Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr\"obner Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that a 6-DOF arm whose consecutive joint axes intersect only two at a time still admits a closed-form inverse kinematics solution, obtained by solving three polynomial equations for the intersection point of two consecutive

desk verdict The algebraic derivation is real and the paper is honest about its limits, but the uncompleted CGS means the advertised global IK solver isn't actually demonstrated. read the letter →

arxiv 2509.00823 v2 pith:A44IGKG4 submitted 2025-08-31 cs.RO cs.SCmath.AC

Takumu Okazaki , Akira Terui , Masahiko Mikawa This is my paper · ORCID
classification cs.ROcs.SCmath.AC MSC 68W3013P1070B15
keywords inversekinematics6-DOFmanipulatorComprehensiveGröbnerSystembasisparametricpolynomialequationstwo-intersecting-axesdecouplingtrajectoryplanning
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 tries to establish that a six-joint robot arm can have its inverse kinematics solved in closed form even when no three consecutive joint axes meet at a single point, the condition the classical decoupling method requires. Its move is to pivot the solution on the intersection point of two consecutive joint axes: once the coordinates of that point are known, every joint angle can be written as an explicit formula involving those coordinates and the desired end-effector pose. The paper derives three polynomial equations for the pivot coordinates, with the pose as parameters, and uses a Comprehensive Gröbner System to precompute specialized polynomial solving data so that each new pose only requires substitution. Experiments on ten thousand pose-and-configuration samples return about six arm configurations per pose, recover the original configuration in roughly 99.7 percent of trials, keep position errors near a ten-thousandth of a millimeter, and cut per-pose solve time from about half a second to about a tenth of a second when the precomputed data is used.

What carries the argument

The load-bearing object is P, the intersection point of the rotation axes of Joints 4 and 5, which is the origin of frame 5. Every joint angle is routed through P: comparing P's coordinates in the end-effector frame gives sinθ6 and cosθ6; axis-vector comparisons give sinθ5, cosθ1, and the remaining angles; the wrist-center position gives sinθ3, cosθ2, and cosθ4. Eliminating the angles leaves three polynomial equations in x,y,z -- a plane from the third component of P, a sphere from the distance to the end-effector, and a squared quartic from the wrist-center geometry -- with the end-effector pose as parameters. The Comprehensive Gröbner System (CGS) is the computational mechanism: it partiti

What would settle it

Complete the CGS computation of system (34). If a reachable end-effector pose in any segment other than the computed first one is fed to Algorithm 1 and the returned joint angles fail the forward-kinematics check or omit the known generating configuration, while direct numerical solving of (34) finds a matching solution, then the claim of a global explicit solver fails for that segment.

Watch

Extended reading notes

Core claim

The paper claims that myCobot 280's inverse kinematics can be solved globally and explicitly even though no three consecutive joint axes intersect. It relaxes the classical three-axis decoupling to a two-axis version: take P, the intersection of the Joint 4 and Joint 5 axes, express every joint angle's sine and cosine in terms of P=(x,y,z) and the desired end-effector pose, then eliminate the angles to get three polynomial equations in x,y,z with the pose as parameters. A Comprehensive Gröbner System -- a finite partition of parameter space into segments, each with a Gröbner basis -- lets the solver substitute a pose into the correct segment's basis instead of recomputing bases. Experiments

Load-bearing premise

The implementation assumes every end-effector pose fed to the solver falls in the one parameter region for which the polynomial system was actually precomputed; no test checks which region a pose belongs to, so the precomputed solving data is taken to be valid after substitution.

