Kinodynamic Model Predictive Control for Energy Efficient Locomotion of Legged Robots with Parallel Elasticity

TL;DR

This paper tackles energy efficiency in dynamic legged locomotion by coupling a kinodynamic MPC with unidirectional parallel springs (UPS) in a hierarchical control framework. The approach combines a SLIP-based motion sketch, a convex SRB MPC for fast warm-starts, and a full kinodynamic MPC that explicitly models UPS torque and joint constraints, enabling real-time, energy-aware planning. Key contributions include explicit UPS integration within the NLP formulation, a hierarchical warm-start strategy to maintain real-time performance, and extensive simulation and hardware results showing significant CoT reductions (up to 38.8% in simulation) and reduced knee torque in hardware (about 14.8% energy savings). The work demonstrates practical energy savings for a monoped with UPS and points to scalable extensions to more complex legged robots, potentially informing designs and controllers for bipedal and humanoid systems in energy-constrained applications.

Abstract

In this paper, we introduce a kinodynamic model predictive control (MPC) framework that exploits unidirectional parallel springs (UPS) to improve the energy efficiency of dynamic legged robots. The proposed method employs a hierarchical control structure, where the solution of MPC with simplified dynamic models is used to warm-start the kinodynamic MPC, which accounts for nonlinear centroidal dynamics and kinematic constraints. The proposed approach enables energy efficient dynamic hopping on legged robots by using UPS to reduce peak motor torques and energy consumption during stance phases. Simulation results demonstrated a 38.8% reduction in the cost of transport (CoT) for a monoped robot equipped with UPS during high-speed hopping. Additionally, preliminary hardware experiments show a 14.8% reduction in energy consumption. Video: https://youtu.be/AF11qMXJD48

Figures (6)

• Figure 1: Simulation results showcasing the capability of the proposed kinodynamic MPC on a hopping robot with UPS. The Cost of Transport (CoT) is plotted w.r.t. hopping frequency for 10 continuous jumps at $v_x=1$ m/s. (left) CoT of the robot without UPS. (right) CoT of the robot with UPS.
• Figure 2: The system overview of the proposed hierarchical controller, where the orange arrows indicate the improvement of model fidelity. (a) The SLIP motion sketch synthesis stage generates the CoM trajectory, touchdown location and GRF. (b) The SRB MPC stage introduces rotational dynamics and friction cone constraints. (c) The kinodynamic MPC stage incorporates joint angle and joint torque to reason about UPS. (d) CAD render of the custom monoped robot. (e) The parallel spring torque as a function of knee joint angle. (f) The cross-section view of the UPS mechanism.
• Figure 3: Monoped with Unidirectional Parallel Spring (MUPS) is a custom robot platform to validate our proposed kinodynamics MPC framework.
• Figure 4: Simulation results for velocity tracking. The robot starts from $v_x=0$ m/s and hops to $v_x^\text{des}=1.0, 2.0, 1.5$ m/s. (a) Desired and measured forward velocity; (b) Desired and measured torso pitch angle.
• Figure 5: Simulation study that investigates the effect of UPS on CoT with increasing commanded forward velocity. CoT is the average value for 10 consecutive jumps.