pith. sign in

arxiv: 2403.12235 · v2 · submitted 2024-03-18 · 💻 cs.RO · cs.SY· eess.SY

IKSPARK: Obstacle-Aware Inverse Kinematics via Convex Optimization

Liangting Wu , Roberto Tron This is my paper

Pith reviewed 2026-05-24 02:56 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords inverse kinematicssemidefinite programmingrank minimizationobstacle avoidanceconvex optimizationrobot motion planningkinematic chains
0
0 comments X

The pith

Inverse kinematics reduces to a semidefinite program whose relaxation is solved first and then recovered to exact rank-1 solutions via iteration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a formulation that casts inverse kinematics for robots with open or closed chains and mixed joint types as a semidefinite program subject to rank-1 constraints. It solves the convex relaxation of this program, whose infeasibility directly proves the original problem has no solution, then applies iterative rank minimization to extract a rank-1 matrix. Obstacle avoidance enters through a convex reformulation of the associated mixed-integer constraints. A reader would care because the approach is reported to deliver accurate solutions without post-processing and markedly higher success rates than standard nonlinear solvers when many obstacles are present.

Core claim

IKSPARK expresses the inverse kinematics task as a semidefinite program on symmetric matrices with fixed traces and explicit rank-1 constraints. The relaxed SDP is solved first; its infeasibility certifies that the original IK problem is infeasible. A rank-1 solution is then recovered by iterative rank-minimization procedures that carry local convergence guarantees. Mixed-integer constraints for obstacle avoidance are replaced by a convex surrogate. Experiments across kinematic structures and environments show that the recovered solutions are highly accurate and that success rates in obstacle-rich fixed workcells exceed those of conventional nonlinear optimization.

What carries the argument

SDP relaxation of the IK equations together with iterative rank-minimization recovery of rank-1 matrices and a convex surrogate for obstacle constraints.

If this is right

  • Infeasibility of an IK query is certified by the SDP solver without further search.
  • Accurate solutions are obtained directly, without separate post-processing steps.
  • The same formulation applies uniformly to open chains, closed chains, and mixtures of spherical, revolute, and prismatic joints.
  • Higher success rates are obtained in obstacle-dense fixed workcell scenes compared with traditional nonlinear methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The convex obstacle encoding could be reused inside sampling-based planners to prune infeasible samples early.
  • If the SDP solve time scales favorably, the method might replace nonlinear IK calls inside model-predictive controllers.
  • The same rank-recovery technique might be applied to other nonconvex kinematic or geometric constraints that admit SDP liftings.
  • Physical-robot experiments would be needed to check whether numerical conditioning of the recovered solutions remains acceptable under sensor noise.

Load-bearing premise

The iterative rank-minimization step is assumed to recover a globally valid rank-1 solution from the SDP relaxation for the robot morphologies and obstacle sets tested.

What would settle it

A robot configuration and obstacle layout for which the original IK problem admits a feasible solution yet the SDP relaxation produces a matrix whose rank cannot be reduced to one by the iterative procedure, or for which the reported success-rate advantage over nonlinear solvers disappears.

Figures

Figures reproduced from arXiv: 2403.12235 by Liangting Wu, Roberto Tron.

Figure 1
Figure 1. Figure 1: Two connected links (i, j) ∈ Er, each associated with a reference frame (Bi and Bj ). A general set of variables to be solved for the IK problem is x = {Ti ∈ R 3 , Rj ∈ SO(3)|i ∈ Vt, j ∈ Vr}. (2) We denote as nt and nr, respectively, the number of free translations and rotations in x. We denote np as the number of prismatic joints. V. KINEMATIC CONSTRAINTS In this section we discuss how to model translatio… view at source ↗
Figure 3
Figure 3. Figure 3: Two links connected by a prismatic joint, where [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: The joint limit between two links (i, j) ∈ Er can be written as an angle limit between two unit vectors wi = RiRee1 and wj = Rje1 (purple sector), which can be further bounded by a ball (painted yellow) on wi − wj . The ball can be then approximated by linear inequalities (orange polygon). D. Prismatic Joints For two links (i, j) connected by a prismatic joint (i, j) ∈ Ep, the physical limit of the joint r… view at source ↗
Figure 4
Figure 4. Figure 4: Suppose nr = 1 and 3 − λ1 is a concave quadratic function, this figure shows the relation in Lemma 3 resulted from the concavity of the function. Proposition 9: Given a sequence {Yk} and a constant c, if the condition W(Yk ) − Xnr i ⟨∇λ1(Yk i ), Yk+1 i − Yk i ⟩ ≤ cW(Yk ) (43) is satisfied for every k, then W(Yk ) ≤ c kW(Y0 ) (44) for every k. Proof: Eq.(43) combined with Lemma 3 implies that W(Yk+1) ≤ cW(Y… view at source ↗
Figure 5
Figure 5. Figure 5: An example posture solved for Baxter, where the two [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Some computational results of the solution in Fig. 5. Figures 6a and 6b show the changes of the largest eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The change of the largest eigenvalues where a sub [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