Editorial extensions

If this is right

  • For any 6R arm with at least one pair of consecutive intersecting joint axes, the inverse kinematics can be reduced from a large coupled system to three polynomial equations in the intersection coordinates, after which all joint angles are obtained by direct formulas.
  • With the CGS precomputation, solving a new pose costs a substitution into a precomputed basis rather than a fresh Gröbner basis computation; the measured five-fold speedup makes real-time trajectory planning more realistic.
  • About six joint configurations solve a typical reachable pose; the forward-kinematics verification step filters the spurious solutions introduced when equations are squared, so the final output is a short list of physical arm poses.
  • For horizontal orientations (n3=0) the method needs extra equations; the paper gives separate subsystems for the two subcases, so the solver still covers these degenerate orientations.
  • Randomly generated poses (position and orientation both arbitrary) produce solutions only about 37% of the time, which the paper attributes to the arm's reachable workspace rather than to a failure of the solver.

Reading between the lines

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

  • The paper attributes the residual failures mostly to numerical error, but the incomplete CGS is a competing explanation: a pose outside the single computed segment would make the substituted basis not a Gröbner basis of the specialized ideal. A segment-membership test before substitution would separate the two causes.
  • The same reduction should transfer to any 6-DOF serial arm whose link-coordinate table contains one intersecting consecutive pair; only the link lengths d4, d5, d6 enter the core system (34), so adapting to another robot is mostly re-deriving the same three equations with new constants.
  • Completing the CGS, or restricting the parameter space with the arm's joint limits, could upgrade the empirical 99.7 percent recovery rate into a completeness statement: every reachable pose would be either solved or certified as unreachable.
  • Algorithm 1's final forward-kinematics check could be replaced by evaluating the CGS segment conditions, which would remove the need to solve forward kinematics for every candidate and make the solver better suited to time-critical trajectory planning.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes an inverse kinematics (IK) solver for the 6-DOF myCobot 280 manipulator. The method extends Pieper's approach to the case where only two consecutive joint axes intersect, using the intersection point of Joints 4 and 5 to decouple the problem. A parametric system of polynomial equations, Eq. (34), is derived for the intersection point coordinates (x,y,z) in terms of the end-effector pose, and a Comprehensive Gröbner System (CGS) is used so that the Gröbner basis does not need to be recomputed for each trajectory point. The authors also handle special orientations with n3=0, present Algorithm 1, and report experiments showing about 99.7% success on forward-kinematics-generated poses and roughly six solutions per pose, with position errors around 1e-4 mm. The paper explicitly acknowledges in §1.3 and §5 that the CGS computation was not completed, and that the implementation assumes inputs belong to the single computed segment.

Significance. If the proposed method were fully realized, it would be a meaningful contribution: a closed-form-style IK solver for a 6-DOF arm without a spherical wrist, enabled by extending Pieper's decoupling to two consecutive intersecting axes, plus a CGS-based precomputation that avoids repeated Gröbner basis computations in trajectory planning. The algebraic derivations in Sections 3.2–3.3 are explicit, internally consistent, and independently checkable; the paper also ships concrete algorithms and numerical experiments. However, the headline claim of a 'global and explicit' solution is currently not supported because the CGS is incomplete, and the experimental validation has a coverage blind spot. The contribution is therefore conditional on completing the CGS or substantially restricting the claims.

