Multiple Shooting Technique Research Articles

Invariant manifolds near L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} in the Sun–Jupiter elliptic restricted three-body problem I

In this paper, we present a way of combining the computation of invariant tori and their stable and unstable manifolds with the multiple shooting technique. We start by showing some of the results of Jorba (Nonlinearity 14(5):943–976, 2001) that should be modified in order to introduce the multiple shooting technique in these computations. After that, by a direct application in the planar elliptic restricted three-body problem (PERTBP), how to modify the equations and methods to compute the above-mentioned objects is introduced. In particular, the structure of the (systems of) equations and matrices involved in these computations is shown. An application of these computations can be found in Duarte and Jorba (Invariant manifolds of tori near L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}_1$$\\end{document} and L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}_2$$\\end{document} in the planar elliptic restricted three-body problem II. The Dynamics of Comet Oterma, Preprint 2023), where the dynamics of comet 39P/Oterma is modelled as a PERTBP.

Time‐optimal trajectory planning of a redundantly actuated planar parallel robot driven with series elastic actuators

AbstractThis paper addresses the time‐optimal trajectory planning of a redundantly actuated parallel robot driven by series elastic actuators. Here, the actuation redundancy allows prestressing the elastic components of the drives and enables more degrees of freedom in the trajectory planning. To this end, a time‐optimal control problem is formulated that takes also the physical constraints into account, that is actuation torques, task and joint space limits and self‐collisions. A flatness‐based parametrization of the dynamics is used in the nonlinear optimization problem and solved with the multiple shooting technique within the CasADi framework using IPOPT as a solver. Results are shown and compared to time‐optimal trajectories, both redundantly and non‐redundantly actuated with rigid drives.

Parallel solution of optimal control problems using the graphics processing unit

AbstractThis paper presents an indirect method for solving optimal control problems (OCPs) using graphics processing unit (GPU). The OCPs considered here include control variable inequality constraints (CVICs), state variable inequality constraints (SVICs), and parameters. The necessary conditions of the minimum for the OCPs are written as a boundary value problem with index‐1 differential algebraic equations (BVP‐DAEs). The complementarity conditions associated with those inequality constraints are approximated using Kanzow's smoothed Fisher‐Burmeister formula. The numerical solution for solving the BVP‐DAEs is based on the multiple shooting technique and the DAEs are solved using a single step linearly implicit Runge–Kutta (Rosenbrock–Wanner, ROW) method. A Newton's continuation method is performed to solve the BVP‐DAEs system and the descent direction is found by solving a sparse bordered almost block diagonal (BABD) linear system with a structured orthogonal factorization algorithm. Parallel computing techniques are used to accelerate the code which is implemented using Python and CUDA on GPU. Numerical examples are presented to illustrate the efficiency of the code. The GPU based parallel implementation is shown to be significantly faster than the implementation using central processing unit (CPU) alone.

Eco-driving control of electric vehicle with battery dynamic model and multiple traffic signals

Limited capacity and short life cycle of a battery are the major impediments in development of practical Electric Vehicles (EVs). Eco-driving is an optimization technique through which a velocity trajectory that consumes minimum energy is advised to the driver. However, presence of traffic signals to control large traffic network degrades the performance of eco-driving; as applying brakes to stop and then maximum re-acceleration to restart a trip consumes lot of energy. Eco-driving problem with multiple traffic signals and static model of battery has been proposed as Two Point Boundary Value Problem (TPBVP). TPBVP fails to solve multi-phase problem as a single phase due to discontinuity of the co-states at the junction, that is, start of a new phase. This paper investigates an optimal solution with both EV and battery dynamics in the presence of multiple traffic signals as Multi Point Boundary Value Problem (MPBVP) using multiple shooting technique. Traffic signals come at some intermediate points of a trip. MPBVP ensures continuity at the junction to solve the multi-phase problem as a single phase through inter dependencies between each phases. Goal of this work is not only to solve constrained eco-driving problem with traffic signals but also include charging and discharging limits on battery that indirectly improves battery’s life cycle. Results indicate that EV has crossed all the traffic signals during their green duration without applying brakes with also satisfying all the other constraints and continuity condition. Moreover, it can be seen that energy consumption using MPBVP is also marginally lesser as compared to TPBVP.

Tuning nonlinear state-space models using unconstrained multiple shooting

A persisting challenge in nonlinear dynamical modelling is parameter inference from data. Provided that an appropriate model structure was selected, the identification problem is profoundly affected by a choice of initialisation. A particular challenge that may arise is initialisation within a region of the parameter space where the model is not contractive. Exploring such regions is not feasible using the conventional optimisation tools for they require a bounded evaluation of the cost. This work proposes an unconstrained multiple shooting technique, able to mitigate stability issues during the optimisation of nonlinear state-space models. The technique is illustrated on simulation results of a Van der Pol oscillator and benchmark results on a Bouc-Wen hysteretic system.

