Direct Trajectory Optimization Research Articles

Costate estimation for a multiple-interval pseudospectral method using collocation at the flipped legendre-Gauss-Radau points

A multiple-interval pseudospectral method is presented for direct trajectory optimization and costate estimation of optimal control problems using collocation at the flipped Legendre-Gauss-Radau points. The recently developed global flipped Radau pseudospectral method is one of the special case. The necessary optimality conditions of the multiple-interval differential scaled optimal control problem are rigorously derived for the first time by applying the calculus of variations. Then, the costate and constraint multiplier estimates for the proposed method are derived in detail by comparing the discretized optimality conditions with the Karush-Kuhn-Tucker conditions of the nonlinear programming problem resulting from collocation. Furthermore, in order to improve the numerical stability and alleviate the computational error when calculating the state pseudospectral differentiation matrix, the barycentric Lagrange interpolation is adopted in the state approximation instead of the classic Lagrange interpolation without losing the computational efficiency. The method presented in this paper is applied to two benchmark optimal control problems and is compared with the de-facto standard pseudospectral methods. Extensive numerical results indicate that it has the ability to obtain highly accurate solutions to a wide variety of complex constrained optimal control problems.

Perturbed‐analytic direct transcription for optimal control (PADOC)

AbstractThe herein coined PADOC (perturbed‐analytic direct transcription for optimal control) stands for a new transcription method for direct trajectory optimization. We construct PADOC on a novel segmented decomposition method that provides series solutions for nonlinear problems. To transcribe the infinite dimensional problem into a finite one, PADOC comes along with a new solution approach, namely, the herein coined average nonlinear programming (aNLP). The aNLP theorem suggests generating a staircase optimal solution (low resolution) and turning it into a distributed optimal solution (high resolution) by exploiting the Hamiltonian of the problem. The analytic architecture provides PADOC with an analytic connection of a truncated order between the discrete nodes of the solution. This renders stability, accuracy within an analytic resolution, robustness, and a cut‐down in the number of the decision variables in the frame of NLP. We also prove that the multipliers associated with the Karush–Kuhn–Tucker optimality conditions in the frame of NLP (as transcribed by PADOC) correspond to a backward analytic solution for the costate equations. We finally show distinct features of PADOC through some examples.

Environment Warped Gait Trajectory Optimization for Complex Terrains

Contact-aware gait trajectory optimization is a challenging non-convex programming problem, especially for complex terrain shapes, where prominent numerical algorithms can fail to find a solution or fall into local minima. To alleviate this issue, we propose an environment warping technique that changes the coordinates for decision variables. Given a terrain of some general shape, our method first generates a locally injective, as-conformal-as-possible function that maps the ambient space around the terrain to a warped space. We then formulate the trajectory optimization in the warped space by remapping all the decision variables. Our method frees the numerical optimizer from tuning the trajectories to fit changing terrain shapes, leading to better numerical conditioning and fewer local minima. Numerical results show that our method outperforms direct trajectory optimization in terms of both success rates and quality of solutions.

Optimizing the Beam Selection for Noncoplanar VMAT by Using Simulated Annealing Approach

Noncoplanar volumetric modulated arc therapy (VMAT) treatment can achieve better organ-at-risk (OAR) avoidance by orienting the radiation beams in a different geometric plane relative to the patient. However, determining the optimal set of beam orientations is challenging due to the additional degrees of freedom. The objective of this study was to use simulated annealing (SA) for beam selection in a noncoplanar VMAT optimization context. The SA method was combined with a direct leaf trajectory optimization approach to obtain a set of globally optimal beams which serve as control points for the treatment trajectory. The proposed method was evaluated through the TG119 benchmark and two clinical cases (prostate and liver cancers). Finally, the SA beam selection method was compared to the standard coplanar and noncoplanar beam selection approaches. The results showed an accurate delivery of the prescription dose to the target tumor volume in all cases. Generally, not on every organ, the noncoplanar SA method showed better OAR sparing compared to the coplanar and noncoplanar greedy method. This work demonstrates that optimized noncoplanar beam orientations using the proposed SA method can be more clinically interesting than the coplanar method in some specific patient cases.

