Predictive Control with Indirect Adaptive Laws for Payload Transportation by Quadrupedal Robots

Abstract

This paper formally develops a novel hierarchical planning and control framework for robust payload transportation by quadrupedal robots, integrating a model predictive control (MPC) algorithm with a gradient-descent-based adaptive updating law. At the framework's high level, an indirect adaptive law estimates the unknown parameters of the reduced-order (template) locomotion model under varying payloads. These estimated parameters feed into an MPC algorithm for real-time trajectory planning, incorporating a convex stability criterion within the MPC constraints to ensure the stability of the template model's estimation error. The optimal reduced-order trajectories generated by the high-level adaptive MPC (AMPC) are then passed to a low-level nonlinear whole-body controller (WBC) for tracking. Extensive numerical investigations validate the framework's capabilities, showcasing the robot's proficiency in transporting unmodeled, unknown static payloads up to 109% in experiments on flat terrains and 91% on rough experimental terrains. The robot also successfully manages dynamic payloads with 73% of its mass on rough terrains. Performance comparisons with a normal MPC and an L1 MPC indicate a significant improvement. Furthermore, comprehensive hardware experiments conducted in indoor and outdoor environments confirm the method's efficacy on rough terrains despite uncertainties such as payload variations, push disturbances, and obstacles.

Figures (6)

• Figure 1: Snapshot of the locomotion of the A1 robot with a payload of $11.34$ (kg) ($91\%$ uncertainty) on wooden blocks using the proposed AMPC.
• Figure 2: Overview of the proposed layered control approach with the AMPC algorithm and the adaptive gradient law at the high level ($160$ Hz) and the nonlinear WBC at the low level ($1$kHz) for robust locomotion with unknown payloads.
• Figure 3: Snapshots of experiments for (a) flat terrain locomotion with a $13.6$ (kg) payload ($109\%$ uncertainty); (b) and (c) rough terrain locomotion with wooden blocks and $11.34$ (kg) ($91\%$ uncertainty) and $8.2$ (kg) payloads; (d) dynamic payload addition; (e) pushing the robot with a $6.48$ (kg) payload; (f), pushing a $3.7$ (kg) object while carrying a $4.5$ (kg) payload; (g) and (h) locomotion on grass and gravel with a $6.48$ (kg) payload; (i) and (j) backward and lateral trot on rough terrain with $9$ (kg) and $6.8$ (kg) payloads and a COM offset. In all of these experiments, the desired forward velocities are $0.5$ (m/s), except for (b) and (d), which are $0.4$ (m/s) and $0.6$ (m/s), respectively. For the backward and lateral trot, the desired velocity is $0.4$ (m/s) and $0.15$ (m/s).
• Figure 4: Plot of the prescribed reduced-order trajectory and vertical GRF for the front right leg by the AMPC planner at the forward speed of $0.4$ (m/s) in (a) and (b). (c) Evolution of the estimated mass parameter. Despite the challenges posed by uneven terrain and an unknown payload of $11.34$ (kg), the robot demonstrates robust locomotion, as depicted in Fig. \ref{['Fig:Snapshots']}(b).
• Figure 5: Plot of the prescribed reduced-order trajectory and vertical GRF for the front right leg by the AMPC planner at the forward speed of $0.6$ (m/s) in (a) and (b). (c) Evolution of the estimated mass parameter. The robot is subject to carrying a payload of $6.48$ (kg) and dynamic additions, as shown in Fig. \ref{['Fig:Snapshots']}(d). The true mass varies in response to these dynamic payload additions.

Theorems & Definitions (7)

• Remark 1
• Remark 2: Real-Time Implementation
• Theorem 1: Stability Guarantees
• proof
• Lemma 1: Convex Stability Guarantees
• proof
• Remark 3