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REVIEW 1 major objections 2 minor 36 references

Dynamical invariants allow design of fast cart trajectories that leave the pole balanced with zero final angle and velocity.

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2026-06-29 09:15 UTC pith:QYOFKAGR

load-bearing objection Applies invariant-based STA to design cart-pole trajectories that stay balanced only inside the linear small-oscillation regime, with post-hoc nonlinear checks to define the domain. the 1 major comments →

arxiv 2605.28177 v1 pith:QYOFKAGR submitted 2026-05-27 physics.class-ph

Invariant-based inverse engineering for balanced displacement of a cart-pole system

I. Lizuain , A. Tobalina , A. Rodriguez-Prieto This is my paper
classification physics.class-ph
keywords cart-pole systemshortcuts to adiabaticitydynamical invariantsinverse engineeringbalanced transportsmall oscillations regimeinverted pendulumtrajectory design
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper applies shortcuts to adiabaticity to an unstable mechanical system by constructing dynamical invariants for the linearized cart-pole equations and then using inverse engineering to obtain the required cart motion. This produces transport protocols that reach the target cart position while ending with the pole upright and at rest. The approach removes the slow-change requirement of ordinary adiabatic processes yet stays inside the regime where the linear approximation holds. Direct comparison of the designed trajectories against numerical integration of the full nonlinear model identifies the practical time and amplitude window where the guarantee remains valid.

Core claim

The trajectories found guarantee a balanced transport of the cart-pole within the small oscillations regime.

What carries the argument

Dynamical invariants of the linearized cart-pole equations, used through inverse engineering to determine the cart-position function of time.

Load-bearing premise

The motion must remain inside the small-oscillations regime so that the linearization and the chosen invariants continue to describe the dynamics throughout the transport.

What would settle it

A simulation or experiment in which the pole angle or angular velocity at the final time is observably nonzero after the designed trajectory is applied would falsify the balanced-transport claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The resulting protocols achieve the target state for times shorter than those required by adiabatic limits.
  • Final pole angle and angular velocity are both zero by construction inside the linear regime.
  • Direct nonlinear simulations bound the domain of validity in time and displacement amplitude.
  • The same invariant construction applies to other systems whose linearized dynamics possess a repulsive potential.

Where Pith is reading between the lines

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

  • The same invariant-based inverse-engineering route could be tested on cart-pole variants that include friction or external forcing.
  • Extension to two-dimensional or multi-link inverted-pendulum systems would require constructing a larger set of invariants.
  • Hardware tests would reveal how sensor noise and actuator limits affect the predicted final balance.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript develops a shortcuts-to-adiabaticity protocol that employs dynamical invariants of the linearized cart-pole system to inverse-engineer cart trajectories. These trajectories are asserted to guarantee balanced transport (final pole angle and angular velocity equal to zero) while remaining inside the small-oscillations regime; nonlinear simulations are used only after design to delineate the practical working domain.

Significance. If the regime-validity claim can be placed on a firmer footing, the work would supply a concrete, invariant-based STA construction for an unstable mechanical system with a repulsive potential, thereby extending adiabatic-invariant techniques into control-engineering contexts where slow adiabatic passage is impractical.

major comments (1)
  1. [Abstract] Abstract: the central claim that the designed trajectories 'guarantee a balanced transport of the cart-pole within the small oscillations regime' is not supported by an a-priori analytic bound on max| heta| (or on the adiabaticity parameter) derived from the invariant construction itself. The manuscript instead relies on post-design nonlinear simulations to 'set the working domain,' which does not establish that the linear model and chosen invariants remain valid throughout the trajectory by construction.
minor comments (2)
  1. The choice and explicit functional form of the dynamical invariants used for the cart-pole linearization should be stated with the corresponding differential equations.
  2. Notation for the cart displacement, pole angle, and control input should be introduced once and used consistently; several symbols appear without prior definition in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment concerns the wording of the central claim in the abstract. We address it below and agree that a clarification is warranted.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the designed trajectories 'guarantee a balanced transport of the cart-pole within the small oscillations regime' is not supported by an a-priori analytic bound on max| heta| (or on the adiabaticity parameter) derived from the invariant construction itself. The manuscript instead relies on post-design nonlinear simulations to 'set the working domain,' which does not establish that the linear model and chosen invariants remain valid throughout the trajectory by construction.

    Authors: We agree that the trajectories are constructed to guarantee balanced transport (final heta = 0 and hetȧ = 0) strictly for the linearized dynamics via the invariant-based inverse engineering. The small-oscillations regime is the domain of validity of that linearization, and the manuscript determines the practical range of parameters for which the designed trajectories remain inside this regime by means of post-design nonlinear simulations. No a-priori analytic bound on max| heta| is derived from the invariant alone. We will revise the abstract to state explicitly that the guarantee holds for the linearized system and that the working domain is established numerically. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation uses dynamical invariants of the linearized cart-pole model to construct trajectories via inverse engineering, then compares results to nonlinear simulations solely to delineate the working domain. No quoted equations or self-citations reduce any central prediction or guarantee to a fitted input, self-definition, or prior author ansatz by construction. The small-oscillations regime is an explicit modeling assumption with post-design numerical checks, leaving the core STA protocol independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the small-oscillations linearization is an implicit domain assumption but cannot be audited without the manuscript.

pith-pipeline@v0.9.1-grok · 5654 in / 977 out tokens · 20982 ms · 2026-06-29T09:15:16.935680+00:00 · methodology

0 comments
read the original abstract

Adiabaticity is a key concept in physics, but its applications in mechanical and control engineering remain underexplored. Adiabatic invariants ensure robust dynamics under slow changes, but they impose impractical time limitations. Shortcuts to Adiabaticity (STA) overcome these limitations by enabling fast operations with minimal final excitations. In this work, we set a STA strategy based on dynamical invariants and inverse engineering to design the trajectory of a cart-pole, a system characterized by its instability and repulsive potential. The trajectories found guarantee a balanced transport of the cart-pole within the small oscillations regime. The results are compared to numerical simulations with the exact non-linear model to set the working domain of the designed protocol.

Figures

Figures reproduced from arXiv: 2605.28177 by A. Rodriguez-Prieto, A. Tobalina, I. Lizuain.

Figure 1
Figure 1. Figure 1: Inverted pendulum on a cartpole. Physical model and relevant parameters. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Color online) Cartpole trajectories (a) and velocities (b) for different total process time [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Color online) (a) Cartpole swing angle θ as a function of time after numerically integrating the exact dynamical equation (3) (blue solid line) and linearized harmonic model (4) (red dashed line). with null initial conditions θ(0) = ˙θ(0) = 0. (b) Phase-space diagram for the same process. Linearized model leads to purely periodic motion and therefore closed orbits in phase-space, whereas nonlinearities of… view at source ↗
Figure 4
Figure 4. Figure 4: (Color online) Ficticious angle Θ as a function of the process time tf in different scenarios. As discussed in the text, this angle quantifies the final deviation from the ideal balanced final state. Numerical results clearly indicate that both excessively fast or slow processes result in undesirably high final angle configurations, whereas an optimal time window minimizes final excitation. Outside this op… view at source ↗
Figure 5
Figure 5. Figure 5: (Color online) Minimum of final time excitations measured by the value of the ficticious angle [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

discussion (0)

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