Fusion of Indirect Methods and Iterative Learning for Persistent Velocity Trajectory Optimization of a Sustainably Powered Autonomous Surface Vessel

arXiv:2503.10884 · cs.RO · Submitted 2025-03-13 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa.

Dev: And I'm Dev, here with Taro.

Rosa: Today's paper: "Fusion of Indirect Methods and Iterative Learning for Persistent Velocity Trajectory Optimization of a Sustainably Powered Autonomous Surface Vessel".

Dev: The gist:

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: We started by looking at the title of this paper, "Fusion of Indirect Methods and Iterative Learning for Persistent Velocity Trajectory Optimization of a Sustainably Powered Autonomous Surface Vessel". It immediately tells you what the paper is trying to achieve.

Dev: It’s a lot packed into that title. They are combining indirect optimal control methods with iterative learning to optimize velocity trajectories for an ASV that uses solar power sustainably.

Taro: So, it sounds like the core challenge they are addressing isn't just controlling the vessel, but doing it in a way that accounts for the unpredictable nature of solar energy and keeps it running reliably over a long period.

Rosa: Exactly. The paper frames this as having two main goals: first, using indirect methods to derive necessary conditions for provably optimal velocity trajectories under time-varying solar resources and state of charge constraints, which is what they did in the first part of the paper.

Dev: And the second goal is using those results from indirect control to create simple control laws that can approximate MPC performance while making tuning much simpler and cutting down on real-time computing needs, all without needing a forecast.

Taro: I’m thinking about what that means for someone trying to build this thing. It suggests you don't need a super complex predictive model for the sun if you want good control.

Rosa: Right. Because they use barrier functions to tighten those state of charge constraints minimally just enough so that the system can still guarantee persistent feasibility, even when the solar resource is unpredictable.

Dev: And then they introduce this iterative learning approach to figure out that constant velocity value using update laws like u i+one = u i + kp(b i(tf) − b i des) and a continuous rate-based update law <ref:2503.10884#pg1>.

Taro: So, the method is essentially a way to find the optimal speed iteratively rather than solving a massive optimization problem all at once upfront.

Rosa: Right. And this iterative learning approach converges to an optimal costate value, which they found corresponds to a speed of one point eight three meters per second <ref:2503.10884#pg1>.

Dev: That specific number is the output of that estimation process, and it’s what drives the subsequent control law derivation based on indirect methods.

Taro: It shows that even with these constraints in place, there is a theoretically optimal speed they can achieve under those conditions.

Rosa: It suggests a certain level of control quality can be achieved by focusing on these theoretical tools rather than just brute-forcing the optimization problem.

Dev: It’s about finding a way to get high performance with reduced computational demands, which is always a win for real-time loops.

The paper's summary: Rosa: Now let’s talk about what this paper actually summarizes, focusing on the methodology they used to tackle the problem of optimizing velocity for this solar-powered ASV.

Dev: They summarize their approach by first using indirect optimal control to find necessary conditions for optimal velocity trajectories in the presence of a time-varying solar resource and pointwise-in-time state of charge constraints.

Taro: So they are using Pontryagin’s Minimum Principle, which is the standard way to look at these kinds of problems, and they derive those necessary conditions.

Rosa: Right. And then they use those necessary conditions to benchmark MPC based results and try to better understand what the MPC optimization is actually deciding when it makes its choices.

Dev: The second part of their summary is using the results from indirect optimal control to derive simple control laws that can approximate the performance of MPC while making tuning much simpler and cutting down on real-time computational demands.

Taro: So, they are aiming for a simplified version of an optimal controller that still gets close to the performance of a more complex one but is easier to run in real time.

Rosa: And they do this all while eliminating the need for a forecast entirely, which is really the key selling point here.

Dev: They also use barrier functions to tighten those state of charge constraints by the minimal amount necessary to achieve persistent feasibility, which prevents running out of power when solar input is zero for some time.

Taro: So it’s a layered approach: first, constrain the energy states with barriers, then use indirect methods to find a simple control law that works under those constraints.

Rosa: Exactly. It’s a systematic way to handle the uncertainty in the solar profile and still get an optimal trajectory without needing perfect foresight.

Dev: The whole summary boils down to deriving a control law that is piecewise constant during each constraint-inactive time interval, which is derived from the indirect methods.

