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Time-Optimal Operation of a Load-Hoisting Gantry Crane

Eric Mountain, Tarunraj Singh

arXiv:2609.18872v1eess.SYcs.SY

TL;DR

The paper addresses time-optimal point-to-point motion of a gantry crane that also hoists its load. It develops a Bessel-function-based control design and reports variable switching structures, experimental residual-energy reduction, and a maneuver-time tradeoff for robustness to cable-length uncertainty.

  • Problem

    The paper studies time-optimal point-to-point crane motion with hoisting, a setting in which simultaneous cart and hoisting motion makes the model nonlinear in prior work.

    Source: §1 · p. 2

  • Method

    The authors model commanded cart velocity with constant-rate hoisting, derive Bessel-function-based motion solutions, and add state sensitivities to design robust profiles.

    Source: §2 · p. 3, §3 · p. 5, §7 · p. 14

  • Results

    98.98% reduction in average residual energy was measured for time-optimal versus rigid-body experiments; the paper also reports 2-, 3-, 4-, or 6-switch profiles and a maneuver-time tradeoff for robustness.

    Source: §6 · p. 11, §7 · p. 14

  • Takeaways & Limitations

    The optimal switch structure varies with cart displacement, and sensitivity constraints can trade maneuver time for robustness to initial cable-length uncertainty.

    Source: §7 · p. 14

    The model assumes constant hoisting velocity and that the cable reaches its terminal length within the cart’s maneuver time.

    Source: §2 · p. 3

Abstract

from arXiv · show

This paper addresses the problem of designing time-optimal control profiles for point-to-point control of a gantry crane moving in a two dimensional plane. It is assumed that the hoisting motor completes the hoisting maneuver at a constant rate and completes its transition in the same time that it takes for the cart to reach its terminal position. This results in a linear time-varying model and a closed form solution to the time-optimal control problem is shown to be parameterized with Bessel functions. A comprehensive analysis of the structure of the time-optimal control profile is studied by examining the switching function which illustrates the mechanism of introduction and decimation of switches in the bang-off-bang control profile. The variation of the number of switches in the optimal control profile is presented and a non-intuitive control profile structure is noted, one which initiates with a stationary cart, while the hoisting cable length is changed prior to the initiation of motion of the cart. To address the issue of uncertainties in the initial cable length, a model with the sensitivity of the system states with respect to the initial cable length is used to augment the system model and is used for the design of robust time-optimal controllers. Experimental results validate the time-optimal and robust time-optimal control profiles.

1. Introduction

The paper develops time-optimal point-to-point control for a gantry crane that moves and hoists simultaneously. It distinguishes this goal from prior vibration-focused approaches and studies how the optimal profile changes with cart displacement.

1. Introduction

  • Automation is framed as a source of productivity and safety gains, with cited estimates spanning reduced injuries, labor time, labor costs, and crane retrofit payback.The cited estimates include 40% fewer injuries, 45–55% less labor time per crane, 33% lower labor costs per automated crane, and a one-year return on investment for retrofits.Source: §1 · p. 1
  • Prior input-shaping approaches target residual vibration rather than maneuver time.The introduction also reviews work on constant hoisting velocity and concurrent cart-hoist motion.Source: §1 · p. 2
  • The work uses a simplified linear time-varying model for time-optimal design, unlike prior formulations using nonlinear equations for simultaneous cart and hoisting motion.The model assumes cart velocity can be commanded.Source: §1 · p. 2
  • The paper analyzes how cart displacement changes the time-optimal switching structure and addresses robustness to uncertainty in initial cable length.It also reports experimental validation of its numerical results and modeling assumptions.Source: §1 · p. 2

2. System Model

The system model treats cart velocity as the control input and assumes constant-rate hoisting completed over the cart’s maneuver time. Under these assumptions, the crane is modeled with time-varying dynamics.

