A Bi-Level Optimization Method for Redundant Dual-Arm Minimum Time Problems

Jonathan Fried; Santiago Paternain

REVIEW 3 major objections 4 minor 34 references

A Bi-Level Optimization Method for Redundant Dual-Arm Minimum Time Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The redundant dual-arm minimum-time problem is reformulated as a bi-level optimization with a convex, closed-form lower level, yielding efficient local solutions with lower traversal time and Cartesian error than a single-arm baseline.

desk verdict Dual-arm bi-level extension is real and clean, but the numerical section has an unresolved error-tolerance mismatch that undermines the reported feasibility claims as written. read the letter →

arxiv 2506.03982 v1 pith:GF6IMBSL submitted 2025-06-04 cs.RO math.OC

Jonathan Fried , Santiago Paternain This is my paper

classification cs.ROmath.OC

keywords bi-leveloptimizationdual-armrobottime-optimaltrajectoryredundancyresolutionconvexconstantpathspeedCartesianerrorpolynomialparameterization

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 the redundant dual-arm minimum-time problem—two arms tracing a prescribed relative Cartesian path at constant path speed under joint position, velocity, and acceleration limits—can be reformulated as a bi-level optimization that is efficiently solvable. The lower level is a convex, closed-form subproblem that, for any fixed joint trajectory, returns the maximum constant path speed and the resulting traversal time. The upper level updates the joint trajectories by subgradient descent, using a single-chain kinematic description that makes the Cartesian error independent of joint poses. The authors report that in a cold-spray-inspired simulation with two planar 3R arms, the dual-arm method attains lower traversal times and lower Cartesian errors than a single-arm baseline, while a general-purpose nonlinear solver fails to converge. This matters for industrial tasks like spray coating or additive manufacturing, where uniform material deposition forces a constant path speed and shorter traversal time means higher productivity without sacrificing accuracy.

What carries the argument

The load-bearing mechanism is the bi-level decomposition of problem (12) into the lower-level problem (13)—a convex, closed-form maximization of the path-speed square for a fixed trajectory, whose value V(θ_A, θ_B) is a max over arms, joints, and path points of squared velocity ratios and acceleration ratios (Theorem 1)—and the upper-level problem (14) that updates the trajectory parameters θ by subgradient descent. The second piece is the single-chain kinematic formulation (16)–(17) expressed in the tooltip frame of robot B, which makes the desired relative path independent of joint positions and produces the relative Jacobian J(q) = [−ψ_{tB,tA} Ω_{tB,bB} J_B, Ω_{tB,bA} J_A] (Proposition 2). The error gradient in Corollary 1 follows from this Jacobian and the chain rule. Together, these components let the algorithm exploit the redundancy of the dual arm: nullspace directions of the relative Jacobian change the arm pose without increasing the Cartesian error, and the convex lower level supplies a cheap subgradient for the upper-level descent.

What would settle it

Take a path whose exact minimum-time constant-speed joint trajectories are known or can be computed by a high-resolution brute-force method, and run the proposed algorithm with the 9th-degree polynomial basis; if the returned traversal time is measurably above the exact optimum and a higher-degree basis reduces the gap, the polynomial restriction is the limiting factor. A specific test is to use a path whose optimal joint trajectory contains a mid-path velocity reversal or a very sharp corner, which a smooth polynomial in s cannot represent faithfully; the method should fail to match the known optimum, showing that the reported efficiency is tied to the restricted search space.

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Extended reading notes

Core claim

The core discovery is that the minimum-time constant-path-speed problem for a redundant dual-arm system has a bi-level structure whose lower level is convex and admits a closed-form solution. Theorem 1 gives V(θ_A, θ_B), the square of the optimal traversal time for a given trajectory, as a maximum over all arms, joints, and path points of squared ratios of basis-function derivatives to the corresponding velocity and acceleration limits; this value is convex in the trajectory parameters. With this, the upper-level problem can be minimized by subgradient descent on V even though V is not differentiable everywhere, extending earlier bi-level time-optimal methods that required differentiability. The paper also shows that writing the relative kinematics as a single chain from the second arm's tooltip to the first arm's tooltip fixes the desired relative path in that frame and yields a compact relative Jacobian (Proposition 2), whose nullspace gives directions that change the pose without changing the Cartesian error. The numerical experiments claim the dual-arm formulation reaches average traversal times of 0.723 s versus 0.865 s for the single-arm baseline, with maximum Cartesian error 3.3 ± 1.4 mm (within the 5 mm tolerance) versus 5.3 ± 3.1 mm for the baseline.