Inverse kinematics (IK) is central to robot control and motion planning, yet its nonlinear kinematic mapping makes it inherently nonconvex and particularly challenging under complex constraints. We present IKSPARK (Inverse Kinematics using Semidefinite Programming And RanK minimization), an obstacle-aware IK solver for robots with diverse morphologies, including open and closed kinematic chains with spherical, revolute, and prismatic joints. Our formulation expresses IK as a semidefinite programming (SDP) problem with additional rank-1 constraints on symmetric matrices with fixed traces. IKSPARK first solves the relaxed SDP, whose infeasibility certifies infeasibility of the original IK problem, and then recovers a rank-1 solution using iterative rank-minimization methods with proven local convergence. Obstacle avoidance is handled through a convexified formulation of mixed-integer constraints. Extensive experiments show that IKSPARK computes highly accurate solutions across various kinematic structures and constrained environments without post-processing. In obstacle-rich settings, especially fixed workcell environments, IKSPARK achieves substantially higher success rates than traditional nonlinear optimization methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper presents IKSPARK, an inverse kinematics solver that casts the nonconvex IK problem (including for open/closed chains with various joint types) as an SDP with rank-1 constraints on trace-fixed symmetric matrices. The relaxed SDP is solved first (with infeasibility certifying original infeasibility), followed by iterative rank-minimization to recover a rank-1 solution; obstacle avoidance uses a convexified mixed-integer formulation. The abstract asserts proven local convergence of the recovery step and substantially higher success rates than nonlinear optimizers in obstacle-rich settings, with accurate solutions obtained without post-processing across diverse morphologies.

Significance. If the rank-1 recovery is reliable across tested cases, the approach would provide a convex-optimization route to IK that can certify infeasibility and incorporate obstacles without nonconvex post-processing, which is potentially valuable for reliable motion planning in constrained environments such as fixed workcells.

major comments (2)
  1. [Abstract] Abstract (recovery step): The claim that IKSPARK 'computes highly accurate solutions ... without post-processing' rests on the iterative rank-minimization always recovering an exact rank-1 matrix satisfying the original nonconvex constraints. Local convergence of the recovery procedure does not guarantee this when the SDP relaxation gap is nonzero, which remains possible after convexification of the obstacle constraints; no analysis bounding the recovery failure rate (e.g., via low-DoF exhaustive checks or known tight instances) is referenced.
  2. [Abstract] Abstract (experiments paragraph): The assertion of 'substantially higher success rates than traditional nonlinear optimization methods' in obstacle-rich settings is presented without any quantitative metrics, baseline algorithms, trial counts, or dataset details, preventing verification of the performance advantage that underpins the central claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract claims. We address each major comment below with honest assessment of the manuscript's content and propose revisions where the points identify overstatements.

read point-by-point responses

  1. Referee: [Abstract] Abstract (recovery step): The claim that IKSPARK 'computes highly accurate solutions ... without post-processing' rests on the iterative rank-minimization always recovering an exact rank-1 matrix satisfying the original nonconvex constraints. Local convergence of the recovery procedure does not guarantee this when the SDP relaxation gap is nonzero, which remains possible after convexification of the obstacle constraints; no analysis bounding the recovery failure rate (e.g., via low-DoF exhaustive checks or known tight instances) is referenced.

    Authors: We agree that proven local convergence of the iterative rank-minimization procedure does not guarantee exact rank-1 recovery in all cases, particularly if the SDP relaxation gap is nonzero after convexification of the mixed-integer obstacle constraints. The manuscript references the local convergence property of the recovery method but does not include theoretical analysis or empirical bounds on recovery failure rates (such as exhaustive low-DoF checks). In the reported experiments the procedure recovered accurate rank-1 solutions satisfying the original constraints without post-processing. We will revise the abstract to state that the recovery step employs iterative rank-minimization with proven local convergence and that it produced accurate solutions in the evaluated scenarios, removing the implication of guaranteed exact recovery without post-processing. revision: yes

  2. Referee: [Abstract] Abstract (experiments paragraph): The assertion of 'substantially higher success rates than traditional nonlinear optimization methods' in obstacle-rich settings is presented without any quantitative metrics, baseline algorithms, trial counts, or dataset details, preventing verification of the performance advantage that underpins the central claim.

