inverted pendulum

Swing low

Whenever I give my “Why we walk the way we do” lecture as I did again last week at our gait course (click here to read some of the delegates comments from the Course Evaluation Forms) it makes me think in more detail of some of the biomechanics. This time it made me think more about the swing phase pendulum. In a sense this mechanism is too obvious and we perhaps don’t think enough about it.

There is a historical perspective. The Weber brothers, writing in 1836, assumed a simple pendulum action and came up with a lot of calculations based on elementary physics to support their assumption. Shortly afterwards, however, Guillame Duchenne, noted that he had several patients with isolated flaccid paralysis of the hip flexors who had to walk with circumduction to clear their limb. This suggested an active rather than passive mechanism. It was this question that drove the amazing work of Braune and Fischer in the late 19th century. They confirmed Duchenne’s clinical observation that initiation of swing is an active process.

moments hip

In my lecture I gloss over this a bit leaving the assumption that the pendulum motion during swing is an energy conserving process. It struck me this year that an indication of whether this is the case is to look at the hip moment.  A good definition of a simple, energy conserving pendulum is one in which no moment is exerted at the pivot (hip). Looking at the hip moment  (above, taken from the notes I prepared for the course) it appears to have three phases in swing. In early swing a marked hip flexor moment is exerted accelerating the limb, through middle swing the moment is essentially zero (we do have a simple pendulum movement) and in late swing the hip extensors are active to decelerate the limb.

In a paper that I’ve only just discovered, Holt et al. (1990) do some calculations to estimate natural frequency of the lower extremity and suggest that the actually frequency of swing is about 40% higher. Doke et al. (2005) confirmed this and also showed that if someone just stands and swings their leg then energy consumption is indeed very low at this frequency and considerably higher at frequencies more normally associated with those of self-selected walking speeds.

I also seem to remember that when I’ve seen scaled up passive dynamic walking machines (which genuinely do use minimal energy through using a free pendulum mechanism, see video above) they walk much slower than we do. Given the rigour with which these guys normally do their work I suspect that they’ll have some theoretical calculations that suggest that walking could be even more efficient if we walked closer to this resonant frequency. I think the Dynamic Walking group are meeting in Zurich this week, it would be interesting to see if any of them could comment.

We thus reach the same conclusion as Braune and Fischer, that this is a forced pendulum, i.e. that it is being forced to swing considerably faster than its natural frequency. This takes energy and thus the simplistic assumption that having something that looks like a pendulum gives us movement for free is seen to be misleading. It could but it doesn’t.

It reminds us that whilst minimising energy cost is an important factor in determining how we walk it is not the only factor. Using the language of Jim Gage (2009) there are a number of attributes of walking. Energy efficiency is one of these but the dynamics of the double pendulum are also critical to at least two others, clearance in swing and appropriate step length. To understand walking we need to understand the inter-relationships between all these attributes rather than focussing on just one. Maybe one day I will!


PS It is also worth noting that it is only a simple or compound pendulum that has a natural frequency. A double pendulum, which is a better model of the swing limb, does not generally have a cyclic motion and has a period that varies somewhat from cycle to cycle. The inverted pendulum is the mechanism by which the mass of the whole body is moved forward  and is thus probably more important for the energetics of walking. It does not have a natural frequency at all.


Doke, J., Donelan, J. M., & Kuo, A. D. (2005). Mechanics and energetics of swinging the human leg. Journal of Experimental Biology, 208, 439-445.

Gage, J. R., Schwartz, M. H., Koop, S. E., & Novacheck, T. F. (2009). The identification and treatment of gait problems in cerebral palsy. London: Mac Keith Press.

Holt KG, Hamill J and Andres RO (1990). The forced harmonic oscillator as a model of human locomotion. Human Movement Science 9:55-58


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.

Inverted pendulum

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.


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.

Kinetics of the inverted pendulum

One of my first posts was about the inverted pendulum and in it I promised a follow-up that I never delivered. So here it is. I commented that for all there is a lot of talk about the inverted pendulum there is little understanding of what it is and what it’s characteristics are. I’ll focus on the kinetics today.

