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

Minhyeong Kang, Sanghyun Kim

arXiv:2610.06863v1cs.RO

TL;DR

Conventional sampling-based planners often treat equality constraints through projection and inequalities through separate validity checks, leaving inequality-induced geometry out of exploration. RMRRT constructs a barrier-informed tangent-space metric and uses it in steering and neighbor selection; experiments report improved reliability and efficiency over representative baselines.

  • Problem

    Conventional sampling-based planners often separate equality-manifold projection from inequality validity checks, rather than using one local geometry that accounts for both during exploration.

    Source: §I · p. 1, A. Proposed Framework · p. 5

  • Method

    RMRRT builds an ambient barrier metric, induces a tangent-space metric through a metric-compatible projection, and uses it for steering and nearest-neighbor selection.

    Source: B. Riemannian Barrier Metric… · p. 5, D. Metric-Aware Nearest-Neighbor… · p. 9

  • Results

    Across constrained manipulation scenarios, RMRRT demonstrated high planning reliability and improved efficiency over representative constrained-planning baselines, with real-world hardware experiments also reported.

    Source: §V · p. 16

  • Takeaways & Limitations

    RMRRT constructs local geometry online from planning-time quantities, unlike approaches requiring problem-specific offline roadmaps or learned representations.

    Source: B. Computational Overhead · p. 14

    The current implementation primarily uses signed-distance-based geometric proxy inequalities; future work will investigate learned models and surrogates for inequality information.

    Source: §V · p. 16

Abstract

from arXiv · show

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

I. INTRODUCTION

Sampling-based planning must handle equality constraints that restrict configurations to a manifold and inequality constraints that further narrow admissible regions. RMRRT couples both in a local geometry used for exploration, addressing a gap in existing approaches.

I. INTRODUCTION

  • Equality constraints define a manifold, while inequality constraints further restrict admissible configurations; conventional planners often separate manifold projection from binary validity checks.This decoupling can cause sample rejection and inefficient exploration in narrow or fragmented regions.Source: §I · p. 1

1) Manifold-Based Motion Planning:

  • Manifold-based planners preserve equality feasibility with charts or tangent-space motions, but often leave inequality boundaries to post-hoc validity checks.Repeated unproductive extensions can result in narrow or cluttered environments.Source: 1) Manifold-Based Motion Planning · p. 2

2) Metric-Based Motion Planning:

  • Metric-based planners encode inequality-aware local geometry, but generally do not directly generate equality-consistent motion on implicit manifolds.Applying them to equality-constrained steering therefore requires additional projection or correction.Source: 2) Metric-Based Motion Planning · p. 2

3) Data-Driven Motion Planning:

  • Data-driven planners use learned priors or offline roadmaps to accelerate exploration, with performance depending on prior quality and whether the target problem changes.The relevant changes include the environment, robot model, or constraints.Source: 3) Data-Driven Motion Planning · p. 2

4) Trajectory Optimization on Constraint Manifolds:

  • Trajectory-optimization methods jointly handle both constraint classes while locally refining an initialized trajectory, unlike sampling-based planners that explore for feasible connections.The paper presents the two paradigms as complementary.Source: 4) Trajectory Optimization on… · p. 2

5) Structural Gap Across Existing Approaches:

  • Existing approaches generally lack one local geometry that couples equality-consistent motion with sensitivity to nearby inequality boundaries during sampling-based exploration.The paper identifies a mismatch between manifold methods, ambient-space metrics, learned priors, and trajectory refinement.Source: 5) Structural Gap Across Existing… · p. 2

B. Overview of Our Approach

  • RMRRT builds an ambient barrier metric and uses a metric-compatible projection to induce a tangent-space metric for equality-constrained planning.The evaluated instantiation uses signed-distance-based geometric proxy inequalities; standard validity checks separately enforce hard feasibility.Source: B. Overview of Our Approach · p. 3
  • RMRRT uses the induced metric in both steering and nearest-neighbor selection, and experiments report improved efficiency and reliability over representative constrained-planning baselines.The experiments include simulation and real-world tasks involving multi-arm and humanoid systems.Source: B. Overview of Our Approach · p. 3

II. PRELIMINARIES

The preliminaries define the feasible set under equality and inequality constraints, then introduce the manifold and tangent-space tools used for constraint-consistent planning.

