Hamiltonian-Guided Diffusion Fields for Variable-Length Rigid-Arm Trajectory Generation

Generating rigid-arm trajectories is fundamental for planning, imitation learning, and proposal generation. In many deployment settings, a simulator may be unavailable, so trajectory generation must rely on offline data while remaining faithful to rigid-body dynamics. We use diffusion probabilistic fields (DPFs) to model trajectories as continuous-time fields, enabling variable-length generation by querying arbitrary time coordinates. Purely data-driven DPF samples can violate Hamiltonian structure, particularly when trajectories must satisfy Hamilton’s equations under actuation and remain energy-consistent in unforced regimes. We therefore learn a system Hamiltonian with a Hamiltonian neural network (HNN) and use it to define a physics inconsistency residual over entire trajectories. We incorporate two guidance mechanisms that condition DPF sampling on this residual: a one-step, gradient-based guidance applied during denoising, and a multi-sample self-normalized importance sampling scheme that reweights candidate trajectories toward physics-consistent outcomes. On 2-DoF and 3-DoF MuJoCo arms across diverse torque policies, Hamiltonian-guided DPF sampling reduces error relative to simulator rollouts and improves generalization to unseen trajectory lengths and torque policies, with importance sampling providing stronger consistency at increased sampling cost.