The Segmented Leg Model

Fundamentals of the compliant Leg with Two Segments.

Running is one of the major gaits of humans and legged animals. Although, the complexity of the leg with bones, muscles and tendons is very high, the total leg function during running can be described by the simple spring-mass model. This model is also known as spring-loaded inverted pendulum (SLIP). It represents the leg force as a linear function with respect to leg compression. An interesting fact is the self-stability of the spring-mass model at medium and high running speeds, which is an important feature when running over rough terrain.

The two-segmented leg model and the basic parameters.

To build a technical application of the spring-mass model, it might be useful to set up the leg with segments and a rotational joint. The leg spring will be replaced by a torsional spring at the joint. From my own personal experience, the production of such a leg is fairly easy. The major question is here, how would the setup of the leg with segments and joints change the leg function compared to the standard spring-mass model?

In order to answer this question, a model is created where two massless segments are connected with a rotational joint and a torsional spring. Similar to the spring-mass model, the body is represented as point mass located at the hip.

The fundamental property of the usual leg during running is the spring-like behaviour with constant leg stiffness. The segmented leg model has a torsional spring with constant stiffness where the leg force is produced in conjunction with the segments when the leg is compressed. For this analysis, we concentrate on that part of the force that is directed from the foot point toward the centre of mass because this force is the only one influencing the point mass motion.

Leg force as function of leg compression $\Delta l$ and the resting angle $\beta_0$.

The theoretical analysis shows that the leg force increases continuously with leg compression, but the increase becomes flatter. The linear torsion spring generates a nonlinear leg spring function. The leg stiffness decreases with increasing leg compression. The larger the resting angle $\beta_0$ of the joint is, the more the leg spring’s nonlinearity is pronounced. Experiments with human subjects running on a treadmill showed that the knee angle is approximately 170° slightly before ground contact. Here, the two-segmented leg model reveals a very prominent nonlinearity of the leg function.

This analysis reveals the fundamental properties of a compliant segmented leg. These are the basis for further investigations on the behaviour of segmented legs during running.


Featured Paper

J. Rummel, A. Seyfarth.
Stable running with segmented legs.
The International Journal of Robotics Research, 27(8): 919-934, 2008.
DOI: 10.1177/0278364908095136