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Induced Riemannian Metrics for Motion Planning with Constraints

Phone Thiha Kyaw, Thomas Cohn, Miguel Angel Rogel Garcia, Jonathan Kelly

arXiv:2609.25695v1cs.RO

TL;DR

Constraints confine feasible robot motion to a curved submanifold, where representation choices can affect the path length a planner optimizes. The paper derives a shared induced metric for implicit and explicit representations and uses it in constrained planners; across six bimanual problems, flat parameter-domain planning gives longer motions under both induced metrics.

  • Problem

    Existing approaches do not explicitly account for constraint-manifold geometry or the effects of non-Euclidean ambient metrics, although these determine the cost of feasible motion.

    Source: §I · pp. 1–2

  • Method

    The paper proves that implicit and explicit representations induce the same metric for any ambient Riemannian metric, then applies it in a sampling-based planner and trajectory optimizer.

    Source: §I · p. 2

  • Results

    Across six bimanual problems, planners return the shortest motion under their chosen metric in both representations, while flat parameter-domain planning gives longer motions under both induced lengths.

    Source: §VII · p. 8

  • Takeaways & Limitations

    The shared induced geometry lets planners choose the metric and representation independently while optimizing the same constrained-motion length.

    Source: §VII · pp. 7–8

Abstract

from arXiv · show

In constrained motion planning problems, task and loop-closure constraints restrict a robot's motion to a curved, lower-dimensional submanifold of its configuration space. Planners measure path length with a metric, which sets the cost of moving in each direction. Under the Euclidean metric, this cost is the same everywhere, whereas under a general Riemannian metric, such as the kinetic-energy metric, the cost can vary with direction and configuration. Existing methods often describe the submanifold either implicitly, as a constraint level set, or explicitly, through a parameterization. The implicit representation is typically combined with the Euclidean metric of the configuration space, and the explicit representation with the parameter domain, so the path length that a planner minimizes depends on the representation. Instead, we measure path length with the induced metric, which the submanifold inherits from a Riemannian metric on the configuration space. The implicit and explicit representations yield the same induced metric, expressed in different coordinates, and hence the same geometry. This result holds for any Riemannian metric on the configuration space, not only the Euclidean one. The choice of metric is therefore independent of the choice of representation. Using this result, we extend planning under a Riemannian metric from unconstrained spaces to constraint submanifolds by applying the induced metric in both a sampling-based planner and a trajectory optimizer. For an explicit representation, the induced metric also accounts for the distortion that the parameterization introduces. In experiments on a bimanual manipulation setup with two Franka arms under end-effector task constraints, we compare the Euclidean and kinetic-energy metrics.

I. INTRODUCTION

Task and loop-closure constraints confine motion to a curved submanifold, where intrinsic geometry determines path cost. The paper proves that implicit and explicit representations yield the same induced metric and applies it in two planning methods.

I. INTRODUCTION

  • Earlier approaches did not explicitly address constraint-manifold geometry or the effects of non-Euclidean ambient metrics.Prior methods often used the ambient Euclidean metric for implicit representations or corrected parameterization distortion without identifying the induced metric.Source: §I · pp. 1–2
  • For any ambient Riemannian metric, implicit and explicit constraint representations induce the same metric and therefore the same geometry on the submanifold.The metrics are related by an invertible change of basis.Source: §I · p. 2
  • A sampling-based planner and a trajectory optimizer use the induced metric so both minimize the same Riemannian length on the constraint manifold.The paper describes this as extending planning under Riemannian metrics to constrained spaces.Source: §I · p. 2
  • The approach is demonstrated in simulation and on a real bimanual setup, comparing Euclidean and kinetic-energy metrics under a loop-closure constraint.The setup uses two Franka arms.Source: §I · p. 2

II. RELATED WORK

Prior constrained-planning methods use implicit constraints or explicit parameterizations, while related metric-based methods generally plan on unconstrained spaces. The preliminaries frame motion costs through Riemannian geometry and its coordinate-invariant length.