Load-bearing premise

The load-bearing premise is that a fixed low-degree polynomial basis in the path parameter can represent the true time-optimal joint trajectories well enough that optimizing the polynomial coefficients yields meaningful minima; if the true optimal trajectories need features outside that family, the algorithm is confined to a suboptimal subset of trajectories.

Editorial extensions

If this is right

For a fixed joint trajectory, the traversal time is computed in closed form, so the algorithmic cost per iteration is dominated by evaluating the max in (15) rather than solving a trajectory optimization.
Because the desired relative path is expressed in the tooltip frame of the second arm, changing the arm poses does not require recomputing the desired Cartesian path or its gradient.
The method only needs subgradients of the lower-level value, so it remains applicable when the upper-level problem is nondifferentiable, unlike previous bi-level time-optimal approaches.
In the reported cold-spray simulations, the dual-arm formulation reduces average traversal time from 0.865 s to 0.723 s and keeps maximum Cartesian error within the 5 mm tolerance while the single-arm baseline violates it.
The approach produces locally optimal trajectories in the chosen polynomial class for any initial condition, with final times varying by initialization.

Reading between the lines

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

The bi-level decomposition is likely to extend to multi-arm systems beyond two, since the lower-level max in (15) scores each joint independently and the single-chain Jacobian can be chained through additional tooltip-base frames.
A natural stress test the authors do not report is to vary the polynomial degree: if the optimal traversal time keeps falling as the degree rises, the 9th-degree basis is an active restriction, and the reported gains over the baseline may partly come from the search-space choice rather than the bi-level structure alone.
The convexity of the lower level and its closed form suggest the method could be warmed-up for online replanning: a small change in the desired path should keep the upper-level parameters in a good region, and the subgradient step would adjust without a full restart.
The constant-path-speed assumption is a proxy for uniform deposition; the same bi-level architecture could be adapted to piecewise-constant speeds by solving the lower level as a small convex program per segment, which would extend the approach to deposition tasks that allow variable speed.

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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,

The takeaway: the dual-arm bi-level extension is a real, incremental contribution with a clean kinematic formulation, but the numerical experiment has an unresolved error-tolerance mismatch that makes the reported feasibility claims unverifiable. The authors need to fix this before the evidence supports their conclusions.

Where the paper earns credit: it takes the single-arm bi-level method from their own prior work and extends it to dual-arm systems by modelling both arms as a single kinematic chain. The chosen frame—robot B's tool frame—makes the desired relative path independent of both robots' joint configurations, which is a nice trick. The lower-level problem (max path speed for a given trajectory) remains convex with a closed-form solution, and they show that the upper-level subgradient can be computed via the relative Jacobian. The derivations in Propositions 1 and 2 and Corollary 1 are careful, and the algebraic form of Eq. (15) is plausible. The dual-arm simulation shows lower traversal times and lower max per-point errors than the single-arm baseline, so the core idea—that extra redundancy can shave time—is supported qualitatively.

The soft spots are non-trivial. The paper defines E in (11) as an RSS error over the path, states the constraint E ≤ ε with ε = 10^{-5} in Section IV, but then reports maximum Cartesian errors of 1.3–4.9 mm. Since E is at least as large as any per-point norm, these reported solutions cannot satisfy the stated constraint by orders of magnitude. At the same time, the text says the industry tolerance is ±5 mm on the Cartesian error, which suggests the actual enforced constraint was per-point error in mm, not the E in Eq. (11). The paper needs to either fix the metric definition, the epsilon value, or the reported errors; as written, the comparison to the single-arm baseline and CasADi is comparing solutions to a different problem than the one solved. A second concern: Theorem 1, the convexity and closed-form result, is deferred to their prior reference [24] with no sketch here. This is common practice, but it makes the paper not self-contained, and a referee should verify that the proof transfers to the dual-arm case rather than just asserting it. The polynomial basis (9th degree) is a strong restriction, and the paper doesn't discuss whether higher-degree or other basis functions would give meaningfully lower times. The CasADi comparison is presented as a failure without adequate detail on initialization, scaling, and feasibility tolerances (the reported errors of 1485 mm suggest the solver was not set up to enforce the error constraint at all). No code or data is provided.