    Authors: The abstract is intended as a concise summary of results rather than a self-contained report of all experimental details. The full manuscript contains an Experiments section that reports quantitative success rates, the specific nonlinear optimization baselines used for comparison, trial counts, robot morphologies tested, and descriptions of the obstacle-rich environments. These details enable verification of the performance claims. No revision to the abstract is required, as the level of detail is appropriate for an abstract while the supporting evidence appears in the body of the paper. revision: no

Circularity Check

0 steps flagged

No circularity: new SDP formulation with independent recovery step

full rationale

The paper formulates IK as an SDP with rank-1 constraints, solves the relaxation, then applies iterative rank-minimization (with stated local convergence). This chain does not reduce any claimed performance metric or feasibility certificate to a quantity defined by the same fitted parameters, self-citation, or ansatz smuggled from prior work. Experiments on external kinematic structures and obstacle sets provide independent validation; no load-bearing step collapses to self-definition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated beyond the general claim of proven local convergence for the rank-minimization step.

axioms (1)
  • domain assumption Iterative rank-minimization methods have proven local convergence for this class of IK problems
    Referenced in the recovery step description but not derived or cited with external verification in the abstract.

pith-pipeline@v0.9.0 · 5715 in / 1262 out tokens · 20838 ms · 2026-05-24T02:56:07.195702+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Aristidou and J

    A. Aristidou and J. Lasenby. Fabrik: A fast, iterative solver for the inverse kinematics problem. Graphical Models, 73(5):243–260, 2011

    work page 2011

  2. [2]

    A. S. Bandeira, N. Boumal, and A. Singer. Tightness of the maxi- mum likelihood semidefinite relaxation for angular synchronization. Mathematical Programming, 163:145–167, 2017

    work page 2017

  3. [3]

    Beeson and B

    P. Beeson and B. Ames. Trac-ik: An open-source library for improved solving of generic inverse kinematics. In 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids) , pages 928–935, 2015

    work page 2015

  4. [4]

    H. Dai, G. Izatt, and R. Tedrake. Global inverse kinematics via mixed- integer convex optimization. The International Journal of Robotics Research, 38(12-13):1420–1441, 2019

    work page 2019

  5. [5]

    R. Diankov. Automated construction of robotic manipulation programs. 2010

    work page 2010

  6. [6]

    Dietmaier

    P. Dietmaier. The stewart-gough platform of general geometry can have 40 real postures. In Advances in robot kinematics: Analysis and control, pages 7–16. Springer, 1998

    work page 1998

  7. [7]

    Giamou, F

    M. Giamou, F. Mari´c, D. M. Rosen, V . Peretroukhin, N. Roy, I. Petrovi´c, and J. Kelly. Convex iteration for distance-geometric inverse kinematics. IEEE Robotics and Automation Letters , 7(2):1952–1959, 2022

    work page 1952

  8. [8]

    M. W. Griffis and J. Duffy. Method and apparatus for controlling geometrically simple parallel mechanisms with distinctive connections, Jan. 12 1993. US Patent 5,179,525

    work page 1993

  9. [9]

    M. L. Husty, M. Pfurner, and H.-P. Schr ¨ocker. A new and efficient algorithm for the inverse kinematics of a general serial 6r manipulator. Mechanism and machine theory , 42(1):66–81, 2007

    work page 2007

  10. [10]

    Kenwright

    B. Kenwright. Inverse kinematics–cyclic coordinate descent (ccd). Journal of Graphics Tools , 16(4):177–217, 2012

    work page 2012

  11. [11]

    Le Naour, N

    T. Le Naour, N. Courty, and S. Gibet. Kinematics in the metric space. Computers & Graphics , 84:13–23, 2019. 0 1 2 3 4 5 1.6 1.8 2 2.2 2.4 2.6 2.8 (a) λ1, rotation matrices 0 1 2 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 (b) λ1, quaternions 1 2 3 4 5 6 7 10-12 10-10 10-8 10-6 10-4 10-2 100 102 (c) eigenvalues, rotation matrices 1 2 3 4 10-20 10-15 10-10 10-5 100 ...

