Direct Multiple-shooting Method Research Articles

Risk-Based Model Predictive Control for Autonomous Ship Emergency Management

Control for semi- and fully-autonomous ships is a broad and complex field. Making autonomous high-level decisions in place of the captain is considered difficult, partly due to the risks and uncertainties involved. Though human operators located in onshore control centers are still needed for safety and regulatory reasons, there is a growing demand for complex decisions to be made by the onboard control system itself, both during normal operations and extraordinary circumstances. Model predictive control (MPC) is a promising approach to tackle this problem. In this paper, a dynamic risk-based decision-making algorithm is constructed through the use of heuristic objectives, capable of planning suitable vessel trajectories in emergency situations. Nonlinear programming using the direct multiple-shooting method implemented with the CasADi framework is considered, and the resulting control performance of several emergency scenarios is analyzed using simulation. The developed algorithm proved capable of both generating suitable trajectories below a certain risk threshold, and to engage the safety systems appropriately. It is concluded that MPC with independent risk cost terms is a promising method for autonomous ship trajectory planning and emergency management.

On the integration of singularity-free representations of $$\varvec{SO(3)}$$ S O ( 3 ) for direct optimal control

In this paper we analyze the performance of different combinations of: (1) parameterization of the rotational degrees of freedom (DOF) of multibody systems, and (2) choice of the integration scheme, in the context of direct optimal control discretized according to the direct multiple-shooting method. The considered representations include quaternions and Direction Cosine Matrices, both having the peculiarity of being non-singular and requiring more than three parameters to describe an element of the Special Orthogonal group $$\textit{SO}(3)$$ . These representations yield invariants in the dynamics of the system, i.e., algebraic conditions which have to be satisfied in order for the model to be representative of physical reality. The investigated integration schemes include the classical explicit Runge–Kutta method, its stabilized version based on Baumgarte’s technique, which tends to reduce the drift from the underlying manifold, and a structure-preserving alternative, namely the Runge–Kutta Munthe-Kaas method, which preserves the invariants by construction. The performances of the combined choice of representation and integrator are assessed by solving thousands of planning tasks for a nonholonomic, underactuated cart-pendulum system, where the pendulum can experience arbitrarily large 3D rotations. The aspects analyzed include success rate, average number of iterations and CPU time to convergence, and quality of the solution. The results reveal how structure-preserving integrators are the only choice for lower accuracies, whereas higher-order, non-stabilized standard integrators seem to be the computationally most competitive solution when higher levels of accuracy are pursued. Overall, the quaternion-based representation is the most efficient in terms of both iterations and CPU time to convergence, albeit at the cost of lower success rates and increased probability of being trapped by higher local minima.

Output-Constraint Handling and Parallelization for Oil-Reservoir Control Optimization by Means of Multiple Shooting

Summary We propose to formulate and solve the reservoir-control optimization problem with the direct multiple-shooting method. This method divides the optimal-control problem prediction horizon in smaller intervals that one can evaluate in parallel. Further, output constraints are easily established on each interval boundary and as such hardly affect computation time. This opens new opportunities to include state constraints on a much broader scale than is common in reservoir optimization today. However, multiple shooting deals with a large number of variables because it decides on the boundary-state variables of each interval. Therefore, we exploit the structure of the reservoir simulator to conceive a variable-reduction technique to solve the optimization problem with a reduced sequential quadratic-programming algorithm. We discuss the optimization-algorithm building blocks and focus on structure exploitation and parallelization opportunities. To demonstrate the method's capabilities to handle output constraints, the optimization algorithm is interfaced to an open-source reservoir simulator. Then, on the basis of a widely used reservoir model, we evaluate performance, especially related to output constraints. The performance of the proposed method is qualitatively compared with a conventional method.

Optimal Lunar Orbit Insertion from a Variable Symmetric Free-Return Trajectory

A method for the optimization of a multiple-impulse lunar orbit insertion sequence from a symmetric free-return trajectory is presented. Both the inclination and altitude at lunar periapsis along the free return are variable. The algorithm systematically optimizes the orbit insertion sequence while simultaneously generating a free-return outbound trajectory. With a direct multiple-shooting method, optimization is performed in the circular restricted three-body model with a constrained sequential quadratic programming algorithm. The necessary gradients are derived with linear perturbation theory, and a closed-form expression is derived for computing gradients along a general n-impulse trajectory. Examples are presented for two classes of symmetric free returns, and a range of lunar orbit orientations is examined to assess the impact on velocity impulse requirements.

Finding Optimal Controls for Helicopter Maneuvers Using the Direct Multiple-Shooting Method

The purpose of this paper deals with direct multiple-shooting method (DMS) to resolve helicopter maneuver problems of helicopters. The maneuver problem is transformed into nonlinear problems and solved DMS technique. The DMS method is easy in handling constraints and it has large convergence radius compared to other strategies. When parameterized with piecewise constant controls, the problems become most effectively tractable because the search direction is easily estimated by solving the structured Karush-Kuhn-Tucker (KKT) system. However, generally the computation of function, gradients and Hessian matrices has considerably time-consuming for complex system such as helicopter. This study focused on the approximation of the KKT system using the matrix exponential and its integrals. The propose method is validated by solving optimal control problems for the linear system where the KKT system is exactly expressed with the matrix exponential and its integrals. The trajectory tracking problem of various maneuvers like bob up, sidestep near hovering flight speed and hurdle hop, slalom, transient turn, acceleration and deceleration are analyzed to investigate the effects of algorithmic details. The results show the matrix exponential approach to compute gradients and the Hessian matrix is most efficient among the implemented methods when combined with the mixed time integration method for the system dynamics. The analyses with the proposed method show good convergence and capability of tracking the prescribed trajectory. Therefore, it can be used to solve critical areas of helicopter flight dynamic problems.