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Chung-huang Yu
07-02-1996, 09:46 PM
Dear all,

Last month, I requested your help about the measurement accuracy in
anthropometic data, kinematic measurement, and ground reactions. Here
is the collective replies. I really thank those you answered my request.
Your valuable suggestions are sincerely appreciated. Since the replies
are very clear and there are few numbers given, there is no summary or
number table in this post.

However, I still need your help to give me the actual error numbers of
measurement in practice (e.g. +/- 2 N for vertical ground reaction by X
force plate, +/- 0.005kg for the shank mass by Y method, etc.). I hope
I can gather enough numbers, table them, and post them. I think this
table would be very helpful for knowing the availability and limition of
current measurement techniques.

Thanks for you help again.

---------------- Original Post ------------------------------------
Dear All,

As you already know, "inverse dynamics" is a widely used method to
estimate the (net) leg-joint moments (and forces) for human gait. The
inverse dynamic calculation is an indirct method, i.e. instead of
measuring the joint moment directly,(1) ground reaction forces,
(2)kinematic variable (postions, velocities, accelerations of segments
or leg joints), and (3) anthropometric data of individuals are measured.
>From these three differnet kinds of independent variable, the net
legjoint moments can be calculated.

Theoretically, this approach seems fine to me. But, there is a question
bothers me when we want to use the joint moments for paraplegic FES
(functional electrical stimulation) feedback control : whether or not
the estimated joint moments are accurate enough as feedback signals.
Unfortunately, after searching literature, I found little discussion
about the accuracy of estimated net joint moment by inverse dynamic
calculation.

Surely, the accuracy of estimated net joint moments by inverse dynamic
calculation is influenced by many factors. They include the measuring
device/approach for independent variables, the drift of markers (or
misaliament of goniometer, etc), soft tissues, moving axes of joints,
unaccuracy of anthropometric data, motion mode (quick or slow, etc),
quantization of sampling, properties of signal filter, etc, etc.

----- (Ignore this paragraph, if you hate equations as I do :-) ----

I managed to derive the error equations to relate the resultant errors
to independent variable, for example :

dM_k = dM_a - (dCP_s x F_k) - (dCP_s x dF_k) - (CP_s x dF_k)
+ (dCD_s x F_a) + (dCD_s x dF_a) + (CD_s x df_a)
+ (dw_s x I_s . w_s) + (w_s x dI_s . w_s) + (w_s x I_s . dw_s)
+ (dw_s x I_s . dw_s) + (dw_s x dI_s . dw_s) + (dI_s . a_s)
+ (I_s . da_s) + (dI_s . da_s)

where d= delta (error of); M= moment; F=force; CD= distance from center
of mass (CM) to Distal joint; CP= distance from CM to Proxmal joint;
_a=ankle joint; _k=knee joint; _s=shank, a=angular acceleration;
w=angular veloccity; I=momnet and product of inertia matrix;
x=cross product; . = dot product.
----- end of equation example ---

To understand the accuary of estimated joint moments, among others, it
seems necessary to find the accuracy of the systems for measruing
indepent variables. However, I have no idea in practice what the
possible accuracy of independent variables are. Please, would any one
help to give me the numbers from your experience or your measurement
systems in ...

(1) the accuracy / resolution of force plate (ground fixed or shoe
insole type) in Fx, Fy, Fz, COP (centre of pressure) / Mx, My, Mz
(2) accuracy of kinematic variables, i.e. accuracy of linear and/or
angular position, velocity, acceleration of segments and/or joints
from camara system, goniometers, accelerometer, etc.
(3) the accuracy of anthropometric data from look up table, regression
equations, volumetric measurement, etc.
(4) the moving range of "joint centre" (if it can be defined)

Furthermore, it seems to me that there is no practical way to calibrate
the calculated joint moments. Hope I am wrong and tell me please.

Sorry for long post and poor English but thanks very much for your
attention and help in advance.

P.S. I will post the collected replies and their summary in due course.

----------------- Collective Replies ------------------
--
From: Ton van den Bogert

Dear Chung-huang,

You wrote to Biomch-L regarding error propagation in inverse
dynamics, and proposed the following equation:

>dM_k = dM_a - (dCP_s x F_k) - (dCP_s x dF_k) - (CP_s x dF_k)
> + (dCD_s x F_a) + (dCD_s x dF_a) + (CD_s x df_a)
> + (dw_s x I_s . w_s) + (w_s x dI_s . w_s) + (w_s x I_s . dw_s)
> + (dw_s x I_s . dw_s) + (dw_s x dI_s . dw_s) + (dI_s . a_s)
> + (I_s . da_s) + (dI_s . da_s)

This is not entirely correct if errors are stochastic variables
(random noise), i.e. they can be positive or negative. Every
input variable will then increase the output error, so you
should never get minus-signs in front of any terms. The correct
error propagation for random errors is calculated as follows.