II. RELATED WORK

  • Implicit planners commonly use ambient Euclidean costs, while explicit parameterizations can distort distances; the paper identifies the correction as the pullback of the ambient metric.Earlier constrained trajectory optimizers also use Euclidean length or smoothness on a chart or parameter domain rather than the induced length.Source: §II · p. 2
  • Prior work applies general Riemannian metrics to unconstrained spaces, whereas this work uses the ambient metric to define the geometry inherited by a constraint manifold.Examples include kinetic-energy metrics from the mass matrix and metrics shaped around obstacles or singularities.Source: §II · p. 2

III. PRELIMINARIES

  • A Riemannian metric assigns a smoothly varying inner product to each tangent space, providing the geometric basis for measuring motion.The paper assumes the configuration manifold is an open subset of R^n.Source: A. Riemannian Manifolds and Metrics · p. 2

A. Riemannian Manifolds and Metrics

  • Riemannian length is invariant under coordinate changes, and the induced distance and geodesics provide geometric notions of separation and locally shortest motion.The metric transforms as a (0, 2)-tensor under a change of coordinates.Source: A. Riemannian Manifolds and Metrics · p. 3

B. Constraint Submanifolds

Task and loop-closure constraints define feasible configurations as a zero level set. Under the stated full-rank assumption, this set is a smooth lower-dimensional submanifold whose feasible-motion costs come from restricting the ambient metric.

B. Constraint Submanifolds

  • A smooth constraint map h defines the feasible configurations as its zero level set Q = {q ∈ M : h(q) = 0}.The map encodes task and loop-closure constraints and takes values in R^k, with k < n.Source: B. Constraint Submanifolds · p. 3
  • Assuming the constraint Jacobian has full row rank on Q excludes singular configurations and makes Q a smooth submanifold of dimension d = n − k.The tangent space at q is the kernel of the constraint Jacobian.Source: B. Constraint Submanifolds · p. 3
  • Feasible velocities lie in ker Dh(q), and restricting the ambient metric to those tangent spaces gives the induced metric that determines feasible-motion cost.The velocity condition follows by differentiating the constraint along a curve that stays on Q.Source: B. Constraint Submanifolds · p. 3

C. Problem Formulation

The problem is to find a collision-free path on the constraint manifold that minimizes its induced Riemannian length. The section introduces the manifold geometry and explains how implicit projection and tangent-space operations support feasible motion.

C. Problem Formulation

  • The planner seeks a feasible path from a collision-free start to a goal that minimizes length under the induced metric; the minimum is assumed attained, as it is when Qfree is compact.Because Q is generally curved, the solution need not be straight in ambient coordinates.Source: C. Problem Formulation · p. 3
  • A planner measuring distance in ambient coordinates optimizes the wrong quantity, so its subroutines must follow Q’s induced geometry.The problem is formulated over piecewise smooth paths that remain collision-free.Source: C. Problem Formulation · p. 3

IV. INDUCED GEOMETRY OF CONSTRAINT SUBMANIFOLDS

  • The section derives the inherited metric from both the implicit level-set representation and an explicit parameterization, then proves their equivalence.The equivalence is stated for any ambient Riemannian metric.Source: §IV · p. 3

A. Implicit Representation

  • In the implicit representation, the constraint Jacobian H defines the tangent space as ker H, while ambient operations are projected back onto Q.The tangent space gives feasible directions; the normal space is used for return to the manifold.Source: A. Implicit Representation · p. 3
  • Iterating the projection update to convergence defines ΠQ, which maps an ambient configuration to a nearby point on Q.A damped pseudoinverse and Tikhonov regularization keep the update well conditioned when H loses rank.Source: A. Implicit Representation · p. 3
  • A tangent step followed by projection back onto Q gives a valid retraction, and the induced metric is represented using an orthonormal basis of ker H.The metric is positive definite and varies through both Gq and the changing basis B.Source: A. Implicit Representation · p. 3
  • The projection correction is Euclidean-normal rather than Gq-normal, so the retraction is second order under the Euclidean metric but generally first order under Gq.The Gq-normal space is im Gq^-1 H^T.Source: A. Implicit Representation · p. 3

B. Explicit Parameterization

An explicit parameterization satisfies the constraint by construction and provides coordinates for feasible configurations. Pulling the ambient metric back through its differential preserves path lengths on Q, whereas treating the parameter domain as flat distorts distances.

