Trajectory Optimization Approach Research Articles

A New Heuristic Approach for Low-Thrust Spacecraft Trajectory Optimization

Electrical propulsion is gaining popularity in the satellite industry. Their high efficiency however comes at the cost of decreased thrust and longer transfer times. As such the interest in solving the large optimization problems associated with these long transfers has re-surged. This paper presents a heuristic approach to solving such low-thrust satellite trajectory optimization problems, based on control law blending and averaging. We derive the nonlinear program resulting from this approach and introduce PANOC, a recent optimization algorithm well suited for optimal control problems. The resulting controller is considerably faster than more classical, non-heuristic approaches, without a significant loss in optimality.

A New Midcourse Guidance Design for Near Space Target Interceptor

This study proposes a new midcourse guidance method, combining the trajectory optimization approach and the optimal sliding mode method, to decrease the larger relative velocity between the near space target and the interceptor. Firstly, the midcourse trajectory optimization model is established. Then, the trajectory optimization guidance law based on a sampling scheme is proposed to decrease the larger relative velocity and satisfy multiple constraints at the end time of the midcourse guidance phase. Thirdly, the optimal sliding mode guidance law is introduced to deal with the target maneuverability. Finally, numerical simulations were performed to verify the performance of the midcourse guidance law.

Robust Trajectory Design for Rendezvous and Proximity Operations with Uncertainties

A robust trajectory optimization approach for rendezvous and proximity operations in perturbed elliptical orbits with uncertainties is proposed. For robust trajectory design, a new discrete-time linear dynamics model describing the relative equations of motion for powered flight in perturbed elliptical orbits is developed. The linear dynamics model is used to formulate a stochastic trajectory optimization problem that takes into account navigation sensor errors, maneuver execution errors, and trajectory and control dispersion. The objective of the stochastic optimization problem is to minimize the sum of the expected maneuver and the maneuver dispersion subject to a constraint on the final state dispersion. A genetic algorithm is used to solve the stochastic trajectory optimization problem, and a series of examples demonstrate that the stochastic maneuver solution can be significantly better than the solution corresponding deterministic trajectory optimization problem. Nonlinear Monte Carlo simulations are used to validate the results.

Planning Approximate Exploration Trajectories for Model-Free Reinforcement Learning in Contact-Rich Manipulation

Recent progress in deep reinforcement learning has enabled simulated agents to learn complex behavior policies from scratch, but their data complexity often prohibits real-world applications. The learning process can be sped up by expert demonstrations but those can be costly to acquire. We demonstrate that it is possible to employ model-free deep reinforcement learning combined with planning to quickly generate informative data for a manipulation task. In particular, we use an approximate trajectory optimization approach for global exploration based on an upper confidence bound of the advantage function. The advantage is approximated by a network for Q-learning with separately updated streams for state value and advantage that allows ensembles to approximate model uncertainty for one stream only. We evaluate our method on new extensions to the classical peg-in-hole task, one of which is only solvable by active usage of contacts between peg tips and holes. The experimental evaluation suggests that our method explores more relevant areas of the environment and finds exemplar solutions faster-both on a real robot and in simulation. Combining our exploration with learning from demonstration outperforms state-of-the-art model-free reinforcement learning in terms of convergence speed for contact-rich manipulation tasks.

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.

Viability of an Electrically Driven Pump-Fed Hybrid Rocket for Small Launcher Upper Stages

An electrically driven pump-fed cycle for a hybrid rocket engine is proposed and compared to a simpler gas-pressurized feed system. A liquid-oxygen/paraffin-based fuel hybrid rocket engine which powers the third stage of a Vega-like launcher is considered. Third-stage ignition conditions are assigned, and engine design and payload mass are defined by a proper set of parameters. Uncertainties in the classical regression rate correlation coefficients are taken into account and robust design optimization is carried out with an approach based on an epsilon-constrained evolutionary algorithm. A mission-specific objective function, which takes into account both the payload mass and the ability of the rocket to reach the required final orbit despite uncertainties, is determined by an indirect trajectory optimization approach. The target orbit is a 700 km altitude polar orbit. Results show that electrically driven pump-fed cycle is a viable option for the replacement of the conventional gas-pressurized feed system. Robustness in the design is granted and a remarkable payload gain is achieved, using both present and advanced technologies for electrical systems.

