Approximate Optimal Trajectories Research Articles

Direct trajectory optimization of macro-micro robotic system using a Gauss pseudospectral framework

Trajectory planning is a crucial aspect of macro-micro robotic systems (MMRSs), especially when the system has high degrees of freedom (DOFs). In the field of robotic polishing, the MMRS is usually composed of an industrial robot and an end-effector, which is responsible for polishing force control. Therefore, the compliance of the macro-robot can be minimized by trajectory planning to reduce its impact on the micro-robot. This study proposes a trajectory planning strategy based on Gauss pseudospectral method for a 9-DOF MMRS. Different from traditional sequential solution strategies, it can be used to obtain an approximate global optimal trajectory. Firstly, the velocity-level kinematics model of MMRS is built, which comprehensively considers the workpiece placement pose and task redundancy. Secondly, an optimal control model for trajectory planning is developed through an effective variable allocation. On the premise of considering traditional trajectory smoothness constraints, a constraint on manipulability is additionally analyzed to avoid reaching a singular configuration during compliance optimization. Thirdly, a Gauss pseudospectral framework based on the optimal control model is proposed, and the costate mapping theorem is proved. The latter provides a theoretical basis for the efficiency and accuracy of the proposed method. Finally, comparison results demonstrate the effectiveness of the proposed method.

Robust Approximate Optimal Trajectory Tracking Control for Quadrotors

This paper uses the adaptive dynamic programming (ADP) method to achieve optimal trajectory tracking control for quadrotors. Relying on an established mathematical model of a quadrotor, the approximate optimal trajectory tracking control, which consists of the steady-state control input and the approximate optimal feedback control input, is designed for a nominal system. Considering the compound disturbances in position and attitude dynamic models, disturbance observers are introduced. The estimated values are used to design robust compensation inputs to suppress the effect of the compound disturbances for good trajectory tracking performance. Theoretically, the Lyapunov theorem demonstrates the stability of a closed-loop system. The robustness and effectiveness of the proposed controller are confirmed by the simulation results.

Adaptive Speed Planning of Connected and Automated Vehicles Using Multi-Light Trained Deep Reinforcement Learning

Through shared real-time traffic information and perception of complex environments, connected and automated vehicles (CAVs) are endowed with global decision-making capabilities far beyond that of human drivers. Given more traffic lights information, this planning ability can be greatly strengthened. This study proposed an adaptive speed planning method of CAVs based on multi-light trained deep reinforcement learning (DRL), aiming to improve the fuel economy and comfort of CAVs. By setting a reasonable reward function, the training algorithm takes the key environmental information received by the vehicle as inputs, and finally outputs the acceleration that maximizes the cumulative reward. The results show that the trained DRL agent can adapt to variable scenarios with traffic lights, and quickly solve the approximate optimal speed trajectory. Multi-light DRL models save 6.79% fuel compared with single-light ones and have better performance on fuel economy and computational efficiency than a non-RL method using multi-light optimization.

Shape-based approach for solar sail trajectory optimization

The analysis of the optimal control law that steers a solar sail-based spacecraft from a given initial condition toward a final target state is typically carried out using either indirect or direct approaches. Both these methods are usually time-consuming and require good initial guesses of costates or state vector. This paper presents a procedure requiring minimum user-computer interaction to compute an approximate three-dimensional optimal trajectory using a shape-based approach. To that end, novel shaping functions are introduced to describe the time evolution of the spacecraft state vector. The optimization problem is solved using a genetic algorithm, in which a set of shape coefficients and the initial and final spacecraft position are computed while enforcing suitable constraints on the magnitude and direction of the propulsive acceleration vector. Numerical simulations of transfers from Earth to potentially hazardous asteroids show that this method provides good estimates of solar sail trajectories, which can be used as guesses for more refined direct optimization approaches.

Optimal solar sail trajectory approximation with finite Fourier series

The heliocentric transfer of a solar sail-based spacecraft is usually studied from an optimal perspective, by looking for the control law that minimizes the total flight time. The optimal control problem can be solved either with an indirect approach, whose solution is difficult to obtain due to its sensitivity to an initial guess of the costates, or with a direct method, which requires a good estimate of a feasible (guess) trajectory. This work presents a procedure to generate an approximate optimal trajectory through a finite Fourier series. The minimum time problem is solved using a nonlinear programming solver, in which the optimization parameters are the coefficients of the Fourier series and the positions of the spacecraft along the initial and target orbits. Suitable constraints are enforced on the direction and magnitude of the sail propulsive acceleration vector in order to obtain feasible solutions. A comparison with the numerical results from an indirect approach shows that the proposed method provides a good approximation of the optimal trajectory with a small computational effort.

