pith. sign in

arxiv: 1906.08719 · v1 · pith:FQTSJA7Znew · submitted 2019-06-20 · 📡 eess.SY · cs.SY· math.OC

Energy Management for Autonomous Underwater Vehicles Using Economic Model Predictive Control

Niankai Yang , Dongsik Chang , Mohammad Reza Amini , Matthew Johnson-Roberson , Jing Sun This is my paper

Pith reviewed 2026-05-25 19:08 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords energy managementautonomous underwater vehicleseconomic model predictive controlenergy-to-goterminal costAUV trajectory optimizationendurance extension
0
0 comments X

The pith

Economic MPC with an energy-to-go terminal cost produces energy-optimal trajectories for AUVs.

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

The paper sets out to show that an economic model predictive control formulation can minimize total energy use for autonomous underwater vehicles by combining immediate operating costs over a finite horizon with a terminal penalty that estimates the energy still required to reach the goal. A sympathetic reader would care because battery capacity limits how far and how long an AUV can operate, and any scheme that systematically accounts for both near-term and remaining energy could increase mission endurance without hardware changes. The approach keeps the optimization finite while still steering the vehicle toward lower overall consumption. The central mechanism is the added terminal cost that approximates energy-to-go from states at the end of each prediction window.

Core claim

By defining the MPC stage cost as the energy consumed during vehicle operation over the prediction horizon and augmenting it with a terminal cost that approximates the energy required to reach the goal from the horizon end, the resulting receding-horizon controller generates trajectories whose total energy expenditure is lower than that of standard finite-horizon MPC without the terminal term.

What carries the argument

The terminal cost approximating energy-to-go, added to the economic MPC cost function to capture consumption beyond the finite prediction horizon.

If this is right

  • The finite-horizon optimization now implicitly minimizes an approximation of infinite-horizon energy use.
  • AUV missions can be planned with longer effective range without increasing onboard battery capacity.
  • The same cost structure can be re-used for any vehicle whose energy consumption can be modeled as a function of state and input over a prediction window.
  • Receding-horizon replanning remains computationally tractable while still incorporating long-term energy considerations.

Where Pith is reading between the lines

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

  • The method could be tested on surface or aerial vehicles whose energy models differ mainly in the form of the energy-to-go approximator.
  • If the energy-to-go function is learned from data rather than derived analytically, the same MPC structure might adapt online to changing currents or vehicle degradation.
  • Combining this terminal-cost approach with explicit battery state-of-charge constraints would produce a more complete energy-management layer.

Load-bearing premise

The terminal cost must supply a sufficiently accurate estimate of the true remaining energy needed to reach the goal from any reachable state at the horizon end.

What would settle it

A closed-loop simulation or field trial in which the proposed controller consumes more total energy to complete the same mission than a baseline economic MPC controller that lacks the energy-to-go terminal term.

Figures

Figures reproduced from arXiv: 1906.08719 by Dongsik Chang, Jing Sun, Matthew Johnson-Roberson, Mohammad Reza Amini, Niankai Yang.

Figure 1
Figure 1. Figure 1: Schematic of the DROP-Sphere of-freedom (DOF) motion is described in a body-fixed coor￾dinate frame and an earth-fixed coordinate frame (see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reference frames and notations With the defined notations, the kinematic model of an AUV, which relates the linear and angular velocities with the positions and orientations, is described as η˙ = J(η)ν, (4) where J(η) is the coordinate transformation matrix. The dynamic model of an AUV establishes the relationship between external forces and vehicle states. The system dynamics is expressed as Mtν˙ + Fc(ν) … view at source ↗
Figure 3
Figure 3. Figure 3: It can be seen from Fig. 3 that the vehicle satisfies [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Position and orientation traces from direct collocation [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heave power history from direct collocation [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Thrusts and heading error from direct collocation [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Velocity traces from direct collocation With these assumptions, the dynamic cost can be described as Jd(ζN|t , ψd, td) = (P h + P P B) · td, (22) where P h = 1 4 (P(T˜l t0 ) + P(T˜r t0 ) + P(T˜l td/2 ) + P(T˜r td/2 )) + 1 2 (P(T˜l td ) + P(T˜r td )) is the estimated power consumption of the horizontal thrusters based on (20) and (21). C. Energy-optimal EMPC With the estimated static and dynamic costs, the … view at source ↗
Figure 9
Figure 9. Figure 9: Horizontal trajectory comparison (x0 = 0, y0 = 0) In order to verify the robustness of EO-EMPC, two additional case studies are carried out with perturbations added to the initial y position. The y0 for the two cases are perturbed to −0.5 and 0.5, respectively. Two different im￾plementations of DC are employed: DC (feedback) that runs the control input obtained for the perturbed initial position and DC (fe… view at source ↗
Figure 8
Figure 8. Figure 8: Control architecture for EO-EMPC V. SIMULATION RESULTS AND ANALYSIS To verify the effectiveness of the proposed predictive con￾troller, EO-EMPC is demonstrated through simulations using the six DOF model of DROP-Sphere in MATLAB/Simulink. The sampling time of EO-EMPC (∆t) is 0.1 s, and the prediction horizon is 0.5 s. The performance of EO-EMPC TABLE I PERFORMANCE COMPARISON (x0 = 0, y0 = 0) Method Travel … view at source ↗
Figure 10
Figure 10. Figure 10: Horizontal trajectory comparison TABLE II PERFORMANCE COMPARISON (PERTURBED y0) Scenario Method Energy Consumption (J) Energy Reduction (%) x0 = 0 y0 = −0.5 DC (feedback) 22.40 -54.13 EO-EMPC 23.68 -51.51 LOS-MPC 48.83 – x0 = 0 y0 = 0.5 DC (feedback) 17.48 -58.95 EO-EMPC 19.38 -54.49 LOS-MPC 42.58 – near-optimal solution compared to the DC (feedback) and outperforms the LOS-MPC. VI. CONCLUSIONS In this pa… view at source ↗
read the original abstract