Trajectory planning method based on transitional segment optimization of spray painting robot on complex-free surface

PurposeThis paper aims to present a new trajectory optimization approach targeting spray painting applications that satisfies the paint thickness requirements of complex-free surfaces.Design/methodology/approachIn this paper, a new trajectory generation approach is developed to optimize the transitional segments at the junction of adjacent patches for straight line, convex arc and concave arc combinations based on different angles between normal vectors of patches. In addition, the paint parameters including the paint gun velocity, spray height and the distance between adjacent trajectories have been determined in the generation approach. Then a thickness distribution model is established to simulate the effectiveness of trajectory planning.FindingsThe developed approach was applied to a complex-free surface of various curvatures, and the analysis results of the trajectory optimization show that adopting different transitional segment according to the angle between normal vectors can obtain the optimal trajectory. Based on the simulation and experimental validation results, the proposed approach is effective at improving paint thickness uniformity, and the obtained results are consistent with the simulation results, meaning that the simulation model can be used to predict the actual paint performance.Originality/valueThis paper discusses a new trajectory generation approach to decrease the thickness error values to satisfy spray paint requirements. According to the successfully performed simulation and experimental results, the approach is useful and practical in overcoming the challenge of improving the paint thickness quality on complex-free surface.

Dynamic analysis with optimum trajectory planning of multiple degree-of-freedom surgical micro-robot

The present work aims to analyze kinematics and dynamics accompanied with an optimum trajectory planning, of a multiple degree-of-freedom positioning surgical micro-robot. The kinematic model was developed using Denavit–Hartenberg algorithm, while dynamic model was developed using Lagrange technique. The trajectory optimization was implemented using different local, global and hybrid optimization techniques. For local optimization, “Fmincon” was employed. Genetic Algorithm (GA), Pattern Search (PS), and Particle Swarm (PSO) were utilized as global optimization techniques. In hybrid optimization approach, GA was used for global optimization while PS was utilized as a local optimization technique for further results refinement. Polynomials of sixth order and fourth order were assumed during all trajectory optimization approaches. The objective function was assumed to minimize the total consumed energy by the positioning manipulator. Non-optimized analysis was investigated as well, using several conventional trajectory planning techniques including fifth order polynomial, third order polynomial, cycloidal formula and elliptical formula. The angle of rotation for all joints ranged from zero to one and half radian, and the motion duration was five seconds for all the presented results. MATLAB codes were created for simulation and optimization processes.

A robust homotopic approach for continuous variable low-thrust trajectory optimization

This paper presents an improved understanding of the interaction of hybrid optimization method with variable low-thrust trajectory optimization requirements. To analyze fuel-optimal bang-bang control problem, a new version of homotopic algorithm, termed robust homotopic method, is investigated with the prospect of improving the efficiency and automation of the homotopic approach to achieve a high-level of robustness, and consequently enlarge its range of application. Such desired characteristics are promoted via a combination of several techniques. As an effective approach, a modified methodology of the switching detection process is presented for the bang-bang optimal-control problem. Moreover, the value of unknown costates and switching functions are mapped to new normalized intervals throughout the computational process. As a result, the optimal solution is rapidly designed to obtain the global robust-convergence to satisfy all constraints without any ambiguity. The fitting process of all iterations robustly find the unknown variables with the percent of converged solutions to maximum, and the penalty terms are quickly satisfied with predetermined high-accuracy, from the energy-optimal to the fuel-optimal solution, especially close to zero point as a critical point. Accordingly, two advanced interplanetary trajectories are optimized using two dynamic modeling approaches for the instantaneous and constant maximal thrust magnitude as a way to analyze and substantiate the robustness of the proposed algorithm. Results and performances are compared with existing solutions of the same mission problem.

Optimization of Vehicle Mounting Motions and Its Application to Full-Sized Humanoid, DRC-Hubo

This paper describes optimization of humanoid’s whole body motion for vehicle mounting task. To accomplish the goal, a trajectory optimization framework based on the reinforcement learning agent is used in this study. The guideline input trajectory is planned and optimized as regards various dynamic and kinematic limitations of humanoid in the framework. In previous studies, the authors demonstrated test-and-evaluation process of the framework using a full-sized humanoid, Hubo+. Experimental testing however presented several problems like overheating and self-collisions. To resolve those issues and to validate the trajectory optimization approach, another humanoid, called DRC-Hubo, is newly designed. Keeping a main structure of the optimization framework, the cost value functions are revised to meet dynamic and kinematic changes. Experimental test and verification process using a simulation model and a physical prototype is demonstrated to confirm the efficacy of the trajectory optimization approach. For both processes, two different types of ground vehicle are used. Last, analytical comparisons with other techniques are also conducted for validation of the proposed framework.

Efficient Reinforcement Learning via Probabilistic Trajectory Optimization.

