Analysis of optimization strategies for solving space manoeuvre vehicle trajectory optimization problem

In this paper, two types of optimization strategies are applied to solve the Space Manoeuvre Vehicle (SMV) trajectory optimization problem. The SMV dynamic model is constructed and discretized applying direct multiple shooting method. To solve the resulting Nonlinear Programming (NLP) problem, gradient-based and derivative free optimization techniques are used to calculate the optimal time history with respect to the states and controls. Simulation results indicate that the proposed strategies are effective and can provide feasible solutions for solving the constrained SMV trajectory design problem

Improved gradient-based algorithm for solving aeroassisted vehicle trajectory optimization problems

Space maneuver vehicles (SMVs) [1,2] will play an increasingly important role in the future exploration of space because their on-orbit maneuverability can greatly increase the operational flexibility, and they are more difficult as a target to be tracked and intercepted. Therefore, a well-designed trajectory, particularly in the skip entry phase, is a key for stable flight and for improved guidance control of the vehicle [3,4]. Trajectory design for space vehicles can be treated as an optimal control problem. Because of the highly nonlinear characteristics and strict path constraints of the problem, direct methods are usually applied to calculate the optimal trajectories, such as the direct multiple shooting method [5], direct collocation method [5,6], or hp hp -adaptive pseudospectral method [7,8]

Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method

Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.delayed differential equations, delayed optimal control, numerical optimization, time-to-build

Fast numerical methods for mixed--integer nonlinear model--predictive control

This thesis aims at the investigation and development of fast numerical methods for nonlinear mixed--integer optimal control and model- predictive control problems. A new algorithm is developed based on the direct multiple shooting method for optimal control and on the idea of real--time iterations, and using a convex reformulation and relaxation of dynamics and constraints of the original predictive control problem. This algorithm relies on theoretical results and is based on a nonconvex SQP method and a new active set method for nonconvex parametric quadratic programming. It achieves real--time capable control feedback though block structured linear algebra for which we develop new matrix updates techniques. The applicability of the developed methods is demonstrated on several applications. This thesis presents novel results and advances over previously established techniques in a number of areas as follows: We develop a new algorithm for mixed--integer nonlinear model- predictive control by combining Bock's direct multiple shooting method, a reformulation based on outer convexification and relaxation of the integer controls, on rounding schemes, and on a real--time iteration scheme. For this new algorithm we establish an interpretation in the framework of inexact Newton-type methods and give a proof of local contractivity assuming an upper bound on the sampling time, implying nominal stability of this new algorithm. We propose a convexification of path constraints directly depending on integer controls that guarantees feasibility after rounding, and investigate the properties of the obtained nonlinear programs. We show that these programs can be treated favorably as MPVCs, a young and challenging class of nonconvex problems. We describe a SQP method and develop a new parametric active set method for the arising nonconvex quadratic subproblems. This method is based on strong stationarity conditions for MPVCs under certain regularity assumptions. We further present a heuristic for improving stationary points of the nonconvex quadratic subproblems to global optimality. The mixed--integer control feedback delay is determined by the computational demand of our active set method. We describe a block structured factorization that is tailored to Bock's direct multiple shooting method. It has favorable run time complexity for problems with long horizons or many controls unknowns, as is the case for mixed- integer optimal control problems after outer convexification. We develop new matrix update techniques for this factorization that reduce the run time complexity of all but the first active set iteration by one order. All developed algorithms are implemented in a software package that allows for the generic, efficient solution of nonlinear mixed-integer optimal control and model-predictive control problems using the developed methods

Path planning for autonomous buses based on optimal control

This thesis presents an algorithm to generate trajectories for an autonomous bus approaching a bus stop. The path planning algorithm is formulated as an Optimal Control Problem (OCP) which is solved by means of nonlinear programming (NLP) using the direct multiple shooting method. This method has shown to be a good choice for solving nonlinear Boundary Value Problems (BVP) like this one -where there are constraints such as the limits of the road, the model dynamics or passengers comfort- due to its highly accurate solution and faster convergence and stability than other methods like direct single shooting methods. It uses a kinematic bicycle model with a coordinate transformation which uses the vehicle position along the path as independent variable instead of using time which permits the definition of the constraints independently of the vehicle’s speed. The OCP is solved in MATLAB using CasADi, a symbolic tool for solving nonlinear optimization problems that provides high level interfaces to make the problem writing easier, in addition of having better performance than similar tools. The proposed algorithm is evaluated in multiple scenarios like different kinds of bus stops and paths inside confined areas, giving as a result a trajectory that meets with the imposed constraints successfully. Experimental tests on a real autonomous bus are carried out, resulting in a smooth bus stop manoeuvre that the passengers evaluated as fully acceptable.Outgoin

Fast Direct Multiple Shooting Algorithms for Optimal Robot Control

International audienceIn this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control. In the survey part, we discuss three direct optimal control approaches in detail: (i) single shooting, (ii) collocation, and (iii) multiple shooting, and we specify why we believe the direct multiple shooting method to be the method of choice for nonlinear optimal control problems in robotics. We couple it with an efficient robot model generator and show the performance of the algorithm at the example of a five link robot arm. In the real-time optimization part, we outline the idea of nonlinear model predictive control and the real-time challenge it poses to numerical optimization. As one solution approach, we discuss the real-time iteration scheme