Differentiable Motion Manifold Primitives for Reactive Motion Generation under Kinodynamic Constraints

Under review

Yonghyeon Lee

Massachusetts Institute of Technology

TL;DR: We propose a two-step method for fast kinodynamic motion planning: (i) learning a manifold of task-relevant trajectories that satisfy kinodynamic constraints offline, and (ii) searching for a trajectory within this manifold online, enabling the system to adapt to dynamic environments.

Example: Throwing with a 7-DoF Robot Arm

Throwing to a nearby target

Throwing to a distant target

DMMP: Fast Generation of Throwing Trajectories
(Gray → Deep Red → Transparent Red)

Online Adaptation to Dynamic Target Changes

Abstract

Real-time motion generation -- which is essential for achieving reactive and adaptive behavior -- under kinodynamic constraints for high-dimensional systems is a crucial yet challenging problem. We address this with a two-step approach: offline learning of a lower-dimensional trajectory manifold of task‑relevant, constraint‑satisfying trajectories, followed by rapid online search within this manifold. Extending the discrete‑time Motion Manifold Primitives (MMP) framework, we propose Differentiable Motion Manifold Primitives (DMMP), a novel neural network architecture that encodes and generates continuous‑time, differentiable trajectories, trained using data collected offline through trajectory optimizations, with a strategy that ensures constraint satisfaction -- absent in existing methods. Experiments on dynamic throwing with a 7‑DoF robot arm demonstrate that DMMP outperforms prior methods in planning speed, task success, and constraint satisfaction.

Summary

1 Challenging Kinodynamic Motion Planning

• Planning problems that require the robot to fully utilize its kinodynamic limits to complete a task are particularly challenging, taking a long time to solve.

The left GIF is adopted from the Timur Garipov’s project video on Robotic Arm Weightlifting via Trajectory Optimization in YouTube.

2 Planning in Dynamic Environments

• As a concrete example, we will work through the dynamic throwing task.

Trajectorie Optimization: Given an analytic inverse dynamics equation, we have a nonlinear and nonconvex objective function \( J(q(t);\tau) \) — where the task variable \( \tau \) is the target box position and the joint trajectory \( q(t) \) is an optimization variable — and constraints \( C(q, \dot{q}, \ddot{q}, \dddot{q}) \leq 0 \) for all \( t \).

Adam: Adaptive Moment Estimation (Kingma et al., International Conference on Learning Representations 2015)
COBYLA: Constrained Optimization BY Linear Approximation (Powell., Advances in Optimization and Numerical Analysis 1994)
SLSQP: Sequential Least SQuares Programming (Kraft., Tech. Rep. DFVLR-FB 88-28 1988)

!Conventional trajectory optimization algorithms are too slow for online replanning and sometimes fail depending on the initialization in challenging kinodynamic planning problems.

3 How can humans perform fast planning?

• Through experience, we have learned motor patterns relevant to throwing motions, and these are encoded in our muscle memory.

Images are generated by ChatGPT 4o.

4 Key Idea: Learning a trajectory manifold offline, Planning within it online

Four-Step Training Method:
1. Data collection via trajectory optimizations
2. Differentiable motion manifold learning
3. Latent flow model learning via flow matching
4. Trajectory manifold optimization