II. PRELIMINARIES

  • A proxy point is a body-fixed point attached to a kinematic link; its world position and Jacobian, together with its signed obstacle distance, feed the ambient barrier metric.The notation also identifies configuration-space and constraint dimensions, regularization, and proxy-point counts.Source: §II · pp. 3–4
  • Inequalities can remove regions from the equality manifold, creating holes in the admissible subset that tangent-space steering alone does not capture.RMRRT represents this inequality-induced geometry with a tangent-space Riemannian metric.Source: Figure 2 · p. 4

A. Problem Formulation

  • The feasible set consists of configurations satisfying g(q) = 0 and strict inequality margins hi(q) > 0, with valid starts and goals assumed.The open inequality set suits the log barrier, which is finite in the interior and diverges at the boundary; separate validity checks enforce hard feasibility.Source: A. Problem Formulation · p. 4

B. Equality-Constrained Manifolds and Tangent Spaces

  • When the equality-constraint Jacobian has full row rank, the equality set is a smooth manifold whose tangent space contains infinitesimal motions preserving equality constraints to first order.The Euclidean projector maps vectors onto this tangent space, supporting constraint-consistent expansions and retraction when needed.Source: B. Equality-Constrained Manifolds and… · p. 4

C. Inequality Constraints and Log-Barrier Functions

RMRRT combines equality-manifold admissibility with inequality-aware local geometry, using a barrier metric restricted to the equality-consistent tangent space. The resulting geometry informs planner operations while hard feasibility remains subject to separate validity checks.

C. Inequality Constraints and Log-Barrier Functions

  • The log-barrier formulation smoothly encodes proximity to inequality boundaries, whose curvature can inform a geometry-aware planning metric.The barrier grows rapidly as a configuration approaches the admissible-region boundary.Source: C. Inequality Constraints and… · pp. 4–5

A. Proposed Framework

  • RMRRT constructs an ambient barrier metric and restricts it to the equality-consistent tangent space, using the induced geometry for planner operations.The metric informs steering, nearest-neighbor selection, and manifold correction.Source: §III · p. 5, A. Proposed Framework · p. 5

B. Riemannian Barrier Metric Construction

  • A Gauss–Newton surrogate retains the dominant positive-semidefinite barrier-curvature contribution because exact barrier Hessians can be expensive and numerically fragile in cluttered scenes.Signed-distance fields may become nonsmooth under closest-feature changes and contact switching.Source: B. Riemannian Barrier Metric… · p. 5
  • The evaluated metric uses signed-distance-based geometric proxy inequalities, while full-resolution state- and edge-validity checks enforce hard feasibility independently of proxy placement.The metric is an exploration bias, not a feasibility certificate; the main empirical claims are restricted to this proxy-based instantiation.Source: B. Riemannian Barrier Metric… · p. 6
  • The framework can incorporate other differentiable inequality margins when they yield symmetric positive-semidefinite metric contributions.The paper gives a manipulability margin as an example, while keeping its main empirical claims limited to signed-distance-based geometric proxy inequalities.Source: B. Riemannian Barrier Metric… · p. 6
  • The G-orthogonal projection induces a tangent-space metric used in subsequent planner operations, and its construction assumes the equality Jacobian has full row rank.At singular configurations, the projector and minimum-G-norm correction are not uniquely defined, and traversal of kinematic singularities is not guaranteed.Source: B. Riemannian Barrier Metric… · p. 7

C. Metric-Induced Tangent-Space Steering

Metric-induced steering chooses a tangent direction that makes first-order progress toward a target while minimizing local metric energy. It uses only the tangent component of the ambient target displacement, then takes a fixed-length step after metric normalization.

