# The last post – on the inverted pendulum

I think this will be my last post focussing on the inverted pendulum. In the first I gave a brief overview and in the second I looked at the vertical component of the ground reaction. The anterior component is also very interesting (well at least I think so).

You’ll remember that the inverted pendulum is a mechanism that allows a mass (body) that has some initial kinetic energy to move in a circular arc over the pivot  (foot). Early on the centre of mass is rising, gaining potential energy and thus, in a conservative system, must be slowing down. If it is decelerating in the horizontal direction then a force must be acting in the horizontal direction to cause this. The only force acting on the mechanism in this direction is the ground reaction so it must be directed posteriorly. As the mass approaches its high point it gains height, and thus loses speed, more slowly so this force must reduce and will be zero when the mass is at its high point. After this it starts to descend, loses potential energy and must speed up. If the mass is accelerating in the horizontal direction then a force must be causing this. During this phase the horizontal component of the ground reaction must be anterior. In qualitative terms, therefore, the horizontal component of the ground reaction under a passive inverted pendulum is predicted to be the same as that under the foot during walking.

Curve in top half is vertical component and lower down is the horizontal components

The graph above shows the results of a quantitative analysis using sensible figures for mass (the dashed line shows the effect of a including a non-zero moment of inertia), leg length and initial velocity. I’ve only plotted this from middle of first double support to the middle of second double support as this is the period of the gait cycle that the inverted pendulum models.

Although (as commented on in the previous post) the vertical component of the ground reaction is quite different from the predictions of the inverted pendulum the horizontal component is nearly identical. We thus reach the conclusion that a completely passive mechanism (a stick with a weight on one end) generates almost exactly the same horizontal forces as we do when we are walking.

This is quite interesting in the context of the argument about whether the foot is “lifted off” or “pushed off” in second double support. On the basis of the horizontal component of the ground reaction it is clearly pushed off, but only to the extent that it would be if the leg was a completely passive mechanism.

It’s also interesting to think about this in the context of induced acceleration analysis. Because the underlying skeleton is unstable any induced acceleration analysis (e.g. Liu et al., 2006) will attribute the majority of the ground reaction to muscle forces. Interpreting what each muscle is doing and what clinical implications this has is quite complex. Thinking about the kinetics of the inverted pendulum, however, leads to the conclusion that the muscles are acting primarily to maintain the length of the limb and enable it to perform as an inverted pendulum would. It may be that this understanding leads to clearer clinical interpretation.

It certainly helps with the interpretation of the rather counter –intuitive finding of Liu et al. that the gluteus medius contributes to forward progression. In order for the body to move as an inverted pendulum it is important that trunk is not allowed to fall in relation to the hip and it is the gluteus medius that contributes that stability. The gluteus medius thus contributes to forwards progression by maintaining stability and allowing the passive dynamics of the inverted pendulum to do its business.

At the ankle and knee during late single support and second double support there is the added complexity of preserving the integrity of the inverted pendulum at the same time as allowing knee flexion to start in preparation for swing. Flexing of the knee alone would allow partial collapse of the pendulum but plantarflexing the ankle (reducing dorsiflexion) at the same time allows the overall length of the limb to be maintained. It is the plantarflexors that are required for this and, as might be expected, the induced acceleration shows these muscles as the primary contributors to the anterior component of the ground reaction through this period.

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Liu, M. Q., Anderson, F. C., Pandy, M. G., & Delp, S. L. (2006). Muscles that support the body also modulate forward progression during walking. J Biomech, 39(14), 2623-2630.

# Mind your language

I’m here in Cincinnati for the Gait and Clinical Movement Analysis Society Annual Meeting. Lovely sunshine makes a change from damp old Manchester.

Anyway today was pre-conference tutorial day and started with a really interesting session with  Art Kuo trying to help us understand induced acceleration analysis. He was particularly concerned to try and demystify the subject using a number of worked examples to show it is possible to get a qualitative feel for the accelerating effect that different joint torques will have on different segments.  He used these to help us understand the sometimes counter-intuitive conclusions that these analyses can lead us to. I found the approach fascinating and will go away and work through some examples myself. I’ll need to think a bit more before I commit any reflections to this blog.

Right at the end he volunteered some fascinating thoughts on terminology that I think are worth passing on immediately. He commented on how some of the terminology we use for accelerations tends to have inappropriate positive and negative connotations and that we need to be very careful that this doesn’t lead us to inappropriate conclusions.

One pair of phrases was “propulsion” and “braking”. We tend to think that propulsion is good and braking is bad but in cyclic walking this is not the case. If  we haven’t changed our speed over a complete gait cycle then, following Newton’s laws, we will have propulsive and braking forces that match exactly (or  more technically propulsive and braking impulses match). All that increasing the propulsive forces does is require an increased demand for braking forces to be applied. To understand how we walk the way we do we really need to have a more nuanced understanding of why braking and propulsive forces are required at all. I agree with Art that using words that suggest that one is beneficial and the other detrimental is not useful.

The other pair was “support” and “falling” (or equivalent ). Again joint torques that apply an upwards (supporting) force to the centre of mass are generally considered to be good whereas those that accelerate the body downwards are considered bad. Again, however, if walking is cyclic then there is no net acceleration of the centre of mass in either direction. I’m less sold on this argument as there is a requirement for the upward forces to average bodyweight over the gait cycle and thus I think there is a sense in which the support mechanisms are more important than those that allow downward accelerations – but I do agree with Art again that if the body accelerates upwards in one part of the gait cycle it must fall in another. Considering one of these as good and the other as bad is not likely to help our understanding.

What Art didn’t propose was alternative words that don’t have these associations. Anyone any ideas?