Primal-Dual iLQR for GPU-Accelerated Learning and Control in Legged Robots

This paper introduces a novel Model Predictive Control (MPC) implementation for legged robot locomotion that leverages GPU parallelization. Our approach enables both temporal and state-space parallelization by incorporating a parallel associative scan to solve the primal-dual Karush-Kuhn-Tucker (KKT) system. In this way, the optimal control problem is solved in $\mathcal{O}(n\log{N} + m)$ complexity, instead of $\mathcal{O}(N(n + m)^3)$, where $n$, $m$, and $N$ are the dimension of the system state, control vector, and the length of the prediction horizon. We demonstrate the advantages of this implementation over two state-of-the-art solvers (acados and crocoddyl), achieving up to a 60\% improvement in runtime for Whole Body Dynamics (WB)-MPC and a 700\% improvement for Single Rigid Body Dynamics (SRBD)-MPC when varying the prediction horizon length. The presented formulation scales efficiently with the problem state dimensions as well, enabling the definition of a centralized controller for up to 16 legged robots that can be computed in less than 25 ms. Furthermore, thanks to the JAX implementation, the solver supports large-scale parallelization across multiple environments, allowing the possibility of performing learning with the MPC in the loop directly in GPU.

@article{amatucci2025_2506.07823,
  title={ Primal-Dual iLQR for GPU-Accelerated Learning and Control in Legged Robots },
  author={ Lorenzo Amatucci and João Sousa-Pinto and Giulio Turrisi and Dominique Orban and Victor Barasuol and Claudio Semini },
  journal={arXiv preprint arXiv:2506.07823},
  year={ 2025 }
}