Overall, the method is plausible and the extension is worth pursuing, but the numerical evidence in this version is not convincing. If I were refereeing it, I would ask for a corrected error metric, an explicit statement of the constraint actually enforced, a sanity check on the CasADi baseline, and ideally code/data. I'd still send it out for review rather than desk-reject it—the formulation and the single-chain handling of the relative error are useful, and the tolerance issue is fixable with careful re-running.

Recommendation: engage with the paper, but don't rely on the reported numbers until the error tolerance is clarified.

Referee Report

3 major / 4 minor

Summary. The paper proposes a bi-level reformulation of the minimum-time, constant-path-speed trajectory optimization problem for redundant dual-arm manipulators. The upper level optimizes joint-trajectory coefficients subject to a Cartesian path-accuracy constraint, while the lower level maximizes the constant path speed for a fixed trajectory and is claimed to have a convex, closed-form value function. The authors introduce a single-chain kinematic formulation with a relative Jacobian to compute the path error and its gradient, and they evaluate a primal-dual subgradient method on a simulated cold-spray task against a single-arm baseline and CasADi.

Significance. If the bi-level decomposition and the closed-form convex lower-level value function were fully established and the numerical evaluation were internally consistent, the paper would offer a practically attractive way to exploit redundancy in coordinated dual-arm tasks; the proposed TCP-frame formulation is a clean idea, and the numerical comparison suggests that the extra redundancy can reduce traversal time. The paper is also transparent about hyperparameters and initial conditions. However, the central theorem is imported from the authors' prior work rather than proved or precisely reduced to the present setting, and the reported Cartesian errors violate the stated path-constraint tolerance by orders of magnitude, so the empirical claims are not currently supported.

major comments (3)

[§IV, Table I and §II, Eq. (12)] The reported numerical results do not correspond to feasible solutions of the stated problem. Section IV sets the path constraint tolerance to ε = 10^{-5}, and problems (12) and (14) require E(θ_A, θ_B) ≤ ε. By definition (11), E is the ℓ_p norm of the per-point Cartesian errors, so E is at least as large as the maximum pointwise Euclidean error. Table I reports maximum Cartesian errors of 1.3–4.9 mm for the proposed dual-arm solutions, and Figure 3 shows millimeter-scale pointwise errors. If ε is interpreted in meters, the allowed error is 0.01 mm; if interpreted in millimeters, it is 10^{-5} mm. Either way, the reported errors exceed the tolerance by orders of magnitude. The paper must report the actual value of E(θ_A, θ_B) at termination, correct the stated ε (for example, to the ±5 mm industry tolerance mentioned in Section IV), or explain why the accuracy constraint was not enforced. Without this, the time savings relative to the single-arm baseline cannot be attributed to solving the constrained problem (12).
[§III-A, Theorem 1] The paper's central structural claim is Theorem 1, which asserts that the lower-level value function has the closed form (15) and is convex. The proof is deferred with the single sentence "Proof is analogous to the single arm in [24]." This theorem is load-bearing: it justifies the convexity claim and supplies the objective used in the upper-level subgradient updates. Since the lower-level problem (13) is not literally identical to the earlier single-arm problem—it involves the coupled dual-arm parameter vector and a different sign/squared notation—the paper should either prove the theorem or state explicitly how (13) reduces to the setting of [24]. As written, the convexity and closed-form claims are not self-contained.
[§IV, Table I and CasADi comparison] The comparison with CasADi is not usable as evidence in its current form. Table I lists CasADi results with final times from 0.272 s to 285 s and Cartesian errors of 420–1645 mm, while the text says that CasADi failed to converge. Presenting non-converged iterates side-by-side with converged solutions, without marking them as such, makes the running-time and trajectory-quality comparisons uninterpretable. If the purpose is to show that a general-purpose solver struggles on this problem, the failure should be described quantitatively—for example, via solver exit status and constraint violation—rather than reported as if the entries were solved trajectories.

minor comments (4)