If X1,X2,..Xn are n input variables, with errors E1,E2,...En, and
you have a function f which describes how the output variable Y
(in your case: M_k) depends on the input:

Y = f(X1,X2,...Xn)

The error Ey in Y can be estimated by:

2 2 df 2 2 df 2 2 df 2
Ey = E1 (---) + E2 (---) + ... + En (---)
dX1 dX2 dXn

So, you need the partial derivatives of f with respect to the
input variables, which should be easy for the inverse dynamics
equations.

These calculations can become quite involved. Personally, I
prefer to use Monte-Carlo simulations to evaluate error
propagation. You use random numbers with a certain Gaussian
distribution to perturb the input data, and see what effect this
has on the output. This has to be repeated many times, and the
results analyzed statistically.

I just finished such a study on 2-D inverse dynamics, and you can
find the information on the ISB website:

http://www.kin.ucalgary.ca/isb/data/invdyn

You asked for some error estimates:

>(1) the accuracy / resolution of force plate (ground fixed or shoe
> insole type) in Fx, Fy, Fz, COP (centre of pressure) / Mx, My, Mz

Force plate: error in forces are neglegible. There is an error
in COP of several cm, but this is *not* a stochastic error! It
depends on landing position. Since the error is has opposite
sign on opposite sides of the center of the plate, the error
becomes almost stochastic (zero-mean, random error) if you
collect a large number of trials. See Bobbert & Schamhardt,
J. Biomech. 23,705-710.

>(2) accuracy of kinematic variables, i.e. accuracy of linear and/or
>angular
> position, velocity, acceleration of segments and/or joints from
> camara system, goniometers, accelerometer, etc.

Position error: is random and depends highly on field of view and
measuring technique. Try 1 mm for a standard video gait analysis
system. There's also a non-random error due to lens distortion.

>(3) the accuracy of anthropometric data from look up table, regression
> equations, volumetric measurement, etc.

No idea, but you can probably find some estimate in papers where
regression models were developed from cadaver data. The 'fit
error': E = SQRT(SUM(model-measurement)/(N-M)), where N is the number of
data points and M is the number of parameters in the regression
model, will be a good error estimate. Be aware that these are
not random errors, so use your version of the error propagation
rather than mine.

>(4) the moving range of "joint centre" (if it can be defined)

Joint centre moves with respect to external markers, due to
movement between skin and bone. Typical: 2 cm at knee and hip,
probably less at ankle.

>Furthermore, it seems to me that there is no practical way to calibrate
>the calculated joint moments. Hope I am wrong and tell me please.

You are correct, and a big problem if you want to assess accuracy
of inverse dynamics analysis methods. That's why I applied my
own error analysis to data generated by computer simulation,
where the actual joint moments are exactly known. See the web
page I mentioned before.

>P.S. I will post the collected replies and their summary in due course.

-- Ton van den Bogert
Human Performance Laboratory
The University of Calgary

-----------------------------------------------------------------------
From: D Tabakin
Date: Thu, 13 Jun 1996 11:27:00 GMT+2
Subject: Re: Accuracy of calculated joint moment by invese dynamics

On the question of anthropometric data from a look up table.
I used data from a published article by Zatsiorsky (excuse the
spelling). Unfortunately I do not have the reference with me. But if
you are interested I can send it to you.

The article describes a method of using gamma rays to calculate the
segment masses , centres of gravity and moments of inertia of the
body. As you move further away from the centre of gravity, errors
increase. Therefore if we are dealing with the whole body, when
calculating the moment of inertia of the feet, the error can be large.
However as the feet weigh much less than the trunk or other segments,
the error can be ignored. (This was relevant for my research last
year, as I was viewing the complete body. in motion.). I do not know
if it is relevant to your research but if you are interested I can
send the reference.

Good Luck

Dudley Tabakin
Wits University South Africa

-----------------------------------------------------------------------
Date: Thu, 13 Jun 1996 07:51:19 -0400 (EDT)
From: cheng cao
To: Chung-huang YU
Subject: Re: Accuracy of calculated joint moment by invese dynamics

Hi, Chung-huang:

I think it is a good point for discussion. The accuracy seems to be
influenced mostly by the approximations of anthropometry and soft tissue
as well as intermediate joint movements. For example many publications
calculated the ankle joint moments by assuming foot as a rigid body and
assuming an anthropometry table. Maybe it's fine for their purposes.
Furthermore, an assumption of a spherical joint may ignore the
translations within the joint for example.