High-Fidelity Trajectory Optimization with Application to Saddle-Point Transfers

A method to optimize space trajectories subject to impulsive controls is presented. The method employs a high-fidelity model and a multiple-shooting technique. The model accounts for an arbitrary number of gravitational attractions, their corrections due to celestial bodies oblateness, and solar radiation pressure. The peculiarity of this paradigm is that the equations of motion are written in a rotopulsating frame, where two primaries are at rest despite the fact that their motion and the one of the perturbers are given by a real ephemeris model. Direct transcription of the dynamics coupled with a multiple-shooting technique and an efficient computation of the Jacobian of the defects are used to optimize trajectories subject to a finite number of impulsive controls. The method has been applied to find a multitude of solutions to the problem of transferring a spacecraft from the sun–Earth collinear Lagrange points to the sun–Earth gravitational saddle point, where a theoretical zero background acceleration allows testing possible deviations from general relativity. The problem of targeting the sun–Earth saddle point with high accuracy represents a major flight dynamics challenge. The applicative scenario encompasses the possible mission extension option for LISA Pathfinder as special case.

Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle

In this article, the minimum time control problem of an electric vehicle is modeled as a Mayer problem in optimal control, with affine dynamics with respect to the control and with state constraints. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle. An analysis, with the techniques of geometric control, is used first to reduce the set of candidates and then to construct the numerical methods. This leads to a numerical investigation based on indirect methods using the HamPath software. Multiple shooting and homotopy techniques are used to build a synthesis with respect to the bounds of the boundary sets.

Discrete-Time Optimal Attitude Control of a Spacecraft with Momentum and Control Constraints

An algorithm is proposed to solve optimal control problems arising in attitude control of a spacecraft under state and control constraints. First, the discrete-time attitude dynamics are derived by employing discrete mechanics. Then, the orientation transfer, with initial and final values of the orientation and momentum and the time duration being specified, is posed as an energy-optimal control problem in discrete time, subject to momentum and control constraints. Using variational analysis directly on the Lie group (the set of special orthonormal matrices), first-order necessary conditions for optimality are derived, leading to a constrained two-point boundary value problem. This two-point boundary value problem is solved via a novel multiple shooting technique that employs a root-finding Newton algorithm. Robustness of the multiple shooting technique is demonstrated through a few representative numerical experiments.

Trajectory refinement of three-body orbits in the real solar system model

In this paper, an automatic algorithm for the correction of orbits in the real solar system model is described. The differential equations governing the dynamics of a massless particle in the n-body problem are written as perturbation of the circular restricted three-body problem in a non-uniformly rotating, pulsating frame by using a Lagrangian formalism. The refinement is carried out by means of a modified multiple shooting technique, and the problem is solved for a finite number of trajectory states at several time instants. The analysis involves computing the dynamical substitutes of the collinear points, as well as several Lagrange point orbits, for the Sun–Earth, Sun–Jupiter, and Earth–Moon gravitational systems.

NMPC-based Trajectory Tracking and Collison Avoidance of Underactuated Vessels with Elliptical Ship Domain

Abstract: This paper presents a Nonlinear Model Predictive Control (NMPC) for position and velocity tracking of underactuated surface vessels of elliptical ship domains with a collision avoidance scheme. A constrained nonlinear optimization problem is formulated to minimize vessel states’ deviation from a time varying reference generated by a virtual vessel over a finite horizon. Linear constraints is imposed on input force and moment to be within the physical limits. Collision avoidance is represented as separation of each pair of the elliptic disks, representing the ship domain of our vessel and each encountered one, and is formulated as nonlinear time-varying constraint over the prediction horizon. A real-time C-code is generated using the ACADO toolkit and qpOASES solver with multiple shooting technique for discretization and Gauss-Newton iteration algorithm. This algorithm is computationally efficient, thus enabling real-time implementation of the proposed technique. MATLAB simulations are used to assess the validity of the proposed technique.

Multiple Shooting Technique for Optimal Control Problems with Application to Power Aware Networks

This paper presents a shooting method for a class of optimal control problems where the input u(t) is expressed analytically in terms of its associated state x(t) and costate (adjoint) p(t) along the optimal trajectory. The problem is formulated in terms of computing the initial costate that drives the final costate to a prescribed value. We solve this problem by a combination of gradient descent and multiple shooting. The proposed technique is applied to a power-aware problem of co-optimizing motion and transmission power in networks of mobile agents, and fast convergence is discerned from simulation of a fifty-agent system.

New Algorithm of Two-Point Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

Optimization of low-energy resonant hopping transfers between planetary moons

In response to the scientific interest in Jupiter's Galilean moons, NASA and ESA have plans to send orbiting missions to Europa and Ganymede, respectively. The inter-moon transfers of the Jovian system offer obvious advantages in terms of scientific return, but are also challenging to design and optimize due in part to the large, often chaotic, sensitivities associated with repeated close encounters of the planetary moons. The approach outlined in this paper confronts this shortcoming by exploiting the multi-body dynamics with a patched three-body model to enable multiple “resonant-hopping” gravity assists. Initial conditions of unstable resonant orbits are pre-computed and provide starting points for the elusive initial guess associated with the highly nonlinear optimization problem. The core of the optimization algorithm relies on a fast and robust multiple-shooting technique to provide better controllability and reduce the sensitivities associated with the close approach trajectories. The complexity of the optimization problem is also reduced with the help of the Tisserand–Poincaré (T–P) graph that provides a simple way to target trajectories in the patched three-body problem. Preliminary numerical results of inter-moon transfers in the Jovian system are presented. For example, using only 59m/s and 158 days, a spacecraft can transfer between a close resonant orbit of Ganymede and a close resonant orbit of Europa.

