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Previous works on Cassie:

• combined reference trajectories and DRL

  → based off only a single manipulated reference and thus were limited in application [18]

• using SLIP models to create libraries of reference trajectories to guide DRL [19]

In this paper:

• proposes a method to generate reduced-order model reference trajectories, Single Rigid-Body Models (SRBM), for general classes of highly dynamic maneuvers for bipedal robots

  → SRBM have been applied to quadruped robots to great success (Mini cheetah [11])

  → also bipeds (Atlas [15] ~ [17])

• SRBM’s simplicity allows for fast iteration and refinement of behaviors

• robustness of learning-based controllers allows for highly dynamic motions to be transferred to hardware

• by using TO, creates libraries of optimal trajectories offline that utilize the SRBM to plan a variety of highly dynamic behaviours

• These trajectories are then incorporated as part of a policies reward function and trained using PPO to develop controllers

Contents:

• SRBM Formulation
• Trajectory Optimization Formulation

SRBM Formulation

where

• $\bold P_c$ : 3D position of the body’s center of mass (CoM)
• $\bold q$: body’s attitude as a quaternion
• $\bold v_c$ : CoM linear velocity
• $\omega$ : body frame angular velocity
• $\bold x$ : state vector
• $\bold u$ : control vector
• $m$ : mass
• $J$ : body frame rotational inertia matrix
• $F_w$ : external forces in the world frame
• $\tau_B$ : external moments in the body frame
• ⊗ : quaternion multiplication
• $\hat \omega$ : angular velocity as a quaternion with zero scalar part

SRBM are actuated by ground reaction forces.

Trajectory Optimization Formulation

Optimization method: Direct collocation in COALESCE

Solver: IPOPT