Analysis of the limb function - Treadmill experiments

3.6 Treadmill experiments

3.6.4 Analysis of the limb function

It is easy enough to represent the dynamics of gait over a small time period by fitting a Fourier series to the limb angles. We assume an arbitrary, constant subject velocity over each experimental sample period. There are then a number of important questions that arise, which may preclude the use of gait as a suitable biometric for identification at a distance.

• How can we compare two reconstructed limb angle functions from a single subject, taken at two different times?

• Though we may be able to compare two similar gaits, is it possible to match gait motion across a range of different speeds?

• Is there an underlying biometric motion function that is unique to the individual, which we can subtly alter through a number of parameter modifiers in order to generate the range of possible gait motions?

• Clearly gait is not a simple one to one function, since the range of gait speeds can be achieved by varying both cadence and stride length. How many factors, both bio-mechanical and psychological, influence the total pattern of gait motion?

Let the Fourier series representation of the limb angle function θ(t) be defined by the equation: θ(t) =a0+ n X k=1 akcos(2πkf0t+φk) (3.14)

A time shifted signalθ(t₋ts) only updates the coefficients of phase within the Fourier

θ(t₋ts) = a0+ n X k=1 akcos(2πkf0·(t−ts) +φk) (3.15) θ(t₋ts) = a0+ n X k=1 akcos(2πkf0t+ψk) (3.16) ψk = φk−2πkf0ts (3.17)

It is unclear which features of gait represent the origin pose position within a gait cycle. It is then natural to align the Fourier signals by computing the time shift that zeros the first coefficient of phase ψ1 = 0, i.e. ts=φ1/2πf0. The normalized set of phase coefficients ψk can then be written:

ψk=φk−kφ1 (3.18)

The normalized set of coefficientsψkare then restricted to lie within the range (−π:π)

by finding the suitable corresponding set of angles. Transformation of the captured motion dynamics into a consistent coordinate framework then allows us to compare two representations of limb motion taken at different times. Other researchers use prominent features of gait itself, such as the heel strike with the floor, to determine the onset of the gait cycle. This entails looking for key frames within the gait sequence to determine the required position. The disadvantage of these methods is that they are often prone to the effects of noise and occlusion. The zero phase alignment method uses a simple property of the Fourier series to determine the start of the gait cycle.

A change in walking speed affects the set of computed Fourier coefficients, corresponding to the limb angle function θ(t) of a test subject. Most notably, the cadence of gait is proportionally similar to the fundamental frequency term f0 of the Fourier function. As subjects increase their speed, the rate of reciprocal foot contact with the floor also increases. Subjects also lengthen their stride to increase their speed. An increase in stride length denotes an increase in limb swing amplitude, which is proportional to the first harmonic Fourier amplitude term a1. These two factors, rate of foot contact with the floor (_∼ f0) and subject stride length (∼ a1), can be simultaneously altered to achieve the desired progressional velocity.

We examine the relationship between controlled walking speed vx, the fundamental

frequency f0 and amplitude coefficients a1 for walking motions that the test subjects report as natural. Figure 3.24 shows the result of subject motion at different walking speeds on the fundamental frequency termf0 of the Fourier reconstruction function.

Figure 3.24:_{Change in fundamental frequency} _f0 for each of the four test subjects

over a range of walking speeds.

We can see that the relationship between fundamental frequencyf0 and walking speed is approximately linear. The gradients and offsets corresponding to each of the linear plots differ between subjects, and similarly correspond to the results of cadence / speed within [103, 102, 5, 6]. The mapping between walking speed, cadence and stride length is influenced by the size of a subject’s legs. Subjects with smaller legs need higher cadences to achieve the required walking speeds, thus accounting for the different line offsets between subject plots.

Figures 3.25 and 3.26 show the resulting behaviour of subject upper and lower leg motion at different walking speeds on thea1 terms of the Fourier reconstruction functionsθ(t). The results for all four subjects are plotted, and show a linear trend with increasing walking speed. The results for subject 01 are significantly different to the others. As previously indicated, poor reconstruction for the trend curve corresponding to subject 01 can be accounted for by the misalignment of marker positions. However, it is interesting to see how much of a difference marker position can make to the reconstruction trend. It may then be worth investigating the potential sensitivity that marker placement has on the accuracy of reconstruction.

The amplitude plots for the lower leg segments shown in figure 3.26 are similar, most notably in initial offset. This may indicate that the lower leg arc of motion is more constrained and similar between people. On the other hand, the amplitude plots for the upper leg segments shown in figure 3.25 are significantly different in initial offset. We can attribute these differences in magnitude offset between subjects to the variation in the size of their limbs.

The apparent natural coupling between walking speed and cadence/stride is evident in the differing line gradients within both the fundamental frequencyf0 and amplitudea1 plots. The trends for both cadence and stride length are both linear, thus we can make the first order approximation that all limb angle reconstructions θ(t) are similar, though have different temporal and angular scalings that are dependent on the speed

Figure 3.25:_{Change in upper leg fundamental amplitude}_a1 for each of the four test

subjects over a range of walking speeds.

