An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

Akira Terui; Masahiko Mikawa; Natsumi Oka; Yusuke Shirato

REVIEW 4 major objections 4 minor 22 references

An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a CGS-based quantifier-elimination reachability check combined with Dijkstra's algorithm over inverse-kinematics solutions yields smooth, feasible trajectories with lower total joint movement than greedy selection.

desk verdict A clear, honest extension of the authors' CGS-QE pipeline, but the graph-based optimization ignores IK branch continuity, so the main optimality claim needs qualification. read the letter →

arxiv 2412.18294 v1 pith:IBZ7Y2ZU submitted 2024-12-24 cs.RO cs.SC

Yusuke Shirato , Natsumi Oka , Akira Terui , Masahiko Mikawa This is my paper · ORCID

classification cs.ROcs.SC MSC 68W3013P1013P2568U0768R10

keywords InversekinematicsComprehensiveGröbnerSystemsRealquantifiereliminationCubicsplineinterpolationPathplanningShortestalgorithmDijkstra'sRobotmanipulator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a complete pipeline for a three-joint robot arm: certify that a smooth spline path lies inside the reachable workspace, then pick, among the several joint-angle solutions at each sampled point, the sequence that minimizes the total joint movement. The certification step uses a Comprehensive Gröbner System and real quantifier elimination, so feasibility is checked symbolically before any numerical trajectory is computed. The selection step models the four joint solutions at each point as vertices of a weighted graph and applies Dijkstra's algorithm. The paper reports that this global shortest-path choice reduces the average total joint movement from 10.1371 rad to 8.0023 rad over five test paths compared with a greedy nearest-solution method, at computation times practical for real-time use.

What carries the argument

The load-bearing object is the weighted graph $G=(V,E)$ whose vertices are the $d$ inverse-kinematics solutions $p_{i,j}=(\theta_1^{(i,j)},\theta_3^{(i,j)},\theta_4^{(i,j)})$ at each sampled path point $p_i$, and whose edge weight from $p_{i,j}$ to $p_{i+1,k}$ is $s_{i,j,i+1,k}=|\theta_1^{(i+1,k)}-\theta_1^{(i,j)}|+|\theta_3^{(i+1,k)}-\theta_3^{(i,j)}|+|\theta_4^{(i+1,k)}-\theta_4^{(i,j)}|$. The graph is supported by two other mechanisms: the precomputed Comprehensive Gröbner System of the inverse-kinematics polynomials, which turns the reachability question into a CGS-based real quantifier elimination check over the spline parameter $s\in[0,1]$, and natural cubic spline interpolation, which generates the smooth candidate path through the given points. Dijkstra's algorithm on this graph selects the globally minimal total joint movement.

What would settle it

Take a path that passes through or near a workspace boundary or singularity where the number of real inverse-kinematics solutions changes, run the proposed pipeline, and test whether the graph construction fails or the reported optimal trajectory is not actually executable; alternatively, for a small number of sample points, enumerate all $4^n$ joint-configuration sequences by brute force and check whether Dijkstra's result equals the true minimum.

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Extended reading notes

Core claim

The central claim is that a CGS-based real-quantifier-elimination check can certify that every point of a cubic spline path is reachable, and that the remaining multi-solution ambiguity of inverse kinematics can then be resolved optimally by a shortest-path computation. Specifically, at each sampled point the four inverse-kinematics configurations form a vertex set, edges connect configurations at consecutive points with weight equal to the sum of absolute joint-angle differences, and Dijkstra's algorithm finds the sequence from start to end with the smallest total joint variation. The paper's experiments on a myCobot 280 modeled as a 3-DOF arm show that this procedure yields smooth trajectories with smaller total joint motion than the greedy rule that minimizes only the jump between consecutive points.

Load-bearing premise

The load-bearing premise is that every point on the path has the same number, $d=4$, of inverse-kinematics solutions, so each point contributes exactly four vertices to the graph; if the solution count varies along the path, the graph construction and the shortest-path result no longer apply.

