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arxiv: 2604.26910 · v1 · submitted 2026-04-29 · cs.RO

Bi-Level Optimization for Contact and Motion Planning in Rope-Assisted Legged Robots

Ruben Malacarne , Ioannis Tsikelis , Enrico Mingo Hoffman , Michele Focchi This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-05-07 11:42 UTCgrok-4.3open to challenge →

classification cs.RO
keywords bi-level optimizationcontact planningmotion planningrope-assisted robotslegged climbingmixed-integer optimizationCross-Entropy MethodALPINE robot
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The pith

Bi-level optimization selects landing regions while tuning rope tensions and leg forces for rope-assisted climbing robots.

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

The paper presents a planning pipeline for locomotion in rope-assisted robots climbing vertical surfaces using a bi-level optimization scheme. This formulation handles a mixed-integer problem by selecting feasible terrain regions for landing at the outer level and optimizing control inputs such as rope tensions, leg forces, and landing locations at the inner level. The outer level applies the Cross-Entropy Method while the inner level uses gradient-based nonlinear optimization to generate dynamically feasible motions. The approach is validated on the ALPINE robot platform across challenging terrain configurations. A sympathetic reader would care because it offers a way to manage discrete contact decisions alongside continuous force and motion planning without exhaustive enumeration of possibilities.

Core claim

The proposed framework is formulated as a bi-level optimization scheme that addresses a mixed-integer problem: selecting feasible terrain regions for landing while simultaneously optimizing the control inputs, namely rope tensions and leg forces, and landing location. The outer level of the optimization is solved using the Cross-Entropy Method, while the inner level relies on gradient-based nonlinear optimization to compute dynamically feasible motions. The approach is validated on a novel climbing robot platform, ALPINE, across a variety of challenging terrain configurations.

What carries the argument

Bi-level optimization scheme with outer Cross-Entropy Method selecting landing regions and inner gradient-based nonlinear optimization computing forces and motions.

If this is right

  • Enables simultaneous optimization of discrete landing region choices and continuous rope tensions plus leg forces.
  • Produces dynamically feasible motions suitable for vertical surface climbing with the ALPINE robot.
  • Handles the mixed-integer character of contact planning through decomposition rather than direct solution of the full problem.
  • Supports validation across varied terrain configurations on the target platform.

Where Pith is reading between the lines

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

  • The decomposition may scale to other hybrid locomotion systems that combine legs with cables or external supports.
  • If the optimizers prove fast enough in practice, the method could support online replanning during actual climbs.
  • Similar outer-inner splits might reduce computational burden in other robotics problems involving both discrete and continuous variables.

Load-bearing premise

The outer Cross-Entropy Method reliably identifies feasible landing regions fast enough and the inner nonlinear optimizer consistently finds dynamically feasible solutions without getting stuck in poor local minima for the ALPINE robot across varied terrains.

What would settle it

Running the framework on the ALPINE robot over multiple challenging terrains and checking whether valid plans are generated in reasonable time or whether the inner optimizer frequently fails to converge would settle whether the central claim holds.

Figures

Figures reproduced from arXiv: 2604.26910 by Enrico Mingo Hoffman, Ioannis Tsikelis, Michele Focchi, Ruben Malacarne.

Figure 1
Figure 1. Figure 1: (left) Dynamic simulation of the ALPINE robot performing a multi-jump maneuver on a rough wall; (right) close-up of the robot during the first intermediate (downward) jump. The reference trajectory is shown in white, while the green marker indicates the intermediate landing target. The ropes are depicted in green, and the red arrows represent the instantaneous pulling forces. Hybrid methods seek to bridge … view at source ↗
Figure 2
Figure 2. Figure 2: The Cost map is generated from the height map of a rocky wall. The color represents the slope intensity at 10 cm resolution. The cost values are represented using a color scale, with green indicating the minimum cost and red the maximum. Base link x y z Right anchor x z y Left anchor Rope plane Rock wall (a) Kinematic model of the ALPINE robot. A prismatic leg initiates the jump. Two ropes guide the motion… view at source ↗
Figure 3
Figure 3. Figure 3: The ALPINE robot. On top, the kinematics model, and on the bottom the real prototype. • wsl (Slope filter): filter associated with first-order derivatives, to penalize the slope. • wsd (Roughness filter): filter associated with second￾order derivatives to penalize abrupt changes in curva￾ture. • wd (Deep filter): heavily penalizes regions inside the anchors’ vertical view at source ↗
Figure 5
Figure 5. Figure 5: Computed trajectory for three types of maps. With view at source ↗
Figure 6
Figure 6. Figure 6: (upper plot) Combination jump selected during all itera￾tions, (lower plot) fitness trend for all 3 use cases. would expect the robot to perform fewer jumps thanks to the help of gravity, but the presence of a complex pattern of obstacles on the way requires at least 3 jumps. The total duration of the multi-jump sequence is 9.45s. C. Use Case 3: Rocky terrain Gazebo Simulation Finally, we evaluate the effe… view at source ↗
read the original abstract

This paper presents a planning pipeline framework for locomotion in rope-assisted robots climbing vertical surfaces. The proposed framework is formulated as a bi-level optimization scheme that addresses a mixed-integer problem: selecting feasible terrain regions for landing while simultaneously optimizing the control inputs, namely rope tensions and leg forces, and landing location. The outer level of the optimization is solved using the Cross-Entropy Method, while the inner level relies on gradient-based nonlinear optimization to compute dynamically feasible motions. The approach is validated on a novel climbing robot platform, ALPINE, across a variety of challenging terrain configurations.

