Variable-Frequency Model Learning and Predictive Control for Jumping Maneuvers on Legged Robots

TL;DR

This work tackles the challenge of achieving target accuracy and robustness in long-flight legged jumps by learning a residual dynamics model directly from hardware and integrating it into a variable-frequency model predictive control framework. By employing phase-specific neural residuals and synchronized variable time steps for both learning and MPC, the approach captures complex leg dynamics, contact switching, and disturbances across the entire jump sequence. Hardware validation on a Unitree A1 demonstrates substantial improvements in jumping accuracy and robustness, including up to an $8\times$ reduction in final distance error and reliable performance on uneven terrain, while maintaining real-time computation. The combination of a data-driven residual model with a multi-phase, real-time MPC offers a practical pathway for robust, high-precision jumping maneuvers in legged robots.

Abstract

Achieving both target accuracy and robustness in dynamic maneuvers with long flight phases, such as high or long jumps, has been a significant challenge for legged robots. To address this challenge, we propose a novel learning-based control approach consisting of model learning and model predictive control (MPC) utilizing a variable-frequency scheme. Compared to existing MPC techniques, we learn a model directly from experiments, accounting not only for leg dynamics but also for modeling errors and unknown dynamics mismatch in hardware and during contact. Additionally, learning the model with variable-frequency allows us to cover the entire flight phase and final jumping target, enhancing the prediction accuracy of the jumping trajectory. Using the learned model, we also design variable-frequency to effectively leverage different jumping phases and track the target accurately. In a total of 92 jumps on Unitree A1 robot hardware, we verify that our approach outperforms other MPCs using fixed frequency or nominal model, reducing the jumping distance error 2 to 8 times. We also achieve jumping distance errors of less than 3 percent during continuous jumping on uneven terrain with randomly placed perturbations of random heights (up to 4 cm or 27 percent the robot standing height). Our approach obtains distance errors of 1 to 2 cm on 34 single and continuous jumps with different jumping targets and model uncertainties. Code is available at https://github.com/DRCL-USC/Learning MPC Jumping.

Figures (8)

• Figure 1: A Unitree A1 robot performs continuous jumps on unknown uneven terrain, achieving both target accuracy and robustness. The target distance for each jump is $0.6$ m. The flight phase covers a vertical height of up to $4 \times$ the robot's normal height. MPC with a nominal single rigid-body model is used to collect data for training a neural-network residual dynamics model following using a variable-frequency scheme. The learned model is then used in a variable-frequency MPC to execute jumping motions. The variable-frequency scheme varies the time step sizes in the contact phase ($\Delta t_{{\@fontswitch\mathcal{C}}}$) and flight phase ($\Delta t_{{\@fontswitch\mathcal{F}}}$) to dedicate more model capacity and more MPC optimization steps to the contact phase, which improves the jumping robustness and accuracy. The green dashed line is the actual robot trajectory. Supplemental video: https://youtu.be/yUqI_MBOC6Q.
• Figure 2: System Architecture. The learning procedure and MPC execution are paired with the same integration timesteps$\lbrace \Delta t_{{\@fontswitch\mathcal{C}}}, \Delta t_{{\@fontswitch\mathcal{F}}} \rbrace$, prediction horizon $K$, and predefined contact schedule ($cs$). The MPC and low-level joint PD controllers are updated at $40Hz$ and $1kHz$, respectively.
• Figure 3: Training loss and testing loss (Log scale).
• Figure 4: State prediction comparison on a random testing dataset. It starts at $650$ ms, consisting of $K_c = 6$ steps in the contact phase and $K_f = 4$ steps covering the flight period of $400$ ms. The blue and yellow areas represent the contact and flight periods, respectively. We compare variable-frequency learned model (our method), fixed-frequency learned model, variable-frequency nominal model, and the ground-truth trajectories. The ground-truth data is directly obtained from hardware experiments.
• Figure 5: MPC solving time ($t_{MPC}^{sol}$) with various horizons $K=10$ and $K=22$. The red dashed line denotes the MPC update time ($1/f_{MPC}^{upd}$) of $25$ ms. Real-time performance is achieved if $t_{MPC}^{sol}<<1/f_{MPC}^{upd}$.