B. Explicit Parameterization

  • The parameterization maps an open domain U into Q while satisfying the constraint by construction, allowing feasible configurations to be sampled without projection.Its differential spans the same tangent space as ker H.Source: B. Explicit Parameterization · p. 4
  • The differential maps a parameter velocity to a tangent velocity on Q, and the induced metric in parameter coordinates measures that motion’s ambient-metric length.The resulting matrix is symmetric positive definite.Source: B. Explicit Parameterization · p. 4
  • Parameter-domain curves measured with the induced metric have exactly their image curves’ lengths on Q, while a flat parameter-domain metric distorts distances.The bimanual task uses a parameterization derived from an analytic inverse-kinematics solution.Source: B. Explicit Parameterization · p. 4

C. Equivalence of the Two Representations

The implicit and explicit constructions express the same induced metric in different bases. An invertible basis change preserves tangent-vector lengths and therefore the submanifold’s inner product and geometry.

C. Equivalence of the Two Representations

  • The columns of the parameterization differential and kernel basis span the same tangent space, so one basis is related to the other by an invertible change of basis.The matrix R is invertible because both bases have full column rank.Source: C. Equivalence of the Two… · p. 4
  • The theorem states that the implicit induced metric and explicit pullback metric are equivalent, with an invertible matrix relating them.The equivalence holds for any positive definite ambient metric, including constant and smoothly varying metrics.Source: C. Equivalence of the Two… · p. 4
  • For a tangent vector represented in either basis, the coordinate transformation makes its squared length identical under the two metrics.The coordinates satisfy ξ = R u_dot.Source: C. Equivalence of the Two… · p. 4
  • Because the metrics assign every tangent vector the same length, they define the same inner product and hence the same lengths, angles, and geodesics on Q.The relation is the tensor transformation law for the change of basis.Source: C. Equivalence of the Two… · p. 4

V. PLANNING WITH THE INDUCED METRIC

The section applies the induced metric within existing motion-planning algorithms without changing their search strategies.

V. PLANNING WITH THE INDUCED METRIC

  • Existing planning algorithms can use the induced metric without changing their underlying search strategies; the section treats sampling-based planning and trajectory optimization.It develops the geometry-dependent sampling-based subroutines first, then the optimizer treatment.Source: §V · p. 4

A. Sampling-Based Planning

The planner must use the constraint submanifold’s induced geometry for sampling, connections, and path costs. A midpoint approximation makes induced distances practical to evaluate, and the resulting cost is equivalent across implicit and explicit representations.

A. Sampling-Based Planning

  • All planner operations must respect the submanifold’s intrinsic geometry, and asymptotic optimality recovers the desired solution only when the planner optimizes induced Riemannian length.Sampling, local connections, and candidate-path comparison all depend on the search-space geometry.Source: A. Sampling-Based Planning · p. 4
  • Implicit sampling projects ambient samples onto the constraint manifold and rejects failed projections, whereas explicit parameter-domain sampling satisfies the constraint by construction.Implicit methods may project samples from either the ambient manifold or a tangent space.Source: A. Sampling-Based Planning · p. 5
  • Connections remain feasible using representation-specific curves, while their costs use the induced metric rather than a constant ambient metric.Explicit planning maps parameter-domain chords to feasible curves; implicit planning projects ambient chords, and the cheaper connections reserve planner budget for sampling and rewiring.Source: A. Sampling-Based Planning · p. 5
  • A midpoint approximation evaluates the induced metric once per edge and agrees with true induced distance to third order in the separation between configurations.The implicit inverse retraction projects differences onto the tangent space and iteratively corrects them; explicit inverse retraction is the difference between parameters. The resulting edge cost yields the same optimization problem in either representation.Source: A. Sampling-Based Planning · p. 5

B. Trajectory Optimization

The trajectory optimizer represents paths in the explicit parameterization’s minimal coordinates, lifts them to configuration space, and minimizes an approximation of Riemannian path energy.