C. Metric-Induced Tangent-Space Steering

  • The steering rule selects a tangent direction that makes first-order progress toward the target while minimizing local metric energy.The induced metric biases the direction away from nearby active inequality boundaries.Source: C. Metric-Induced Tangent-Space Steering · p. 8
  • The target sample need not satisfy equality constraints because steering uses only the tangent projection of its ambient displacement.The finite-step state is subsequently corrected back onto the manifold.Source: C. Metric-Induced Tangent-Space Steering · p. 8
  • With a full-row-rank equality Jacobian and positive-definite ambient metric, the induced tangent-space metric is positive definite and the steering minimizer is unique.The procedure reprojects for numerical robustness, normalizes in the induced metric, and takes a fixed-length step.Source: C. Metric-Induced Tangent-Space Steering · p. 8

D. Metric-Aware Nearest-Neighbor Selection

RMRRT selects expansion parents using the same induced tangent-space metric used for steering, favoring nodes by locally equality-consistent motion cost. To limit search cost, it reranks a Euclidean-preselected candidate set.

D. Metric-Aware Nearest-Neighbor Selection

  • The nearest-neighbor rule ranks nodes by the local cost of tangent-projected displacement under M(q), favoring locally well-connected nodes over merely Euclidean-nearest ones.The sampled configuration need not lie on the manifold; projection makes the comparison depend on locally equality-consistent first-order motions.Source: D. Metric-Aware Nearest-Neighbor… · pp. 8–9
  • Metric-aware reranking follows Euclidean preselection of K candidates, avoiding an exhaustive metric-aware search over all tree nodes.The candidate pool grows logarithmically with tree size as a practical balance between parent-selection quality and evaluation cost.Source: D. Metric-Aware Nearest-Neighbor… · p. 9
  • The distance is evaluated with the metric at the tree node, aligning parent selection with the local geometry used for subsequent steering.The implementation chooses K adaptively as the maximum of 4 and the floor of 2 log2(|V| + 1).Source: D. Metric-Aware Nearest-Neighbor… · p. 9

E. Overall RMRRT Algorithm

RMRRT alternates metric-aware expansion and validity checks in a bidirectional planning loop, with local correction for finite-step equality drift. Under stated regularity, clearance, and coverage conditions, its geometric components are compatible with standard path-covering arguments, not an independent completeness proof.

E. Overall RMRRT Algorithm

  • Each iteration samples in ambient space, selects a metric-aware neighbor, steers, checks state and edge validity, and extends the opposite tree toward a valid new node.The planner extracts a path when the connect phase reaches the new node.Source: E. Overall RMRRT Algorithm · p. 9
  • A minimum-G-norm correction removes first-order equality violation caused by finite-step drift, while tangent-space geometry determines admissible motion.The correction is a local numerical recovery mechanism, and hard feasibility remains enforced by state and edge validity checks.Source: E. Overall RMRRT Algorithm · pp. 9–10
  • The metric-aware components are compatible with standard path-covering arguments under regularity, positive path clearance, and local-neighborhood coverage assumptions.This is compatibility with completeness arguments, not an independent completeness proof for the implemented planner.Source: E. Overall RMRRT Algorithm · p. 10
  • The implementation uses a regularized Gauss–Newton surrogate rather than requiring the exact barrier Hessian for the compatibility argument.The stated conditions include positive regularization, full-row-rank equality Jacobian along the path, positive clearance, and local-neighborhood coverage for finite K.Source: E. Overall RMRRT Algorithm · p. 10

F. Computational Complexity and Practical Overhead

The section examines metric-aware overhead and proxy-density effects. The deployed proxy density balances metric informativeness and planning efficiency, while metric construction remains inexpensive relative to validity checking in the tested range.

F. Computational Complexity and Practical Overhead

  • m_proxy = 11 achieved the lowest planning time, fewest added nodes and iterations, and shortest path among tested densities while maintaining a high valid-extension ratio.The density sweep characterizes the trade-off for a fixed geometric link-coverage heuristic rather than tuning proxy density separately for each scene.Source: Table I · p. 11, F. Computational Complexity and… · p. 11
  • When the number of equality constraints is much smaller than the configuration dimension, metric-aware overhead is dominated by local metric construction and scales linearly with proxy points.Nearest-neighbor reranking adds cost over the K Euclidean-preselected candidates.Source: F. Computational Complexity and… · pp. 10–11
  • Metric-aware computations reuse signed distances and point Jacobians from validity evaluation, making their marginal overhead largely algebraic.In cluttered environments, reduced rejection-driven validity-check cycles can partially offset this cost.Source: F. Computational Complexity and… · p. 11
  • Ambient-metric build time increases approximately linearly with proxy-point count and remains sub-millisecond even at 10× the deployed density.Within the tested range, collision and validity checking—not metric construction—remain the dominant per-iteration cost.Source: F. Computational Complexity and… · p. 11, Figure 4 · p. 11

