Ruprecht-Karls-Universität Heidelberg
Optimization in Robotics and Biomechanics

Open-loop stable walking, running and jumping

Katja Mombaur, Hans-Georg Bock, Johannes Schlöder - IWR
Richard Longman
- Columbia University, New York

In this research, we investigate periodic walking and running motions of one- and two-legged robots. By definition, running motions involve flight phases, in contrast to walking motions where always at least one foot is in contact with the ground (for monopod robots, running/hopping is obviously the only possible form of locomotion). Our work focuses on the dynamic stability aspect of walking and running.

The classical control concept for walking and running robots is feedback control. A real-time closed-loop control of these walking robots requires sophisticated and expensive sensory systems and feedback-controllers. The computation of appropriate feedback is time critical and often a limitation for making motions faster, hence requiring high computational capacities on the robot.

Our approach to the stability control issue is to determine, in a first step,what can be achieved without any active feedback, and to search in fact for purely open-loop controlled, self-stabilizing system configurations and running motions.

This implies

  • that there is no feedback and no sensors at all, not even a detection of contact times
  • that the robots response is only based on its natural dynamics, i.e. its inherent kinematic in kinetic properties
  • that the system input (i.e. the joint torques etc.) which is continuously active over the full cycle and not impulse-like is never modified (not even the actuation cycle frequency)
  • that the robot motion must always stay synchronized with the invariable external exciting frequency and that there is no possibility of time shifts.

These rules may seem very strict, since some types of feedback (like a phase adjustment at contact events) are easily implementable, but we consider these purely open-loop stable solutions to be good starting points for an experimental phase in which the robots motion can then be made more robust by simple feedback measures.

Over the course of our research, we have determined a variety of different robot cofigurations that are able to perform open-loop stable motions in the sagittal plane (see also the picture galery below)

  • a one-legged hopping robot with a telescopic leg and with a trunk
  • a two-legged running robot with two telescopic legs and with a trunk

  • the same two-legged robot performing somersaults

  • the same robot again, but one leg turned upwards, performing flip-flops

  • a bipedal walking robot with knees, but without torso

  • a human-like running robot

We have also treated passive-dynamic walking (i.e. gravity powered walking on an inclides slope without actuators) as a special case, optimizing the stability of Mike Coleman's 3D tinkertoy robot



K. Mombaur, Using optimization to create self-stable human-like running, Robotica, Vol. 27, 2009, p. 321-330, published online June 2008

K. D. Mombaur - Performing Open-loop Stable Flip-Flops - An Example for Stability Optimization and Robustness Analysis of Fast Periodic Motions, Springer LNCIS, 2006

K. D. Mombaur , H.G.Bock, J. P. Schlöder and R. W. Longman - Self-stabilizing somersaults, IEEE Transactions on Robotics, Vol. 21, No.6 , Dec. 2005

K. D. Mombaur, R. W. Longman, H. G. Bock, and J. P. Schlöder - Open-loop stable running, Vol. 23, No. 1, Robotica, 2005

K.D. Mombaur, H.G. Bock, P. Schlöder and R. W. Longman - Open-loop stability -- A new paradigm for periodic optimal control and  analysis of walking mechanisms,  Proceedings of IEEE CIS-RAM 04, Singapore, Dec. 2004

K. D. Mombaur, H.G.Bock, J. P. Schlöder and R. W. Longman - Stable Walking and Running Robots Without Feedback, Proceedings of Clawar 04, Madrid, September, 2004, Springer Verlag

K. D. Mombaur, H.G.Bock,  J. P. Schlöder and R. W. Longman  - Open-loop Stable Control of Periodic Multibody Systems, in H. G. Bock, E. Kostina, H. X. Phu and R. Rannacher (eds.): "Modeling, Simulation and Optimization of Complex Processes", Lecture Notes in Scientific Computing, Springer, 2004

K. D. Mombaur, H.G.Bock,  J. P. Schlöder and R. W. Longman - Stability Optimization of Periodic Mechanical Systems without Feedback, Proceedings of Applied Mathematics and Mechanics (GAMM Annual Conference),  Padua, Italy, March 2003

K. D. Mombaur, R. W. Longman, H. G. Bock, and J. P. Schlöder - Stable one-legged hopping without feedback and with a point foot. In Proceedings of IEEE International Conference on Robotics and Automation, Washington D.C., 2002.

K. D. Mombaur, H. G. Bock, J. P. Schlöder, and R. W. Longman - Human-like actuated walking that is asymptotically stable without feedback. In Proceedings of IEEE International Conference on Robotics and Automation, Seoul, 2001.

M. J. Coleman, M. Garcia, K. D. Mombaur, and A. Ruina, Prediction of stable walking for a toy that cannot stand. Physical Review E, 2001.

K. D. Mombaur - Stability Optimization of Open-loop Controlled Walking Robots. Dissertation, Universität Heidelberg, 2001. VDI Fortschritt-Bericht, Reihe 8, Nr. 922. also published at


K. Mombaur,
Last Update: 19.10.2011 - 16:03

The photographs in the header of this webpage have been taken at the Musee de l'Automate in Souillac, France