A WAVELET COLLOCATION SCHEME FOR SOLVING SOME OPTIMAL PATH PLANNING PROBLEMS

We consider an approximation scheme using Haar wavelets for solving optimal path planning problems. The problem is first expressed as an optimal control problem. A computational method based on Haar wavelets in the time domain is then proposed for solving the obtained optimal control problem. A Haar wavelets integral operational matrix and a direct collocation method are used to find an approximate optimal trajectory of the original problem. Numerical results are also presented for several examples to demonstrate the applicability and efficiency of the proposed method.

Approximate optimal trajectory tracking for continuous-time nonlinear systems

Adaptive dynamic programming has been investigated and used as a method to approximately solve optimal regulation problems. However, the extension of this technique to optimal tracking problems for continuous-time nonlinear systems has remained a non-trivial open problem. The control development in this paper guarantees ultimately bounded tracking of a desired trajectory, while also ensuring that the enacted controller approximates the optimal controller.

Dynamic programming approach to the numerical solution of optimal control with paradigm by a mathematical model for drug therapies of HIV/AIDS

In this paper, we present a new numerical algorithm to find the optimal control for the general nonlinear lumped systems without state constraints. The dynamic programming-viscosity solution (DPVS) approach is developed and the numerical solutions of both approximate optimal control and trajectory are produced. To show the effectiveness and efficiency of new algorithm, we apply it to an optimal control problem of two types of drug therapies for human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS). The quality of the obtained optimal control and the trajectory pair is checked through comparison with the costs under the arbitrarily selected different controls. The results illustrate the effectiveness of the algorithm.

Analytical Multi-Point Trajectory Generation for Differentially Flat Systems with Output Constraints

AbstractIn this paper, an analytical off-line multi-point trajectory generation scheme is presented for differentially flat systems. For control of dynamical systems along a given set of control points, multi-point trajectory generation is required when input and state constraints exist. It is assumed that differential constraints for flat coordinates can be formulated explicitly. The trajectory scheme is based on analytically solving a set of polynomial equations to parameterize n-times continuously differentiable segmented transition polynomials, that approximate time optimal trajectories. The computational effort for determining valid trajectories is low in comparison to numerical optimization. As an example, a multi-point trajectory generation problem for a 3-DOF gantry crane is presented.

Automatic pipe-routing particle swarm optimization algorithm in electromechamical products

Based on particle swarm optimization (PSO) algorithm, a new automatic pipe-routing algorithm is proposed for electromechanical products in 3D space. The algorithm consists of pre-processing phase and searching optimization phase. The pre-processing phase is aimed at making implement the searching optimization phase easily by reducing the searching range efficiently. By the chaos technique, chaos grid pre-processing model (CGPM) is constructed in the pre-processing phase. The main objective of the searching optimization phase is to find an approximate optimal trajectory in the available solution space. In the searching optimization phase, a modified PSO algorithm is designed to search for an optimal pipe trajectory by analyzing PSO algorithm and migration characters of people. Modified PSO gives evolution strategies for overcoming premature convergence of PSO. By CGPM and modified PSO, the automatic pipe-routing algorithm flow chart is put forward. By contrastive experiment and theory analysis, it shows that modified POS algorithm effectively increases diversity of particles and has well performance. The convergence of the modified PSO algorithm is given. The example illustrates effectiveness and practicability of the automatic pipe-routing algorithm.

Real-Time Dynamic Optimization of Controllable Linear Systems

A real-time optimization controller is developed to steer a linear time-varying control system to closed-loop trajectories that optimize a cost functional of interest. When advantage is taken of the differential flatness of the linear systems, a basis function approach is used to parameterize the open-loop trajectories of the system and to approximate the optimal solution of the finite time optimal control problem. An adaptive optimization method is used to formulate the real-time optimization scheme. The problem is posed as a real-time optimal trajectory generation problem in which the approximate optimal trajectories are computed using an extremum-seeking approach. The control algorithm provides tracking of the approximate optimal trajectories. Two optimal control problems are considered to demonstrate the application of the technique. It is shown that the technique can be successfully implemented in real time.