Editorial extensions

If this is right

A path can be certified as feasible before any numerical inverse-kinematics solve, because the CGS-QE check decides existence of real solutions for every point on the spline.
Choosing joint configurations by shortest path over the whole trajectory beats greedy local selection: average total joint movement drops from 10.1371 rad to 8.0023 rad across Tests 1–5.
The shortest-path computation is fast enough for practical use: average runtime is on the order of $10^{-3}$ seconds for the tested paths.
The complexity of the optimization is $O(nd^2\log(nd))$ when each of $n$ points has $d$ inverse-kinematics solutions.
Spline paths are not guaranteed to stay in the workspace: Test 6 failed because part of the generated spline left the feasible region, so the certification step is essential and not merely a formality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the constant-$d$ assumption is the main practical obstacle; a version that allows the number of solutions to vary along the path (and handles singular points where solutions merge) would make the graph construction robust to workspace boundaries.
Inference: the same graph formulation could be applied to other cost functions, such as weighted joint torques, squared displacements, or time, and to higher-degree-of-freedom arms, as long as the solution sets at sampled points stay finite.
Inference: the Test 6 failure suggests an adaptive loop that re-generates the spline (or switches to Bézier segments) until the CGS-QE check passes; the paper names Bézier curves as the next step.
Inference: the claimed 21% average reduction in joint motion is over five specific test paths; a broader benchmark with randomized waypoints would be needed to establish the typical improvement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a modest but real extension of the authors' earlier CGS-QE inverse kinematics pipeline. They replace the straight-line path with natural cubic splines and add two ways to pick among the multiple IK solutions at each sampled point: a greedy per-step method and a Dijkstra shortest-path method that minimizes total joint variation. The spline and graph mathematics are standard and correctly presented; the CGS-QE feasibility certificate for each spline segment is a legitimate use of their prior tooling. The five successful experiments run in practical time, and Dijkstra does cut total joint motion versus the greedy baseline (average 8.0 vs 10.1 rad). That is a genuine, if small, empirical point.

The soft spot is the one the stress-test note flags, and it lands. Section 4.2 builds the graph with all-to-all edges between the four IK solutions at consecutive points and then runs Dijkstra. That implicitly assumes any pair of solutions can be connected by a continuous manipulator motion with the end effector staying on the spline. The paper does not certify that. The CGS-QE check only certifies that at least one real solution exists for each segment; it says nothing about the four solutions forming four distinct smooth branches. The authors list "number and continuity of solutions" as future work in Section 5, so they know. But as it stands, the optimal joint sequences in Table 3 may hop between branches, and such a hop may require passing through a singularity or leaving the curve entirely. That does not invalidate the shorter joint-motion totals as a lower bound on the graph, but it undercuts the claim that the method produces an executable optimum. The fix is conceptually straightforward: track solution branches using local continuity, or add a continuity test to the graph edges before letting Dijkstra cross them.

Other concerns are minor. There is no baseline from the robotics literature, only the greedy comparison. Test 6 fails because the spline leaves the feasible workspace; the failure is reported honestly but the paper stays quiet about whether the abort happened during the CGS-QE certificate or later at a specific time step. The self-citation is heavy but fair, since this is a direct continuation of work the same authors published at CASC 2023; the dependency is explicit. No code or data are shipped, which is a weakness for reproducibility in a systems paper but not a flaw in the math.

Who should read it: people working on certified path planning with polynomial constraints and on computer algebra methods in robotics. It is not a breakthrough, but it is a clear, honest extension with a real limitation named. I would send it to review, expecting that the continuity issue be addressed before publication.

Referee Report

4 major / 4 minor

Summary. The paper extends the authors' earlier CGS-based inverse kinematics and trajectory feasibility work to spline-based paths and joint-configuration optimization. It models the myCobot 280 as a 3-DOF manipulator, constructs a natural cubic spline through four waypoints, uses CGS-QE to certify the existence of real IK solutions along each spline segment, and then builds a layered graph whose vertices are IK solutions at sample points and whose edge weights are summed absolute joint changes; Dijkstra's algorithm selects a sequence minimizing total joint variation. Experiments on five of six tests show moderate computation times and lower joint movement for Dijkstra than for a greedy nearest-neighbor method; Test 6 fails because the spline leaves the feasible workspace.