Taro: So, when the solar input changes suddenly, this control law just switches between a few different predefined constant velocities.

The paper's improvements: Rosa: Let’s discuss the specific improvements they propose in this paper compared to previous work on this problem.

Dev: One major improvement is their use of barrier functions to tighten the pointwise-in-time state of charge constraints by the minimal amount necessary to achieve persistent feasibility, which is a significant way they improve upon earlier approaches that might have failed when solar input was zero for a period.

Taro: So they are ensuring that even if the anticipated future solar resource is zero for some time, the system stays feasible, which addresses a specific failure mode.

Rosa: That’s right. And then they use an iterative learning approach to estimate this unknown constant velocity using update laws like u i+one = u i + kp(b i(tf) − b i des) and a continuous rate-based update law <ref:2503.10884#pg1>.

Dev: This iterative learning part is key because it lets them estimate that constant velocity value, which corresponds to the optimal costate value, and it converges to a solution.

Taro: So they’ve solved the issue of not having a straightforward way to solve for the co-state value p1 by using this iterative learning approach instead.

Rosa: And finally, they achieve a control law that is shown based on indirect optimal control methods to be piecewise constant during each constraint-inactive time interval, which makes it simple and easy to tune compared to other complex solutions.

Dev: So the improvements are three things: persistent feasibility guarantee via barrier functions, a closed-form control law derived from indirect optimal control that is piecewise constant, and the iterative learning approach for velocity estimation.

Taro: It’s a lot of machinery put together to solve a tricky problem in real time, but it sounds like it provides a robust framework for dealing with uncertain energy inputs.

Rosa: It does sound like they've made real progress by showing that you can get good performance without needing an environmental forecast or the associated computational burden.

Conclusion: Dev: So to wrap up, this paper, "Fusion of Indirect Methods and Iterative Learning for Persistent Velocity Trajectory Optimization of a Sustainably Powered Autonomous Surface Vessel", presents a method that combines indirect optimal control with iterative learning for real-time velocity optimization.

Rosa: The main implication is that you can achieve good performance without needing an environmental forecast by using barrier functions to ensure persistent feasibility and an iterative learning loop to estimate the constant velocity.

Taro: From my side, it’s interesting that they focused on moving toward actual information maximization in future work, which suggests a direction for making the system even smarter by optimizing what it gathers.

Dev: And from an engineering standpoint, we have a method that works well with reduced computational demands compared to MPC, and they validated it performs nearly as well in simulation without needing the forecast or the associated computational burden.

Rosa: This work shows that you can get comparable performance to MPC without needing a forecast or the associated computational burden for real-time operation.

Taro: I just want to say that their focus on experimental validation on their actual ASV is something I’m keen to see next, because seeing it handle real world misbehavior is the ultimate test of any control strategy.

Dev: Agreed, and the method provides a robust framework for dealing with uncertain energy inputs by making sure the vessel stays feasible even when power input is zero for some time.

Rosa: That’s it for this paper on this fusion of indirect methods and iterative learning to persistent velocity trajectory optimization of a sustainably powered autonomous surface vessel.

University of Michigan

cs.RO

Submitted: 2025-03-13

Updated: 2025-03-13

DOI: 10.1109/CCTA53793.2025.11151452

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 73/100

The gist: The gist: The methodology presents a closed-form control law derived from indirect optimal control methods combined with an iterative learning approach to achieve real-time velocity trajectory

Key concepts

Indirect Optimal Control: This technique uses mathematical conditions derived from calculus (like necessary conditions) to find the best possible control inputs. In this paper, it helps determine that the optimal velocity remains constant during periods when battery constraints are not active.
Barrier Functions: These are mathematical tools used to enforce constraints, specifically the state of charge limits. The authors use them minimally to tighten these constraints just enough so that a solution is guaranteed to exist over a long time, ensuring 'persistent feasibility' despite unpredictable weather.
Iterative Learning: Since traditional methods struggle to find the exact optimal control value (costate), this method updates an estimate of that value repeatedly. It uses two types of updates—one based on current error and one based on past errors—to converge on the correct speed required for optimal performance.
Persistent Feasibility: This means ensuring that the vessel's battery state will remain within safe limits indefinitely, even when facing unpredictable disturbances like changing solar patterns or weather. The barrier functions are specifically designed to guarantee this long-term stability.

