As shown above, the neural-network based estimation of step length using step features is very accurate. This begs the question, “Why?”.
In Astrand’s classic on work physiology [24], the relationship between animal size and locomotion speed is discussed. Very generally, the frequency of limb mo-tion is inversely propormo-tional to limb length. While an important relamo-tion between preferred step length and frequency of individuals is suggested, no information is given about the exact functional relationship. The energy cost of various activities is also discussed. With regards to walking, the energy cost varies widely depending on the individual and circumstances. However, the freely chosen step rate is said to be the one that requires the least energy expenditure (i.e., oxygen uptake) at any given speed.
It is known from many decades of biomechanical research that many parameters such as
- the effective value of the vertical body accelerations - the step length
- the forward velocity change in each step, and - the stride time
are all correlated to walking speed [45, 191, 44]. In Kuo’s work [118, 119], several biomechanical models for bipedal walking are discussed and a relationship between speed and step length is proposed. A surprisingly simple “inverted pendulum” model is shown to be very accurate in terms of energy expenditure at different walking speeds [58], see Figure 3.10. In [36], it is shown that the preferred speed to step length relationship arises because humans are trying to minimize the total metabolic
Discussion
(a) Skeletal view
(b) Work at foot strike
Figure 3.10: Inverted Pendulum Walking Model, adapted from [58] and [119]
cost at a given walking speed. Using data gathered from treadmill experiments, it was demonstrated that the relationship between walking speedv and step length d proposed by Kuo
d=Cv1−b or in terms of step frequencyf,
f =Cvb
fits very accurately to the empirical data. The scalar constant C was confirmed to depend mostly on body size and can be determined for each individual through testing. The optimal value of the exponent, b = 0.58 or 1−b = 0.42, was found to be valid over a wide range of walking speeds and across individuals. Taking data gathered during walking tests with a belt-mounted sensor and doing a manual adjustment toC, a very good fit to the empirical data was obtained7, see Figure 3.11.
The fit using noisier data from head-mounted sensors would also likely be very good.
In a practical system, a linear fit using the log-transformed data could easily be automated and made to run in real time. For outlier rejection, the RANdom SAmple Consensus, or RANSAC method could be used. This would likely give superior results to those reported in [57], where an unconstrained step length/frequency look-up table was used.
Looking at the recorded data from our experiments, one can make the following observations. In Figure 3.12, a clear increase in the acceleration magnitude variance with walking speed can be seen. This is not surprising as the hips, and hence the head-mounted IMU, follow an approximate circular arc during each stride, as shown in Figure 3.10(a). For a given stride length, the acceleration minimum measured near the center of gravity or at the head will decrease with increasing speed since the circular arc will be traversed faster, resulting in a higher “centrifugal” force which is
7This work was part of a Master’s student project done under the supervision of the author [54].
(a)
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(b)
Figure 3.11: Fit of Empirical Speed/Step Length Model for Belt-mounted IMU (adapted from [54]).
measured by the accelerometers8. At the center of the stance phase, this centrifugal force is aligned with the vertical as is gravity but is working in the opposite direction.
Consequently, there is a maximum cancelation, or “unweighting”, in the middle of the stance phase. Due to the velocity change at heel strike (depicted in Figure 3.10(b)), the acceleration maximum, which is in fact experienced at the heel stike, will increase with increasing speed. The net effect is that the acceleration magnitude variance increases as walking speed is increased, which is what was observed in the experiments.
It is worth noting that, contrary to measurements made at the pelvis, head ac-celerations are not affected by the quality (i.e., hardness and irregularity) of the walking surface [146]. This is likely because the human locomotor system is try-ing to optimize the stability of the head. Consequently, the head-mounted sensors might actually yield more reliable step length estimates than their body-mounted counterparts.
It would be possible to extend the neural-network prediction model by looking at the variation of the detected features with changes in slope. The author observed over many experiments that the step features from the helmet-mounted sensor defi-nitely change with the angle of walked inclines. Extensive analysis on this data was not carried out as the expected phenomena have already been thoroughly studied by other researchers [129, 170, 22, 81].
Finally, Alvarez et al. [20] showed that there are several simple closed-form step feature to step length relationships that can be derived from the inverted pendulum
8Accelerometers sense are specific force sensors and as such detect the sum of all forces acting upon them. In force rebalance accelerometers, a small servo loop maintains a small proof mass at a null point by supplying a restoring force. The value of this restoring force is directly proportional to acceleration. In the present case, the accelerometer measures the sum of a “centrifugal” force and the gravitational force.
Discussion
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0 0.5 1 1.5 2
Measured acceleration magnitude variance (g)
Measured average speed over step (m/s)
Acceleration Magnitude Variance vs Step Speed Relationship
Rejected data Data used in model training
Figure 3.12: Step Acceleration Magnitude Variance vs Speed
model and that these have different predictive power. Kuo has also derived more complex relationships for walking energetics and dynamics with the addition of toe-off mechanics to the simple inverted pendulum model [119]. In effect, the neural-network configuration described above is simply finding a mapping between the input and output features in the training data that is very similar to the closed-form, theoritical ones. The expected smooth, continuous functional relationship (which has a large extent linear component) between stride features and walking speed is simply being learnt by the NN from the training data.
At first glance, the technique would appear to address the target application do-main requirements. However, from extensive experiments carried out by the author [34, 33], it was determined that this and similar occurrential PDR approaches are likely to provide good results only if the user’s locomotion isregular and essentially rectilinear. Step dynamics (i.e, the step time interval and the acceleration profile) change during manoeuvres, see Figure 3.14. In the author’s opinion, occurrential PDR techniques are likely to break down when the user walks an irregular, curved or halting trajectory. It is exactly this type of locomotion that will predominate when rescue personnel are "sweeping" buildings or manoeuvring through a fire, for example. A possible solution to this problem would be to build an explicit model for every possible locomotor pattern. However, this would rapidly become unwieldy and likely be prone to error. Consequently, occurrential PDR based positioning will likely perform poorly during real world tests. How to overcome this major pitfall is the subject of the next chapter.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5
Measured step frequency (Hz)
Measured average speed over step (m/s)
Empirical Step Frequency vs Step Speed Relationship
Rejected data Data used in model training
Figure 3.13: Step Frequency vs Speed