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Optimally Trading Energy for Information through Speed Trajectory Optimization of Sustainably Powered Mobile Robots

Kavin M. Govindarajan, Dimitra Panagou, Chris Vermillion

arXiv:2610.20875v1cs.ROcs.SYeess.SY

TL;DR

Persistent mobile-robot missions must allocate harvested energy to propulsion and sensing while information both accumulates and decays. The paper optimizes periodic speed trajectories using clarity dynamics and derives optimality conditions and related properties. Its simulations report higher weighted clarity for information-value-of-energy speed adaptation than for uniform speed under identical power constraints.

  • Problem

    Persistent missions must balance harvested energy with propulsion and sensing while accounting for information gain and decay, a setting not captured by discrete routing or typically finite-horizon trajectory formulations.

    Source: §I · p. 1

  • Method

    The paper formulates periodic speed optimization on a fixed path, coupling clarity dynamics with energy expenditure and deriving necessary conditions and analytical properties.

    Source: §I · p. 2, A. Information Model · p. 2

  • Results

    Information Value of Energy speed adaptation achieves significantly higher weighted clarity than a uniform-speed baseline under identical power constraints.

    Source: §VI · pp. 7–8

  • Takeaways & Limitations

    The optimized strategy decelerates in high-importance regions and accelerates through lower-weighted zones in the reported simulations.

    Source: §VI · p. 7

Abstract

from arXiv · show

Sustainably powered mobile robots enable long-duration observational missions by harvesting energy from the environment. However, effective autonomy requires the robots to make decisions as to how to allocate this energy to maximize information quality over time. In this paper, we study a persistent (infinite-horizon) speed trajectory optimization problem that is posed as a periodic optimal control problem in which a robot repeatedly traverses a fixed closed path, coupling a super-linear propulsion cost with dynamic information gain and decay. Using the information-theoretic quantity of clarity as an information metric, we first derive the necessary conditions for optimal control. We then derive a number of specialized follow-on properties, which characterize (i) the nature of the optimal control when points on the path are of varying importance and when they are of equal importance, along with (ii) a high-speed information limit that is approached when energy reserves approach infinity. We illustrate the information-optimal speed trajectories on a simple example in order to showcase key trends that are followed under the optimized speed profile. This is followed by a simulation study based on the persistent characterization of salinity in the Chesapeake Bay, illustrating the practical merit of applying the results from this work.

I. INTRODUCTION

Persistent monitoring couples harvested energy, propulsion, sensing, and information that both accumulates and decays. The paper formulates speed selection as a continuous, periodic optimization problem and analyzes clarity as its information measure.

I. INTRODUCTION

  • Persistent missions must balance harvested energy against propulsion and sensing while repeatedly refreshing information that decays over time.The paper frames energy and information as coupled reservoirs.Source: §I · p. 1
  • Discrete routing and finite-horizon trajectory methods do not capture how continuous observations affect information across locations during persistent missions.The paper identifies this as a reason to optimize a continuous speed trajectory rather than a discrete routing objective.Source: §I · p. 1
  • The proposed problem optimizes speed on a prescribed path under limited power, unequal spatial importance, and periodic energy and clarity states.The analysis derives necessary optimality conditions and properties of the resulting control strategy.Source: §I · p. 2
  • The paper combines analytical results with a toy example and Chesapeake Bay salinity simulation; its stated contributions include numerical validation and an application case study.The simulations illustrate speed-profile changes with information importance and the value of optimizing the ASV’s speed profile.Source: §I · p. 2

A. Information Model

  • Clarity measures certainty at a location and evolves through information gain and decay; the paper couples these dynamics with nonlinear propulsion cost.The resulting objective rewards information over the entire path over time.Source: §I · pp. 1–2, A. Information Model · p. 2

B. Energy Model

The energy model links speed-dependent propulsion to battery state of charge, while the mission formulation discretizes a closed path and imposes periodicity for persistent operation.