major comments (3)
  1. [§1.3, §4.3, Algorithm 1] The central claim of a 'globally and explicitly determine the solution' is not supported by the evidence. Table 2 and §4.3 state that the CGS of Eq. (34), Case 1, was not completed after more than two months; only the first segment was found. Algorithm 1, lines 14–16, instructs the solver to choose an appropriate segment from the CGS, but §4.3 states the implementation simply assumes the input pose belongs to the only calculated segment. No membership test is performed. For a pose outside that segment, the precomputed generic basis need not specialize to a Gröbner basis of the substituted ideal, so the computed (x,y,z), and hence all joint angles, can be wrong or missing. The experiments in Tables 3–4 cannot detect this because the test poses are generated by forward kinematics from random joint angles, which may all fall inside the one segment. Please either compute the complete CGS, im
  2. [Remark 3, Algorithm 1 line 22] The verification step in Algorithm 1, line 22, is supposed to check that the forward kinematics reproduce the given (l,m,n,p). Remark 3 admits that the implementation only checks the position part p, not the orientation. This is load-bearing because Eq. (34) was obtained by squaring equations (compare Remark 2), so extraneous solutions with correct position but wrong orientation are expected. The success rates in Tables 3–4, and the error statistics in §4.5, therefore overstate the validity of the returned joint angles. The verification must include the orientation components (l,m,n), or at least the rotation-matrix orthogonality constraints, before a solution is counted as successful.
  3. [§4.4, Table 5] The experiment with randomly assigned positions and orientations (Table 5) reports only about 37.3% success, which the authors attribute to unreachable regions. Because the CGS is incomplete and only one segment is used, this failure rate confounds two distinct causes: poses that are genuinely unreachable, and poses that lie outside the single computed CGS segment. No analysis is given of how many of the 10,000 random poses actually belong to the precomputed segment. Without that separation, the 37.3% figure cannot be interpreted as evidence about the manipulator's workspace, nor can it validate the global claim. Please report segment-membership information for the test poses, or otherwise distinguish the two causes.
minor comments (4)
  1. [§3.3] Typo: 'we wee that' should be 'we see that' in the paragraph following Eq. (27).
  2. [§4.3] Typo: 'Gr¨ ober basis' should be 'Gröbner basis' in the paragraph after Table 2.
  3. [§4.4] The notation in Step 4 of the accuracy assessment, 'θ_i = (θ_{i,1}, ..., θ_{i,6})', uses the subscript i for both the sample index and the joint index, which is confusing. Use a different index, e.g., 'θ^{(k)}' for the k-th sample.
  4. [Figure 3] The figure lacks axis labels and a legend explaining the color scale. The text refers to bright and dark colors, but the figure should be self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: joint-angle formulas are algebraic consequences of DH kinematics; incomplete CGS is a completeness gap, not a circular reduction.

full rationale

The paper's derivation chain is self-contained and non-circular. Sections 3.2–3.4 derive sinθi/cosθi (i=1..6) purely as algebraic consequences of the DH transformation matrices (1)–(3) and the orthonormality relations (8); no parameter is fitted to any experimental target. The parametric system (34) for the wrist-center coordinates (x,y,z) is obtained by comparing components of kinematically fixed vectors (7P, w4, P3, 5O1) and eliminating the joint angles, so solving (34) is a genuine function of the input pose, not an input–output tautology. Algorithm 1's line 22 verification re-substitutes recovered angles into the forward-kinematics model; this is a consistency filter, not a construction of the answer from the desired output. The self-citations [16,26] supply the CGS-QE computational strategy (an external algorithm from [4,21,7]) and do not contain the target 6-DOF IK result, so they are not load-bearing evidence for the paper's central claim. The honest limitations — §1.3 and §4.3/Table 2 state the CGS of (34) was not completed (only one segment computed), §4.3 states the implementation 'assume[s] that the position and orientation of the end effector in the input belong to the only calculated segment', and §5 admits uncovered segments — undermine the 'globally and explicitly' headline claim but are completeness/correctness concerns, not circular reductions. Similarly, Remark 3's note that only position (not orientation) is verified in the implementation weakens the experimental validation but does not make any derived quantity identical to an input by construction. Finally, the validation protocol (random joints → forward kinematics → inverse kinematics) is retrospective testing of derived formulas; no fitted parameter is re-labeled as a prediction. No equation in the paper reduces to another by construction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The ledger is intentionally short: no data-fitting constants are introduced and no new physical entities are postulated. The derivation rests on standard rotation-matrix identities, the manufacturer's DH table, and the as-yet-unproven coverage of the single computed CGS segment over the experimental inputs, which is the item that carries the most risk.