Model based control of a liquid swelling constrained batch reactor subject to recipe uncertainties

This work presents the application of nonlinear model predictive control (NMPC) to a simulated industrial batch reactor subject to safety constraint due to reactor level swelling, which can occur with relatively fast dynamics. Uncertainties in the implementation of recipes in batch process operation are of significant industrial relevance. The paper describes a novel control-relevant formulation of the excessive liquid rise problem for a two-phase batch reactor subject to recipe uncertainties. The control simulations are carried out using a dedicated NMPC and optimization software toolbox OptCon which implements efficient numerical algorithms. The open-loop optimal control problem is computed using the multiple-shooting technique and the arising nonlinear programming problem is solved using a sequential quadratic programming (SQP) algorithm tailored for large-scale problems, based on the freeware optimization environment HQP. The fast response of the NMPC controller is guaranteed by the initial value embedding and real-time iteration technologies. It is concluded that the OptCon implementation allows small sampling times and the controller is able to maintain safe and optimal operation conditions, with good control performance despite significant uncertainties in the implementation of the batch recipe.

Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.

Periodic waveform Krylov subspace method of circuit systems by multiple shooting

A multiple shooting technique is presented for determining the periodic responses of a linear circuit system by waveform relaxation (WR). Two ways to compute the unknown multiple shooting points are also provided. The new WR approach, namely a Krylov subspace algorithm in function space for a periodic system, is an accelerating process of classic WR with multiple shooting. The waveform Krylov subspace algorithm can enlarge the convergence region of function iterations. Further, the multiple shooting technique can numerically solve a periodic problem with a long interval in parallel.

Optimal Control with State Constraints and the Space Shuttle Re-entry Problem

In this article, we initialize the analysis under generic assumptions of the small time optimal synthesis for single input systems with state constraints. We use geometric methods to evaluate the small time reachable set and necessary optimality conditions. Our work is motivated by the optimal control of the atmospheric arc for the re-entry of a space shuttle, where the vehicle is subject to constraints on the thermal flux and on the normal acceleration. A multiple shooting technique is finally applied to the computation of the optimal longitudinal arc.

Dynamical modelling of measured time series from a Q-switched CO 2 laser

The transient dynamics of a Q-switched CO 2 laser is modelled quantitatively on the base of the four level model, a five dimensional nonlinear system of ordinary differential equations. Using the multiple shooting technique, internal parameters of the laser are estimated and the unobserved time courses of the population densities are constructed. For excitations barely above the laser threshold large pulse variations are identified as an effect of small variations of the pump parameter.

Identifying physical properties of a CO2 laser by dynamical modeling of measured time series.

We estimate internal parameters of a Q-switched CO2 laser by fitting trajectories of the four-level model to measured scalar time series. The four-level model is a five-dimensional nonlinear system of ordinary differential equations. A multiple shooting technique is used to construct the unobserved time courses of the population densities and to reveal the dependence of the parameters on the excitation current. For excitations barely above the laser threshold large pulse variations are identified as an effect of small variations of the pump parameter.

Analysis of fatigue crack growth behaviour in polymers and their composites based on ordinary differential equations parameter estimation

Traditionally, the fatigue behaviour of polymers and polymer composites is analyzed by performing Wöhler experiments, a well-known technique for the investigation of metals. These experiments are very expensive and time consuming. The application of fracture mechanics provides a much faster way to analyze the fatigue crack growth behaviour of polymers and polymer composites. This analysis is based on the Paris–Erdogan equation describing a stable phase of crack growth. With this method the fatigue crack growth diagram is derived simultaneously with the reading of the measurements. Care must be taken however, to minimize error arising from variations in initial crack length, the uncertainty in the determination of the elastic modulus and estimation of the region of stable crack growth. To overcome some of the systematic problems mentioned above, the following procedure is applied: The fracture mechanical parameter estimation problem is mapped onto a nonlinear least squares problem restricted by a differential equation. This problem is numerically solved by a boundary value problem approach which consists of a multiple shooting technique combined with a damped Gauss–Newton procedure. In this procedure, the only data to be measured is the compliance as a function of time or number of cycles. After the problem of least squares is solved and the material parameters determined, the distribution of residuals of the fit is used to estimate the region of stable crack growth. Since the analysis is based on well defined statistical foundations (maximum likelihood estimation), we gain a methodological advantage of the increased objectivity of the analysis. Considering the increasing importance of lifespan prognoses for plastic components based on fatigue crack propagation experiments, this method is preferred to the conventional, rather subjective procedure.