## Relaxed Walking with Compliant Legs

### The choice of Leg Stiffness for stable and robust Walking

When planning to build a legged robot that should be able to walk, it is important to think about the general structure of the leg. So far, there exist three types of robots or models applying walking. One type is the Passive Dynamic Walker who have a rigid leg during the stance phase, no flexibility exists. The second type of robots are fully activated, which means each joint is driven by a single motor. The third category of robots have legs with less motors than joints, using some passive structures like springs to generate motion. This article addresses the latter two robot legs where the leg as a whole could act like a spring. That means, the motors of a fully actuated leg could represent the force of a spring. In an under-actuated leg the single springs could act together like a total leg spring.

How should the virtual leg stiffness be adjusted to allow for stable walking?

Periodic and stable pattern of symmetric walking with ground reaction forces.

This question was tried to be answered using the fundamental model for compliant legged walking, the bipedal spring-mass model. This model is able to show the dynamics of human walking while the major verification is the reproduction of the ground reaction forces. The bipedal spring-mass model revealed that a kind of self-stability exists in walking with compliant legs. That means, there is a simple strategy without feedback for generating stable walking, i.e. the strategy of a fixed angle of attack.

Periodic and stable pattern of asymmetric walking.

In a study at the Lauflabor Locomotion Lab in Jena, the model was investigated in order to identify all periodic walking solutions. For this study a novel methodology, i.e. the VLO return map was applied, which is describe in another article. The identified periodic solutions of walking were tested regarding stability. Local stability means that mathematically small deviations from the periodic solution would lead to come back to the solution in any time. In a mathematical sense, the stability is represented by the maximum of the absolute eigenvalues. The model does have symmetric and asymmetric walking solutions with local stability as feature. According examples of stable walking patterns are shown in the figures.

Local stability of walking solutions.

The local stability of a walking solution is not enough for the robots success as it describes the behaviour when mathematically small disturbances occurred. It is more helpful if the compliant leg and the controller are able to compensate realistic or larger disturbances. For that reason, the basin of attraction of a stable walking solution is analysed, which determines how large a deviation could be to compensate passively. The larger the basin of attraction is, the more useful is the according walking solution including the parameter settings for the robot. The size of the basin of attraction can be understand as a measure for the robustness. In a comprehensive study both values for stability and robustness were calculated. The number for stability should be small with 1.0 as upper limit. In the case of robustness, the number, which is here an area, should be large. The simulation study showed that symmetric gaits with stable behaviour exist at smaller angles of attack $\alpha_0$. With increasing angle of attack a change from symmetric to asymmetric is observed. The point of the change is a so-called transcritical bifurcation and at this point the stability disappeared with a number of 1.0. However, with further increasing angle of attack the stability has changed to the asymmetric walking patterns. For a fixed leg stiffness, the numbers representing the stability reveal that there is a quasi optimum closed to the bifurcation point and the stability decreases toward the borders.

Robustness of walking measured as the area of the basin-of-attraction.

Analysing robustness of the walking solutions, the results are quite different. In the case of symmetric walking, left side of the thick line, the size of the basin of attraction increases in direction to the border, respectively with decreasing angle of attack. In asymmetric walking, right side of the thick line, there exists of course a maximum. In a total view, the robustness is greatest when applying medium leg stiffnesses. However, there is no optimum for the combination of stability and robustness in the area of symmetric walking. Let us focusing on a parameter setting where high robustness is identified. Some deviations can be compensated but there is actually no chance to come back to a periodic walking solution. From a technical point of view, this could still be a helpful setting and behaviour. In the section of asymmetric walking exists a kind of optimum when mainly considering robustness. However, one should take into account that the area of the basin of attraction does have a curious form.

In conclusion, the leg stiffness of a robot should be adjusted within a range of $\tilde{k} = 10 … 20$ in order to generate stable and robust walking. The controller does not need to work perfect as there exists a relatively large range of angles of attack where stability still exists. Hence, a relaxed leg and a relaxed controller can be installed in a robot for successful walking.

#### Featured Paper

J. Rummel, Y. Blum, H.M. Maus, C. Rode, A. Seyfarth.
Stable and robust walking with compliant legs.
IEEE International Conference on Robotics and Automation, May 3-8, Anchorage, Alaska: 5250-5255, 2010.
DOI: 10.1109/ROBOT.2010.5509500
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