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 / 3 minor

Summary. The paper presents a bi-level optimization pipeline for locomotion planning in rope-assisted legged robots on vertical surfaces. The outer level employs the Cross-Entropy Method to select feasible landing regions on the terrain, while the inner level uses gradient-based nonlinear programming to optimize rope tensions, leg forces, and landing locations for dynamically feasible motions. The framework is demonstrated on the ALPINE climbing robot platform in various terrain configurations.

Significance. If the bi-level decomposition reliably produces dynamically feasible climbing motions, the work would provide a practical engineering solution to mixed-integer contact planning problems in rope-assisted legged robotics. The separation of discrete region selection (via CEM) from continuous control optimization is a standard and effective approach for this class of problems, and the use of a physical platform like ALPINE for validation adds relevance to vertical climbing applications.

major comments (2)
  1. [§5] §5 (Experimental Validation): The manuscript claims validation of the bi-level scheme on the ALPINE platform across challenging terrains, yet supplies no quantitative results such as success rates, planning times, error metrics, failure cases, or comparisons to baselines. This absence makes it impossible to assess whether the outer CEM and inner NLP consistently yield feasible solutions as asserted.
  2. [§3.2] §3.2 (Inner-Level Optimization): The formulation assumes the gradient-based NLP will consistently avoid poor local minima for the ALPINE robot across varied terrains, but no initialization strategy, convergence analysis, or robustness discussion is provided. This assumption is load-bearing for the central claim that the decomposition produces dynamically feasible motions.
minor comments (3)
  1. [§2] Clarify the exact formulation of the mixed-integer problem in §2, including how the outer-level discrete decisions interface with the inner-level continuous variables.
  2. [Results] Add timing benchmarks for the CEM outer loop in the results to address the practical feasibility of real-time planning.
  3. [Figures] Ensure all figures include clear labels for rope tensions, leg forces, and selected landing regions to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our bi-level optimization framework for rope-assisted legged robots. We address each major comment below and describe the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§5] §5 (Experimental Validation): The manuscript claims validation of the bi-level scheme on the ALPINE platform across challenging terrains, yet supplies no quantitative results such as success rates, planning times, error metrics, failure cases, or comparisons to baselines. This absence makes it impossible to assess whether the outer CEM and inner NLP consistently yield feasible solutions as asserted.

    Authors: We agree that the experimental validation section would be strengthened by explicit quantitative metrics. While the manuscript demonstrates successful hardware and simulation trials on the ALPINE platform, we will revise §5 to report success rates over repeated trials for each terrain configuration, average planning times for the outer CEM and inner NLP stages, quantitative error metrics on force feasibility and motion tracking, and a summary of observed failure modes. Where feasible, we will also include comparisons against simpler baseline planners. revision: yes

  2. Referee: [§3.2] §3.2 (Inner-Level Optimization): The formulation assumes the gradient-based NLP will consistently avoid poor local minima for the ALPINE robot across varied terrains, but no initialization strategy, convergence analysis, or robustness discussion is provided. This assumption is load-bearing for the central claim that the decomposition produces dynamically feasible motions.

    Authors: We acknowledge that further details on the inner-level solver are warranted to support the reliability claim. In the revised manuscript we will expand §3.2 with an explicit initialization strategy (warm-starting from the previous time-step solution together with terrain-geometry heuristics), report convergence statistics such as iteration counts and KKT residual norms across the tested terrains, and add a short robustness discussion addressing sensitivity to initial guesses and terrain variations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard external optimizers applied to robot planning

full rationale

The paper formulates a bi-level scheme with CEM (outer, for discrete region selection) and gradient-based NLP (inner, for continuous tensions/forces/location) to handle mixed-integer contact planning. These are standard, externally defined algorithms with no parameters fitted from the paper's own data and re-labeled as predictions. No self-definitional equations, no load-bearing self-citations that reduce the central claim to prior author work, and no ansatz smuggled via citation. The derivation chain consists of applying known optimization methods to the robot's dynamics and terrain constraints, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the method relies on standard optimization primitives whose internal parameters are not detailed.

pith-pipeline@v0.9.0 · 5395 in / 1113 out tokens · 39856 ms · 2026-05-07T11:42:25.247688+00:00 · methodology

discussion (0)

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Reference graph

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