Significance. If the underlying assumptions were certified, the paper would offer a useful combination of symbolic feasibility certification and graph-based IK selection for low-DOF manipulators, with a clear complexity estimate and a reproducible experimental setup. The spline and Dijkstra mathematics are standard and are, in themselves, correctly presented. The main numerical claim, however, rests on an unverified constancy and continuity of the number of IK solutions, and the evaluation contains no comparison with existing planners. The current evidence supports a qualified algorithmic demonstration rather than a general claim of optimized path planning.

major comments (4)

[Section 4.2, Eq. (11)] The graph E includes edges (p_{i,j}, p_{i+1,k}) for all j,k, but the paper never certifies that an edge corresponds to a continuous joint-space motion while the end effector follows the spline. The CGS-QE check in Section 3.2 certifies only that the system has a real solution for each parameter value; it does not establish that the four roots at consecutive sample points lie on the same analytic branch, nor that a selected pair can be connected without passing through a singularity. As a result, the totals in Table 3 may sum over edges that are not physically executable, so the reported advantage of Section 4.2 over Section 4.1 is not yet a guaranteed trajectory cost.
[Sections 4.3 and 5] The graph construction assumes d=4 at every point p_i, but the only evidence is the observation that the number of solutions was 4 'from the start to the end of the path.' Section 5 itself acknowledges that the number may vary and lists the determination of the number and continuity of solutions as future work. The optimality of Dijkstra on this graph is therefore conditional; the method needs either a CGS-QE-based certification that d is constant and that the branches persist over each whole segment, or an extension of the graph to variable d.
[Sections 3.4 and 5] One of the six tests fails because the spline leaves the feasible workspace, yet Section 5 states that the proposed method 'can efficiently generate smooth trajectories and optimize joint configurations.' This is an overstatement: as written, the method detects infeasibility but cannot repair the path. The claim should be restricted to paths whose spline segments are certified feasible, or the method should include a path-repair mechanism.
[Tables 3 and 4] The experimental evaluation contains no baseline from existing path planning or IK selection methods; the only comparison is between the greedy method of Section 4.1 and Dijkstra's method in Section 4.2. Without comparison to, for example, sampling-based planners, optimization-based planners, or continuous branch-tracking methods, the practical significance of the claimed improvement cannot be assessed.

minor comments (4)

[Section 3.1, Eq. (6)] The text says each C_j has s in [0,1], but the interpolation conditions are written as X_j(j)=x_j and X_j(j+1)=x_{j+1}; these should be X_j(0)=x_j and X_j(1)=x_{j+1}, or one should define a global parameter over [0,N]. As printed, the conditions are inconsistent with the domain of C_j.
[Section 4.2] The index sets are inconsistent: V is defined with i in {0,...,n} and E with i in {0,...,n-1} and j,k in {0,...,d}, but the starting vertex is p_{1,m} and configurations are p_{i,j} for i=1,...,n and j=1,...,d. The ranges should be i in {1,...,n} and j,k in {1,...,d} throughout.
[Throughout] There are typographical errors such as 'T able 1', 'T able 2', and 'T able 3' in the text and table captions; these should be corrected.
[Section 4.3] The paper reports T=n=15 but does not state how the sample points p_i are obtained from the spline curve; please specify the sampling rule explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spline-plus-Dijkstra derivation is self-contained; reliance on prior CGS-QE software is an implementation dependency, not an assumed conclusion.

full rationale

The paper's new contribution is the combination of cubic-spline path generation with a graph shortest-path selection among discrete inverse-kinematics solutions. The optimality claim in Section 4 is direct: edge weights (eq. 11) are the exact objective being minimized, and both Section 4.1 (greedy) and Section 4.2 (Dijkstra) evaluate the same edge-weight sums, so the comparison in Table 3 is not a fitted prediction. The feasibility certification (Section 3.2) uses CGS-QE from prior work [9,11,17]; although self-cited, this is an external algorithmic tool rather than an assumption of the paper's new conclusions, and no uniqueness or existence theorem is imported to force the graph construction. The d=4 solution count is an observation acknowledged in Section 5 as needing future treatment (number and continuity of solutions), which is a stated limitation, not a circular step. No parameter is fitted to the reported output and no result is defined in terms of the quantity it purports to predict, so no circular reduction is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The main load-bearing premises are the kinematic model of the robot, the correctness of the CGS-QE software, and the constant-solution-count assumption for the graph optimization.