assumptions (3)
  • standard math Rotation-matrix orthonormality identities (8) hold for the end-effector orientation.
    Invoked repeatedly to simplify (14)-(18) and in the expansion of (28). This is a standard property of rotation matrices.
  • domain assumption The DH parameters in Table 1 for myCobot are exact, specifically a4=0 and a5=0, so the axes of joints 4 and 5 intersect at the origin of Σ5.
    The entire construction uses P as the intersection point of the joint 4 and 5 axes (Section 3.2, eq. (7) and (9)). If the manufacturer's geometry differs (e.g., nonzero a4), equations (19), (30), (32), and (34) do not follow.
  • ad hoc to paper Each tested end-effector pose lies in the single computed CGS segment of (34), so the precomputed generic Gröbner basis remains valid after numerical substitution.
    The full CGS computation exceeded two months and was not completed (Table 2, Case 1); only the first segment (2.59 s) is available. Section 4.3 states the implementation 'assume[s] that the position and orientation of the end effector in the input belong to the only calculated segment', but no test for segment membership is provided or used. This premise is load-bearing for Tables 3-5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr\"obner Systems." pith.science (2026). https://pith.science/paper/A44IGKG4

@misc{pith2026250900823,
  author       = {Pith},
  title        = {Pith review of: Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr\"obner Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A44IGKG4}},
  note         = {Machine review of arXiv:2509.00823}
}
read the original abstract

We propose an effective method for solving the inverse kinematic problem of a specific model of 6-degree-of-freedom (6-DOF) robot manipulator using computer algebra. It is known that when the rotation axes of three consecutive rotational joints of a manipulator intersect at a single point, the inverse kinematics problem can be divided into determining position and orientation. We extend this method to more general manipulators in which the rotational axes of two consecutive joints intersect. This extension broadens the class of 6-DOF manipulators for which the inverse kinematics problem can be solved, and is expected to enable more efficient solutions. The inverse kinematic problem is solved using the Comprehensive Gr\"obner System (CGS) with joint parameters of the robot appearing as parameters in the coefficients to prevent repetitive calculations of the Gr\"obner bases. The effectiveness of the proposed method is shown by experiments.

Figures

Figures reproduced from arXiv: 2509.00823 by the authors.

Figure 1
Figure 1. myCobot 280 [2] Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6 The end-effector (Joint 7) d1 a2 a3 d4 d5 d6 θ1 θ2 θ3 θ4 θ5 θ6 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Distribution of error rates of the solutions output of Algorithm 1, most of them had an error evaluation function E of 10−5 or more. Considering the reproducibility of myCobot, the accuracy of the output of Algorithm 1 cannot be considered good for these samples. From both the evaluation of the errors in the forward and inverse kinematic solutions, investigating the causes, and proposing solutions for the phenomenon… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An Effective Trajectory Planning and an Optimized Path Planning for a 6-Degree-of-Freedom Robot Manipulator

    cs.RO 2025-08 conditional novelty 5.0 of 10

    For the myCobot 280 arm, trajectories along line segments with fixed orientation are planned by solving inverse kinematics at sampled points, and the optimal sequence of joint configurations is found by Dijkstra's alg...

Reference graph

Works this paper leans on

26 extracted references · 15 canonical work pages · cited by 1 Pith paper

  1. [1]

    Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

    Denavit, J., Hartenberg, R.: A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices. Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

    doi:10.1115/1.4011045 1955

  2. [2]

    elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2025-05-09

    Elephant Robotics Co., Ltd.: myCobot 280 M5 (2023),https://www. elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2025-05-09

    2023

  3. [3]

    Research Report RR-5923, INRIA (2006),https://hal.inria

    Faug` ere, J.C., Merlet, J.P., Rouillier, F.: On solving the direct kinematics problem for parallel robots. Research Report RR-5923, INRIA (2006),https://hal.inria. fr/inria-00072366

    work page 2006

  4. [4]

    In: Proceedings of the 2015 ACM on Inter- national Symposium on Symbolic and Algebraic Computation