A computational method for optimal control problem of time-varying state-delayed systems by Haar wavelets

We present a computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system. The method is simple and computationally advantageous. The approximate optimal trajectory and optimal control are calculated using the Haar wavelet integral, product, and delay operational matrices. An illustrative example is included to demonstrate the validity and applicability of the technique.

Iterative dynamic programming for solving linear and nonlinear differential equations

In this paper we solve a wide range of nth order ordinary nonlinear differential equations with initial conditions by using iterative dynamic programming. First we transform the problem to a first order system of ordinary nonlinear differential equations, and then transform it to an optimal control problem. By the optimal solution of the problem we obtain a piecewise-constant optimal control function and then by using the approximate control signals, we obtain the approximate optimal trajectory. Finally, we calculate the error functional E.

Solving some optimal path planning problems using an approach based on measure theory

In this paper we solve a collection of optimal path planning problems using a method based on measure theory. First we consider the problem as an optimization problem and then we convert it to an optimal control problem by defining some artificial control functions. Then we perform a metamorphosis in the space of problem. In fact we define an injection between the set of admissible pairs, containing the control vector function and a collision-free path defined on free space and the space of positive Radon measures. By properties of this kind of measures we obtain a linear programming problem that its solution gives rise to constructing approximate optimal trajectory of the original problem. Some numerical examples are proposed.

Dynamic boundary evaluation method for approximate optimal lunar trajectories

Introduction R ESEARCH into low-thrust Earth-moon trajectories has increased recently. The time and effort required to obtain an optimal low-thrust lunar trajectory depends greatly on the initial estimate of the trajectory used in the optimizing process. Pierson and Kluever present an effective method for finding an initial estimate for a class of Earth-moon trajectories. However, their method requires the setup of tabular data before the initial estimate can be found. This data table must be recomputed for each new initial thrust-to-weight ratio, radius of the initial low Earth orbit (LEO), or radius of the final low lunar orbit (LLO). This Note describes a dynamic boundary evaluation (DBE) method that replaces the data table construction and subsequent interpolation with an on-line iterative numerical process. The optimal low-thrust lunar trajectory problem treated here involves a planar transfer from a circular LEO to a circular LLO. A specified thrust-coast-thrust sequence is assumed using constant thrust. The spacecraft thrusts away from the LEO, in an increasing spiral, until enough energy is gained to make the lunar rendezvous. The spacecraft then coasts to the vicinity of the lunar sphere of influence, where the spacecraft thrusts, in a decreasing spiral, to the desired LLO. The minimum-fuel solution is the one that accomplishes this transfer with the minimum time spent in the two thrusting periods, because the assumed constant thrust implies a constant mass flow. The Pierson-Kluever method uses maximum-energy spirals, described later, to approximate both the Earth-departure and the lunar-capture thrusting arcs. This composite trajectory, consisting of the maximum-energy spirals plus an optimal translunar coast, is then used to estimate the fully minimum-fuel transfer from a circular LEO to a circular LLO. This estimate is very accurate in terms of performance; i.e., the fuel spent is very close to that for the fully optimal case, and the control time histories for the thrust-steering angle are adequate to ensure convergence to a solution of the full problem.

Optimal inlet temperature trajectories for adiabatic packed reactors with catalyst decay

Abstract An algorithm for simulation and optimization of adiabatic fixed-bed reactors subject to catalyst deactivation is developed. It is applied to the low-temperature CO converter, the kinetic model for which is deduced from some industrial data and considering chlorine as the main cause of catalyst poisoning. Sintering is not considered due to the short range of variation of temperature in the catalyst bed. Optimal inlet temperature trajectories are determined to maximize the obtained total amount of reaction over a fixed operation time. Approximate optimal trajectories in discrete steps of temperature are proposed to be followed in industrial practice.

Optimal control with a functional averaged along the trajectory

A set of infinite optimal trajectories (IOT) is defined. It is shown that in an arbitrary fixed time interval any optimal trajectory of a system for a problem with fairly large control time (and arbitrary initial conditions) can be uniformly approximated to some IOT with the desired accuracy. Sufficient conditions are presented which ensure the existence of IOT, and the structure of the IOT set is investigated, using the rearrangement operator. The set of main trajectories is defined, and the correctness of that definition is proved. A chain of approximations is obtained: IOT approximate optimal trajectories of finite length, and the main trajectories approximate the IOT.