IV. EXPERIMENTS

The experiments assess RMRRT's reliability, efficiency, and practical applicability under coupled equality and inequality constraints across robotic systems and manipulation scenarios.

IV. EXPERIMENTS

  • The evaluation spans single-arm, multi-arm, and humanoid systems, reporting planning success, runtime, computational overhead, ablations, and hardware experiments.The metric encodes distance- and proximity-based geometric inequalities, while other feasibility constraints are enforced separately by state- and edge-validity checks.Source: §IV · p. 11

A. Simulation Benchmarks

RMRRT was evaluated across constrained manipulation scenarios using fixed geometric proxies and comparisons with manifold-, guidance-, learning-, and precomputation-based planners. It achieved 100% success in every scenario while maintaining low mean planning time and low inter-trial variability.

A. Simulation Benchmarks

  • The evaluation used 100 independent feasible queries per scenario, a 300-second success limit, identical validity checks, and baselines spanning constrained, guidance-based, learning-based, and precomputed planners.The benchmark scenarios range from single-arm to triple-arm systems; Fig. 5 depicts four representative tasks and planned trajectories.Source: A. Simulation Benchmarks · pp. 11–12, Figure 5 · p. 12
  • Proxy points were fixed offline using a link-coverage heuristic, with object-coupling proxies for multi-arm transport and clusters or corner proxies for larger humanoid links and carried objects.Single points represented host links in manipulator scenarios; the Unitree H1-2 used local point clusters and corner proxies for its carried object.Source: A. Simulation Benchmarks · p. 12
  • 100% success in all scenarios, with low mean planning time and inter-trial variability; Scenario 4 involved a 21-DoF triple-arm system with 12 equality constraints.RMRRT maintained this performance while traversing a narrow admissible region in Scenario 4.Source: A. Simulation Benchmarks · p. 12
  • RMRRT combines an inequality-aware ambient metric, equality-consistent tangent projection, metric-aware node selection, and sampling-based nonlocal exploration without task-specific training data or precomputed roadmaps.Manifold-based planners generally do not account directly for nearby inequality boundaries, while sequentially combining ambient guidance with equality projection can weaken obstacle-avoidance bias.Source: A. Simulation Benchmarks · p. 13

B. Computational Overhead

RMRRT adds online geometric computation, but its steering stage is much cheaper than projection-based CBiRRT steering, and its local geometry can be reused without problem-specific offline representations.

B. Computational Overhead

  • 17.56 ms for RMRRT steering versus 999.35 ms for CBiRRT; total per-iteration runtime was 18.83 ms for RMRRT versus 1004.22 ms for CBiRRT.RMRRT generates its extension direction within the tangent space and uses lightweight local correction, while CBiRRT repeatedly projects during local steering.Source: B. Computational Overhead · p. 14
  • 18.83 ms per iteration for RMRRT versus 4.86 ms for LJCMP, but Scenario 4 averaged approximately 13.6 ± 17 iterations for RMRRT versus 77.0±270 for LJCMP.The passage attributes fewer expansion cycles to more effective progress per expansion, offsetting RMRRT's higher per-iteration cost.Source: B. Computational Overhead · p. 14
  • RMRRT constructs local geometry online and avoids problem-specific offline representations that generally must be rebuilt or retrained when the robot, environment, or constraints change.The online costs include metric construction, nearest-neighbor evaluation, steering, and validity checking; metric construction adds only a small fraction of total runtime.Source: B. Computational Overhead · pp. 13–14

C. Ablation Study

The ablation separates metric-induced steering from metric-aware nearest-neighbor selection and tests their combination on a narrow-passage constrained transfer task. Full RMRRT achieves the best reported path-length outcomes, while the individual components improve different aspects of exploration.

