### 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.

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.

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.

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.

### Understand Jerboa Gait Transitions Using a Template Model

Egyptian jerboas are non-cursorial rodents that live in Arabia, Africa, and Asia. In the previous research, we found that the evolution of bipedalism of Egyptian jerboas enhanced their predator evasion ability, and they were able to frequently switch between gait rather than other desert rodents. To investigate the underlying mechanisms of jerboas’ frequent gait transitions, we first made the assumptions that jerboas would not shift their center of mass (COM) during locomotion and the COM located at the midpoint of their eyes and tail bases since their legs were almost massless. Then we extracted the trajectories of the torso of jerboas as well as the leg angles using a markless-position-tracking neural network called DeepLabCut (DLC). Based on the assumptions we made, we applied an extended SLIP model to reproduce the motion of jerboas.

In the above figures: (A) shows how the proposed model relates to a jerboa. The COM location is approximated as the midpoint between the eye and the tail-base. The leg angles are estimated by the orientation of the line segments connecting the COM to the feet. (B) illustrates the proposed SLIP model with passive swing leg motion. There are four continuous states (shown in blue) including the position of the torso (x , y) as well as the leg angles (α_l, α_r). Model parameters are highlighted in red, including total body mass, m, uncompressed leg length, l_o, gravity, g, and leg stiffness, k. Adding a torsional spring to a SLIP model enables motion of passive swing leg. The rotational speeds of both swing legs are determined by ω and the neutral leg angle are φ_l and φ_r respectfully. Note that the neutral leg swing angle for the right leg, φ_r, is different from that of the left leg, demonstrating an uncoupled model. A simplified version of the model showing the range of the swing leg motion is also shown in the top-right corner.

The the apex transitions, touchdowns, and liftoffs for one stride of four different gait patterns are demonstrated by jerboas on the left, with inset gait diagrams showing footfall patterns, the corresponding simulated gait patterns using our model are shown on the right. The right leg of jerboa is shown in white and the left leg is in the same color as the corresponding gait branches shown in the inset gait diagram. The left leg of the model is shown in grey and the right leg is in white. (A) shows hopping in which both feet strike and lift off simultaneously; (B) shows skipping with overlapping but non-simultaneous foot strikes; (C) shows asymmetrical running with two different aerial phases; (D) shows symmetrical running which contains two aerial phases with approximately the same duration. Blue curved arrows indicate leg touchdown (t_td) and the gray curved arrows denote liftoff events (t_lo).

### 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.

### Bounding and Pronking Controller Design for Quadrupedal robots A1 and Aliengo

We have been working on developing bounding and pronking gait controllers for two quadrupedal robots A1 and Aliengo from Unitree Robotics.