Direct Trajectory Optimization of Free-Floating Space Manipulator for Reducing Spacecraft Variation

This letter investigates the direct trajectory optimization of the free-floating space manipulator (FFSM). The main purpose is to plan the joint space trajectories to reduce the spacecraft motion due to the joint rotation during the FFSM performing tasks. To improve the calculation efficiency, the adaptive Radau pseudospectral method (A-RPM) is applied to discretize the system dynamics and transform the formulated optimal problem into a nonlinear programming problem (NLP). By adaptively subdividing the current segment and assigning collocation points according to the solution error, high-degree interpolation polynomials are avoided. To verify the effectiveness of the proposed method, a ground micro-gravity platform of the FFSM system is designed by using the air-bearing technique, on which experiments are carried out. The results show that the variation of the base spacecraft is dramatically reduced if the joints rotate along the optimized trajectories.

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function (RBF) interpolation with Galerkin projection to efficiently solve general optimal control problems. The goal is to develop a highly flexible solution to optimal control problems, especially nonsmooth problems involving discontinuities, while accounting for trajectory accuracy and computational efficiency simultaneously. The proposed solution, called the RBF-Galerkin method, offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points. The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker (KKT) conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem, if a set of discrete conditions holds. The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem. In addition, the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

Analysis of driving styles of a GP2 car via minimum lap-time direct trajectory optimization

This paper addresses the problem of the link between the driving style of an ideal driver, modelled as an optimal controller, and fundamental set-up parameters of a vehicle in the GP2 motorsport class. The aim is to evaluate quantitatively how set-up parameters, like distribution of aerodynamic loads, weight and roll stiffness between front and rear axles, affect the driving style, encoded in the shape of the optimal trajectory and in the acceleration, brake and steer inputs. To this aim, we develop an optimization code that includes a double-track vehicle model capable of solving the minimum lap-time problem (MLTP) on a given track. The track is represented via NURBS curves and the MLTP is framed and solved as an optimal control problem by transcription into a nonlinear program using direct collocation. To assess the accuracy of the vehicle model and the optimization pipeline, we also validate our results against real telemetry data. The developed software framework lends itself to easily perform both sensitivity analysis and concurrent trajectory planning and set-up parameter optimization: this is obtained by simple promotion of static parameters of interests to variables in the optimal control problem. Some results along these lines are also included.

OPTIMAL CONTROL OF NONLINEAR TIME-DELAY FRACTIONAL DIFFERENTIAL EQUATIONS WITH DICKSON POLYNOMIALS

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations. Finally, the unknown coefficients and control parameters in the original problem can be easily estimated by resolving the new system of equations. Given the high efficiency of the Dickson polynomials to deal with time-delay fractional systems, the proposed strategy involves a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The convergence analysis of the proposed method is presented, along with some illustrative examples which demonstrate its most relevant features.

Direct Policy Optimization Using Deterministic Sampling and Collocation

We present an approach for approximately solving discrete-time stochastic optimal-control problems by combining direct trajectory optimization, deterministic sampling, and policy optimization. Our feedback motion-planning algorithm uses a quasi-Newton method to simultaneously optimize a reference trajectory, a set of deterministically chosen sample trajectories, and a parameterized policy. We demonstrate that this approach exactly recovers LQR policies in the case of linear dynamics, quadratic objective, and Gaussian disturbances. We also demonstrate the algorithm on several nonlinear, underactuated robotic systems to highlight its performance and ability to handle control limits, safely avoid obstacles, and generate robust plans in the presence of unmodeled dynamics.

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm’s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

Contact-Implicit Trajectory Optimization Using an Analytically Solvable Contact Model for Locomotion on Variable Ground

This paper presents a novel contact-implicit trajectory optimization method using an analytically solvable contact model to enable planning of interactions with hard, soft, and slippery environments. Specifically, we propose a novel contact model that can be computed in closed-form, satisfies friction cone constraints and can be embedded into direct trajectory optimization frameworks without complementarity constraints. The closed-form solution decouples the computation of the contact forces from other actuation forces and this property is used to formulate a minimal direct optimization problem expressed with configuration variables only. Our simulation study demonstrates the advantages over the rigid contact model and a trajectory optimization approach based on complementarity constraints. The proposed model enables physics-based optimization for a wide range of interactions with hard, slippery, and soft grounds in a unified manner expressed by two parameters only. By computing trotting and jumping motions for a quadruped robot, the proposed optimization demonstrates the versatility for multi-contact motion planning on surfaces with different physical properties.