The graph below shows the vertical component of the ground reaction under an inverted pendulum (Anderson and Pandy, 2006). You can work out the shape the curve must have from basic physics. Early on the pendulum is rising as it moves towards the vertical. As it is does so it gains potential energy and must be losing kinetic energy. Its upward velocity is thus reducing so it has a downward acceleration (i.e. an upward deceleration). If the overall force is acting downwards then the ground reaction (up) must be less than bodyweight (down). As the pendulum moves towards its highest point along a circular arc it rises less slowly, decelerates less, so the ground reaction must get closer to bodyweight.

inv pen GRF

Once it is over the highest point it starts to lose height, and accelerate downwards. Again this requires a downwards force so again the ground reaction (up) must be less than bodyweight (down). The further the mass goes around the circular arc the more quickly it loses height, the more it accelerates, so the ground reaction must be a decreasing fraction of the ground reaction. Easy eh! Appliance of science and we can predict the curve above.

The interesting thing here is that the vertical component of the ground reaction under an inverted pendulum is always less than its own weight. The inverted pendulum may be an excellent mechanism for carrying a mass from one point to another but its a pretty hopeless one for supporting that mass. On reflection this should be obvious because the vertical component of velocity is upwards at the start and downwards at the end and thus the nett acceleration during the movement is downwards and the average force must be less than bodyweight.

If the average force is less than bodyweight then you can’t possibly have a viable walking pattern simply by stringing a series of inverted pendulums together no matter how good the drawings of the kinematics look.

There are two mechanisms by which we get over this. The first is that we use our muscles so that the the ground reaction does not just mimic the mechanics of a passive inverted pendulum. In the figure below the ground reaction is under an inverted pendulum (solid line) is plotted against Winter’s data (1991)  for the vertical component of the ground reaction (grey band) from the middle of one double support phase to the next. The characteristics double bumps of the ground reaction clearly increase the average vertical force (all forces are plotted as % bodyweight).


This isn’t the whole story however. If you look more critically at this data you will see that the average vertical component of the ground reaction under the body is still considerably less than bodyweight (about 10% less) for most of us. The peaks aren’t much higher than bodyweight and they don’t last that long. How can we walk around and not support out own bodyweight?

The answer lies in two words, “double support”. During double support the forces  under both limbs combine to exceed bodyweight. The largest total vertical component of the ground reaction is actually in mid double support when relatively modest looking ground reactions under both limbs combine (you can see a graph of this in an earlier blog). By allowing the two ground reactions to combine like this we are able to rely on an inverted pendulum like movement to move the body forwards whilst achieving an average total ground reaction equal to bodyweight – a fundamental pre-requisite of cyclic walking.

A double support phase is thus an essential requirement of a gait pattern based on an inverted pendulum. It’s interesting that modelling the body as a simple inverted pendulum leads to a prediction that double support needs to last for 15% of the gait cycle. The actual value is, of course, 10%. That’s not a bad guess for such a simple model.

I put these ideas in a slightly wider context in one of the screencasts in the series “Why we walk the way we do. The whole series is linked to on my Videos page.


Anderson, F. C., & Pandy, M. G. (2003). Individual muscle contributions to support in normal walking. Gait Posture, 17(2), 159-169.

Buczek, F. L., Cooney, K. M., Walker, M. R., Rainbow, M. J., Concha, M. C., & Sanders, J. O. (2006). Performance of an inverted pendulum model directly applied to normal human gait. Clin Biomech (Bristol, Avon), 21(3), 288-296.

Winter, D. (1991). The biomechanics and motor control of human gait: Normal, Elderly and Pathological (2nd ed.). Waterloo:: Waterloo Biomechanics.

What is an inverted pendulum?

“Inverted pendulum” is one of those terms that seems to have crept up on me over my time in biomechanics. I don’t remember it being commonly used or taught when I was a student but now it seems to be everywhere. I suspect it is one of those terms that is not understood anywhere nearly as well as it should be. I’m not aware, for instance, of any biomechanics text book that properly explains what an inverted pendulum is or what its mechanical characteristics are. This is particularly important because in mechanics the “inverted pendulum” is more often studied as a classic example of dynamics and control theory (see the Wikipedia article for example). Anyone looking at these descriptions but wanting insight into the biomechanics of walking is going to end up very confused.