B. Trajectory Optimization

  • The optimizer represents the trajectory in minimal parameter coordinates and lifts it to configuration space to apply costs and constraints.It uses a B-spline path and imposes collision avoidance, joint limits, reachability constraints at sampled points, and fixed endpoints.Source: B. Trajectory Optimization · p. 5
  • For configuration-space Riemannian metrics, the objective approximates path energy, encouraging evenly spaced control points and more reliable pointwise constraints.Path length can be nonsmooth when adjacent control points coincide.Source: B. Trajectory Optimization · p. 5
  • The metric is evaluated on a lifted configuration-space chord rather than pulled back to parameter coordinates, avoiding the need for parameterization second derivatives in gradient-based optimization.The scale factor is set so the initial guess has cost 1, making the solver’s optimality tolerance consistent across trials and metric choices.Source: B. Trajectory Optimization · pp. 5–6
  • A feasible parameter-space initial trajectory is sampled at control-point locations, and trajectory timing is determined after optimization using TOPP-RA.The initial guess may already be feasible or sufficiently close for the optimizer to recover feasibility.Source: B. Trajectory Optimization · p. 6

VI. EXPERIMENTS

Across six bimanual manipulation problems, planners and optimizers using induced metrics produce shorter motions under their chosen metric than the flat parameter-domain metric. Experiments also validate agreement between implicit and explicit induced lengths and compare optimizer convergence across metric-based initial guesses.

VI. EXPERIMENTS

  • The experiments use two 7-DoF Franka arms holding a book under loop closure, with six transfers between three shelf compartments and Euclidean, kinetic-energy, and flat metrics.The planners run for 10 seconds; each problem uses 50 trials with different pseudorandom seeds, and results are reported as medians.Source: §VI · p. 6

A. Empirical Studies

  • The implicit and explicit measurements of induced path length match, while the parameterization is not an isometry and its distortion increases under the kinetic-energy metric.The comparison uses the same solution paths, and the reported pullback-metric eigenvalue ratio summarizes distortion across 10,000 sampled configurations.Source: A. Empirical Studies · p. 6, Table I · p. 6

B. Constrained Bimanual Manipulation Problems

  • The flat metric produces the longest motions under both induced lengths, while each induced metric ranks motions differently and determines which motion the planner returns.The results cover both representations and six bimanual problems; Table II reports median induced lengths over 50 trials.Source: B. Constrained Bimanual Manipulation… · p. 6, Table II · p. 7
  • The optimizer converges fastest on most problems from an initial guess generated under the same metric it minimizes, while flat-metric guesses are slowest under both induced metrics on every problem.The optimized motions are the same across initial guesses; only solve time differs. Table III reports median solve times over 50 trials.Source: B. Constrained Bimanual Manipulation… · p. 6, Table III · p. 7

C. Execution on the Real Bimanual Setup

In real bimanual trials, Euclidean plans have shorter joint-space paths and faster execution, while kinetic-energy plans minimize integrated kinetic energy. The conclusion is that induced geometry lets planners choose metric and representation independently, with metric-weighted projection left for future work.

C. Execution on the Real Bimanual Setup

  • Kinetic-energy plans achieve the lowest integrated kinetic energy for every task, while taking 17% to 30% longer than Euclidean plans, which have the shortest joint-space paths and fastest execution.Each task was run five times under each objective, with median results reported.Source: C. Execution on the Real Bimanual Setup · p. 7, Table IV · p. 8

VII. CONCLUSION

  • Implicit and explicit representations yield one induced geometry, allowing planners to choose their metric and representation independently while minimizing the same submanifold length functional.Across six bimanual problems, planners return the shortest motion under their chosen metric in both representations; treating the parameter domain as flat produces longer motions under both induced lengths.Source: §VII · pp. 7–8
  • The retraction projects along the Euclidean normal, so it is only first order under a general ambient metric; metric-weighted projection is left for future work.The authors also identify trajectory optimization directly in the implicit representation as future work.Source: §VII · p. 8
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