Global Trajectory Optimization Research Articles

Multi-fidelity approach for global trajectory optimization using GPU-based highly parallel architecture

A flexible and robust design space search method based on a systematic approach as well as effective optimization algorithms is required to accomplish challenging space missions. This paper presents a global trajectory optimization framework via a multi-fidelity approach that utilizes a graphics processing unit (GPU) for low-fidelity initial solution search and a central processing unit (CPU) to determine high-fidelity feasible solutions compliant with imposed constraints. The proposed framework consists of the following specific processes: (1) identifying a multitude of feasible trajectories with the aid of highly parallelized trajectory propagation using single-precision GPU cores; and then (2) determining accurate trajectories by means of gradient-based optimization incorporating double-precision propagation using CPU cores. The resultant trajectories have been verified in light of the primer vector theory that evaluates local optimality, thereby demonstrating the effectiveness of the proposed framework by robustly identifying optimum trajectories in the vast design space. Surrogate models have been constructed based on the solution database obtained in the course of optimization, enabling accurate prediction of trajectory attributes for given design parameters. Global sensitivity analysis has been performed using surrogate prediction to evaluate the influence of the decision variables on the trajectory parameters, which can be employed in multidisciplinary system design optimization.

Evolutionary Rao algorithm

This paper proposes an evolutionary Rao algorithm (ERA) to enhance three state-of-the-art metaheuristic Rao algorithms (Rao-1, Rao-2, Rao-3) by introducing two new schemes. Firstly, the population is split into two sub-populations based on their qualities: high and low, with a particular portion. The high-quality sub-population searches for an optimum solution in an exploitative manner using a movement scheme used in the Rao-3 algorithm. Meanwhile, the low-quality one does in an explorative fashion using a new random walk. Secondly, two evolutionary operators: crossover and mutation, are incorporated to provide both exploitation and exploration strategies. A fitness-based adaptation is introduced to dynamically tune the three parameters: the portion of high-quality individuals, mutation radius, and mutation rate throughout the evolution, based on the improvement of best-so-far fitness. In contrast, the crossover is implemented using a standard random scheme. Comprehensive examinations using 38 benchmarks: twenty-three classic functions, ten CEC-C06 2019 benchmarks, and five global trajectory optimization problems show that the proposed ERA generally outperforms the four competitors: Rao-1, Rao-2, Rao-3, and firefly algorithm with courtship learning (FA-CL). Detailed investigations indicate that both proposed schemes work very well to make ERA evolves in an exploitative manner, which is created by a high portion of high-quality individuals and the crossover operator, and avoids being trapped on the local optimum solutions in an explorative manner, which is generated by a high portion of low-quality individuals and the mutation operator. Finally, the adaptation scheme effectively controls the exploitation-exploration balance by dynamically tuning the portion, mutation radius, and mutation rate throughout the evolution process.

Spanning tree trajectory optimization in the galaxy space

The 10th edition of the Global Trajectory Optimization Competition considered the problem of the galaxy settlement wherein competitors from all over the world were expected to design the trajectories of different settler vessels to maximize the given multi-faceted merit function. The synthesis methods used by the winning team, led jointly by the National University of Defense Technology (NUDT) and Xi’an Satellite Control Center (XSCC), are described along with a greedy search method and the improved solution obtained by University of Jena. Specifically, we presented a layout-first topology-second approach that allows an efficient settlement tree search guided by the pre-specified ideal spatial distribution. We also explained how the problem of constructing settlement trees can be modeled as the widely studied minimum spanning tree problem. Furthermore, University of Jena explored the possibility that a greedy search can generate even better settlement trees, based on the same initial conditions, when compared to that of the winning solution.

A Multi-Objective, Multi-Agent Transcription for the Global Optimization of Interplanetary Trajectories

Distributed Spacecraft Missions present challenges for current trajectory optimization capabilities. When tasked with the global optimization of interplanetary Multi-Vehicle Mission (MVM) trajectories specifically, state-of-the-art techniques are hindered by their need to treat the MVM as multiple decoupled trajectory optimization subproblems. This shortfall blunts their ability to utilize inter-spacecraft coordination constraints and may lead to suboptimal solutions to the coupled MVM problem. Only a handful of platforms capable of fully-automated multi-objective interplanetary global trajectory optimization exist for single-vehicle missions (SVMs), but none can perform this task for interplanetary MVMs. We present a fully-automated technique that frames interplanetary MVMs as Multi-Objective, Multi-Agent, Hybrid Optimal Control Problems (MOMA HOCP). This framework is introduced with three novel coordination constraints to explore different coupled decision spaces. The technique is applied to explore the preliminary design of a dual-manifest mission to the Ice Giants: Uranus, and Neptune, which has been shown to be infeasible using only a single spacecraft anytime between 2020 and 2070.