B. Energy Model

  • Battery state of charge changes with harvested power input and speed-dependent power consumption.The power-consumption function depends on the platform and its propulsion characteristics.Source: B. Energy Model · p. 2
  • The assumed propulsion draw grows monotonically and super-linearly with speed, and the analysis assumes energy reserves remain sufficient to keep the battery nonnegative.Constant speed-independent hotel loads are not precluded.Source: B. Energy Model · p. 2

A. Optimal Control Problem Formulation

  • The robot repeatedly traverses a fixed closed path, with speed as the control and lap duration determined by the chosen profile.The problem is posed with free final time to give the optimization freedom over the speed profile.Source: A. Optimal Control Problem Formulation · p. 2
  • The finite-dimensional transcription represents the path with N nodes, each assigned a clarity state, importance weight, and localized sensing intensity.This discretization supports a finite-dimensional state-space representation.Source: A. Optimal Control Problem Formulation · p. 3, Figure 2 · p. 3
  • Periodicity constraints match end-of-lap clarity to initial clarity and require energy use to be supported over each period, enabling indefinite periodic operation.The finite node representation is stated to apply for arbitrarily large N.Source: A. Optimal Control Problem Formulation · p. 3

B. Conditions for Optimality via PMP

The paper applies Pontryagin’s Maximum Principle to derive necessary conditions for the periodic speed-control problem. Its stationarity condition links the optimal speed to energy and information costates.

B. Conditions for Optimality via PMP

  • The formulation uses a single trajectory-wide average speed and periodic boundary conditions for clarity and energy states.The state and costate vectors collect path position, node clarity, battery state, and their associated costates.Source: 1) Necessary Conditions · p. 3
  • Pontryagin’s Maximum Principle yields costate dynamics for clarity states and a constant costate for battery state of charge.The battery costate is constant because the energy state does not explicitly appear in the Hamiltonian.Source: 1) Necessary Conditions · p. 3
  • PMP requires pointwise Hamiltonian maximization, yielding a stationarity condition that defines unconstrained optimal speed through the energy and information costates.The paper interprets this relationship as the information value of energy.Source: 1) Necessary Conditions · p. 3

C. Theoretical Properties and Analysis of Optimal Solutions

This section establishes properties of the optimal control solution, including how available power relates to optimal clarity and when the Hamiltonian has a unique pointwise maximizer. The problem’s non-convexity prevents a blanket sufficiency result.

1) Sufficiency Considerations:

2) Continuity of the Optimal Control:

With continuously differentiable sensing profiles and strictly convex propulsion power, the optimal unconstrained speed trajectory is continuous over a lap. The authors connect this property to avoiding abrupt, potentially infeasible speed changes.

2) Continuity of the Optimal Control:

3) Speed Trajectory Analysis:

The optimal speed response to spatially varying information importance is governed by weighted sensing sensitivity: the robot slows where important sensing potential rises and accelerates as it falls. In a uniformly important, shift-invariant environment, the optimal speed is constant.

3) Speed Trajectory Analysis

  • Weighted spatial sensing sensitivity is a weighted sum of how each node’s sensing function changes with position along the path.The clarity costates supply weights proportional to node importance.Source: 3) Speed Trajectory Analysis · p. 5
  • The speed derivative opposes the weighted spatial sensing sensitivity under the strictly energy-constrained assumption.The result follows from the positive clarity costates and strict convexity of the power-draw model.Source: 3) Speed Trajectory Analysis · p. 5
  • Higher-importance nodes carry greater weight in the sensing-sensitivity sum, so the robot tends to decelerate on approach and accelerate as sensing potential falls on departure.The clarity costate is positive and directly proportional to each node's importance weight.Source: 3) Speed Trajectory Analysis · p. 5
  • When all nodes are equally important in a uniform, shift-invariant environment, the optimal speed is strictly constant.The aggregate spatial gradient vanishes, and strict convexity makes the marginal-power mapping injective.Source: 3) Speed Trajectory Analysis · pp. 5–6

4) High-Speed Information Limit:

The clarity achieved at a node approaches a finite saturation limit as traversal speed grows without bound. The paper derives this limit using averaging and establishes convergence through stability of the averaged equilibrium.

4) High-Speed Information Limit

  • As constant traversal speed u tends to infinity, time-averaged clarity at each node converges to a finite limit.The result reflects saturation in the nonlinear clarity dynamics, despite reward increasing with available energy.Source: 4) High-Speed Information Limit · p. 6
  • The high-speed analysis changes from time to spatial coordinates and applies averaging to the periodic clarity dynamics.With ε = 1/u, the trajectory converges uniformly to the autonomous averaged system as u grows.Source: 4) High-Speed Information Limit · p. 6
  • The averaged equilibrium is exponentially stable, supporting uniform convergence of periodic clarity to the high-speed limit.The Jacobian is strictly negative under the stated positive sensing and decay conditions.Source: 4) High-Speed Information Limit · p. 6

IV. NUMERICAL VALIDATION ON AN ILLUSTRATIVE EXAMPLE

The illustrative study optimizes a piecewise-constant speed profile over a fixed path with known spatial information distribution. The computed profile slows in high-information regions and accelerates in low-information regions, with the objective converging around 50 segments.