Q-Law Aided Direct Trajectory Optimization of Many-Revolution Low-Thrust Transfers

Low-thrust spacecraft trajectory optimization for the many-revolution orbital transfer problem is especially challenging due to the high problem dimension and perturbing accelerations that prevent or complicate accurate analytical solutions. This analysis seeks to simplify the calculation of the nonaveraged spiral trajectory through the use of the well-known Q-Law guidance algorithm. Here, Q-Law is used to seed phases of the direct trajectory optimization problem. We explore the trade space between the fast calculation speed of Q-Law and the time or mass optimality of the full optimization problem using the common example of a geostationary transfer orbit to geostationary transfer. We find that a computationally efficient, near-optimal solution can be achieved by using Q-Law for an initial fraction of a spiral trajectory and direct optimization for the remainder of the trajectory. This reduces the direct optimization problem to a lower dimension and enables the spacecraft to reach a specific final state. Q-Law is particularly effective for portions of the trajectory where there are many eclipses, which can challenge direct optimization methods.

Beam flatness modulation for a flattening filter free photon beam utilizing a novel direct leaf trajectory optimization model.

Flattening filter free (FFF) linear accelerators produce a fluence distribution that is forward peaked. Various dosimetric benefits, such as increased dose rate, reduced leakage and out of field dose has led to the growth of FFF technology in the clinic. The literature has suggested the idea of vendors offering dedicated FFF units where the flattening filter (FF) is removed completely and manipulating the beam to deliver conventional flat radiotherapy treatments. This work aims to develop an effective way to deliver modulated flat beam treatments, rather than utilizing a physical FF. This novel optimization model is an extension of the direct leaf trajectory optimization (DLTO) previously developed for volumetric modulated radiation therapy (VMAT) and is capable of accounting for all machine and multileaf collimator (MLC) dynamic delivery constraints, using a combination of linear constraints and a convex objective function. Furthermore, the tongue and groove (T&G) effect was also incorporated directly into our model without introducing nonlinearity to the constraints, nor nonconvexity to the objective function. The overall beam flatness, machine deliverability, and treatment time efficiency were assessed. Regular square fields, including field sizes of 10 × 10 cm2 to 40 × 40 cm2 were analyzed, as well as three clinical fields, and three arbitrary contours with "concave" features. Quantitative flatness was measured for all modulated FFF fields, and the results were comparable or better than their open FF counterparts, with the majority having a quantitative flatness of less than 3.0%. The modulated FFF beams, due to the included efficiency constraint, were able to achieve acceptable delivery time compared to their open FF counterpart. The results indicated that the dose uniformity and flatness for the modulated FFF beams optimized with the DLTO model can successfully match the uniformity and flatness of their conventional FF counterparts, and may even provide further benefit by taking advantage of the unique FFF beam characteristics.

King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems

Abstract In this study, a new optimization algorithm, called King, is introduced for solving variable order fractional optimal control problems (VO-FOCPs). The variable order fractional derivative is portrayed in the Caputo sense through the dynamics of the system as variable order fractional differential equation (VO-FDE). To this end, firstly, the VO-FOCP is converted into a system of VO-FDEs. Then, according to the fact that the VO-FDE is equivalent to a Volterra integral equation, the system of VO-FDEs is transformed into an equivalent system of variable order fractional integro-differential equations. In the next step, both the context of minimization of total error and a joint application of Banach’s fixed-point theorem are used to solve a nonlinear optimization problem. Actually, using the existing mechanism, the synchronization problem is recast to an optimization problem. Due to the discretization and its board range of arbitrary nodes, the proposed method provides a very adjustable framework for direct trajectory optimization. Finally, the proposed algorithm is verified using several common optimization functions and a chaotic financial system. Also, through simulation results, the proposed method is compared with some popular methods. Simulation results demonstrate the appropriate performance of the introduced method.

