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.

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3 comments

  1. Thank you very much.. I was struggling with this term as well. after reading your explanation, it makes sense to me now 🙂

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