Multi-sensor fusion localization algorithm for outdoor mobile robot

In this paper, the multi-sensor fusion positioning technology of outdoor mobile robot is studied, and a real-time outdoor positioning algorithm combining GPS and lidar odometry is proposed. Lidar odometry based on 3D lidar for point cloud feature matching is realized. At the same time, a covariance matrix reflecting the local positioning information of the lidar odometry is constructed in real time according to the matching distance and the absolute median difference, and the accuracy factor, the circular probability error, etc. are used as the standard GPS. The positioning effect is evaluated to measure the respective positioning effects of the lidar odometry and GPS. On this basis, the GPS is used as the global factor in the pose structure, and the lidar odometry is used as the local factor to construct the pose observation constraint, and the global nonlinear trajectory optimization is carried out to realize the outdoor real-time positioning of the mobile robot. Experimental results and data analysis verify the effectiveness and practicability of the proposed method.

LOW-COST WHEELED ROBOT-BORNE LASER SCANNING SYSTEM FOR INDOOR AND OUTDOOR 3D MAPPING APPLICATION

Abstract. Aiming to accomplish automatic and real-time three-dimensional mapping in both indoor and outdoor scenes, a low-cost wheeled robot-borne laser scanning system is proposed in this paper. The system includes a laser scanner, an inertial measurement unit, a modified turtlebot3 two-wheel differential chassis and etc. To achieve a globally consistent map, the system performs global trajectory optimization after detecting the loop closure. Experiments are undertaken in two typical indoor/outdoor scenes that is an underground car park and a road environment in the campus of Wuhan University. The point clouds acquired by the proposed system are quantitatively evaluated by comparing the derived point clouds with the ground truth data collected by a RIEGL VZ 400 laser scanner. The results present an accuracy of 90% points below 0.1 meter error in the tested scene, showing that its applicability and potential in indoor and mapping applications.

Success-history based biology-inspired algorithms for global trajectory optimization

Biology-inspired algorithms are computationally efficient for real-parameter optimization. However, the search efficiency of such algorithms depends significantly on their ability in keeping the balance between exploration and exploitation when solving complex multimodal problems. A new technique for generating potential solutions in biology-inspired algorithms is proposed. The stated technique uses a historical memory of successful positions found by individuals to guide them in different directions, thereby improving their exploration and exploitation abilities. Thus, this paper describes the application of modified biology-inspired algorithms, namely the Firefly Algorithm, the Cuckoo Search Algorithm and the Bat Algorithm to global trajectory optimization problems. The problems are provided by the European Space Agency and represent trajectories of several well-known spacecraft, such as Cassini and Messenger. Firstly, modified versions of the listed heuristics as well as their original variants were evaluated on a set of various test functions. Then their performance was evaluated on two global trajectory optimization problems: Cassini-1 and Messenger. The experimental results obtained by them are presented and compared. It was established that success-history based position adaptation allows better solutions to be found with the same computational effort while solving complex real-world problems. Thus, the usefulness of the proposed position adaptation technique was demonstrated.

Trajectory optimization using quantum computing

We present a framework wherein the trajectory optimization problem (or a problem involving calculus of variations) is formulated as a search problem in a discrete space. A distinctive feature of our work is the treatment of discretization of the optimization problem wherein we discretize not only independent variables (such as time) but also dependent variables. Our discretization scheme enables a reduction in computational cost through selection of coarse-grained states. It further facilitates the solution of the trajectory optimization problem via classical discrete search algorithms including deterministic and stochastic methods for obtaining a global optimum. This framework also allows us to efficiently use quantum computational algorithms for global trajectory optimization. We demonstrate that the discrete search problem can be solved by a variety of techniques including a deterministic exhaustive search in the physical space or the coefficient space, a randomized search algorithm, a quantum search algorithm or by employing a combination of randomized and quantum search algorithms depending on the nature of the problem. We illustrate our methods by solving some canonical problems in trajectory optimization. We also present a comparative study of the performances of different methods in solving our example problems. Finally, we make a case for using quantum search algorithms as they offer a quadratic speed-up in comparison to the traditional non-quantum algorithms.