Figure 3.26: _{Change in lower leg fundamental amplitude}_a1for each of the four test

subjects over a range of walking speeds.

and mode of walking motion. This allows us to make the reconstructed limb angle functions invariant to walking speed by applying scalings that map the fundamental frequency f0 and amplitude coefficients a1, in both articulated leg segments, to unity. The corresponding set of normalized amplitudes bk are given by bk = ak/a1. The modified Fourier series representation of the original limb angle function θ(t) can then be written. θ(t) = a0+a1cos(2πf0·(t+ts)) + a1· n X k=2 bkcos(2πkf0·(t+ts) +ψk) (3.19)

The set of normalized coefficientsv_e= (b2, ψ2,· · · , bn, ψn)⊤then form the basis for a bio-

metric parameter vector. The remaining parametersw_e = (f0, a0, a1, ts)⊤of the modified

The normalized limb angle functionθe(t) formed from the set of biometric parameters _ev alone describes the unique underlying limb dynamics of gait motion, and is invariant to initial subject position, stride length and cadence.

e θ(t) = cos(2πt) + n X k=2 bkcos(2πkt+ψk) (3.20)

Figures 3.27 and 3.28 show the reconstructed normalized leg angle functions θe(t) corresponding to subject 00 over a range of walking speeds. The set of reconstructed plots are almost identical, even though the captured image sequences correspond to subject motion at different speeds, with different initial poses.

Figure 3.27: _{Reconstructed normalized upper leg angle function}_{θ(t) corresponding}e

to subject 00.

Figure 3.28:_{Reconstructed normalized lower leg angle function}e_{θ(t) corresponding to}

subject 00.

Figures 3.29 to 3.32, corresponding to all four test subjects, show the original amplitudes, normalized amplitudes and aligned phases for the captured gait dynamics of upper and lower legs over a range of walking speeds.

We can clearly see the linear trend corresponding to amplitude / speed changes within the original amplitude plots. The set of amplitude harmonics have an exponential trend,

(a) Upper leg amplitudes :ak (b) Lower leg amplitudes : ak

(e) Upper leg aligned phases : ψk (f) Lower leg aligned phases : ψk

Figure 3.29: _{Subject 00: Reconstructed leg angle functions for amplitude, normalized}

amplitude and aligned phase plots of both upper (a,c,e) and lower legs (b,d,f) at a number of different walking speeds.

(a) Upper leg amplitudes :ak (b) Lower leg amplitudes : ak

(e) Upper leg aligned phases : ψk (f) Lower leg aligned phases : ψk

Figure 3.30: _{Subject 01: Reconstructed leg angle functions for amplitude, normalized}

amplitude and aligned phase plots of both upper (a,c,e) and lower legs (b,d,f) at a number of different walking speeds.

(a) Upper leg amplitudes :ak (b) Lower leg amplitudes : ak

(e) Upper leg aligned phases : ψk (f) Lower leg aligned phases : ψk

Figure 3.31: _{Subject 02: Reconstructed leg angle functions for amplitude, normalized}

amplitude and aligned phase plots of both upper (a,c,e) and lower legs (b,d,f) at a number of different walking speeds.

(a) Upper leg amplitudes :ak (b) Lower leg amplitudes : ak

(e) Upper leg aligned phases : ψk (f) Lower leg aligned phases : ψk

Figure 3.32: _{Subject 03: Reconstructed leg angle functions for amplitude, normalized}

amplitude and aligned phase plots of both upper (a,c,e) and lower legs (b,d,f) at a number of different walking speeds.

such that the higher order coefficient magnitudes are comparatively smaller than the fundamental. The detail within the higher order harmonics is not visible within the linear scale shown. Figure 3.33 shows the set of normalized Fourier components plotted for each of the test subjects, with a logarithmic scale. We can clearly see the poor reconstruction for subject 01, since the second normalized harmonic terms b2 for the upper leg are substantially different across all walking speeds. Reconstruction of the lower leg angle function is though reasonable, leading us to believe that the knee joint marker may be placed similarly to that shown in figure 3.23.

The plots for the other subjects approximate a log-linear relationship, illustrated by the straight line trends within the logarithmic plots in figure 3.33. The uncertainty within the higher order harmonics becomes more apparent as k increases. This first order approximation for the dynamics of gait motion over the range of walking speeds gives us a fairly accurate feature vector for the second harmonic normalized amplitude and phase terms (b2, ψ2)⊤. The remaining coefficients are less accurate, leading us to believe that the relationship between leg functions over different walking speeds is more complicated than just a simple scaling of a baseline waveform.

In document Pose Invariant Gait Analysis And Reconstruction (Page 102-111)