Automated Longitudinal Control Based on Nonlinear Recursive B-Spline Approximation for Battery Electric Vehicles

This works presents a driver assistance system for energy-efficient ALC of a BEV. The ALC calculates a temporal velocity trajectory from map data. The trajectory is represented by a cubic B-spline function and results from an optimization problem with respect to travel time, driving comfort and energy consumption. For the energetic optimization we propose an adaptive model of the required electrical traction power. The simple power train of a BEV allows the formulation of constraints as soft constraints. This leads to an unconstrained optimization problem that can be solved with iterative filter-based data approximation algorithms. The result is a direct trajectory optimization method of which the effort grows linearly with the trajectory length, as opposed to exponentially as with most other direct methods. We evaluate ALC in real test drives with a BEV. We also investigate the energy-saving potential in driving simulations with ALC compared to MLC. On the chosen reference route the ALC saves up to 3.4% energy compared to MLC at same average velocity, and achieves a 2.6% higher average velocity than MLC at the same energy consumption.

Contact-implicit trajectory optimization using variational integrators

Contact constraints arise naturally in many robot planning problems. In recent years, a variety of contact-implicit trajectory optimization algorithms have been developed that avoid the pitfalls of mode pre-specification by simultaneously optimizing state, input, and contact force trajectories. However, their reliance on first-order integrators leads to a linear tradeoff between optimization problem size and plan accuracy. To address this limitation, we propose a new family of trajectory optimization algorithms that leverage ideas from discrete variational mechanics to derive higher-order generalizations of the classic time-stepping method of Stewart and Trinkle. By using these dynamics formulations as constraints in direct trajectory optimization algorithms, it is possible to perform contact-implicit trajectory optimization with significantly higher accuracy. For concreteness, we derive a second-order method and evaluate it using several simulated rigid-body systems, including an underactuated biped and a quadruped. In addition, we use this second-order method to plan locomotion trajectories for a complex quadrupedal microrobot. The planned trajectories are evaluated on the physical platform and result in a number of performance improvements.

An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization

The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for NWLS approximation are restricted to a bounded definition range. We present an algorithm termed NRBA for an iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a MPF, in which a KF solves the linear subproblem optimally while a PF deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that, regardless of the initially chosen definition range, all data points can be processed. In numerical experiments, NRBA achieves approximation results close to those of the Levenberg–Marquardt algorithm. An NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentially with the trajectory length. We demonstrate how NRBA can be applied for a multiobjective trajectory optimization for a BEV in order to determine an energy-efficient velocity trajectory. With NRBA, the effort increases only linearly with the processed data points and the trajectory length.

Two Direct Low Thrust Trajectory Optimization Techniques

Direct low thrust trajectory optimization techniques that reduce the optimization problem in the class of continuous functions to a nonlinear programming problem are described. Two direct low thrust trajectory optimization methods are implemented; one of them is based on the optimization of thrust acceleration and the other optimizes the impulses approximating the thrust arcs. These methods are compared using a sample problem of transfer from Earth to Mars in terms of runtime, the number of iterations, and the size of the convergence domains. The computations are performed on a personal computer using MATLAB and on a multiprocessor computer.

Direct trajectory optimization framework for vertical takeoff and vertical landing reusable rockets: case study of two-stage rockets

ABSTRACTThis study presents a direct trajectory optimization framework for a vertical takeoff and vertical landing (VTVL) reusable rocket with a two-stage structure for a launch mission to simultaneously deliver a payload to the desired orbit and recover the rocket. A two-phase trajectory optimization problem of VTVL reusable rockets is established, with the main features of the flight process. The finite-element collocation approach with Radau collocation is utilized to transcribe the established problem into a nonlinear programming (NLP) problem solved by Interior Point OPTimizer (IPOPT). Novel internal–external-growth initialization strategies are designed to enhance the convergence of solving the NLP problem. The proposed direct trajectory optimization framework is applied to two representative scenarios of VTVL reusable rockets: return to a launch site and return to a drone ship. Simulation results illustrate that the proposed trajectory optimization framework has the adaptability to deal effectively with the complex missions of VTVL reusable rockets.

Сравнение двух прямых методов оптимизации межпланетных траекторий

Сравнение двух прямых методов оптимизации межпланетных траекторий