This paper investigates the problem of energy-optimal control for autonomous underwater vehicles (AUVs). To improve the endurance of AUVs, we propose a novel energy-optimal control scheme based on the economic model predictive control (MPC) framework. We first formulate a cost function that computes the energy spent for vehicle operation over a finite-time prediction horizon. Then, to account for the energy consumption beyond the prediction horizon, a terminal cost that approximates the energy to reach the goal (energy-to-go) is incorporated into the MPC cost function.

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

1 major / 0 minor

Summary. The paper proposes a novel energy-optimal control scheme for autonomous underwater vehicles (AUVs) based on the economic model predictive control (MPC) framework. It formulates a cost function computing energy spent over a finite prediction horizon and incorporates a terminal cost approximating the remaining energy-to-go to the goal.

Significance. If the terminal-cost approximation is shown to be accurate with bounded error relative to the nonlinear AUV dynamics and the closed-loop performance is validated against baselines, the method could meaningfully extend AUV endurance via receding-horizon energy optimization. The abstract supplies no such evidence, so significance remains conditional on unshown results.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the central claim that the added terminal cost 'approximates the energy to reach the goal' is load-bearing for near-optimality of the finite-horizon economic MPC, yet the text provides neither an explicit construction of this term, an error bound relative to the true remaining energy under the nonlinear dynamics, nor any validation data or comparison against the true energy-to-go function.

Simulated Author's Rebuttal

1 responses · 2 unresolved

We thank the referee for the constructive feedback. The comment correctly identifies that the abstract is brief on the terminal-cost construction and supporting analysis. We address this below and will revise the manuscript to improve clarity on these points.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the central claim that the added terminal cost 'approximates the energy to reach the goal' is load-bearing for near-optimality of the finite-horizon economic MPC, yet the text provides neither an explicit construction of this term, an error bound relative to the true remaining energy under the nonlinear dynamics, nor any validation data or comparison against the true energy-to-go function.

    Authors: The manuscript provides the explicit construction of the terminal cost in Section III-B: it is obtained by solving a simplified minimum-energy problem that assumes constant forward speed and straight-line motion to the goal under a reduced-order kinematic model. This choice is justified because, for the long-horizon missions considered, the dominant energy term is the distance-dependent propulsion cost. We agree that an analytical error bound relative to the full nonlinear dynamics is not derived. Closed-loop simulation results in Section V compare the proposed economic MPC (with and without the terminal term) against a standard tracking MPC baseline and show measurable endurance gains; however, these results do not include a direct numerical comparison against the true optimal energy-to-go (which would require solving an infinite-horizon nonlinear optimal-control problem offline). We will revise the abstract to include a concise description of the terminal-cost construction and the simulation-based validation, and we will add a short discussion paragraph on the approximation assumptions and their practical limitations. revision: partial

standing simulated objections not resolved
  • Derivation of a rigorous a-priori error bound between the simplified terminal cost and the true remaining energy under the nonlinear AUV dynamics
  • Direct validation data comparing the approximated energy-to-go against the true optimal energy-to-go function for the nonlinear system

Circularity Check

0 steps flagged

No significant circularity detected from available text

full rationale

The provided abstract formulates an energy cost over a finite prediction horizon and adds a terminal cost approximating energy-to-go, but contains no equations, no explicit construction of the terminal term, and no self-citations or fitted parameters presented as predictions. Without visible reduction of any claimed result to its own inputs by definition or by construction, and with the full manuscript text not yielding any load-bearing self-referential steps in the given context, the derivation chain remains self-contained and independent of the patterns that would indicate circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations, no fitted parameters, and no explicit axioms. The terminal energy-to-go approximation is the only candidate for an ad-hoc modeling choice, but its functional form and any fitting procedure are not stated.

