FlowMP: Learning Motion Fields for Robot Planning with Conditional Flow Matching

TL;DR

FlowMP addresses motion planning by learning second-order dynamics through conditional flow matching, enabling direct, rapid generation of smooth and dynamically feasible trajectories without iterative denoising. By modeling velocity, acceleration, and jerk fields via a conditional motion field and grounding expert paths with B-spline via-points, FlowMP produces high-quality trajectories with improved feasibility and execution smoothness. Empirical results show FlowMP outperforms diffusion-based MPD and Stoch-GPMP in trajectory smoothness, planning feasibility, and inference speed, including successful real-robot demonstrations on a Kinova Gen3. The approach offers scalable, real-time capable planning suitable for complex, multimodal tasks and long-horizon trajectories in both 2D and 3D spaces.

Abstract

Prior flow matching methods in robotics have primarily learned velocity fields to morph one distribution of trajectories into another. In this work, we extend flow matching to capture second-order trajectory dynamics, incorporating acceleration effects either explicitly in the model or implicitly through the learning objective. Unlike diffusion models, which rely on a noisy forward process and iterative denoising steps, flow matching trains a continuous transformation (flow) that directly maps a simple prior distribution to the target trajectory distribution without any denoising procedure. By modeling trajectories with second-order dynamics, our approach ensures that generated robot motions are smooth and physically executable, avoiding the jerky or dynamically infeasible trajectories that first-order models might produce. We empirically demonstrate that this second-order conditional flow matching yields superior performance on motion planning benchmarks, achieving smoother trajectories and higher success rates than baseline planners. These findings highlight the advantage of learning acceleration-aware motion fields, as our method outperforms existing motion planning methods in terms of trajectory quality and planning success.

Figures (3)

• Figure 1: Executions of a smooth, dynamically feasible motion on a Kinova Gen3 manipulator. The same start and goal configurations can yield multiple valid solutions when sampling from our flow matching framework, demonstrating its ability to capture different modes of the trajectory distribution.
• Figure 2: Motion planning at denoising steps along the time horizon $t$ from $0$ to $1$, inferred by the trained conditional motion field as outlined in the generate_motion function (Alg. \ref{['alg:flow_motion_planning']}). The paths correspond to different task objectives within the same maze environment: Path 1 and Path 2 start from different initial positions but share the same goal, while Path 3 has distinct start and goal points. The start and goal positions are indicated as ● and ✖, respectively, and are encoded as the same color of the paths. Their corresponding velocity (blue and orange) and acceleration (green and yellow) profiles of the simultaneously inferred paths are shown accordingly.
• Figure 3: The conditional motion field is extended to 3D space, where collision-free paths are denoised along the time horizon $t$ from $0$ to $1$ within a obstacle map from the initial noises covering the workspace. The model is trained on expert motion distributions with various task objectives in the pre-defined 3D space using Alg. \ref{['alg:flow_motion_planning']}. Three example motions are sampled with the generate_motion function, where Path 1 and Path 3 start from different initial positions but go to the same goal, whereas Path 2 follows a distinct trajectory with unique start and goal points. Velocity (blue, orange, and violet) and acceleration (green, yellow, and apricot) profiles on three axes are sampled from their respective noise distributions. At $t = 1$, the trajectories and their derivatives are smooth.