2. System Model

  • The cart’s stepper motor permits commanded velocity under a zero-acceleration assumption, and cart velocity is the sole control input.The hoisting motor is also assumed to move at constant velocity.Source: §2 · p. 3
  • Hoisting velocity is constant, with the cable moving from its initial to terminal length over the same maneuver time as the cart.This assumption yields a closed-form cable-length representation.Source: §2 · p. 3
  • The model assumes hoisting velocity is small enough that its impact on the dynamics can be ignored.The paper then presents a time-varying state-space model.Source: §2 · p. 3

3. Time-Optimal Control

The time-optimal design uses bang-off-bang cart velocity and represents the suspended-mass motion with Bessel functions. The resulting profiles vary in switch count, including a case where hoisting begins before cart motion.

3. Time-Optimal Control

  • PMP yields a bang-off-bang optimal cart-velocity profile.The switching function determines the profile’s switches.Source: §3 · p. 4
  • Bessel-function solutions for delayed-ramp inputs allow the authors to formulate switch times and maneuver time as a constrained parameter optimization problem.The delayed-ramp responses are matched at each switch time.Source: §3 · pp. 5–6
  • The illustrated profiles have two, three, and four switches at cart displacements of 1.5, 2.7, and 4 m, respectively.The displacement and velocity profiles are shown in Figure 2, alongside the associated control and switching-function profiles in Figure 3.Source: §3 · p. 6, Figure 2 · p. 7, Figure 3 · p. 7
  • At a 2.7 m terminal displacement, the time-optimal profile holds the cart stationary before motion begins, while hoisting has already started.The pendular motion begins at 0.05 s rather than at t = 0 s.Source: §3 · p. 6

4. Switching Profile Transition

The switching function’s Bessel-function structure permits changes in switch count as cart displacement and hoisting distance vary. The paper traces individual and simultaneous switch transitions and maps their evolution across these dimensions.

4. Switching Profile Transition

  • The Bessel-function switching function has infinitely many zero crossings, allowing switch counts to vary; switches can be added or removed one at a time at maneuver boundaries.The analysis holds total hoisting distance constant while varying terminal cart displacement.Source: §4 · p. 7
  • Across xf=2–6 m with L0=1 m and Lf=3 m, the optimal profile changes through 2-, 3-, 4-, 2-, 3-, 4-, and 6-switch structures.The sequence describes the profile as cart displacement varies.Source: §4 · p. 7
  • Interior simultaneous switch transitions require both the switching function and its slope to be zero at the transition time.The transition time lies between the maneuver’s initial and terminal times.Source: §4 · p. 7
  • Near cart displacements of 2.7 m and 2.8 m, single switches are introduced into the control profile.Figure 5 marks active cart-control periods as shaded and pauses as blank.Source: §4 · p. 8
  • Across hoisting and cart displacements, switch-time surfaces show smooth transitions; clamshell pairs represent switch birth and collapse, and Figure 5 is a slice at Lf=3 m.The surfaces also depict switch births at the start of the maneuver.Source: §4 · p. 8

5. Robust Control

The robust controller augments the model with sensitivities to initial cable-length uncertainty and targets zero terminal sensitivity. The desensitized profile incurs a time penalty relative to the time-optimal profile.

5. Robust Control

  • The robust design augments the system model with state sensitivities to initial cable-length uncertainty and forces those sensitivities to zero at terminal time.The augmented model is presented after the sensitivity-based design objective.Source: §5 · pp. 8, 10
  • Desensitizing terminal residual energy to errors in initial cable length incurs a time penalty.Figure 7 compares the time-optimal profile with the desensitized time-optimal profile.Source: §5 · p. 10, Figure 7 · p. 10

6. Experimental Results

Experiments validate varied time-optimal profiles and show substantially lower residual energy than rigid-body control, while robust profiles reduce sensitivity to initial cable-length errors. The tested profiles also reveal a trajectory that begins hoisting while the cart remains stationary.