    Fukasaku, R., Iwane, H., Sato, Y.: Real Quantifier Elimination by Computation of Comprehensive Gr¨ obner Systems. In: Proceedings of the 2015 ACM on Inter- national Symposium on Symbolic and Algebraic Computation. pp. 173–180. IS- SAC ’15, ACM, New York, NY, USA (2015).https://doi.org/10.1145/2755996. 2756646

    work page doi:10.1145/2755996 2015

  5. [5]

    Husty, M.L., Pfurner, M., Schr¨ ocker, H.P.: A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator. Mech. Mach. Theory 42.1 pp. 66–81 (2007).https://doi.org/10.1016/j.mechmachtheory.2006.02.001

    work page doi:10.1016/j.mechmachtheory.2006.02.001 2007

  6. [6]

    Kalker-Kalkman, C.M.: An implementation of Buchbergers’ algorithm with ap- plications to robotics. Mech. Mach. Theory28(4), 523–537 (1993).https://doi. org/10.1016/0094-114X(93)90033-R Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator 23

    doi:10.1016/0094-114x(93)90033-r 1993

  7. [7]

    Kapur, D., Sun, Y., Wang, D.: An efficient method for computing comprehensive Gr¨ obner bases. J. Symbolic Comput52, 124–142 (2013).https://doi.org/10. 1016/j.jsc.2012.05.015

    work page 2013

  8. [8]

    Kim, C.H., H. R. Kam, S.H.L., Park, T.: R Viz: a toolkit for real domain data vi- sualization. In: Telecommunication Systems, vol. 60, pp. 337–345. Springer (2015). https://doi.org/10.1007/s11235-015-0034-5

    doi:10.1007/s11235-015-0034-5 2015

Show all 26 references
  1. [9]

    Science robotics7(66), eabm6074 (2022).https://doi.org/10.1126/scirobotics.abm6074

    Macenski, S., Foote, T., Gerkey, B., Lalancette, C., Woodall, W.: Robot Operat- ing System 2: Design, architecture, and uses in the wild. Science robotics7(66), eabm6074 (2022).https://doi.org/10.1126/scirobotics.abm6074

    2022 doi

  2. [10]

    IEEE Transactions on Robotics and Automation10(5), 648–657 (1994).https: //doi.org/10.1109/70.326569

    Manocha, D., Canny, J.F.: Efficient inverse kinematics for general 6R manipulators. IEEE Transactions on Robotics and Automation10(5), 648–657 (1994).https: //doi.org/10.1109/70.326569

    1994 doi

  3. [11]

    Nabeshima, K.: CGS: a program for computing comprehensive Gr¨ obner systems in a polynomial ring [computer software] (2018),https://www.rs.tus.ac.jp/ ~nabeshima/softwares.html, accessed 2025-02-05

    2018

  4. [12]

    Nabeshima, K.: Generic Gr¨ obner basis of a parametric ideal and its application to a comprehensive Gr¨ obner system. Appl. Algebra Eng. Commun. Comput.35(1), 55– 70 (2024).https://doi.org/10.1007/S00200-023-00620-8,https://doi.org/ 10.1007/s00200-023-00620-8

    2024 doi

  5. [13]

    Advances in Linear Algebra & Matrix Theory8(1), 33–40 (2018)

    Ni, Z., Wu, R.: 6R robot inverse solution algorithm based on quaternion matrix and Gr¨ obner base. Advances in Linear Algebra & Matrix Theory8(1), 33–40 (2018). https://doi.org/10.4236/alamt.2018.81004

    2018

  6. [14]

    In: Proc

    Noro, M., Takeshima, T.: Risa/Asir — A Computer Algebra System. In: Proc. ISSAC ’92. pp. 387–396. Association for Computing Machinery, New York, NY, USA (1992).https://doi.org/10.1145/143242.143362

    1992

  7. [15]

    In: Computer Algebra in Scientific Computing, CASC 2025

    Okazaki, T., Terui, A., Mikawa, M.: An Effective Trajectory Planning and an Opti- mized Path Planning for a 6-Degree-of-Freedom Robot Manipulator. In: Computer Algebra in Scientific Computing, CASC 2025. Lecture Notes in Computer Science, Springer (2025),https://doi.org/10.485...