C. Ablation Study

  • The four variants selectively use the barrier-informed metric for both steering and nearest-neighbor selection, steering only, nearest selection only, or neither; other major settings remain identical.The study targets narrow-passage constrained transfer with a Unitree H1-2 single-humanoid dual-arm system.Source: C. Ablation Study · p. 14
  • Both variants using metric-induced steering reach a 0.349 valid extension ratio, the highest among the four variants, and reduce validity-checking cost relative to No barrier.The comparison is reported in Table IV and illustrated by Fig. 6, where Full RMRRT takes a more direct passage trajectory than No barrier.Source: C. Ablation Study · p. 15, Table IV · p. 15, Figure 6 · p. 15
  • Nearest-only barrier significantly reduces average path length relative to No barrier, indicating better parent-node quality and path formation under the induced geometry.Its valid extension ratio is not fully specified in the supplied passage.Source: C. Ablation Study · p. 15
  • Improved parent selection alone remains insufficient for local extension quality: the resulting extension can still move in a locally unproductive direction.The reported local extension quality is comparable to No barrier when steering remains unaware of inequality-induced geometry.Source: C. Ablation Study · p. 15
  • Full RMRRT achieves the shortest average path length and smallest path-length standard deviation among the four variants.The passage reports these as improvements in mean trajectory quality and inter-trial consistency.Source: C. Ablation Study · p. 15, Table IV · p. 15

D. Real-World Quantitative Evaluation

The hardware evaluation tests RMRRT on triple-arm cooperative transport under noisy signed-distance inputs; the conclusion frames its tangent-space barrier metric as a unified steering and nearest-neighbor geometry.

Robustness to hardware noise

V. CONCLUSION

  • RMRRT induces a tangent-space metric from an ambient barrier metric for steering and nearest-neighbor selection, biasing expansion away from infeasible regions while preserving first-order equality consistency.The conclusion reports high planning reliability and improved efficiency over representative constrained-planning baselines across single-arm, multi-arm, and humanoid scenarios.Source: §V · p. 16
  • The current implementation primarily uses signed-distance geometric proxy inequalities, whose repeated metric evaluations must remain informative, inexpensive, and numerically stable.Future work will investigate learned SDF/RDF models and constraint surrogates to approximate inequality information efficiently.Source: §V · p. 16

APPENDIX A PROXY AND METRIC PARAMETERS

The appendices specify per-scenario proxy settings and examine ambient-space performance, a manipulability extension, and adaptive metric-aware nearest-neighbor selection.

APPENDIX A PROXY AND METRIC PARAMETERS

  • Scenarios 1–4 use δact = 0.15 m and δsafe = 0.02 m; the H1-2 humanoid narrow-passage task uses δact = 0.03 m and δsafe = 0.01 m with milder barrier weights.All scenarios share λ = 1.0 and ϵd = 10^-3; the humanoid setting avoids overly conservative expansion in the narrow passage.Source: Appendix A · p. 17

APPENDIX B AMBIENT-SPACE COMPARISON ON MOTIONBENCHMAKER

  • RMRRT and CUROBO each achieved 100% success in both MotionBenchMaker scenes, while CUROBO produced the shortest average paths.CHOMP performed well in the open box but degraded in the more enclosed cage; APF and RMP showed similar sensitivity to local minima.Source: Table VII · p. 17, Appendix B · p. 17

METRIC TERM

  • With the manipulability term, minimum translational manipulability increased from approximately 0.023 to 0.040 along the near-singular reaching path.The experiment is a qualitative proof of concept and does not establish singularity traversal or restore the regularity assumptions for tangent-space projection and completeness compatibility.Source: METRIC TERM · p. 18, Figure 10 · p. 18, Figure 9 · p. 18

APPENDIX D CANDIDATE-SIZE ANALYSIS FOR METRIC-AWARE

  • All candidate-size regimes solved every Scenario 4 instance, but the adaptive rule yielded the lowest and most consistent planning time without exhaustive reranking.It uses a compact candidate set for small trees and increases K logarithmically as the tree grows.Source: Table VIII · p. 18, Appendix D · p. 18
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