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Approximating High Dimensional Self-Motion Manifolds via Deep Generative Models

Haitao Gao, Yang Song, Liao Wu

arXiv:2609.18169v1cs.RO

TL;DR

Existing SMM methods are costly or built around curves, limiting their reach to higher-dimensional solution sets. The paper learns a conditional distribution over robot configurations and clusters samples to recover SMM components. It reports 1-D performance comparable to existing approaches and demonstrates 4-D SMM approximation for a 7R position task.

Abstract

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Self-motion manifold (SMM) characterizes the geometric structure of the infinite inverse kinematic solutions set of a redundant manipulator at a fixed end-effector pose, and its efficient recovery underpins feasible and global optimal motion planning. Existing methods such as null-space continuation and learning-based methods are formulated around the assumption that an SMM is a curve, and do not extend to higher redundancy orders. We instead adopt a probabilistic view: SMMs are the support of the conditional posterior over configurations given a target pose, so that recovering it reduces to sampling from a learned distribution and separating its disjoint components by clustering. The formulation is independent of the manifold dimension and requires no architectural change as the redundancy order grows. In this work, we demonstrate that our method can approximate 1-D SMMs with performance comparable to the latest null-space continuation and learning-based approach, and that it is the first method capable of approximating highly redundant 4-D SMMs in a 7R manipulator for position tasks. Project website: \href{https://github.com/accuracy-maker/high-dimenstional-self-motion-manifold-approximation}{https://github.com/accuracy-maker/high-dimenstional-self-motion-manifold-approximation}

I. INTRODUCTION

As manipulators grow more redundant, their fixed-pose inverse-kinematics solutions form higher-dimensional self-motion manifolds that are difficult to represent with methods designed for curves. The paper models these manifolds through conditional sampling and clustering, with a formulation that extends to higher dimensions and enforces joint limits by construction.

I. INTRODUCTION

  • The paper frames the need for higher-dimensional SMM methods in the growth of redundant designs, including hands and humanoids with redundancy often exceeding 1-D.Redundant manipulators have more joint DoF than the workspace task requires, yielding infinitely many solutions for a fixed target pose.Source: §I · p. 1
  • Existing Fourier- and ODE-based approaches focus on 1-D manifolds and do not extend trivially to higher-dimensional SMMs.A cellular-automata search tested r = 2 but is computationally expensive.Source: §I · p. 1
  • The method learns p(q | x) and clusters generated configurations to recover distinct self-motion manifold components.Here q is the robot configuration and x the desired end-effector pose.Source: §I · pp. 1–2
  • The method generates configurations satisfying joint limits by construction, unlike ODE- and Fourier-based representations that do not directly accommodate them.Enforcing joint limits can violate the periodic assumptions underlying those representations.Source: §I · p. 2
  • The approach is presented as the first generative method for approximating high-dimensional SMMs, including 4-D manifolds.The paper also describes integration of alternative generative models and clustering algorithms as plug-and-play.Source: §I · p. 2

II. RELATED WORKS

Self-motion manifolds collect the inverse-kinematics solutions for a fixed target, but their structure and number of components vary with the target. Existing approaches face computational costs or are limited to curves, motivating a learned sampling-and-clustering alternative.

II. RELATED WORKS

  • For a unit-length 3R manipulator, the inverse-kinematics preimage has two disjoint closed curves when ρ < 1 and one when 1 < ρ < 3.The disc ρ < 3 is the reachable workspace; SMM branches vary with target and may be disjoint.Source: §II · p. 2
  • Null-space continuation traces a closed curve by stepping along the Jacobian null space, but is sensitive near singularities because the Jacobian pseudoinverse can blow up.An ODE-based initial-value method for 1-D SMMs uses standard high-order explicit integrators [9].Source: §II · p. 2
  • Sampling-based approaches generate configurations and cluster them; prior examples use IK-driven random samples or grid exploration of cells intersected by neighboring null spaces.The grid-search approach is described in [10].Source: §II · p. 2
  • Study [11] integrates neural-network classification with cellular automata search: the networks identify SMM homotopy classes and regress Fourier-series coefficients, yielding a closed-form curve representation of 1-D SMMs.Source: §II · p. 2
  • Existing methods can be prohibitively costly or fail to extend readily to high-dimensional SMMs, where exhaustive tracing becomes intractable.Cellular-automata search is cited as costly [10], [11], while the ODE method does not extend readily [9].Source: §II · p. 2
  • The proposed probabilistic view treats an SMM as support of p(q | x), then clusters generated configurations to identify disjoint components.Flow matching is adopted for simplicity and stability relative to diffusion, with a fixed number of network evaluations independent of manifold dimension.Source: §II · p. 2

III. BACKGROUND

The background defines the configuration and task spaces and relates redundancy to the dimension and connected components of the inverse-kinematics solution set. For regular target poses, each connected component is a self-motion manifold.