We present a trajectory optimization approach to reinforcement learning in continuous state and action spaces, called probabilistic differential dynamic programming (PDDP). Our method represents systems dynamics using Gaussian processes (GPs), and performs local dynamic programming iteratively around a nominal trajectory in Gaussian belief spaces. Different from model-based policy search methods, PDDP does not require a policy parameterization and learns a time-varying control policy via successive forward-backward sweeps. A convergence analysis of the iterative scheme is given, showing that our algorithm converges to a stationary point globally under certain conditions. We show that prior model knowledge can be incorporated into the proposed framework to speed up learning, and a generalized optimization criterion based on the predicted cost distribution can be employed to enable risk-sensitive learning. We demonstrate the effectiveness and efficiency of the proposed algorithm using nontrivial tasks. Compared with a state-of-the-art GP-based policy search method, PDDP offers a superior combination of learning speed, data efficiency, and applicability.

A Robot Trajectory Optimization Approach for Thermal Barrier Coatings Used for Free-Form Components

This paper is concerned with a robot trajectory optimization approach for thermal barrier coatings. As the requirements of high reproducibility of complex workpieces increase, an optimal thermal spraying trajectory should not only guarantee an accurate control of spray parameters defined by users (e.g., scanning speed, spray distance, scanning step, etc.) to achieve coating thickness homogeneity but also help to homogenize the heat transfer distribution on the coating surface. A mesh-based trajectory generation approach is introduced in this work to generate path curves on a free-form component. Then, two types of meander trajectories are generated by performing a different connection method. Additionally, this paper presents a research approach for introducing the heat transfer analysis into the trajectory planning process. Combining heat transfer analysis with trajectory planning overcomes the defects of traditional trajectory planning methods (e.g., local over-heating), which helps form the uniform temperature field by optimizing the time sequence of path curves. The influence of two different robot trajectories on the process of heat transfer is estimated by coupled FEM models which demonstrates the effectiveness of the presented optimization approach.

Trajectory Optimization Through Contacts and Automatic Gait Discovery for Quadrupeds

In this letter, we present a trajectory optimization framework for whole-body motion planning through contacts. We demonstrate how the proposed approach can be applied to automatically discover different gaits and dynamic motions on a quadruped robot. In contrast to most previous methods, we do not prespecify contact-switches, -timings, -points or gait patterns, but they are a direct outcome of the optimization. Furthermore, we optimize over the entire dynamics of the robot, which enables the optimizer to fully leverage the capabilities of the robot. To illustrate the spectrum of achievable motions, we show eight different tasks, which would require very different control structures when solved with state-of-the-art methods. Using our trajectory optimization approach, we are solving each task with a simple, high level cost function and without any changes in the control structure. Furthermore, we fully integrate our approach with the robot's control and estimation framework such that we are able to run the optimization online. Through several hardware experiments, we show that the optimized trajectories and control inputs can be directly applied to physical systems.

Terminal Control Area Aircraft Scheduling and Trajectory Optimization Approaches

Aviation authorities are seeking optimization methods to better use the available infrastructure and better manage aircraft movements. This paper deals with the realtime scheduling of take-off and landing aircraft at a busy terminal control area and with the optimization of aircraft trajectories during the landing procedures. The first problem aims to reduce the propagation of delays, while the second problem aims to either minimize the travel time or reduce the fuel consumption. Both problems are particularly complex, since the first one is NP-hard while the second one is nonlinear and a combined solution needs to be computed in a short-time during operations. This paper proposes a framework for the lexicographic optimization of the two problems. Computational experiments are performed for the Milano Malpensa airport and show the existing gaps between the performance indicators of the two problems when different lexicographic optimization approaches are considered.

Multi-objective reentry trajectory optimization method via GVD for hypersonic vehicles

In the constrained reentry trajectory design of hypersonic vehicles, multiple objectives with priorities bring about more difficulties to find the optimal solution. Therefore, a multi-objective reentry trajectory optimization (MORTO) approach via generalized varying domain (GVD) is proposed. Using the direct collocation approach, the trajectory optimization problem involving multiple objectives is discretized into a nonlinear multi-objective programming with priorities. In terms of fuzzy sets, the objectives are fuzzified into three types of fuzzy goals, and their constant tolerances are substituted by the varying domains. According to the principle that the objective with higher priority has higher satisfactory degree, the priority requirement is modeled as the order constraints of the varying domains. The corresponding two-side, single-side, and hybrid-side varying domain models are formulated for three fuzzy relations respectively. By regulating the parameter, the optimal reentry trajectory satisfying priorities can be achieved. Moreover, the performance about the parameter is analyzed, and the algorithm to find its specific value for maximum priority difference is proposed. The simulations demonstrate the effectiveness of the proposed method for hypersonic vehicles, and the comparisons with the traditional methods and sensitivity analysis are presented.

