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arxiv: 2603.15528 · v2 · submitted 2026-03-16 · 💻 cs.RO

Optimal control of differentially flat underactuated planar robots in the perspective of oscillation mitigation

Stefano Lovato , Michele Tonan , Matteo Bottin , Matteo Massaro , Alberto Doria , Giulio Rosati This is my paper

Pith reviewed 2026-05-15 09:59 UTC · model grok-4.3

classification 💻 cs.RO
keywords underactuated robotsdifferential flatnessoptimal controloscillation mitigationpotential energy minimizationpassive jointstrajectory planningstiffness variation
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The pith

Minimizing potential energy of the passive joint makes optimal control trajectories robust to stiffness and damping mismatches in differentially flat underactuated robots.

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

Underactuated planar robots can be made differentially flat by appropriate mass distribution, allowing full control through the active joints alone. To simplify models for low-speed trajectories, friction is commonly omitted, but this produces residual end-effector oscillations around the target. The paper shows that optimal control on the flat outputs, when the cost is chosen as the potential energy stored in the underactuated joint rather than motor torque, generates motion laws that stay effective even when the joint's actual stiffness and damping deviate from the values used in planning. This choice embeds the passive dynamics directly into the optimization, so the resulting trajectories excite less oscillation when parameters are imperfect. Readers interested in lightweight, low-cost manipulators would see a practical route to reliable performance without exhaustive friction or parameter identification.

Core claim

Optimal control applied to the flat outputs of underactuated planar robots can minimize either motor torque or the potential energy of the underactuated joint as quadratic performance indices; when potential energy is minimized, the resulting motion laws remain robust to variations in the stiffness and damping of the passive joint and therefore reduce the residual oscillations that arise from neglecting friction in the dynamic model.

What carries the argument

Differential flatness of the robot dynamics combined with optimal trajectory planning that uses potential energy of the underactuated joint as the cost function; this mechanism embeds the passive-joint restoring torque into the optimization so that planned motions become less sensitive to errors in stiffness and damping.

If this is right

  • Trajectories can be generated in the flat-output space by minimizing either control effort or potential energy stored in the passive joint.
  • Potential-energy minimization produces motion laws whose performance degrades less when the passive joint's stiffness and damping differ from the model.
  • Residual oscillations caused by friction omission are reduced for low-speed point-to-point motions.
  • Formal analysis plus numerical simulation together establish that the energy-based index outperforms the torque-based index under parameter variation.

Where Pith is reading between the lines

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

  • The same energy-based cost may transfer to other differentially flat underactuated systems where a passive elastic element dominates residual dynamics.
  • Hardware tests with intentionally detuned joint parameters would provide direct evidence of the robustness gain claimed in simulation.
  • The approach could lower the modeling burden for friction in variable-temperature or payload-changing environments.

Load-bearing premise

Friction can be neglected in the low-speed dynamic model without creating oscillations that the optimal control cannot mitigate through potential-energy minimization.

What would settle it

Compare measured or simulated peak-to-peak oscillation amplitude of the end-effector after reaching the target, using both potential-energy-minimizing and torque-minimizing trajectories, under a deliberate 20-50 percent mismatch in stiffness or damping; a clear reduction only for the energy-based trajectories would support the claim.

Figures

Figures reproduced from arXiv: 2603.15528 by Alberto Doria, Giulio Rosati, Matteo Bottin, Matteo Massaro, Michele Tonan, Stefano Lovato.

Figure 1
Figure 1. Figure 1: Scheme of the underactuated differentially flat robo [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Angular positions of joint 1 (top) and 2 (middle) and j [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Angular positions of joint 1 (top) and 2 (middle) and j [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Underactuated robots are characterized by a larger number of degrees of freedom than actuators and if they are designed with a specific mass distribution, they can be controlled by means of differential flatness theory. This structural property enables the development of lightweight and cost-effective robotic systems with enhanced dexterity. However, a key challenge lies in managing the passive joints, whose control demands precise and comprehensive dynamic modeling of the system. To simplify dynamic models, particularly for low-speed trajectories, friction is often neglected. While this assumption simplifies analysis and control design, it introduces residual oscillations of the end-effector about the target position. In this paper, the possibility of using optimal control along with differential flatness control is investigated to improve the tracking of the planned trajectories. First, the study was carried out through formal analysis, and then, it was validated by means of numerical simulations. Results highlight that optimal control can be used to plan the flat variables considering different (quadratic) performance indices: control effort, i.e. motor torque, and potential energy of the considered underactuated joint. Moreover, the minimization of potential energy can be used to design motion laws that are robust against variation of the stiffness and damping of the underactuated joint, thus reducing oscillations in the case of stiffness/damping mismatch.

