Benchmarking Different QP Formulations and Solvers for Dynamic Quadrupedal Walking

TL;DR

The paper benchmarks multiple QP formulations (sparse, partially condensed, fully condensed) and solvers for dynamic quadrupedal walking using MPC and WBC across different hardware platforms, introducing the Solve Frequency per Watt ($SFPW$) metric for cross-platform efficiency. It finds that sparse formulations with condensing commonly yield the best performance on larger horizons, while dense solvers can excel on smaller problems; in WBC, Eiquadprog consistently offers fast, reliable performance. ARM-based platforms (e.g., Jetson Orin) often provide superior energy efficiency compared to desktop-scale x86 hardware, making them attractive for energy-constrained legged robotics. The study provides practical recommendations for selecting QP formulations, solvers, and hardware to optimize real-time dynamic walking, and it contributes an open-source benchmark for ongoing evaluation.

Abstract

Quadratic Programs (QPs) are widely used in the control of walking robots, especially in Model Predictive Control (MPC) and Whole-Body Control (WBC). In both cases, the controller design requires the formulation of a QP and the selection of a suitable QP solver, both requiring considerable time and expertise. While computational performance benchmarks exist for QP solvers, studies comparing optimal combinations of computational hardware (HW), QP formulation, and solver performance are lacking. In this work, we compare dense and sparse QP formulations, and multiple solving methods on different HW architectures, focusing on their computational efficiency in dynamic walking of four legged robots using MPC. We introduce the Solve Frequency per Watt (SFPW) as a performance measure to enable a cross hardware comparison of the efficiency of QP solvers. We also benchmark different QP solvers for WBC that we use for trajectory stabilization in quadrupedal walking. As a result, this paper provides recommendations for the selection of QP formulations and solvers for different HW architectures in walking robots and indicates which problems should be devoted the greater technical effort in this domain in future.

Figures (5)

• Figure 1: Go2 quadruped unitree_robotics_unitree_nodate used for evaluation in a Volcanic field test in an analogous scenario with limited onboard computing power.
• Figure 2: Block diagram showing dynamic walking controller's architecture
• Figure 3: Mean solution time for all solvers, target platforms and condensation levels for $N=20$ and $N=10$ in the trotting experiment. The qpOASES and DAQP solvers only show single points, as no comparison of condensation levels is possible here. Note that higher $N_p$ means the is sparser.
• Figure 4: Histogram of the ($N=10$) solve times for selected solvers on all target platforms for dynamic trotting compared to standing.
• Figure 5: Computation time for different solvers using the reduced and full TSID in the trotting scenario.