A concurrent optimization approach for energy efficient multiple axis positioning tasks

Automated production systems typically comprise numerous electrical servo drives, many of which conduct positioning motions, e.g. for handling or manipulation tasks. The power electronics of modern multi-axis systems often comprise coupled DC-links, enabling for internal exchange of recuperative brake energy. However, the motion sequences of manipulators are often commanded at maximum dynamics for minimum time motion, neglecting possible optimization potential, e.g. available idle time, leading to inefficient energy management. A robust trajectory optimization approach based on the particle swarm algorithm and well-established path planning methods is presented for the adaption of multi-axis positioning tasks with only two parameters per axis and positioning motion during system run-time. Experimental results prove that, depending on the positioning task and chosen optimization constraints, energy demands are distinctly reduced. The approach is applicable to diverse multi-axis configurations and enables for considerable energy savings without additional hardware invest.

Parsimonious shooting heuristic for trajectory design of connected automated traffic part II: Computational issues and optimization

Advanced connected and automated vehicle technologies enable us to modify driving behavior and control vehicle trajectories, which have been greatly constrained by human limits in existing manually-driven highway traffic. In order to maximize benefits from these technologies on highway traffic management, vehicle trajectories need to be not only controlled at the individual level but also coordinated collectively for a stream of traffic. As one of the pioneering attempts to highway traffic trajectory control, Part I of this study (Zhou et al., 2016) proposed a parsimonious shooting heuristic (SH) algorithm for constructing feasible trajectories for a stream of vehicles considering realistic constraints including vehicle kinematic limits, traffic arrival patterns, car-following safety, and signal operations. Based on the algorithmic and theoretical developments in the preceding paper, this paper proposes a holistic optimization framework for identifying a stream of vehicle trajectories that yield the optimum traffic performance measures on mobility, environment and safety. The computational complexity and mobility optimality of SH is theoretically analyzed, and verifies superior computational performance and high solution quality of SH. A numerical sub-gradient-based algorithm with SH as a subroutine (NG-SH) is proposed to simultaneously optimize travel time, a surrogate safety measure, and fuel consumption for a stream of vehicles on a signalized highway section. Numerical examples are conducted to illustrate computational and theoretical findings. They show that vehicle trajectories generated from NG-SH significantly outperform the benchmark case with all human drivers at all measures for all experimental scenarios. This study reveals a great potential of transformative trajectory optimization approaches in transportation engineering applications. It lays a solid foundation for developing holistic cooperative control strategies on a general transportation network with emerging technologies.

Single- and Multi-UAV Trajectory Control in RF Source Localization

In this paper, two trajectory control approaches are presented for any number of unmanned aerial vehicles (UAVs) in radio frequency (RF) source localization. The UAVs observe the received signal strength (RSS) in distinctive time intervals to localize a stationary RF source. The location of the source is estimated recursively applying the extended Kalman filter. The objective of the optimal trajectory control is to steer the UAVs to the locations which minimize the uncertainty about the target state. The Fisher information matrix (FIM) is inversely proportional to the estimation variance. Since the true target state is unknown, the FIM is approximated by the estimated target state. Two criteria based on the approximated FIM are applied to measure the information content of the observations to optimize the UAV waypoints: The D-optimality and the A-optimality. The objective of the present paper is to propose two trajectory control approaches for any number of UAVs in RSS-based localization to increase the target localization accuracy. The superiority of the trajectory optimization approach based on the D-optimality in terms of mean squared error is illustrated through simulation examples.

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

ABSTRACT(Chemical) kinetic models in terms of ordinary differential equations correspond to finite-dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the so called reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion and exploitation for deeper insight of unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We newly relate both of them within our variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as novel viewpoints on dimension reduction approaches that might open up new deep perspectives and approaches to the problem from dynamical systems theory.

Trajectory Optimization of a Launch Vehicle with Explicit Instantaneous Impact Point Constraints for Various Range Safety Requirements

AbstractThis paper proposes a new trajectory optimization approach that can directly handle the range safety for launch operations. An analytic solution for an instantaneous impact point (IIP) of a vehicle is introduced and used to describe the constraints of the optimization problem in an explicit way. A quadratic formula that can address various types of range safety requirements is proposed. Based on the aforementioned elements, the launch trajectory optimization problem with explicit IIP constraints is mathematically defined. The validity of the proposed trajectory optimization is demonstrated through two case studies using realistic launch vehicle data and range safety issues.