The Virtual Pendulum Concept

Ground Reaction Forces for Stabilizing the Body

The locomotion of humans is typically a bipedal gait while the centre of mass is located close to the hip. Therefore, the human body can be defined as an inverted pendulum. From a technical point of view, the stabilisation of an inverted pendulum is a challenging control. When we take a closer look at the human body, we find the centre of mass above the hip joint. This, however, makes the system more complicated because it is quasi a doubled inverted pendulum, even more difficulty to stabilise.

At the Locomotion Lab in Jena, a concept was created, how the upper body of a human could be stabilised during the bipedal locomotion. The basic idea is to generate a leg force acting on the ground (ground reaction force) that is aligned to a virtual point above hip and centre of mass. In this way, the inverted pendulum becomes a virtual pendulum. Investigations with a spring-mass model upgraded with a rigid body demonstrated that stability in walking and running can be achieved.

Does the concept of the virtual pendulum exist in humans? If true, is it a unique feature of humans who move on two legs in contrast to many other animals?

Experiments with human subjects walking and running on an equipped treadmill revealed that the ground reaction forces intersect the body axis above the centre of mass. The force intersects more or less at an area than on an exact point. The intersections are almost always above the centre of mass. A noticeable exception is identified during the short time when the foot hits the ground. Here, the impact forces point more toward the centre of mass. This could have the effect that during the impact no torque would be applied at the body. An undesired rotation and an inefficient counteraction can be avoided before the real elastic ground contact is performed.

Ground reaction forces in running chickens and walking dogs.

Animals have a more elastic ground contact and here the similar ground reaction force can be observed. As well as in humans, the ground reaction force intersects a vertical line above the centre of mass and transforms the inverted pendulum of the animals into a stable virtual pendulum. In running chickens, who move bipedal like humans, the virtual intersection area is very prominent. Experiments with walking dogs showed that the concept of the virtual pendulum still exists but there is no single point where the force intersects the vertical line. The intersections are large above the centre of mass and the range is much wider. Thanks to the four leg, the dog can use for locomotion and stabilisation, the control of upper body motion can be performed in a very relaxed manner.

In conclusion, we can assume that a concept of a virtual pendulum for stabilising the upper body already existed before the human ancestors erected and learned to move on two legs. It is not a unique feature, however, walking on two legs was probably a main advantage to use the free hands for creative activities.

Featured Paper

H.M. Maus, S.W. Lipfert, M. Gross, J. Rummel, A. Seyfarth.
Upright human gait did not provide a major mechanical challenge for our ancestors.
Nature Communications, 1(6): 70, 2010.
DOI: 10.1038/ncomms1073
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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
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