Controlled symmetries for the compass biped
Citation for published version (APA):Zutven, van, P. W. M., Kostic, D., & Nijmeijer, H. (2011). Controlled symmetries for the compass biped. In Proceedings of the 30th Benelux Meeting on Systems and Control, 15-17 March 2011, Lommel, Belgium (pp. 65-).
Document status and date: Published: 01/01/2011 Document Version:
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Controlled symmetries for the compass biped
Pieter van Zutven, Dragan Kosti´c and Henk Nijmeijer
Eindhoven University of Technology, Dynamics and Control group
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Email:
{p.w.m.v.zutven, d.kostic, h.nijmeijer}@tue.nl
Introduction
In general, it is very difficult to analyze the stability of bipedal walking, since bipeds are hybrid, under-actuated systems with impulsive effects and they commonly have a large number of degrees of freedom. Most commonly used stability methods are too conservative to guarantee stable walking overall [2]. Valuable insights into stability and dy-namical behavior of bipedal walking can be gained by study-ing a compass biped, as shown in Figure 1. A compass biped is the simplest planar biped without knees, feet or a torso. It is well-known that this biped can walk naturally and with-out actuation down a shallow slope [3]. The resulting gait is energy efficient, since gravity is the only source of energy. This makes it an interesting system to analyze, because real-ization of efficient gaits is the main goal in bipedal walking.
θ1+ β θ2− β (x, z) β g θ1 θ2 (x, z) g
Figure 1: Visualization of controlled symmetries Simulation and control
Since the compass biped can walk passively and energy ef-ficient down a shallow slope, we try to induce a similar gait for an actuated biped on a horizontal surface. We use the controlled symmetries approach [1] to achieve this. This controller mimics the behavior of the biped as if it would be on a slope with angle β , which is schematically drawn in Figure 1. Basically, the controller adds the same amount of energy to the system as gravity would do on a shallow slope. To verify the control law in simulations, we derive a model of the compass biped using the framework of systems with unilateral constraints. In this way we can incorporate con-tacts, friction and impulsive forces in a single model and numerically integrate it using a time-stepping instead of an event-detection algorithm [2].
We perform two simulations, one with a compass biped on a shallow slope with angle 0.015 rad, and the other with the same biped on a horizontal surface. In the
sec-ond simulation, the controlled symmetries approach is ap-plied with β = 0.015. The results are shown in Figure 2.
t[s] q 0 0.5 1 1.5 2 -0.2 0 0.2 0.4 0.6 0.8 z[m] x[m] θ1[rad] θ2[rad] t[s] q 0 0.5 1 1.5 2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 z[m] x[m] θ1[rad] θ2[rad]
Figure 2:Simulation results: passive (left) and controlled (right)
Future work
We are in the process of building an experimental version of a compass biped, shown in Figure 3. The biped is able to walk passively down a slope and it has a DC motor to walk on a horizontal surface. On this biped we want to validate the control approach. This biped can already walk stably down a shallow slope. The next step is to apply our control law and realize the similar walking gait on a horizontal sur-face.
Figure 3: Compass biped CAD and experimental set-up References
[1] M.W. Spong and F. Bullo. Controlled symmetries and passive walking. IEEE Transactions on Automatic Control, 50(7):1025 – 1031, 2005.
[2] P.W.M. van Zutven, D. Kosti´c, and H. Nijmeijer. On the stability of bipedal walking. In Simulation, Modeling and Programming for Autonomous Robots, pages 521–532, Nov. 2010.
[3] M. Wisse, A.L. Schwab, R.Q. van der Linde, and F.C.T. van der Helm. How to keep from falling forward: elementary swing leg action for passive dynamic walkers. IEEE Transactions on Robotics, 21(3):393 – 401, 2005.