[§II, Eq. (13)] The constraint "− ˙q^2 ≤ sign(p'_i θ)(p'_i θ)^2 ˙s^2 ≤ ˙q^2" uses sign(x)x^2 = x|x|, which is not a square for negative values; please rewrite the velocity and acceleration constraints in the standard form ˙q_ℓj ≤ p'_i θ_ℓj ˙s ≤ ˙q_ℓj and ¨q_ℓj ≤ p''_i θ_ℓj ˙s^2 ≤ ¨q_ℓj, or explain the intended simplification.
[§IV] The text states both "A Cartesian error tolerance of ±5 mm is set per industry standards" and "path constraint tolerance ε = 10^{-5}" without reconciling the two; please state the relationship between ε and the ±5 mm tolerance.
[§II, Eq. (11)] The norm parameter p in the definition of E is never specified in the numerical section; please state the value used in the experiments (presumably p = 2).
[Title and Abstract] The title/abstract contains a spurious space in "Mini mum"; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bi-level reformulation derives from the stated constraints, and the few self-citations are supporting lemmas rather than smuggled conclusions.

full rationale

The derivation chain is self-contained. The lower-level value function V(θ_A,θ_B) is not defined as the formula in Eq. (15); instead, Eq. (13) is a genuine minimization over the constant path speed, and Theorem 1 characterizes its closed form from the joint velocity and acceleration constraints in Eqs. (9)–(10). The upper-level problem in Eq. (14) minimizes that explicit convex value function while enforcing the Cartesian error and position limits, and the error gradient in Eq. (23) follows from the relative Jacobian of Eq. (21), which is cited to an external source [23]. The proof of Theorem 1 is deferred to the authors' prior work [24], but this is not load-bearing circularity: the theorem is stated in full, is parameter-free, and its assumptions are the joint-limit inequalities, not the paper's target result. The dual-arm extension is a direct combination of the single-arm statement over both robots, so the self-citation is supporting evidence rather than a reduction of the claim to its own input. No fitted parameter is renamed as a prediction: the reported times and errors are simulation demonstrations under fixed hyperparameters, not quantities used to calibrate the model. The polynomial basis ansatz in Eq. (6) and the single-chain kinematic model are explicit assumptions or externally cited results, not hidden equivalences. The discrepancy between the stated path tolerance ε=1e-5 and the millimeter-scale maximum Cartesian errors in Table I and Figure 3 is a correctness or validation concern, but it is not a circularity, because the method's derivation does not rely on that tolerance being satisfied. Overall, the proposed optimization of θ against V(θ) with subgradient updates and a primal-dual enforcement of the error constraint has independent content and is not equivalent by construction to the physical time-minimization objective.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The method relies on a fixed polynomial parameterization, a self-cited theorem for the lower level, a cited relative Jacobian, and an unproven local convergence claim for the primal-dual scheme. Hyperparameters such as step sizes, tolerance, basis degree, and discretization are chosen by hand without sensitivity analysis.

free parameters (5)

step size ηθ = 1e-5
Chosen for the θ update; no theory or sensitivity analysis is provided to justify this value.
dual step size ηd = 0.5
Chosen for the dual variable update; no convergence or sensitivity study is reported.
path constraint tolerance ϵ = 1e-5
Stated as the error constraint tolerance in the optimization, but the reported max Cartesian errors are about 3 mm, which is inconsistent with a 1e-5 bound on the error metric E.
polynomial basis degree = 9
Degree of the basis p(s) used for all joint trajectories; chosen without discussion of whether this subspace can approximate the true time-optimal trajectory.
number of path points N+1 = 500
Discretization of the path; chosen for the simulation, with no convergence study over N.

assumptions (4)

domain assumption Joint trajectories are linearly parameterized as q_{ℓij} = p(s_i) θ_{ℓj} with a twice-differentiable basis p (Eq. 6).
The entire optimization is over θ in this finite-dimensional subspace; optimality is relative to this subspace, not the full trajectory space.
ad hoc to paper Theorem 1 from the authors' prior work [24] gives the closed-form convex value V(θ_A, θ_B) in Eq. 15.
The proof is deferred to a self-cited prior paper, so the central formula of this paper is taken as given without self-contained derivation.
standard math The relative Jacobian formula J(q) = [-ψ Ω_B J_B, Ω_A J_A] from [23] is correct for the single-chain formulation (Eq. 21).
The proof is omitted and cited to Jamisola and Roberts; the paper assumes the formula applies to the coupled dual-arm chain.
domain assumption The primal-dual subgradient algorithm converges to a local minimum of the nonconvex upper-level problem (14).
No convergence result is provided; the method is justified only by numerical demonstration on five initial conditions.