Multi-stage global trajectory optimization for the overlapping shift of a seamless two-speed transmission using Legendre pseudo-spectral method

For a seamless two-speed transmission specially equipped in electric vehicles, the global trajectory optimization problem of the overlapping shift process with multiple stages and complex path/point constraints is investigated in this article. The overlapping shift principle and the dynamic models in each phase are also demonstrated in detail. The parameters to evaluate the shift quality are analyzed and then introduced into the objective function and constraints, respectively. The jerks in process and at the key moments have been distinguished and selected as the path and point constraints, respectively. Moreover, considering the joint characteristics of the torque phase and the inertia phase, the trajectory optimization problems in the above two phases are summarized as a multi-stage global trajectory optimization problem. Then, Legendre pseudo-spectral method is used to transfer the multi-stage global trajectory optimization problem into a nonlinear programming problem for numerical solutions. Finally, the effectiveness and feasibility of the multi-stage global trajectory optimization method have been verified through the comparison of the optimal shift trajectories under different conditions with that obtained by the piecewise trajectory optimization method.

DRIFT-FREE INDOOR NAVIGATION USING SIMULTANEOUS LOCALIZATION AND MAPPING OF THE AMBIENT HETEROGENEOUS MAGNETIC FIELD

Abstract. In the absence of external reference position information (e.g. surveyed targets or Global Navigation Satellite Systems) Simultaneous Localization and Mapping (SLAM) has proven to be an effective method for indoor navigation. The positioning drift can be reduced with regular loop-closures and global relaxation as the backend, thus achieving a good balance between exploration and exploitation. Although vision-based systems like laser scanners are typically deployed for SLAM, these sensors are heavy, energy inefficient, and expensive, making them unattractive for wearables or smartphone applications. However, the concept of SLAM can be extended to non-optical systems such as magnetometers. Instead of matching features such as walls and furniture using some variation of the Iterative Closest Point algorithm, the local magnetic field can be matched to provide loop-closure and global trajectory updates in a Gaussian Process (GP) SLAM framework. With a MEMS-based inertial measurement unit providing a continuous trajectory, and the matching of locally distinct magnetic field maps, experimental results in this paper show that a drift-free navigation solution in an indoor environment with millimetre-level accuracy can be achieved. The GP-SLAM approach presented can be formulated as a maximum a posteriori estimation problem and it can naturally perform loop-detection, feature-to-feature distance minimization, global trajectory optimization, and magnetic field map estimation simultaneously. Spatially continuous features (i.e. smooth magnetic field signatures) are used instead of discrete feature correspondences (e.g. point-to-point) as in conventional vision-based SLAM. These position updates from the ambient magnetic field also provide enough information for calibrating the accelerometer bias and gyroscope bias in-use. The only restriction for this method is the need for magnetic disturbances (which is typically not an issue for indoor environments); however, no assumptions are required for the general motion of the sensor (e.g. static periods).

Optimization of observing sequence based on nominal trajectories of symmetric observing configuration

This paper presents the crucial method for obtaining our team’s results in the 8th Global Trajectory Optimization Competition (GTOC8). Because the positions and velocities of spacecraft cannot be completely determined by one observation on one radio source, the branch and bound method for sequence optimization of multi-asteroid exploration cannot be directly applied here. To overcome this difficulty, an optimization method for searching the observing sequence based on nominal low-thrust trajectories of the symmetric observing configuration is proposed. With the symmetric observing configuration, the normal vector of the triangle plane formed by the three spacecraft rotates in the ecliptic plane periodically and approximately points to the radio sources which are close to the ecliptic plane. All possible observing opportunities are selected and ranked according to the nominal trajectories designed by the symmetric observing configuration. First, the branch and bound method is employed to find the optimal sequence of the radio source with thrice observations. Second, this method is also used to find the optimal sequence of the left radio sources. The nominal trajectories are then corrected for accurate observations. The performance index of our result is 128,286,317.0 km which ranks the second place in GTOC8.

Bio-inspired Collision-free 4D Trajectory Generation for UAVs Using Tau Strategy

Inspired by the general tau theory in animal motion planning, a collision-free four-dimensional (4D) trajectory generation method is presented for multiple Unmanned Aerial Vehicles (UAVs). This method can generate a group of optimal or near-optimal collision-free 4D trajectories, the position and velocity of which are synchronously planned in accordance with the arrival time. To enlarge the shape adjustment capability of trajectories with zero initial acceleration, a new strategy named intrinsic tau harmonic guidance strategy is proposed on the basis of general tau theory and harmonic motion. In the case of multiple UAVs, the 4D trajectories generated by the new strategy are optimized by the bionic Particle Swarm Optimization (PSO) algorithm. In order to ensure flight safety, the protected airspace zone is used for collision detection, and two collision resolution approaches are applied to resolve the remaining conflicts after global trajectory optimization. Numerous simulation results of the simultaneous arrival missions demonstrate that the proposed method can effectively provide more flyable and safer 4D trajectories than that of the existing methods.

