + M3 + M4

So:

ri = R + Ri [1]

M = r1 x F1 + r2 x F2 + r3 x F3 + r4 x F4
= (R + R1) x F1 + (R + R2) x F2 + (R + R3) x F3 + (R + R4) x F4
= R x (F1 + F2 + F3 + F4) + ( R1 x F1 + R2 x F2 + R3 x F3 + R4 x F4)
= R x F + T [2]

T = R1 x F1 + R2 x F2 + R3 x F3 + R4 x F4 [3]

As I stated previously, T is the resultant of the torques produced by the
four forces about the CP.

If I further run down [3]:

T = (x1, y1, z1) x (F1z, F1y, F1z) + ....
= (y1*F1z - z1*F1y, z1*F1x - x1*F1z, x1*F1y - y1*F1x) + ....

Therefore:

Tx = ( y1*F1z + y2*F2z + y3*F3z + y4*F4z ) - ( z1*F1y + z2*F2y + z3*F3y +
z4*F4y ) [4]
Ty = -( x1*F1z + x2*F2z + x3*F3z + x4*F4z ) + ( z1*F1x + z2*F2x + z3*F3x +
z4*F4x ) [5]
Tz = ( x1*F1y + x2*F2y + x3*F3y + x4*F4y ) - ( y1*F1x + y2*F2x + y3*F3x +
y4*F4x ) [6]

We define the CP in such a way that Tx = Ty = 0:

( y1*F1z + y2*F2z + y3*F3z + y4*F4z ) - ( z1*F1y + z2*F2y + z3*F3y + z4*F4y
) = 0 [7]
-( x1*F1z + x2*F2z + x3*F3z + x4*F4z ) + ( z1*F1x + z2*F2x + z3*F3x + z4*F4x
) = 0 [8]

>From [7] and [8], it is clear that the CP is NOT DETERMINED SOLELY by the
Fz's. Or CP is not the balance point for the Fz's only. This misconception
commonly occurs because people tend to think that the four forces were
measured at the surface, which is obviously not the case. This could be the
source of Tomislav's confusion. If one defines CP as the balance point for
the Fz's only, [7] and [8] certainly seem not to be equal to 0 and you need
to know (xi, yi) to compute Tx and Ty.

For the Kistler plates, we know the exact locations of the sensors: (+/-a,
+/-b, c).

If 'i' in Ri stands for the quadrant number that it is located (i.e. sensor
1 is in the first quadrant of the XY plane of the plate reference frame):

r1 = ( x1, y1, z1 ) = ( a, b, c ) [9]
r2 = ( -a, b, c ) [10]
r3 = ( -a, -b, c) [11]
r4 = ( a, -b, c) [12]

where c < 0.

If we plug [9] - [12] into [2]:

Mx = b*( F1z + F2z - F3z - F4z ) - c*Fy
= Y*Fz - 0*Fy [13]
My = -a*(F1z - F2z - F3z + F4z ) + c*Fx
= 0*Fx - X*Fz [14]

>From [13] and [14]:

X = [ a*(F1z - F2z - F3z + F4z ) - c*Fx ] / Fz [15]
Y = [ b*( F1z + F2z - F3z - F4z ) - c*Fy ] / Fz [16]

[15] and [16] are slightly different from what Frank presented. Frank's
equations did not include 'c*Fx' and 'c*Fy'. Since forces F1, F2, F3, and F4
were not measured at the surface, these terms should be included in the
computation of X and Y.

Again, from [2]:

Mz = a*( Fy1 - Fy2 - Fy3 + Fy4 ) - b*( Fx1 + Fx2 - Fx3 - Fx4 ) = X*Fy - Y*Fx
+ Tz [17]

Therefore,

Tz = a*( Fy14 - Fy23 ) - b*( Fx12 - Fx34 ) - X*Fy + Y*Fx [18]

where Fy14 = Fy1 + Fy4, etc.

>From [15] to [18], it is obvious that one needs to know 8 pieces of
information (Fx12, Fx34, Fy14, Fy23, Fz1, Fz2, Fz3, and Fz4) to compute X,
Y, and Tz, and this is why Kistler plates have 8 channels.

With regards,

Young-Hoo
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Young-Hoo Kwon, Ph.D.
Biomechanics Laboratory
Texas Woman's University
Phone & Fax: (940) 898-2598
ykwon@mail.twu.edu / kwon3d@kwon3d.com
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