M$^3$P-R1: Reinforcement Learning for Large Language Model Guided Multi-Modal Motion Planning via MIP Code Generation

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Multi-Modal Motion Planning (M$^3$P) requires joint reasoning over continuous motions and discrete mode transitions, making it difficult to solve efficiently. For instance, a bipedal robot may walk to a target location and then use its arms to grasp an object. This scenario captures both mode transitions and continuous dynamics, yielding feasible paths that neither purely discrete nor continuous planners can handle. While Mixed-Integer Programming (MIP) offers a principled framework, constructing tractable formulations for non-convex problems is typically manual and domain-specific, especially in the approximate, discretization-based MIP regime needed for non-convex robotic tasks. We propose M$^3$P-R1, a reinforcement learning method that fine-tunes large language models (LLMs) to decompose M$^3$P tasks into MIP variables, constraints, and objectives. Instead of directly outputting answers, which are often prone to hallucination, the model generates executable Python code using MIP optimization libraries and constraint interfaces. This enables solver-backed execution for robust and verifiable solutions. Trained with an outcome-driven reward against the solver, M$^3$P-R1 learns to compose modality-level discretization primitives and synthesize cross-modal coupling constraints, producing executable MIP programs for complex M$^3$P tasks.