An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

Single-stage to orbit ascent trajectory optimisation with reliable evolutionary initial guess

In this paper, the ascent trajectory optimization of a lifting body Single-Stage To Orbit (SSTO) reusable launch vehicle is investigated. The work is carried out using a Direct Multiple Shooting method to solve the Optimal Control problem. The crucial initialisation of the optimisation process is performed by using a combination of two evolutionary algorithms, namely a Multi-Objective Parzen-based Estimation of Distribution (MOPED) algorithm and a Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA). Multi-Objective Parzen-based Estimation of Distribution (MOPED) belongs to the class of Estimation of Distribution Algorithms (EDAs) and it is used in the first phase of the initial guess research to explore the search space, then Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA) is used to refine the obtained results, and better fulfill the imposed constraints. The initial guesses obtained with this evolutionary framework were tested on different multiple shooting configurations. The importance of the continuity properties of the employed mathematical models was also quantitatively addressed.

Trajectory Tracking For Quadrotors: An Optimization-Based Planning Followed By Controlling Approach

We present an optimization-based reference trajectory tracking method for quadrotor robots for slow-speed maneuvers. The proposed method uses planning followed by the controlling paradigm. The basic concept of the proposed method is an analogy to Linear Quadratic Gaussian (LQG) in which Nonlinear Model Predictive Control (NMPC) is employed for predicting optimal control policy in each iteration. Multiple-shooting (MS) is suggested over Direct-collocation (DC) for imposing constraints when modelling the NMPC. Incremental Euclidean Distance Transformation Map (EDTM) is constructed for obtaining the closest free distances relative to the predicted trajectory; these distances are considered obstacle constraints. The reference trajectory is generated, ensuring dynamic feasibility. The objective is to minimize the error between the quadrotor’s current pose and the desired reference trajectory pose in each iteration. Finally, we evaluated the proposed method with two other approaches and showed that our proposal is better than those two in terms of reaching the goal without any collision. Additionally, we published a new dataset, which can be used for evaluating the performance of trajectory tracking algorithms.

Trajectory Tracking For Quadrotors: An Optimization-Based Planning Followed By Controlling Approach

We present an optimization-based reference trajectory tracking method for quadrotor robots for slow-speed maneuvers. The proposed method uses planning followed by the controlling paradigm. The basic concept of the proposed method is an analogy to Linear Quadratic Gaussian (LQG) in which Nonlinear Model Predictive Control (NMPC) is employed for predicting optimal control policy in each iteration. Multiple-shooting (MS) is suggested over Direct-collocation (DC) for imposing constraints when modelling the NMPC. Incremental Euclidean Distance Transformation Map (EDTM) is constructed for obtaining the closest free distances relative to the predicted trajectory; these distances are considered obstacle constraints. The reference trajectory is generated, ensuring dynamic feasibility. The objective is to minimize the error between the quadrotor’s current pose and the desired reference trajectory pose in each iteration. Finally, we evaluated the proposed method with two other approaches and showed that our proposal is better than those two in terms of reaching the goal without any collision. Additionally, we published a new dataset, which can be used for evaluating the performance of trajectory tracking algorithms.

Limit Cycle Oscillation Suppression Controller Design and Stability Analysis of the Periodically Time-varying Flapping Flight Dynamics in Hover

The periodically time-varying forces make the equilibrium state of Beihawk, an X-shaped flapping-wing aircraft, to be a periodic limit cycle oscillation. However, traditional controllers based on averaging theory fail to suppress this oscillation and the derived stability result may be inaccurate. In this study, a period-based method is proposed to design the oscillation suppression controller, locate the corresponding cycle and analyze its stability. A periodically time-varying wing–tail interaction model is built and Discrete Fourier Transform is applied to adapt the model for controller design. The harmonics less than quintuple flapping frequency account for more than 96 percent of the total harmonics and are reserved to present a concise model. Based on this model, Active Disturbance Rejection Controller (ADRC) is designed and its Extended State Observer can observe the disturbance to suppress the oscillation. Poincaré map is introduced to convert the stability analysis of the cycle to a fixed point. A multiple shooting method is adopted to locate several points on the cycle and the map is obtained by calculating the submaps between the adjacent points with the Floquet theory. The located points are proved to be accurate compared with the numerical solved cycle and the stability analysis result of the cycle is verified by the dynamic evolution. Compared with the State Feedback Controller, the ADRC performs better in suppressing the limit cycle oscillation and eliminating the attitude control error. The oscillation suppression is meaningful in maintaining a stable flight and capturing high quality images.

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.

MATLAB Implementation of Direct and Indirect Shooting Methods to Solve an Optimal Control Problem With State Constraints

The paper presents a general procedure to solve numerically optimal control problems with state constraints. Itis used in the case, when the simple time discretization of the state equations and expressing the optimal control problem as a nonlinear mathematical programming problem is too coarse. It is based on using in turn two multiple shooting BVP approaches: direct and indirect.The paper is supplementary to the earlier author’s paper on direct and indirect shooting methods, presenting the theory underlying both approaches. The same example is considered here and brought to an end, that is two full listings of two Matlab codes are shown.

Bioptim, a Python framework for Musculoskeletal Optimal Control in Biomechanics

Musculoskeletal simulations are useful in biomechanics to investigate the causes of movement disorder, to estimate non-measurable physiological quantities or to study the optimality of human movement. We introduce bioptim, an easy-to-use Python framework for biomechanical optimal control, handling musculoskeletal models. Relying on algorithmic differentiation and the multiple shooting formulation, bioptim interfaces nonlinear solvers to quickly provide dynamically consistent optimal solutions. The software is both computationally efficient (C++ core) and easily customizable, thanks to its Python interface. It allows to quickly define a variety of biomechanical problems such as motion tracking/prediction, muscle-driven simulations, parameters optimization, multiphase problems, etc. It is also intended for real-time applications such as moving horizon estimation and model predictive control. Six contrasting examples are presented, comprising various models, dynamics, objective functions and constraints. They include data-driven simulations (i.e., a multiphase muscle driven gait cycle and an upper-limb real-time moving horizon estimation of muscle forces) and predictive simulations (i.e., a muscle-driven pointing task, a twisting somersault with a quaternion-based model, a position controller using external forces, and a multiphase torque-driven maximum-height jump motion).