A. Configuration Space and Workspace

  • Configuration space C describes joint configurations, with one degree of freedom contributed by each joint.For revolute manipulators, C has the geometry of a torus manifold.Source: A. Configuration Space and Workspace · pp. 2–3
  • A 3R manipulator’s self-motion examples are shown as curves in joint space, including disjoint solutions for one target and a closed curve for another.The figure also shows a task-space view and a projection onto the q2 − q3 plane.Source: Figure 2 · p. 3
  • Workspace W describes end-effector pose, represented for rigid motion as an element of SE(3).The paper distinguishes workspace from configuration space.Source: A. Configuration Space and Workspace · pp. 2–3

B. Redundant Manipulators and Self-motion Manifold

  • A manipulator is redundant for a task when its configuration-space dimension exceeds the task-space dimension.The redundancy degree is r = n − m, where n and m are the dimensions of configuration and workspace, respectively.Source: B. Redundant Manipulators and… · p. 3
  • For a regular reachable pose, the inverse-kinematics preimage is a smooth r-dimensional submanifold, and each connected component is an SMM.The preimage may be a finite disjoint union of components, whose number depends on the pose.Source: B. Redundant Manipulators and… · p. 3

C. Ordinary differential equation for SMM Computation

The ODE approach computes a one-dimensional SMM by integrating a direction aligned with the Jacobian kernel. Its direction is regularized against sign flips, but the method does not recover higher-dimensional manifolds by tracing one curve.

C. Ordinary differential equation for SMM Computation

  • For a 1-D SMM, the Jacobian kernel supplies a tangent direction, allowing the manifold to be framed as an initial-value problem and integrated with an ODE solver.The internal-motion direction is the unit vector spanning the kernel.Source: C. Ordinary differential equation for… · p. 3
  • The directionally regularized field aligns the velocity with a reference heading to address sign ambiguity in the numerically returned kernel vector.The reference heading is carried over from the previous step.Source: C. Ordinary differential equation for… · p. 3
  • The SMM-IVP uses fixed-step RK5 integration from an unconstrained IK solution and stops when the trace returns near its start.With h = 0.05, solver accuracy keeps pose error below 10^-9 without a separate end-effector error-correction term.Source: C. Ordinary differential equation for… · p. 3
  • For a surface, fixing one tangent direction traces only an integral curve, a measure-zero subset that cannot recover the manifold without a chart and an advancement rule.The formulation is intrinsically one-dimensional because every direction in a two-dimensional tangent plane is admissible.Source: C. Ordinary differential equation for… · pp. 3–4

D. Representing SMMs as Fourier Series

The Fourier representation learns one-dimensional SMM branches through periodic coefficients, while flow matching learns a transport from a source density to the data density. The Fourier approach relies on a known domain, a constraint that does not generalize directly to higher-dimensional manifolds.

D. Representing SMMs as Fourier Series

  • Fourier methods encode each one-dimensional SMM branch as a continuous curve reconstructed from truncated Fourier coefficients.The branch is first represented as discrete joint configurations; joint angles are mapped to the circle group, with phase and traversal-direction ambiguities resolved by fixed conventions.Source: D. Representing SMMs as Fourier Series · p. 4
  • At r = 2, no canonical domain is available, making the Fourier regression target ill-posed as manifold type and parameterization may vary.The coefficient count also grows as (2K +1)r.Source: D. Representing SMMs as Fourier Series · p. 4
  • Fourier and ODE methods ignore joint limits, so applying them to real robotic systems requires post-processing.Joint limits can break periodicity and the property of being diffeomorphic to S1.Source: D. Representing SMMs as Fourier Series · p. 4

IV. FLOW-MATCHING MODELS

  • Flow matching learns a time-dependent velocity field transporting a tractable source density to the data density along a probability path.Rather than fit the intractable marginal field directly, it regresses the velocity field onto a conditional target from interpolated source and data samples.Source: §IV · p. 4
  • The conditional flow-matching objective needs neither ODE simulation nor divergence terms during training, and sampling uses N network evaluations for an N-step solver regardless of data dimension.The learned field is integrated forward to obtain samples from the data density.Source: §IV · p. 4

A. Conditional Flow Matching Model for SMM

The method learns a conditional distribution over configurations given a target pose, treating its support as the inverse-kinematics solution set. It recovers distinct SMM components by filtering generated samples and clustering them using intrinsic configuration-space distances.

A. Conditional Flow Matching Model for SMM

  • The conditional velocity field is trained to transport a source distribution to the inverse-kinematics posterior using configuration–pose pairs labeled by forward kinematics, without an IK solver.Because the labels are exact, the training distribution is supported on the fibers of the forward-kinematics map.Source: A. Conditional Flow Matching Model for… · p. 4
  • The method treats the inverse-kinematics fiber as the posterior's support, with SMMs as connected components of disjoint support rather than local density maxima.This link makes sampling from the learned distribution and clustering its samples the route to recovering the SMMs.Source: A. Conditional Flow Matching Model for… · p. 4

B. Recovering SMMs by sampling and clustering

  • For a query pose, generated configurations are retained when consistent with the query, and a Jacobian-based update can reduce forward-kinematics residuals beyond the application's tolerance.The initial filter evaluates only forward kinematics; the update uses the manipulator Jacobian's Moore–Penrose pseudoinverse.Source: B. Recovering SMMs by sampling and… · p. 4
  • Clustering at a radius larger than the sample tube width but smaller than the gap between components partitions the surviving samples, with cluster count estimating the number of SMMs.Distances use the intrinsic metric of configuration space, wrapping joint differences to avoid splitting one component at a coordinate cut; tracing then requires a batched forward pass and clustering.Source: B. Recovering SMMs by sampling and… · p. 5

C. Evaluation Metrics

The evaluation compares methods by forward-kinematics error and runtime. Error is assessed by recomputing forward kinematics for returned configurations and comparing it with the target; efficiency includes training and inference time.

