[Paper Review] Optimal Trajectory Planning and Model Predictive Control of Underactuated Marine Surface Vessels using a Flatness-Based Approach

Max Lutz, Thomas Meurer|arXiv (Cornell University)|Jan 29, 2021

Adaptive Control of Nonlinear Systems26 references22 citations

TL;DR

This paper presents a flatness-based direct optimal control method for energy-efficient trajectory planning and model predictive control (MPC) of underactuated marine surface vessels. By parameterizing the highest derivative of the flat output using continuous piecewise linear functions, the approach transforms the optimal control problem into a nonlinear programming problem, enabling efficient solution with a novel A*-based initial guess. The method achieves real-time feasibility and robustness under dynamic disturbances, obstacle constraints, and parameter mismatches, demonstrating superior energy efficiency and constraint handling in simulation.

ABSTRACT

This paper demonstrates a refined approach to solving dynamic optimization problems for underactuated marine surface vessels. To this end the differential flatness of a mathematical model assuming full actuation is exploited to derive an efficient representation of a finite dimensional nonlinear programming problem, which in turn is constrained to apply to the underactuated case. It is illustrated how the properties of the flat output can be employed for the generation of an initial guess to be used in the optimization algorithm in the presence of static and dynamic obstacles. As an example energy optimal point to point trajectory planning for a nonlinear 3 degrees of freedom dynamic model of an underactuated surface vessel is undertaken. Input constraints, both in rate and magnitude as well as state constraints due to convex and non-convex obstacles in the area of operation are considered and simulation results for a challenging scenario are reported. Furthermore, an extension to a trajectory tracking controller using model predictive control is made where the benefits of the flatness based direct method allow to introduce nonuniform sample times that help to realize long prediction horizons while maintaining short term accuracy and real time capability. This is also verified in simulation where additional disturbances in the form of environmental disturbances, dynamic obstacles and parameter mismatch are introduced.

Motivation & Objective

• To address energy-optimal point-to-point trajectory planning for underactuated marine surface vessels under input and state constraints.
• To extend the approach to closed-loop trajectory tracking using model predictive control (MPC) with real-time capability.
• To develop a robust initial guess generation strategy using A* pathfinding and mollifier smoothing to improve convergence in complex environments.
• To ensure feasibility and performance under dynamic disturbances, parameter mismatches, and non-convex obstacles.
• To demonstrate the effectiveness of non-uniform sampling and slack constraints in MPC for handling constraint violations and environmental uncertainty.

Proposed method

• The differential flatness of a 3DOF nonlinear vessel model is exploited to parameterize the highest derivative of the flat output using continuous, piecewise linear functions.
• The optimal control problem is transcribed into a finite-dimensional nonlinear programming problem (NLP) via a flatness-based direct collocation method.
• An initial guess for the NLP is generated by solving a discrete path-finding problem with A* on a grid, followed by smoothing using a mollifier to ensure continuity and differentiability.
• For MPC, the OCP is reformulated on a receding horizon with a slack variable in obstacle constraints to maintain feasibility under disturbances.
• The cost function includes a last-waypoint-match (LWM) strategy penalizing end-state deviation, with weights for energy, constraint violation, and terminal state error.
• Non-uniform sampling times are employed in MPC to balance long prediction horizons with short-term accuracy and real-time performance.

Experimental results

Research questions

• RQ1 Can a flatness-based direct method with piecewise linear parameterization of the flat output's highest derivative enable efficient and accurate solution of energy-optimal trajectory planning for underactuated marine vessels?
• RQ2 How can the geometric meaning of the flat output be leveraged to generate a reliable initial guess for optimization in the presence of static and dynamic obstacles?
• RQ3 To what extent does the proposed A*-based initial guess strategy improve convergence robustness compared to relying on previous MPC solutions, especially under sudden environmental changes?
• RQ4 How effective is the use of slack variables and non-uniform sampling in MPC for maintaining feasibility and performance under disturbances and parameter mismatches?
• RQ5 What is the trade-off between energy efficiency and constraint violation in MPC when using a last-waypoint-match strategy versus an all-waypoint-match strategy?

Key findings

• The proposed method achieved an energy cost of 37.2 m in the MPC scenario, outperforming the all-waypoint-match strategy (85.6 m) in energy efficiency.
• The average optimization time for MPC was 250 ms on the reported hardware, confirming real-time feasibility.
• The A*-based initial guess enabled the MPC to react within one iteration to a sudden change in the dynamic obstacle's motion, avoiding the memory effect seen when reusing previous solutions.
• Feasibility was maintained throughout the simulation, even under a −10% parameter mismatch and a 0.04 m/s ocean current, demonstrating robustness.
• The use of slack variables in obstacle constraints prevented infeasibility due to disturbances, while the LWM strategy ensured terminal state convergence with low energy cost.
• The simulation results confirmed that non-uniform sampling in MPC allows for long prediction horizons while preserving short-term accuracy and real-time capability.