ON OPTIMAL FILTERING FOR INVERSE DYNAMICS ANALYSIS
A.J. van den Bogert and J.J. de Koning*
Human Performance Laboratory, University of Calgary, Alberta, Canada
*Faculty of Human Movement Sciences, Vrije Universiteit Amsterdam, The Netherlands
INTRODUCTION
Inverse dynamics analysis, the determination of
intersegmental loads from movement data and ground
reaction forces (GRF), is becoming a standard tool in
gait analysis laboratories. It is well known that move-
ment data must be low-pass filtered in order to pre-
vent excessive noise in the second derivatives which
occur in inertial terms in the equations of motion.
However, low-pass filtering of kinematic data may
remove high-frequency components of the actual
movement, especially in impact movements such as
running. The effects of such filter-induced errors are
difficult to assess, since the true intersegmental forces
and moments are never known. It is the purpose of
this paper to evaluate the effect of low-pass filtering
on calculation of intersegmental loading during run-
ning, and to develop recommendations for optimal fil-
ter parameters. Data from a simulated running
movement, with known intersegmental loading, will
be used for this purpose.
METHODS
A 2-D musculoskeletal model, consisting of four rigid
segments (trunk, thigh, shank, and foot) and seven
muscles was used to simulate the support phase of
running. Details of this model are described else-
where2. A parameter optimization was carried out to
determine muscle stimulation patterns which resulted
in a realistic vertical ground reaction force with the
frequency content of running (Fig. 1). A second simu-
lation was run backwards in time, starting at heel
strike, to obtain movement data for the preceding
swing phase.
Simulated ground reaction force, point of application,
and coordinates of the joint centers were written to an
output file at 5 ms intervals. White noise (0.5 mm
RMS) was added to the kinematic data to simulate
digitization errors. The data were then optionally fil-
tered by a two-pass second order low-pass Butter-
worth filter. Four cut-off frequencies, for the three
body segments and force plate data, could be selected
independently. Filtered data were used as input for a
standard 2-D inverse dynamics analysis4, in which the
same body segment parameters were used as in the
simulation that generated the data.
In order to find the best combination of four filter cut-
off frequences, optimizations were carried out to min-
imize the root-mean-square (RMS) difference
between the inverse dynamics results and the known
intersegmental loads. These optimizations were done
separately for the intersegmental forces and the
intersegmental moments and repeated ten times with
newly generated white noise.
RESULTS
Inverse dynamics results were identical to the known
intersegmental loads, when presented with unfiltered
noise-free data at a high sampling rate. When using
the data with realistic sampling rate and noise, the
results were sensitive to the filtering procedure. Fig. 2
shows the intersegmental loads obtained with a com-
monly used filtering procedure: a 15 Hz low pass filter
for the kinematic data, and no filtering of force plate
data. Large errors in the moments occur during the
impact phase, especially at the proximal joints. Fig. 3
shows the intersegmental loads obtained by filtering
all data (force and kinematics) with a 15 Hz low pass
filter. This improved the hip and knee moments con-
siderably, but removed the impact peak from the
intersegmental forces, thereby increasing the error.
These results suggest that optimal filtering procedures
should be found, depending on the variables of inter-
est. Optimized combinations of cut-off frequencies
for intersegmental forces and moments, and the
resulting errors in all six loading variables, are pre-
sented in Table 1.
DISCUSSION
The large errors in joint moments, shown in Fig. 2,
occur only during the impact phase and are therefore
evidently not related to noise in the kinematic data.
Rather, these errors are caused by a combination of
impact peaks in the horizontal GRF and our inability
to calculate the high-frequency components of seg-
ment accelerations, which would `absorb' external
impact, with sufficient accuracy. Since low-pass filter-
ing of kinematic data cannot be avoided, the proper
method to avoid these artifacts is to filter the GRF
with a similar cut-off frequency. This effectively
removes inconsistencies between kinematics and
forces.
Several studies on inverse dynamics of running have
reported impact peaks in hip joint moments, which
may be artifacts (as in Fig. 2) of the inverse dynamics
analysis. Unfortunately, results are usually presented
in a way that these peaks are not as obvious as in our
simulated results. Either group averages were pre-
sented3 or moments were low-pass filtered1. Both
operations would reduce the amplitude of the artifact.
However, since the artifact is not random noise but
highly correlated to the movement, a systematic error
will remain. We strongly recommend therefore to be
critical of published joint moments, especially for the
hip joint, in impact activities
The results in Table 1 show that different filters must
be applied to the raw data, depending on the purpose
of the analysis. If intersegmental forces and moments
are both required, as for example in a procedure to
estimate joint contact forces, we recommend that the
inverse dynamics should be done twice: once to
obtain moments and once to obtain forces. One might
even optimize the filtering procedure for each of the
six intersegmental loading variables.
The cut-off frequencies listed in Table 1 are optimal
for this specific data set: a typical running movement,
with kinematics sampled at 200 fps and 0.5 mm noise.
Using the same set of simulated data, optimal cut-off
frequencies for other frame rates and noise levels are
easily obtained. The results of this study should not be
directly applied to movements with a different fre-
quency content.
REFERENCES
1. DeVita, P. et al., Hum. Movt Sci. 9:99-115, 1990.
2. Gerritsen, K.G.M. et al., J. Biomech. 28:661-668, 1995.
3. Simpson, K.J. et al., Int. J. Sport Biomech. 6:309-324,
1990.
4. Winter, D.A., Biomechanics of Human Movement, Wiley,
1979.
This work was funded by a grant from the Whitaker founda-
tion and by NSERC of Canada. The data files are available
at http://www.kin.ucalgary.ca/isb/data/invdyn .
Fig. 2: Inverse dynamics results after using a 15 Hz low-
pass filter for kinematics and no filter for ground reaction
forces. Known intersegmental loads are shown in grey.
Table 1: Optimized cut-off frequencies and resulting RMS
errors (average SD of 10 simulated sets of data).
Fig. 1: Rigid-body model (from Gerritsen et al., 1995)
shown at heel strike and simulated ground reaction forces.