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Cite this review

Pith. "Pith review of A Bi-Level Optimization Method for Redundant Dual-Arm Minimum Time Problems." pith.science (2026). https://pith.science/paper/GF6IMBSL

@misc{pith2026250603982,
  author       = {Pith},
  title        = {Pith review of: A Bi-Level Optimization Method for Redundant Dual-Arm Minimum Time Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GF6IMBSL}},
  note         = {Machine review of arXiv:2506.03982}
}

read the original abstract

In this work, we present a method for minimizing the time required for a redundant dual-arm robot to follow a desired relative Cartesian path at constant path speed by optimizing its joint trajectories, subject to position, velocity, and acceleration limits. The problem is reformulated as a bi-level optimization whose lower level is a convex, closed-form subproblem that maximizes path speed for a fixed trajectory, while the upper level updates the trajectory using a single-chain kinematic formulation and the subgradient of the lower-level value. Numerical results demonstrate the effectiveness of the proposed approach.

Figures

Figures reproduced from arXiv: 2506.03982 by the authors.

**Figure 3.** Cartesian errors for the whole trajectory, for singl [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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CasADi – A software framework for nonlinear optimization a nd optimal control,

J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, an d M. Diehl, “CasADi – A software framework for nonlinear optimization a nd optimal control,” Mathematical Programming Computation , 2018

2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [24]

A Bi-Level Optimization Approach to Joint Trajectory Optimization for Redundant Manipulators

J. Fried and S. Paternain, “A bi-level optimization app roach to joint trajectory optimization for redundant manipulators,” 202 4. [Online]. Available: https://arxiv.org/abs/2412.07859

work page Pith review arXiv

[2] [1]

Research on motion algorithm for intelligent operation of rock drill with eight degrees of freedom,

Y . Xu, S. Tong, Z. Ju, Z. Wang, Y . Zhao, J. Y ao, and Y . Zhao, “Research on motion algorithm for intelligent operation of rock drill with eight degrees of freedom,” in 2018 IEEE ISR , 2018, pp. 61–65

work page 2018

[3] [2]

Printing-while -moving: a new paradigm for large-scale robotic 3d printing,

M. E. Tiryaki, X. Zhang, and Q.-C. Pham, “Printing-while -moving: a new paradigm for large-scale robotic 3d printing,” in 2019 IEEE/RSJ IROS, 2019, pp. 2286–2291. Simulation Results Single Arm Dual Arm - Proposed Dual Arm - CasADi Run tf (s) Running Time (s) Max Cartesian Error (mm) tf (s) Running Time (s) Max Cartesian Error (mm) tf (s) Running Time (s) ...

work page 2019

[4] [3]

Mechatronics design and kinematic simulation of 5 dof seri al robot manipulator for soldering tht electronic components in pri nted circuit boards,

O. Mejia, D. Nu˜ nez, J. R´ azuri, J. Cornejo, and R. Paloma res, “Mechatronics design and kinematic simulation of 5 dof seri al robot manipulator for soldering tht electronic components in pri nted circuit boards,” in 2022 ICEEICT , 2022, pp. 1–7

work page 2022

[5] [4]

Robotic deep rolling with iterative learning motio n and force control,

S. Chen, Z. Wang, A. Chakraborty, M. Klecka, G. Saunders, and J. Wen, “Robotic deep rolling with iterative learning motio n and force control,” IEEE Robotics and Automation Letters , vol. 5, no. 4, pp. 5581–5588, 2020

work page 2020

[6] [5]

Automated industrial robot path planning for spray painting process: A review,

H. Chen, T. Fuhlbrigge, and X. Li, “Automated industrial robot path planning for spray painting process: A review,” in 2008 IEEE International Conference on Automation Science and Engine ering, 2008, pp. 522–527

work page 2008

[7] [6]

Clarifying the deﬁnitio n of redundancy as used in robotics,

E. S. Conkur and R. Buckingham, “Clarifying the deﬁnitio n of redundancy as used in robotics,” Robotica, vol. 15, p. 583–586, 1997

work page 1997

[8] [7]

Off-line programming of six-a xis robots for optimum ﬁve-dimensional tasks,

J. L´ eger and J. Angeles, “Off-line programming of six-a xis robots for optimum ﬁve-dimensional tasks,” Mechanism and Machine Theory , vol. 100, pp. 155–169, 2016

work page 2016