Passive Dynamic Locomotion

In this project, we systematically investigated passive dynamic gaits that emerge from the natural mechanical dynamics of a bipedal legged system. To this end, we developed an energetically conservative, yet complete dynamical model of a biped. We achieved this by extending the established Spring-Loaded Inverted Pendulum (SLIP) model to include two legs and by adding a foot mass and a hip spring to enable passive swing leg dynamics. By letting the foot mass and hip stiffness go to zero while keeping their ratio (and thus the leg swing frequency) constant, I prevented energy losses at touchdown. Through a targeted continuation of periodic motions, I showed that a range of different bipedal gaits emerged in this model from a simple bouncing-in-place motion with different discrete footfall patterns. Among others, these passive dynamic gaits included walking, running, hopping, skipping, and galloping.

Click the picture for a short video explanation.

The different gaits arose along with one-dimensional manifolds of solutions. These manifolds bifurcated into different branches with distinctly different types of motions. That is, the gaits were obtained as different oscillatory motions (or nonlinear modes) of a single mechanical system with a single set of parameters. As this biped model has neither actuation nor control, it supports the hypothesis that different gaits are primarily a manifestation of the underlying natural mechanical dynamics of a legged system. The occurrence and prevalence of certain gaits in nature are thus possibly the consequence of animals exploiting passivity based gaits in order to move in an energetically economical fashion. The same argument should hold for legged robots: the passive motions derived in this chapter establish a blueprint of how to move economically. In the absence of losses, the passive dynamic gaits constitute the only feasible way of locomoting without performing any actuator work. As losses are introduced, such as losses due to friction and collision impacts, the motions will have to change and will, of course, require some actuation. However, staying close to the original passive motions might reduce the need for motor torques and for negative actuator work, and might hence reduce the energetic cost of locomotion.


It is also notable, that despite the vast differences in morphology, the gaits of bipedal and quadrupedal animals share some important similarities.  Heglund (1982) investigated the dynamic similarity between walking in bipeds and quadrupeds and hypothesized that they utilize the same mechanism similar to an inverted pendulum in which kinetic energy is exchanged for potential (gravitational) energy and vice versa. This implies that fluctuations in kinetic and potential energy happen out of phase. These energy-based observations can be extended to other gaits: in bipedal running or hopping and in quadrupedal trotting, fluctuations of potential and kinetic energy happen in phase and both are exchanged for elastic energy.  However, this analysis breaks down for asymmetrical gaits of quadrupeds.

Click the picture for a short video explanation.

Due to the lack of the additional pair of legs, a biped cannot move in a fashion that is dynamically similar to a galloping quadruped. In this project, we also explore the dynamic similarity between bipedal gaits and asymmetrical quadrupedal gaits by using simplistic passive models. These models are built on an extensive body of previous work that investigates the passive dynamics of legged locomotion.  In the present work, we employ our two models to reveal potential dynamic relationships between bipedal gaits on the one side and quadrupedal asymmetrical gaits on the other. By letting the inertia of the torso in the quadrupedal model vary from zero to infinitely large, we explicitly connect the two models and link all bounding gaits of the quadrupedal model to the two-legged gaits of the bipedal model.




Walking Controller Design for Bipedal Robot Cassie

We have been working on developing walking controllers for a bipedal robot Cassie built by Agility Robotics. This 3D robot has in total of twenty degrees of freedom and ten electric motors. We have been using the insights gained from the simple conservative templates to create a library of optimal gaits using full-body models implemented in an optimal control framework where the motions of every joint are taken into consideration. The current research projects include:

  • Creating gait library using hybrid trajectory optimization framework C-FROST: based on the previous work at the biped robotics lab, to generate more versatile walking motions rather than moving at a constant speed, we have been building an extended set of periodic gaits that have various forward speeds, turning speeds, stride times, and terrain slopes. These solutions are optimized in parallel in a rapid gait creation framework called C-FROST where solutions are subjected to virtual constraints based on the hybrid zero dynamics.
    All periodic motions are identified offline and optimized trajectories are converted to b polynomials that can be used for the online controller design.
  • Stair climbing controller design with perception: the controller based on the above gait library is sufficient to reject certain amount of disturbance from the uneven terrain. However, for some specific tasks, such as stair climbing, we cannot rely solely on the controller developed for the level ground. To this end, I have been developing and testing controllers to dynamically climbing stairs with the help of LiDAR and stereo cameras.
  • System identification using reinforcement learning: to overcome the obstructions imposed by the unmodelled motor dynamics and model uncertainty, we are applying data-driven methods to systematically identify the parameters in the multibody model of Cassie. This has the potential to ease the difficulty in the manual tuning of low-level controllers and speed up the implementation of specific controller design based on the gait libraries.