Terminology

Benchmark — 7× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
Reward Function — 3× in this paper · explained in 4 episodes
American Control Conference — 2× in this paper
Objective Function — 1× in this paper · explained in 2 episodes

Summary

The gist: The methodology presents a closed-form control law derived from indirect optimal control methods combined with an iterative learning approach to achieve real-time velocity trajectory optimization for a solar-powered autonomous surface vessel without requiring a solar forecast.

Problem Formulation and Constraints

The problem involves controlling the ASV velocity to maximize distance traveled subject to energetic dynamics and constraints imposed by a time-varying resource and pointwise-in-time state of charge (SOC) constraints The ASV exhibits cyclic operation due to the nature of the solar profile, but weather patterns create inevitable disturbances in this profile The goal is to maximize information gathered over a very long (ultimately persistent) time duration To address challenges, barrier functions are used to tighten pointwise-in-time state of charge constraints by the minimal amount necessary to achieve persistent feasibility

Indirect Optimal Control and Barrier Functions

The authors use indirect methods to derive necessary conditions for optimality, which show that the optimal velocity is constant during each constraint-inactive time interval They construct barrier functions to ensure persistent feasibility by reformulating the state constraints into time-varying barrier functions These barriers encode this notion of persistent feasibility, establishing SOC constraints as shown in Fig. 6 Lemma 1 proves that whenever tightened pointwise-in-time SOC constraints are inactive, the optimal velocity is constant The resulting optimal control solution is given by a formula involving the unconstrained velocity when state constraints are satisfied

Iterative Learning for Velocity Estimation

Because indirect methods do not provide a straightforward mechanism for solving for the co-state value p1, an iterative learning approach is employed to estimate this constant velocity The iterative learning approach updates the estimate of the costate value p1 using two parts: a once-periteration update law and a continuous rate-based update law The first part is given by u i+1 = u i + kp(b i(tf) - b i des) The second part is given by u i(t + 1) = u i + kd(b it - b(i-1)(t)) This iterative learning approach converges to the optimal costate value p1, which corresponds to a speed of u = 1.83 m · s−1

Performance and Validation

Simulation-based validation results show that the proposed methodology performs nearly as well as a model predictive control approach that requires an accurate future solar forecast and significantly greater computational capability The ILC-based controller performs nearly as well as an MPC-based approach, without requiring a forecast or the associated computational burden The ILC approach is particularly sensitive to the fluctuations in solar irradiance that are present in the real-world data The ILC approach requires no forecast of the solar resource and does not require the heuristic tuning of a terminal reward function within a real-time optimization Furthermore, the ILC approach took roughly 20 seconds to simulate for a year-long simulation, whereas the MPC strategy took 21 hours and 36 minutes The core contributions include minimally tightened battery SOC constraints based upon a barrier function approach to guarantee persistent feasibility An optimal control law that is shown based on indirect optimal control methods to be piecewise constant An iterative learning approach to estimate the value of the optimal control signal A performance comparison to approaches such as MPC in a case-study simulation that demonstrates the comparable performance of the proposed approach without the need for an environmental forecast or the associated computational burden The presented strategies were implemented in Julia and simulated on a computer with an Intel Xeon W-2125 4.0 GHz CPU and 32GB of RAM

Future Work

In future work, the authors plan to focus on replacing the distance-maximization objective with an actual information maximization mechanism They will also focus on combining velocity trajectory optimization with path planning, all with the goal of maximizing information gathered Finally, they will focus on experimentally validating the developed control algorithms on the SP-48 ASV presently in their possession

References

[1] K. Naik and C. Vermillion, “Integrated physical design, control design, and site selection for an underwater energy-harvesting kite system,” Renewable Energy, vol. 220, p. 119687, 2024

[1] Kavin M. Govindarajan is a Graduate Research Assistant with the Department of Robotics, University of Michigan, Ann Arbor, MI 48109, USA

[2] M. N. Huda, “Machine learning for improvement of ocean data resolution for weather forecasting and climatological research,” Ph.D. dissertation, Virginia Tech, 2023

[3] H. E. Seim, D. Savidge, M. Andres, J. Bane, C. Edwards, G. Gawarkiewicz, R. He, R. E Todd et al., “Overview of the processes driving exchange at cape hatteras program.” Oceanography, vol 35 no 2 2022