I did a kinematic study in finding the center of of rotation of thorax
relative to pelvis. It locates approximate 10 cm above and 8 cm
anterior the first sacrum during voluntary flexion and extension.

Good luck to you.

Cheng Cao. - Department of Mechanical Engineering and Applied Mechanics
The University of Michigan, Ann Arbor. -*-

-----------------------------------------------------------------------
Date: Fri, 14 Jun 1996 10:42:56 +0100 (MET)
From: "A.L.HOF"
Subject: Re: Accuracy of calculated joint moment by invese dynamics
To: Chung-huang YU , c.yu@ucl.ac.UK

Dear Chung-Huang,
The question you posted is very relevant indeed. I suspect many
biomechanists are sleeping badly when they think of it.
If I see it well, your error formula can be simplified. Some time
ago I wrote a short Technical Note in the J. Biomechanics 25: 1209-
1211, in which the joint moment is directly expressed in the measured
variables. The (error in the) moment depends on:
1) distance between joint centre (JC) and ground reaction vector
(GRF) x GRF
2) sum(distances between JC and CoM of segments between ground and
JC x weight of segment)
3) sum(distances between JC and CoM of segments between ground and
JC x mass.acceleration)
4) sum (I.angular acceleration).
My opinions, not strongly founded on evidence but more on
prejudice, are the following:
In stance, thus when a big GRF is present, term 1 is by far the
biggest, and so is the associated error. Mind that 1 cm error in JC x
1000 N GRF = 10 Nm! Errors of several cm in knee or hip JC are hard
to avoid, I think.(Work of Cappozzo, several sources.) Errors in the
GRF vector seem also to be present (paper of vd Bogert and Schamhart
in J.Biomech, 2-3 yr ago).
Term 2) can be relatively accurate, as long as the trunk
is not among the segments.
Term 3) is rather inaccurate, because the acceleration is so
noisy, but it is much smaller than 1) as long as a ground reaction
is present. Term 4), the rotational moment, needs a lot of high
mathematics to be calculated, but as far as I can see, it is in all
practical cases relatively small.
I suppose you have the book "3-D analysis of human movement" by
Allard, Stokes and Bianchi (Human Kinetics, 1995). Woltring also has
discussed the problem in Ch 11 of "Biolocomotion: A century of
research using moving pictures" ISB series no 1 (1992).
I am very interested in your results! Good luck,

At Hof
Department of Medical Physiology
University of Groningen
Bloemsingel 10
NL-9712 KZ GRONINGEN
The Netherlands
Phone: (31) 50 3632645
Fax: (31) 50 3632751

-----------------------------------------------------------------------
Date: Thu, 13 Jun 1996 17:14:27 -0400 (EDT)
From: Louise A Gilchrist
To: c.yu@ucl.ac.uk
Subject: errors posting

In response to your BIOMCH-L post, here are some papers that you may want
to check (if you haven't already!):

Angulo and Dapena (1992). "comparison of film and video techniques for
estimating three-dimensional coordinates within a large field" Int J
Sport Biomech 8: 145-151.

Cappozzo et al. (1975). "A general computing method for the analysis of
human locomotion" J Biomechanics 23: 617-621.

Cappozzo and Gazzani (1990). "Joint kinematic assessment during physical
exercise" in Biomechanics of Human Movement. N.Berme and A. Cappozzo
(eds), Worthington, Ohio, Bertec Corporation, 263-274.

Cappozzo, A. (1991). "Three-dimensional analysis of human walking:
Experimental methods and associated artifacts." Hum Mov Sci 10: 589-602.

Chen et al. (1994). "An investigation on the accuracy of three-dimensional
space reconstruction using the direct linear transformation technique". J
Biomechanics 27(4): 493-500.

Dapena et al (1982). "Three-dimensional cinematography with control object
of unknown shape." J Biomechanics 15(1): 11-19.

DeLuzio et al (1993). "A procedure to validate three-dimensional motion
assessment systems." J Biomechanics 26(6): 753-759.

Kennedy et al (1989). "Comparison of film and video techniques for
three-dimensional DLT repredictions". Int J Sport Biomech 5: 457-460.

These don't address all the issues that you've raised but they do cover
some of them. Good luck!

Louise Gilchrist
Department of Physical Therapy and Exercise Science
SUNY at Buffalo
405 Kimball Tower
Buffalo, NY 14214
ph: 716-829-2941 ext 102
lag@acsu.buffalo.edu

----------- End of Collective Replies ----------------------------

Chung-huang YU
Medical Physics & Bioengineering,
1st floor Shropshire House,
11-20 Capper Street,
University College London,
London WC1E 6JA
tel: (44) 0171-3807777 ext 5715
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