6. Experimental Results

  • At fixed hoisting displacement, the tested optimal profiles changed from four switches to two and then three as cart displacement increased.The three-switch trajectory begins with hoisting while cart motion remains quiescent; all experiments hoisted the payload downward.Source: §6 · p. 11
  • Time-optimal profiles reduced post-maneuver oscillations, with maneuver times of 4.01, 4.17, and 4.40 s versus rigid-body times of 3.80, 4.10, and 4.24 s.The three matched comparisons cover four-, two-, and three-switch profiles, respectively.Source: §6 · p. 11, Figure 9 · p. 12
  • 98.98% lower average residual energy was measured for the four-switch time-optimal trajectory than for the rigid-body trajectory.The comparison averaged ten experiments per controller; energy was evaluated over two oscillation periods.Source: §6 · p. 11, Figure 10 · p. 12
  • Robust time-optimal control produced a reduced range of residual energy across initial cable-length offsets of ±5, ±10, and ±15 cm.The nominal initial length was 40 cm; seven distinct length conditions were tested.Source: Figure 12 · p. 13, §6 · p. 13
  • For a 5 cm initial cable-length error, the robust controller left the suspended mass nearly stationary at maneuver end, unlike the more vibration-sensitive time-optimal controller.The rigid-body, time-optimal, and robust maneuver times were 3.80, 4.01, and 4.35 s, respectively.Source: §6 · p. 13, Figure 11 · p. 13

7. Conclusions

The paper develops time-optimal rest-to-rest gantry-crane profiles that include hoisting, using a linear time-varying model and Bessel-function solutions. It analyzes switching structure and designs robust profiles that trade maneuver time for reduced sensitivity to initial-condition uncertainty.

7. Conclusions

  • The paper develops time-optimal control profiles for rest-to-rest gantry-crane maneuvers that include hoisting.Assuming hoisting finishes in the same time as the cart displacement yields a linear time-varying model.Source: §7 · p. 13
  • Bessel functions parameterize the optimal pendulum motion, converting the time-optimal control problem into parameter optimization.Source: §7 · pp. 13–14
  • Across the considered cart displacements, optimal profiles can have 2, 3, 4, or 6 switches, with constraints identifying switch introduction or removal.Source: §7 · p. 14
  • Sensitivity states support robust profiles that constrain terminal state sensitivity to zero, trading maneuver time for robustness to initial-condition uncertainty.Numerous experiments validate the numerical results.Source: §7 · p. 14

Appendix A. Reduction to Bessel’s Equation

The appendix abstracts the pendulum-crane dynamics as a linear time-varying equation with constant parameters.

Appendix A. Reduction to Bessel’s Equation

  • The pendulum-crane dynamics are abstracted to a linear time-varying system with constants a, b, and c.Source: Appendix A · p. 14
  • The displayed equation presents the system model that is reduced to Bessel form in the appendix.Source: Appendix A · p. 14

Homogeneous Solution

The homogeneous solution is obtained by transforming the equation into standard order-1 Bessel form, solving it, and then undoing the substitutions.

Homogeneous Solution

Particular Solution

The trial solution x_p(t) = t satisfies the stated equation and is identified as a valid particular solution. Combining it with the homogeneous part gives the general solution.

Particular Solution

Appendix B. Derivation of λ3

The derivation substitutes a standard form into Eqs. (14) and (16), then applies successive substitutions and integration to obtain a solution.

Appendix B. Derivation of λ3

  • The standard form used in Appendix A is substituted into Eqs. (14) and (16).Source: Appendix B · p. 15
  • The derivation defines Φ(t) = 2 √k(b + ct) and differentiates it with respect to time.Source: Appendix B · p. 15
  • The derivation plugs Eq. (B.2) into Eq. (B.1), then substitutes Eq. (B.3) into Eq. (B.4) and integrates both sides.Source: Appendix B · p. 15
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