[4] R. He, J. Bane, M. Muglia et al., “Gulf Stream marine hydrokinetic energy resource characterization off Cape Hatteras, North Carolina USA,” in OCEANS 2016 - Shanghai, 2016

[5] M. Muglia, H. Seim, and P. Taylor, “Gulf Stream marine hydrokinetic energy off Cape Hatteras, North Carolina,” Marine Technology Society Journal vol 54 no 6 pp 24–36 2020

[6] S. Haines, H. Seim, and M. Muglia, “Implementing quality control of high-frequency radar estimates and application to Gulf Stream surface currents,” Journal of Atmospheric and Oceanic Technology vol 34 no 6 pp 1207 – 1224 2017

[7] G. Gawarkiewicz, R. E Todd, W. Zhang et al., “The changing nature of shelf-break exchange revealed by the OOI Pioneer array,” Oceanography vol 31 no 1 pp 60–70 2018

[8] R. E Todd, “Export of Middle Atlantic Bight shelf waters near Cape Hatteras from two years of underwater glider observations,” Journal of Geophysical Research: Oceans vol 125 no 4 p e2019JC016006 2020

[9] C. Gentemann, J. P Scott, P. L Mazzini et al., “Adaptively sampling the marine environment,” 2020

[10] “SeaTrac.” [Online]. Available: https://www.seatrac.com/

[11] K. Govindarajan, B Haydon, and C Vermillion, “Predictive velocity trajectory control for a persistently operating solar-powered autonomous surface vessel,” in 2023 American Control Conference (ACC), 2023 pp 2077–2083

[12] M. Miyatake and K.

Improvements for AI systems

Bold header: Persistent Feasibility Guarantee via Barrier Functions

The system can guarantee persistent feasibility by tighten[ing] pointwise-in-time state of charge constraints by the minimal amount necessary to achieve persistent feasibility, preventing violations when the anticipated future solar resource is zero for some period of time.

Bold header: Closed-Form Control Law Derivation

The AI system will derive a simple closed-form control law that does not require a solar forecast by leveraging indirect optimal control methods, which shows that the optimal velocity is undetermined constant value during each constraint-inactive time segment.

Bold header: Iterative Learning for Optimal Velocity Estimation

The system can estimate the unknown constant velocity using an iterative learning approach, specifically employing update laws like u i+1 = u i + kp(b i(tf) − b i des) and a continuous rate-based update law, which converges to an optimal costate value.

Bold header: Real-Time Control Implementation with Buffer

The resulting control strategy can be implemented in a sampled (discrete) time setting by incorporating a buffer shown in Eqn. 16 to mitigate chatter caused by the switching controller when SOC barrier functions change states.

Bold header: Performance Benchmarking Against MPC

The improved AI system can perform nearly as well as a model predictive control approach that requires an accurate future solar forecast and significantly greater computational capability, demonstrating comparable performance without the need for a forecast or the associated computational burden.

Abstract

In this paper, we present the methodology and results for a real-time velocity trajectory optimization for a solar-powered autonomous surface vessel (ASV), where we combine indirect optimal control techniques with iterative learning. The ASV exhibits cyclic operation due to the nature of the solar profile, but weather patterns create inevitable disturbances in this profile. The nature of the problem results in a formulation where the satisfaction of pointwise-in-time state of charge constraints does not generally guarantee persistent feasibility, and the goal is to maximize information gathered over a very long (ultimately persistent) time duration. To address these challenges, we first use barrier functions to tighten pointwise-in-time state of charge constraints by the minimal amount necessary to achieve persistent feasibility. We then use indirect methods to derive a simple switching control law, where the optimal velocity is shown to be an undetermined constant value during each constraint-inactive time segment. To identify this optimal constant velocity (which can vary from one segment to the next), we employ an iterative learning approach. The result is a simple closed-form control law that does not require a solar forecast. We present simulation-based validation results, based on a model of the SeaTrac SP-48 ASV and solar data from the North Carolina coast. These simulation results show that the proposed methodology, which amounts to a closed-form controller and simple iterative learning update law, performs nearly as well as a model predictive control approach that requires an accurate future solar forecast and significantly greater computational capability.