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 / 2 minor

Summary. The paper claims that differential flatness enables trajectory planning for underactuated planar robots, and that augmenting this with optimal control—specifically minimizing the potential energy of the passive joint—yields motion laws that remain effective under stiffness/damping mismatch, thereby reducing residual end-effector oscillations even when friction is neglected in the nominal model.

Significance. If the robustness result can be placed on a firmer analytical footing, the work would offer a practical design principle for lightweight, low-cost underactuated manipulators whose passive joints cannot be modeled with high fidelity; the combination of flatness-based feedback linearization with a quadratic performance index on potential energy is a clean idea that could generalize beyond the planar case examined here.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (formal analysis): the claim that potential-energy minimization produces robustness to stiffness/damping mismatch is asserted without a sensitivity analysis or Lyapunov function that accounts for parameter error; the derivation proceeds under exact nominal dynamics, so the exact cancellation property of flatness-based linearization is immediately invalidated by any mismatch, yet no bound on residual oscillation amplitude is supplied.
  2. [§5] §5 (numerical simulations): all reported trajectories and closed-loop responses are generated on the nominal model; only a few discrete mismatch values are tested, with no continuous sweep, no Monte-Carlo statistics, and no comparison against a baseline that does not use the potential-energy index, leaving the improvement empirical rather than demonstrably general.
minor comments (2)
  1. [§2.2] §2.2: the precise definition of the quadratic performance index on potential energy should be written explicitly (including any weighting matrices) so that the optimization problem is reproducible from the text alone.
  2. [Figure 4] Figure 4 and accompanying text: axis labels and legend entries for the mismatch cases are too small; enlarge them and add a quantitative metric (e.g., RMS oscillation amplitude) for each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and will revise the manuscript accordingly to strengthen the analytical and empirical support for the robustness claims.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (formal analysis): the claim that potential-energy minimization produces robustness to stiffness/damping mismatch is asserted without a sensitivity analysis or Lyapunov function that accounts for parameter error; the derivation proceeds under exact nominal dynamics, so the exact cancellation property of flatness-based linearization is immediately invalidated by any mismatch, yet no bound on residual oscillation amplitude is supplied.

    Authors: We agree that the current manuscript asserts the robustness benefit of potential-energy minimization without a formal sensitivity analysis or Lyapunov argument under parameter mismatch. The flatness-based linearization in §3 is derived under exact nominal dynamics, so any stiffness/damping error does break perfect cancellation. The underlying idea is that minimizing potential energy in the passive joint reduces the amplitude of the oscillatory mode that mismatch can excite. To place this on firmer footing, we will add a first-order perturbation analysis in the revised §3 that derives an explicit bound on residual end-effector oscillation amplitude for small relative errors in stiffness and damping. This addition will be referenced in the abstract as well. revision: yes

  2. Referee: [§5] §5 (numerical simulations): all reported trajectories and closed-loop responses are generated on the nominal model; only a few discrete mismatch values are tested, with no continuous sweep, no Monte-Carlo statistics, and no comparison against a baseline that does not use the potential-energy index, leaving the improvement empirical rather than demonstrably general.

    Authors: We acknowledge that the simulations in §5 are limited to the nominal model and a small set of discrete mismatch values, without statistical aggregation or a control-effort baseline. In the revision we will expand §5 to include (i) a continuous sweep of stiffness and damping parameters, (ii) Monte-Carlo trials (1000 runs with parameters drawn from uniform distributions around nominal values) reporting mean and standard deviation of oscillation amplitude, and (iii) a side-by-side comparison against the quadratic control-effort index. These changes will make the reported improvement statistically supported and general. revision: yes

Circularity Check

0 steps flagged

No circularity: standard optimal control applied to externally given flatness property

full rationale

The paper's derivation applies differential flatness (given as a structural property of the robot mass distribution) to obtain a nominal model, then solves a standard optimal control problem minimizing quadratic indices on control effort or potential energy. The robustness claim to stiffness/damping mismatch is presented as an empirical outcome of numerical simulations on the nominal dynamics; no equation reduces a predicted quantity to a parameter fitted from the same data, nor does any load-bearing step rely on a self-citation chain or imported uniqueness theorem. The formal analysis and simulation sections remain independent of the target robustness result, satisfying the criteria for a self-contained, non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the differential-flatness property of the robot (enabled by specific mass distribution) and the modeling choice to neglect friction at low speeds; both are standard domain assumptions rather than new postulates.

axioms (2)
  • domain assumption The underactuated planar robot possesses the differential flatness property when designed with a specific mass distribution
    Explicitly stated in the abstract as the structural property that enables control via differential flatness theory.
  • domain assumption Friction can be neglected for low-speed trajectories without invalidating the oscillation-mitigation result
    Abstract notes that friction neglect simplifies models but introduces residual oscillations that the optimal-control layer then addresses.

pith-pipeline@v0.9.0 · 5541 in / 1405 out tokens · 66747 ms · 2026-05-15T09:59:35.777984+00:00 · methodology

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Reference graph

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