    2025 doi arXiv

  8. [16]

    Preprint (2021).https://doi.org/10.48550/ arXiv.2111.00384

    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 (2021).https://doi.org/10.48550/ arXiv.2111.00384

    2021 doi arXiv

  9. [17]

    Pieper, D.L.: The Kinematics of Manipulators under Computer Control. Ph.D. Thesis, Stanford University (October 1968),https://apps.dtic.mil/sti/ citations/AD0680036

    1968

  10. [18]

    In: Proceedings of the SCSS 2024 Work in Progress Work- shop

    Shirato, Y., Oka, N., Terui, A., Mikawa, M.: An optimized path planning of ma- nipulator with spline curves using real quantifier elimination based on comprehen- sive gr¨ obner systems. In: Proceedings of the SCSS 2024 Work in Progress Work- shop. pp. 105–112. Sun SITE Central...

    2024

  11. [19]

    Springer (2008).https://doi.org/10.1007/978-1-84628-642-1

    Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G.: Robotics: Modelling, Planning and Control. Springer (2008).https://doi.org/10.1007/978-1-84628-642-1

    2008 doi

  12. [20]

    Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/2021/6680687

    Ricardo Xavier da Silva, S., Schnitman, L., Cesca Filho, V.: A Solution of the Inverse Kinematics Problem for a 7-Degrees-of-Freedom Serial Redundant Manip- ulator Using Gr¨ obner Bases Theory. Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/20...

    2021 doi

  13. [21]

    In: Proc

    Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gr¨ obner bases using Gr¨ obner bases. In: Proc. ISSAC 2006. pp. 326–331 (2006).https://doi. org/10.1145/1145768.1145821 24 Takumu Okazaki, Akira Terui, and Masahiko Mikawa

    2006

  14. [22]

    Bordeaux: PARI/GP version2.13.1(2021), available fromhttps://pari.math.u-bordeaux.fr/

    The PARI Group, Univ. Bordeaux: PARI/GP version2.13.1(2021), available fromhttps://pari.math.u-bordeaux.fr/

    2021

  15. [23]

    Multibody System Dynamics25, 335–356 (2011).https://doi.org/10.1007/s11044-010-9241-8

    Uchida, T., McPhee, J.: Triangularizing kinematic constraint equations using Gr¨ obner bases for real-time dynamic simulation. Multibody System Dynamics25, 335–356 (2011).https://doi.org/10.1007/s11044-010-9241-8

    2011 doi

  16. [24]

    Uchida, T., McPhee, J.: Using Gr¨ obner bases to generate efficient kinematic solu- tions for the dynamic simulation of multi-loop mechanisms. Mech. Mach. Theory 52, 144–157 (2012).https://doi.org/10.1016/j.mechmachtheory.2012.01.015

    2012 doi

  17. [25]

    In: Computer Algebra in Sci- entific Computing, CASC 2024

    Wada, N., Nabeshima, K.: Merging Multiple Algorithms for Computing Compre- hensive Gr¨ obner Systems Using Parallel Processing. In: Computer Algebra in Sci- entific Computing, CASC 2024. Lecture Notes in Computer Science, vol. 14938, pp. 373–391. Springer (2024).https://doi.or...

    2024 doi

  18. [26]

    In: Computer Algebra in Scientific Computing

    Yoshizawa, M., Terui, A., Mikawa, M.: Inverse Kinematics and Path Planning of Manipulator Using Real Quantifier Elimination Based on Comprehensive Gr¨ obner Systems. In: Computer Algebra in Scientific Computing. CASC 2023, Lecture Notes in Computer Science, vol. 14139, pp. 393...

    2023 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.