Tutorial on Embodiment Self-stabilization*


Passive dynamic walkers have shown that locomotion can be realized through pure, but carefully tuned mechanics. However, how stable or adaptive is such a solution? In other words, how does a brainless machine cope with different slopes or with disturbances? The theory of non­linear dynamical systems is often employed to analyze the phenomena involved in the mechanical (and also neural) aspects of locomotion. The walker is an example of a nonlinear dynamical system and walking patterns (which are periodic motions) correspond to limit cycles. Limit cycles in a nonlinear system can display attractive behavior, i.e. nearby trajectories are ‘pulled' toward the limit cycle.

Look at Video first. You see a bicycle being pushed and driving alone without any control, coping with disturbances that arise from the interaction with the ground. In the second half of the video, you can see that the system can compensate even for a major disturbance when it is pushed to the side. A wonderful illustration of self-stabilization! For a formal treatment of the bicycle's dynamics please see Meijaard et al. (2007) and Kooijman et al. (2011).

Video Bicycle stability demonstration video from Cornell Human Power, Biomechanics and Robotics Laboratory.


Mechanical self-stability, i.e. robustness to disturbances through local attractivity of the mechanical system, has been shown in a physical (McGeer, 1990) and mathematical (Coleman et al., 1997) walking model. In hopping or running, the dynamics is even more prolific. Fig. illustrates this pheno­menon schematically. A monopod hopper driven by an open-loop controller compensates for disturbances without any explicit feedback mechanism, that is, without measuring the disturbances or altering the system. Self-stabiliza­tion has been investigated in a monopod (Ringrose, 1997), or quadruped (Poulakakis et al., 2006; Ringrose, 1997), for instance. Kubow and Full (1999) designed a dynamic model of a hexapedal runner and observed the recovery from rotational, lateral, and fore-aft velocity perturbations. Perturbations alte­red the translation and/or rotation of the body that consequently provided mechanical feedback by altering leg moment arms. Koditschek et al. (2004) provide an excellent review of the mechanical aspects of legged locomotion, analyzing cockroaches in particular and showing how this inspired the con­struction of the RHex robot - a robot with unprecedented mobility (Saranli et al., 2001). These studies show that running on rough terrain can be accomplished with simple feed-forward control in concert with a mechanical system that stabilizes passively. In the biological realm, the intrinsic proper­ties of muscles further aid self-stability (Blickhan et al., 2007) and further assist in making the neural contribution to locomotion control simpler.

Fig. Self-stabilization. Adaptivity is part of the mechanical structure itself. (A) Picture of a two-dimensional underactuated monoped hopping robot attached to a central rod with a rotational joint (courtesy of A. Seyfarth and A. Karguth). (B) A schematic representation of the hopping robot in the different phases of locomotion: flight, touchdown (TD) [with angle of attack (AOA)], and takeoff (TO). Only the joint depicted by the black circle (hip joint) is actuated, the knee (white circle) is passive, and the lower limb is attached to the upper limb with a simple spring. (C) Output of a simulation of the robot. The upper part of the panel shows the trajectory of the model over time as a sequence of stick figures; in the lower part, the angle of attack (the angle at which the leg hits the ground) is plotted. The model exhibits a stable hopping gait with a periodic hip motor oscillation, as indicated by the constant AOA at every step in the left side of the panel. At distance d = 0 m, there is a step in the ground that disturbs the robot's movement but to which the robot adapts without the need for any changes in the control. This purely mechanical phenomenon is called self-stabilization (Figure from Pfeifer et al., 2007; there adapted from Blickhan et al., 2007).


*This section has been adapted from Hoffmann and Pfeifer (2011).


Blickhan, R.; Seyfarth, A.; Geyer, H.; Grimmer, S.; Wagner, H. & Guenther, M. (2007), 'Intelligence by mechanics', Phil. Trans. R. Soc. Lond. A 365, 199-220.
Hoffmann, M. & Pfeifer, R. (2011), The implications of embodiment for behavior and cognition: animal and robotic case studies, in W. Tschacher & C. Bergomi, ed., The Implications of Embodiment: Cognition and Communication, Exeter: Imprint Academic, pp. 31-58.
Koditschek, D. E.; Full, R. J. & Buehler, M. (2004), 'Mechanical aspects of legged locomotion control', Arthropod structure and development 33, 251-272.

Kooijman, J. D. G.; Meijaard, J. P.; Papadopoulos, J. M.; Ruina, A. & Schwab, A. L. (2011). 'A bicycle can be self-stable without gyroscopic or caster effects', Science 332(6027), 339-342.
Kubow, T. M. & Full, R. J. (1999), 'The role of the mechanical system in control: a hypothesis of self-stabilization in hexapedal runners', Phil. Trans. R. Soc. Lond. B 354, 849-861.
McGeer, T. (1990), 'Passive Dynamic Walking', The International Journal of Robotics Research 9(2), 62-82.
Meijaard, J.; Papadopoulos, J.; Ruina, A. & Schwab, A. (2007), 'Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review', Proc. Roy. Soc. A 463, 1955-1982.
Poulakakis, I.; Papadopoulos, E. & Buehler, M. (2006), 'On the stability of the passive dynamics of quadrupedal running with a bounding gait', International Journal of Robotics Research 25, 669-687.
Ringrose, R. (1997), Self-stabilizing running, in 'Proc. IEEE Int. Conf. Robotics and Automation (ICRA)'.
Saranli, U.; Buehler, M. & Koditschek, D. (2001), 'RHex: a simple and highly mobile hexapod robot', Int. J. Robotics Research 20, 616-631.