RMRRT: Riemannian Barrier Metric RRT for Inequality-Aware Steering on Equality Manifolds

Summary

This paper presents a motion planning framework that unifies equality and inequality constraints within a single geometric formulation for sampling-based planning in high-dimensional robotic systems.

Abstract Summary

This paper presents a motion planning framework that unifies equality and inequality constraints within a single geometric formulation for sampling-based planning in high-dimensional robotic systems. In conventional sampling-based planners, equality constraints are typically enforced through projection, whereas inequality constraints are handled separately through binary validity checks such as collision testing, often leading to inefficient exploration. To address this limitation, we propose Riemannian Barrier Metric RRT (RMRRT), which constructs a unified local geometry for planning on equality-constrained manifolds. RMRRT first builds an ambient barrier metric from inequality-sensitive barrier terms and then induces a tangent-space metric via a (G)-orthogonal projection associated with the equality constraints. The resulting tangent-space metric is used consistently in both steering and nearest-neighbor selection, biasing exploration away from nearby inequality boundaries while preserving first-order equality consistency. In this work, the metric is instantiated from signed-distance-based geometric proxy inequalities to provide collision-informative tangent-space directions; hard feasibility is enforced separately through standard validity checks. Experimental results show that RMRRT achieves a 100% success rate across diverse constrained manipulation tasks in both simulation and real-world settings, while reducing planning time relative to representative constrained planning baselines. Ablation studies further demonstrate that the proposed metric improves exploration quality by reducing rejected samples and shortening path length. Experiment videos and source code are available at: https://rmrrt-anonymous.github.io

Key Points

This paper presents a motion planning framework that unifies equality and inequality constraints within a single geometric formulation for sampling-based planning in high-dimensional robotic systems.
In conventional sampling-based planners, equality constraints are typically enforced through projection, whereas inequality constraints are handled separately through binary validity checks such as collision testing, often leading to inefficient exploration.
To address this limitation, we propose Riemannian Barrier Metric RRT (RMRRT), which constructs a unified local geometry for planning on equality-constrained manifolds.
RMRRT first builds an ambient barrier metric from inequality-sensitive barrier terms and then induces a tangent-space metric via a (G)-orthogonal projection associated with the equality constraints.
The resulting tangent-space metric is used consistently in both steering and nearest-neighbor selection, biasing exploration away from nearby inequality boundaries while preserving first-order equality consistency.

Preliminaries

This section introduces the notation and geometric preliminaries used throughout the paper. Specifically, we formulate the motion planning problem under equality and inequality constraints, review equality-constrained manifolds and their tangent spaces, and summarize barrier-based representations of inequality constraints used in the metric construction of Section III . Notation.

Riemannian Barrier Metric RRT

In this section, we present Riemannian Barrier Metric RRT (RMRRT), a sampling-based planner that combines equality-manifold admissibility with inequality-aware local geometry. We first construct an ambient barrier metric and the induced tangent-space metric, and then show how the resulting geometry is used consistently in steering, nearest-neighbor selection, and manifold correction.

Experiments

This section evaluates RMRRT under coupled equality and inequality constraints in terms of reliability, efficiency, and practical applicability. Consistent with the scope clarified in Section III , the metric incorporates only distance- and proximity-based geometric inequalities, instantiated primarily from robot–obstacle proximity.

Other feasibility constraints, including joint limits, self-collision, and task-specific validity conditions, are enforced separately through standard state- and edge-validity checks.

Experiments — RMRRT: Riemannian Barrier Metric RRT for Inequality-Aware Steering on Equality Manifolds — (a)

Sources and Demonstrations

Figures are reproduced from Kang et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.