IV. NUMERICAL VALIDATION ON AN ILLUSTRATIVE EXAMPLE

  • The example uses a 100 m path with high-density information regions centered at 30 m and 80 m, discretized into N segments for optimization.The piecewise-constant NLP parameterization approaches the continuous optimal trajectory as N increases.Source: §IV · p. 6, Figure 3 · p. 7
  • IPOPT solves the finite-dimensional nonlinear program that maximizes time-averaged clarity subject to energy dynamics and constraints.The study assumes a fixed path and a spatial information distribution known in advance.Source: §IV · p. 6
  • The objective converges around N = 50 points, while computation time grows super-linearly with segment count.The profile slows near the high-information centers at 30 m and 80 m and accelerates through low-information regions.Source: §IV · p. 6, Figure 3 · p. 7

V. CHESAPEAKE BAY SIMULATION STUDY

The simulation considers a solar-powered autonomous surface vehicle monitoring salinity along a fixed closed path in the lower Chesapeake Bay. The region’s salinity variability makes it a relevant observational setting.

V. CHESAPEAKE BAY SIMULATION STUDY

  • The study considers a solar-powered autonomous surface vehicle monitoring salinity along a fixed closed path in the lower Chesapeake Bay.Source: §V · p. 7
  • The region has substantial spatial salinity variability associated with mixing of freshwater from the bay and saltwater from the ocean.Source: §V · p. 7
  • The authors describe the Chesapeake Bay setting as practically relevant for salinity monitoring.The passage cites and for this application context.Source: §V · p. 7

A. Simulation Setup

The Chesapeake Bay study optimizes an ASV’s speed along a fixed transect using salinity data and target-weighted clarity. The setup compares profiles under different concentrations of target weights.

A. Simulation Setup

  • The ASV follows an approximately 150 km transect across the mouth of Chesapeake Bay, with salinity fields represented using NOAA forecast data.Representative salinity fields and the monitored transect are shown in Fig. 4.Source: A. Simulation Setup · p. 7
  • At each lap’s end, collected salinity measurements update the target-weight profile used to optimize the next lap’s speed.The simulation uses 750 W average power input, 200 path segments, and a constant-speed baseline.Source: A. Simulation Setup · p. 7
  • Target weights are defined from measured salinity relative to a 30 PSU reference, with σζ settings of 2, 3, and 4 PSU.Smaller σζ values concentrate target weights and produce larger speed variations.Source: A. Simulation Setup · p. 7

B. Simulation Results

In the Chesapeake Bay simulation, IVE speed allocation improves weighted clarity over constant-speed traversal, with the gain depending on target-weight spread. The optimized profiles slow in high-weight regions and accelerate in less relevant locations; the conclusion identifies future extensions and field validation as open work.

B. Simulation Results

  • 59.5% is the total-clarity improvement for σζ = 2, while σζ = 3 yields 19.6% and σζ = 4 yields 2.0% over constant speed.These comparisons are interpreted within each common weight function because changing σζ changes the monitoring objective.Source: B. Simulation Results · p. 7, Table I · p. 7
  • The optimizer slows in high-weight regions and accelerates in lower-value locations; σζ = 2 and 3 profiles include near-stationary portions.The σζ = 4 profile remains close to constant speed while retaining a small clarity advantage.Source: B. Simulation Results · p. 7, Figure 5 · p. 8, Figure 6 · p. 8

VI. CONCLUSIONS AND FUTURE WORK

  • The paper derives necessary conditions for periodic speed optimization and analytical properties for varying and uniform weights, including a high-speed information limit.The numerical optimizations and Chesapeake Bay study show deceleration in high-importance regions and acceleration through lower-weighted zones.Source: §VI · p. 7
  • IVE-based speed adaptation achieves higher weighted clarity than uniform-speed traversal under identical power constraints.The conclusion lists joint path-and-speed co-optimization, receding-horizon control, and SeaTrac SP-48 field validation as future work.Source: §VI · p. 8
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