BMP: Bridging the Gap between B-Spline and Movement Primitives

TL;DR

The paper introduces BMPs, a framework that reinterprets B-splines as probabilistic Movement Primitives to jointly satisfy boundary conditions and model trajectory distributions. By using a clamped uniform B-spline basis and linear-Gaussian weight modeling, BMPs inherit ProMP-like probabilistic capabilities while ensuring exact start/end positions and velocities. The approach is demonstrated to be more expressive than existing MPs and capable of satisfying velocity bounds and boundary constraints, with successful applications in imitation learning and episodic reinforcement learning, including specialized training schemes (log-likelihood-based and time-pair sampling). The findings suggest BMPs broaden the applicability of B-splines in robot learning and offer enhanced expressiveness for IL and RL tasks, potentially improving planning, learning efficiency, and constraint satisfaction in real-world robotic systems.

Abstract

This work introduces B-spline Movement Primitives (BMPs), a new Movement Primitive (MP) variant that leverages B-splines for motion representation. B-splines are a well-known concept in motion planning due to their ability to generate complex, smooth trajectories with only a few control points while satisfying boundary conditions, i.e., passing through a specified desired position with desired velocity. However, current usages of B-splines tend to ignore the higher-order statistics in trajectory distributions, which limits their usage in imitation learning (IL) and reinforcement learning (RL), where modeling trajectory distribution is essential. In contrast, MPs are commonly used in IL and RL for their capacity to capture trajectory likelihoods and correlations. However, MPs are constrained by their abilities to satisfy boundary conditions and usually need extra terms in learning objectives to satisfy velocity constraints. By reformulating B-splines as MPs, represented through basis functions and weight parameters, BMPs combine the strengths of both approaches, allowing B-splines to capture higher-order statistics while retaining their ability to satisfy boundary conditions. Empirical results in IL and RL demonstrate that BMPs broaden the applicability of B-splines in robot learning and offer greater expressiveness compared to existing MP variants.

Figures (5)

• Figure 1: (a) The first basis for B-spline of degree 0, 1, 2, 3. (b) Basis functions for a degree 2 uniform B-spline with 4 basis defined on knot vector $[0,\ 0.1, \ 0.2, \ 0.3, \ 0.4, \ 0.5, \ 0.6]$, the valid spans for representing a trajectory are $[0.2,\ 0.4]$, where there are always 3 non-zero bases in each span. (c) Clamped uniform B-spline of degree 3 with 5 Basis defined on knot vector $[0,\ 0,\ 0,\ 0,\ 0.5,\ 1,\ 1,\ 1,\ 1]$. (d)The Clamped B-spline trajectory starts exactly from first control point and ends at the last control point, the whole trajectory stays inside the polygon region of control points
• Figure 2: (a) The log-scaled averaged regression MSE loss on 20000 3-second digit-writing trajectories, by applying B-spline and MPs with different numbers of basis functions. (b) Regressing B-spline and ProDMP on a trajectory with three constant segments, where the grey-shadowed transition segments are unimportant and not considered in regression. ProDMP exhibits obvious wiggles in the all constant segments $[0,\ 0.2],\ [0.4,\ 0.6],\ [0.8,\ 1]$
• Figure 3: Using CMA-ES hansen_cma_2023 algorithm for 50 iterations to generate trajectories from given initial states to reach goal states while avoiding obstacles. (a) Generated best trajectory path within 50 iterations. (b),(c) The velocity profile of the B-spline and ProDMP trajectory, where the red dash lines are the velocity bounds.
• Figure 4: (a) A batch of digit images and reconstructed trajectories using BMP. (b) Sampling trajectories from predicted distribution conditioned on an image of digit '3' and '8'.
• Figure 5: (a) The averaged success rate of 4 random seeds. Both BMP (B-spline) and ProDMP use in total 9 basis functions. The basis functions of ProDMP are scaled into the same value range to make it work. (b) The joint velocity profile of joint 3, where the velocity at 0s and 2s should be 0.