    work page 2019

  12. [12]

    Lee and C.-G

    H.-Y . Lee and C.-G. Liang. Displacement analysis of the general spatial 7-link 7r mechanism. Mechanism and machine theory , 23(3):219–226, 1988

    work page 1988

  13. [13]

    M. Li, G. Liang, H. Luo, H. Qian, and T. L. Lam. Robot-to-robot relative pose estimation based on semidefinite relaxation optimization. In International Conference on Intelligent Robots and Systems (IROS) , pages 4491–4498, 2020

    work page 2020

  14. [14]

    J. R. Magnus. On differentiating eigenvalues and eigenvectors. Econometric Theory, 1:179–191, 1985

    work page 1985

  15. [15]

    Mari´c, M

    F. Mari´c, M. Giamou, A. W. Hall, S. Khoubyarian, I. Petrovi ´c, and J. Kelly. Riemannian optimization for distance-geometric inverse kinematics. IEEE Transactions on Robotics , 38(3):1703–1722, 2021

    work page 2021

  16. [16]

    The MOSEK optimization toolbox for MATLAB manual

    MOSEK ApS. The MOSEK optimization toolbox for MATLAB manual. Version 10.0., 2022. 0 50 100 150 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Fig. 8: The change of the largest eigenvalues where a sub- optimal solution from Algorithm 1 ( k = 1 , . . . ,100) is projected to another point in the relaxed set using Algorithm 2 ( k = 101, . . . ,108) and restarted again with ...

    work page 2022

  17. [17]

    Muller-Cajar and R

    R. Muller-Cajar and R. Mukundan. Triangualation-a new algorithm for inverse kinematics. 2007

    work page 2007

  18. [18]

    L. Peng, M. Fazlyab, and R. Vidal. Semidefinite relaxations of Truncated Least-Squares in robust rotation search: Tight or not. In European Conference on Computer Vision (ECCV) , 2022

    work page 2022

  19. [19]

    J. M. Porta, L. Ros, and F. Thomas. A linear relaxation technique for the position analysis of multiloop linkages. IEEE Transactions on Robotics, 25(2):225–239, 2009

    work page 2009

  20. [20]

    Raghavan and B

    M. Raghavan and B. Roth. Inverse kinematics of the general 6r manipulator and related linkages. 1993

    work page 1993

  21. [21]

    Saunderson, P

    J. Saunderson, P. A. Parrilo, and A. S. Willsky. Semidefinite descriptions of the convex hull of rotation matrices. SIAM Journal on Optimization , 25(3):1314–1343, 2015

    work page 2015

  22. [22]

    Siciliano, L

    B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo. Modelling, planning and control. Advanced Textbooks in Control and Signal Processing. Springer,, 2009

    work page 2009

  23. [23]

    D. Stewart. A platform with six degrees of freedom. proceedings of the institute of mechanical engineers. 1965

    work page 1965

  24. [24]

    S. Umeyama. Least-squares estimation of transformation parameters between two point patterns. IEEE Transactions on Pattern Analysis & Machine Intelligence, 13(04):376–380, 1991

    work page 1991

  25. [25]

    Wu and R

    L. Wu and R. Tron. An sdp optimization formulation for the inverse kinematics problem. In 2023 62nd IEEE Conference on Decision and Control (CDC), pages 4731–4738, 2023

    work page 2023

  26. [26]

    H. Yang. Certifiable Outlier-Robust Geometric Perception. PhD thesis, Massachusetts Institute of Technology, 2022

    work page 2022

  27. [27]

    Yang and L

    H. Yang and L. Carlone. A quaternion-based certifiably optimal solution to the Wahba problem with outliers. In International Conference on Computer Vision (ICCV) , pages 1665–1674, 2019

    work page 2019

  28. [28]

    H. Yang, J. Shi, and L. Carlone. TEASER: Fast and certifiable point cloud registration. IEEE Transactions on Robotics , 37(2):314–333, 2020

    work page 2020

  29. [29]

    Yenamandra, F

    T. Yenamandra, F. Bernard, J. Wang, F. Mueller, and C. Theobalt. Convex optimisation for inverse kinematics. In 2019 International Conference on 3D Vision (3DV) , pages 318–327. IEEE, 2019. APPENDIX I SEPARATE PROOF FOR PROPOSITION 3 In this appendix we provide a separate proof of the claim in Proposition 3 that any rank-1 Yτ i satisfying (24) can be writ...

    work page 2019