An ordinary pendulum is one with the pivot at the top and the mass at the bottom. An inverted pendulum is the opposite way round. The pivot is at the bottom and the mass is on top. Fierljeppen (canal vaulting) is the best example I’ve got of an inverted pendulum (see video below). The pole rotates about its foot (at the bottom of the canal) and transports the vaulter from one side of the canal to the other. “Transports” is the key word here. The inverted pendulum is a mechanism for carrying an object form one place to another and this is how it functions during walking. The “passenger unit” as Perry would call it is carried forward by the outstretched leg as it pivots over the foot.

It should be noted that there are important differences between the two types of pendulum. The inverted pendulum only carries an object in one direction, it doesn’t swing backward and forward like the ordinary pendulum. Another difference is that the inverted pendulum does not have a characteristic frequency like an ordinary pendulum – it would be absolutely useless inside a grandfather clock.

The earliest use of the term as a model of the stance phase of walking that I am aware of was by Cavagna et al. (1976). Earlier workers have used different terms for essentially the same concept. The “compass gait” of the much aligned Saunders, Inman and Eberhart (1953) is essentially a description of the inverted pendulum. A decade later Elftman (1966) suggested that “the body moves forwards as if vaulting on a pole” and a further decade on Alexander used the term “stiff-legged gait” (1976). It is probably the more recent work of the dynamic walking group (best summarised by Kuo, 2007) that has really popularised the use of the term.

Some papers refer to Cavagna as having tested the hypothesis that the leg behaves like an inverted pendulum (e.g. Kuo, 2007, page 619). I’ve never found any evidence of this in Cavagna’s writing or anywhere else. He certainly commented that changes in kinetic and potential energy of the centre of mass correlate so that the total energy remains approximately constant throughout the gait cycle but there are an infinite number of ways this can occur without requiring an inverted pendulum mechanism (I might write more about this in a later post).

“Proving” that walking is based on the inverted pendulum is problematic in that at a very broad level it is obvious that walking involves a similar mechanism. The foot is clearly planted and the passenger unit is carried over it by the outstretched leg. On the other hand it is equally clear that the mechanism is not a simple inverted pendulum. The trunk remains upright, there is stance phase knee flexion and the pivot with the floor changes position and anatomical location through stance (Perry’s rockers). Any study attempting to establish whether stance is like an inverted pendulum will inevitably conclude that it is a bit like one but not exactly. Forming a sensible research question to “prove” the importance of this mechanism is quite a challenge.

Anderson and Pandy (2003) reported briefly on the dynamics of the inverted pendulum as a model of stance phase and Buczek and his team in more detail (2006). Both these papers are worth reading and held a couple of surprises for me but I’ll keep those for a later post.

Alexander, M. (1976). Mechanics of bipedal locomotion. In P. Davis (Ed.), Perspectives in experimental biology (pp. 493-504). Oxford: Pergamon.

Anderson, F. C., & Pandy, M. G. (2003). Individual muscle contributions to support in normal walking. Gait Posture, 17(2), 159-169.

Buczek, F. L., Cooney, K. M., Walker, M. R., Rainbow, M. J., Concha, M. C., & Sanders, J. O. (2006). Performance of an inverted pendulum model directly applied to normal human gait. Clin Biomech (Bristol, Avon), 21(3), 288-296.

Cavagna, G. A., Thys, H., & Zamboni, A. (1976). The sources of external work in level walking and running. J Physiol, 262(3), 639-657.

Elftman, H. (1966). Biomechanics of muscle with particular application to studies of gait. J Bone Joint Surg Am, 48(2), 363-377.

Kuo, A. D. (2007). The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Hum Mov Sci, 26(4), 617-656.

Saunders, J. D. M., Inman, V. T., & Eberhart, H. D. (1953). The major determinants in normal and pathological gait. Journal of Bone and Joint Surgery, 35A(3), 543-728.