assumptions (6)

domain assumption Modified D-H parameters in Table 1 accurately represent the myCobot 280 geometry.
Used to derive forward kinematics eq. (3); any error propagates to all IK solutions and path planning results.
domain assumption Joints 2, 5, 6, and 7 can be fixed so the 6-DOF myCobot behaves as a 3-DOF manipulator.
The paper treats myCobot as 3-DOF by operating only three main joints; this is an approximation of the physical robot.
domain assumption The CGS-QE implementation (refs [9,11,17]) correctly decides real solvability of the polynomial systems.
The feasibility certificates and spline segment checks rely on this software; no independent verification is provided in this paper.
domain assumption Every sampled point along the tested spline paths has exactly four real IK solutions.
Used to build the graph with d=4 in Section 4; the authors note in Section 5 that the number of solutions may vary in practice.
standard math Natural cubic spline interpolation formulas (eq. 8) produce the interpolating curve through the given points.
Standard CAGD result; no new derivation is offered.
standard math Dijkstra's algorithm computes the true shortest path in a nonnegative weighted graph.
Standard algorithm; the edge weights are L1 joint variations, which are nonnegative.

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Cite this review

Pith. "Pith review of An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems." pith.science (2026). https://pith.science/paper/IBZ7Y2ZU

@misc{pith2026241218294,
  author       = {Pith},
  title        = {Pith review of: An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IBZ7Y2ZU}},
  note         = {Machine review of arXiv:2412.18294}
}

read the original abstract

This paper presents an advanced method for addressing the inverse kinematics and optimal path planning challenges in robot manipulators. The inverse kinematics problem involves determining the joint angles for a given position and orientation of the end-effector. Furthermore, the path planning problem seeks a trajectory between two points. Traditional approaches in computer algebra have utilized Gr\"obner basis computations to solve these problems, offering a global solution but at a high computational cost. To overcome the issue, the present authors have proposed a novel approach that employs the Comprehensive Gr\"obner System (CGS) and CGS-based quantifier elimination (CGS-QE) methods to efficiently solve the inverse kinematics problem and certify the existence of solutions for trajectory planning. This paper extends these methods by incorporating smooth curves via cubic spline interpolation for path planning and optimizing joint configurations using shortest path algorithms to minimize the sum of joint configurations along a trajectory. This approach significantly enhances the manipulator's ability to navigate complex paths and optimize movement sequences.

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Reference graph

Works this paper leans on

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Reviewed August 11, 2026 · model on record in the stance chip above.

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Mathematical Problems in Engineering 2021, 6680687 (2021) https://doi.org/10.1155/2021/6680687

Silva, S., Schnitman, L., Cesca Filho, V.: A Solution of the Inverse Kinemat- ics Problem for a 7-Degrees-of-Freedom Serial Redundant Manipulator Using Gr¨ obner Bases Theory. Mathematical Problems in Engineering 2021, 6680687 (2021) https://doi.org/10.1155/2021/6680687

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doi:10.1007/s11044-010-9241-8 2011

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Horigome, N., Terui, A., Mikawa, M.: A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gr¨ obner Bases. In: Bigatti, A.M., Carette, J., Davenport, J.H., Joswig, M., Wolff, T. (eds.) Mathematical Software – ICMS 2020, pp. 3–13. Springer, Cham (2020). https://doi.org/10. 1007/978-3-030-52200-1 1

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Preprint

Otaki, S., Terui, A., Mikawa, M.: A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Real Quantifier Elimination based on Comprehensive Gr¨ obner Systems. Preprint. arXiv:2111.00384 (2021). https: //doi.org/10.48550/arXiv.2111.00384

arXiv doi:10.48550/arxiv.2111.00384 2021