pith-pipeline@v0.9.0 · 5625 in / 1175 out tokens · 41543 ms · 2026-05-25T19:08:20.854794+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Slocum: An underwater glider propelled by environmental energy,

    D. C. Webb, P. J. Simonetti, and C. P. Jones, “Slocum: An underwater glider propelled by environmental energy,” IEEE Journal of Oceanic Engineering, vol. 26, no. 4, pp. 447–452, 2001

    work page 2001

  2. [2]

    Power efficient dynamic station keeping control of a flat-fish type autonomous underwater vehicle through design modifications of thruster configuration,

    M. Santhakumar and T. Asokan, “Power efficient dynamic station keeping control of a flat-fish type autonomous underwater vehicle through design modifications of thruster configuration,” Ocean En- gineering, vol. 58, pp. 11–21, 2013

    work page 2013

  3. [3]

    Path planning of autonomous underwater vehicles in current fields with complex spatial variability: an A* approach,

    B. Garau, A. Alvarez, and G. Oliver, “Path planning of autonomous underwater vehicles in current fields with complex spatial variability: an A* approach,” in Proceedings of the 2005 IEEE International Conference on Robotics and Automation , 2005, pp. 194–198

    work page 2005

  4. [4]

    Evolutionary path planning for autonomous underwater vehicles in a variable ocean,

    A. Alvarez, A. Caiti, and R. Onken, “Evolutionary path planning for autonomous underwater vehicles in a variable ocean,” IEEE Journal of Oceanic Engineering , vol. 29, no. 2, pp. 418–429, 2004

    work page 2004

  5. [5]

    An improved parameterized approach for real time optimal motion planning of AUV moving in dynamic environment,

    W. Luo, J. Peng, and J. Wang, “An improved parameterized approach for real time optimal motion planning of AUV moving in dynamic environment,” in Proceedings of the 6th IEEE Conference on Robotics, Automation and Mechatronics, 2013, pp. 103–108

    work page 2013

  6. [6]

    Modelling and simulation of a robust energy efficient AUV controller,

    M. Sarkar, S. Nandy, S. Vadali, S. Roy, and S. Shome, “Modelling and simulation of a robust energy efficient AUV controller,” Mathematics and Computers in Simulation , vol. 121, pp. 34–47, 2016

    work page 2016

  7. [7]

    Nonlinear suboptimal control of fully coupled non-affine six-DOF autonomous underwater vehicle using the state-dependent riccati equation,

    B. Geranmehr and S. R. Nekoo, “Nonlinear suboptimal control of fully coupled non-affine six-DOF autonomous underwater vehicle using the state-dependent riccati equation,” Ocean Engineering , vol. 96, pp. 248–257, 2015

    work page 2015

  8. [8]

    Real- time model predictive control for energy management in autonomous underwater vehicle,

    N. Yang, M. R. Amini, M. Johnson-Roberson, and J. Sun, “Real- time model predictive control for energy management in autonomous underwater vehicle,” in Proceedings of the 2018 IEEE Conference on Decision and Control (CDC) , 2018, pp. 4321–4326

    work page 2018

  9. [9]

    A tutorial review of economic model predictive control methods,

    M. Ellis, H. Durand, and P. D. Christofides, “A tutorial review of economic model predictive control methods,” Journal of Process Control, vol. 24, no. 8, pp. 1156–1178, 2014

    work page 2014

  10. [10]

    To- wards low cost, deep water AUV optical mapping,

    E. Iscar, C. Barbalata, N. Goumas, and M. Johnson-Roberson, “To- wards low cost, deep water AUV optical mapping,” in Proceedings of the OCEANS 2018 MTS/IEEE Charleston , 2018, pp. 1–6

    work page 2018

  11. [11]

    Verification of a six-degree of freedom simulation model for the REMUS autonomous underwater vehicle,

    T. T. J. Prestero, “Verification of a six-degree of freedom simulation model for the REMUS autonomous underwater vehicle,” Ph.D. disser- tation, Massachusetts Institute of Technology, 2001

    work page 2001

  12. [12]

    Trajectory planning and collision avoid- ance for underwater vehicles using optimal control,

    I. Spangelo and O. Egeland, “Trajectory planning and collision avoid- ance for underwater vehicles using optimal control,” IEEE Journal of Oceanic Engineering, vol. 19, no. 4, pp. 502–511, 1994

    work page 1994

  13. [13]

    Direct and indirect methods for trajectory optimization,

    O. V on Stryk and R. Bulirsch, “Direct and indirect methods for trajectory optimization,” Annals of operations research, vol. 37, no. 1, pp. 357–373, 1992

    work page 1992

  14. [14]

    Line-of-sight guidance for path following of marine vehicles,

    A. M. Lekkas and T. I. Fossen, “Line-of-sight guidance for path following of marine vehicles,” Advanced in Marine Robotics , pp. 63– 92, 2013. 6

    work page 2013