  • Methods are compared using forward-kinematics error and inference time.The evaluation also reports training time, including data generation, for learning-based methods.Source: C. Evaluation Metrics · p. 5
  • The reported forward-kinematics error is the mean across all tested samples.Source: C. Evaluation Metrics · p. 5
  • For returned configurations, forward kinematics is recomputed and compared with the target pose for both the 3R arm and 7R arm.The 3R evaluation uses a position target; the 7R evaluation uses a target pose.Source: C. Evaluation Metrics · p. 5

V. RESULTS

The results section validates the method on a planar 3R manipulator and a 7R KUKA iiwa, including a demonstration of 4-D SMM approximation for a 3D position task. The supplied passages describe the simulation software and parameter tables but provide no comparative result values.

V. RESULTS

  • The evaluation covers a planar 3R manipulator and a 7R KUKA iiwa, with a 4-D SMM demonstration on the 7R robot for a 3D position task.The paper also compares 1-D SMM approximation with ODE and Fourier-series representations.Source: §V · p. 5

A. Simulation Software

  • The planar 3R simulation uses robotics-toolbox-python, while the KUKA iiwa 14 is trained and visualized with the Mujoco physics engine.Source: A. Simulation Software · p. 5
  • The listed tables cover standard Denavit–Hartenberg parameters and flow-matching architecture and training hyperparameters for the two manipulators.Fourier and ODE hyperparameters follow their original works and are listed in the appendix.Source: Table I · p. 5, Table II · p. 5, A. Simulation Software · p. 5

B. 1-D SMM Approximation for Planar 3R

The method approximates 1-D SMMs on Planar3R and KUKA iiwa and extends to 4-D SMMs on a 7R position task; flow-matching training remains efficient as robot complexity grows.

B. 1-D SMM Approximation for Planar 3R

C. 1-D SMM Approximation for 7-DoF Manipulators

  • Three Jacobian-correction iterations reduce KUKA iiwa flow-matching FK error to 1.44 × 10−10%, approaching ODE accuracy of 1.18×10−6% while retaining substantially lower inference time.The uncorrected FM model has 1.39% total FK error versus 3.35×101% for Fourier; FM inference takes 296.89 ms versus 8209.95 ms for ODE.Source: C. 1-D SMM Approximation for 7-DoF… · p. 6, Figure 4 · p. 6

D. High-Dimensional SMM Approximation

  • For 7R position tasks, the model approximates smooth 4-D SMMs with mean FK errors of 0.0479% before and 0.0013% after Jacobian correction.Tests cover 1,000 randomly selected targets, and workspace plots show generated configurations reach the desired positions without a change to the model formulation.Source: D. High-Dimensional SMM Approximation · p. 6, Figure 5 · p. 7, Figure 6 · p. 7

E. Training Efficiency

  • Flow-matching models train in approximately 25 minutes on 2,000,000 samples, while Fourier training reaches nearly 2 hours for high-DoF robots.The Fourier runtime uses a manually optimized implementation; the unmodified version would take approximately six times longer.Source: E. Training Efficiency · p. 7, Figure 7 · p. 7

VI. DISCUSSION

The discussion highlights accuracy, inference, and training advantages while noting task-specification dependence and open questions about convergence and sample complexity.

  • The formulation extends to redundancy orders r ≥2 without architectural changes, representing a distribution supported on the manifold rather than explicitly parameterizing it.Flow matching also supports GPU-parallel large-batch sampling, increasing the probability of a constraint-satisfying sample at small additional inference cost.Source: §VI · p. 7
  • The learned distribution is restricted to the joint-limit-feasible subset, unlike continuation and Fourier methods that represent the full manifold independently of joint limits.Changing the dimension of x requires retraining, although configuration–pose datasets can be reused.Source: §VI · pp. 7–8
  • Convergence guarantees for component separation and how sample complexity scales with redundancy order r remain unestablished.The authors also identify possible use of recovered components or homotopy classes to annotate configuration space for motion planners.Source: §VI · p. 8

APPENDIX

The appendix describes implementation changes made to the Fourier baseline to obtain results for a 7-DoF arm within a reasonable time budget.

  • The Fourier implementation was upgraded with batched forward kinematics, faster high-DoF workspace clustering, and Fibonacci sampling in SO(2) instead of uniform sampling.The upgraded implementation also supports the Franka Panda and KUKA iiwa 14 robots.Source: APPENDIX · p. 8
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