Optimization of Multiple-Rendezvous Low-Thrust Missions on General-Purpose Graphics Processing Units

A massively parallel method for the identification of optimal sequences of targets in multiple-rendezvous low-thrust missions is presented. Given a list of possible targets, a global search of sequences compatible with the mission requirements is performed. To estimate the feasibility of each transfer, a heuristic model based on Lambert’s transfers is evaluated in parallel for each target, making use of commonly available general-purpose graphics processing units such as the Nvidia Tesla cards. The resulting sequences are ranked by user-specified criteria such as length or fuel consumption. The resulting preliminary sequences are then optimized to a full low-thrust trajectory using classical methods for each leg. The performance of the method is discussed as a function of various parameters of the algorithm. The efficiency of the general-purpose graphics processing unit implementation is demonstrated by comparing it with a traditional CPU-based branch-and-bound method. Finally, the algorithm is used to compute asteroid sequences used in a solution submitted to the seventh edition of the Global Trajectory Optimization Competition.

The optimization of global fault tolerant trajectory for redundant manipulator based on self-motion

The redundancy feature of manipulators provides the possibility for the fault tolerant trajectory planning. Aiming at the completion of the specific task, an algorithm of global fault tolerant trajectory optimization for redundant manipulator based on the self-motion is proposed in this paper. Firstly, inverse kinematics equation of single redundancy manipulator based on self-motion variable and null-space velocity array of Jacobian are analyzed. Secondly, the mathematical description of fault tolerance criteria of the configuration of manipulator is established and the fault tolerance configuration group of manipulator is obtained by using iteration traversal under the fault tolerance criteria. Then, considering the joint limits and minimum the energy consumption as the optimization target, the global fault tolerant joint trajectory is achieved. Finally, simulation for 7 degree of freedom (DOF) manipulator is performed, by which the effectiveness of the algorithm is validated.

Tour of Jupiter Galilean moons: Winning solution of GTOC6

The paper presents the trajectory designed by the Italian joint team Politecnico di Torino & Sapienza Università di Roma (Team5), winner of the 6th edition of the Global Trajectory Optimization Competition (GTOC6). In the short time available in these competitions, Team5 resorted to basic knowledge, simple tools and a powerful indirect optimization procedure. The mission concerns a 4-year tour of the Jupiter Galilean moons. The paper explains the strategy that was preliminarily devised and eventually implemented by looking for a viable trajectory. The first phase is a capture that moves the spacecraft from the arrival hyperbola to a low-energy orbit around Jupiter. Six series of flybys follow; in each one the spacecraft orbits Jupiter in resonance with a single moon; criteria to construct efficient chains of resonant flybys are presented. Transfer legs move the spacecraft from resonance with a moon to another one; precise phasing of the relevant moons is required; mission opportunities in a 11-year launch window are found by assuming ballistic trajectories and coplanar circular orbits for the Jovian satellites. The actual trajectory is found by using an indirect technique.

GTOC5: Results from the European Space Agency and University of Florence

This paper describes the methodologies used to tackle the problem of the 5th Global Trajectory Optimization Competition within the team composed by the Advanced Concepts Team of the European Space Agency and theGlobal Optimization Laboratory of the University of Florence. The method pursued is powered by two innovative approaches: a linearized model of the ‘self fly-by’ aiding a first broad tree search of chemical propulsion options and the use of global optimization techniques (Monotonic Basin Hopping, in this case) applied directly to the low-thrust trajectory model.

Choosing promising sequences of asteroids

We consider the problem proposed on the 4th Global Trajectory Optimization Competition (GTOC4).

Multi-objective interplanetary trajectory optimization combining low-thrust propulsion and gravity-assist maneuvers

To expand mission capabilities needed without a proportional increase in cost or risk for exploration of the solar system, the multiple objective trajectory using low-thrust propulsion and gravity-assist technique is considered. However, low-thrust, gravity-assist trajectories pose significant optimization challenges because of their large design space. Here, the planets are selected as primal scientific mission goals, while the asteroids are selected as secondary scientific mission goals, and a global trajectory optimization problem is introduced and formulated. This multi-objective decision making process is transformed into a bi-level programming problem, where the targets like planets with small subsamples but high weight are optimized in up level, and targets like asteroids with large subsamples but low weight are optimized in down level. Then, the selected solutions for bi-level programming are optimized thanks to a cooperative Differential Evolution (DE) algorithm that is developed from the original DE algorithm; in addition, an sequential quadratic programming (SQP) method is used in low-thrust optimization. This solution approach is successfully applied to the simulation case of the multi-objective trajectory design problem. The results obtained are presented and discussed.

Optimal Control Problem for Low-Thrust Multiple Asteroid Tour Missions

The use of low-thrust propulsion systems makes very complex missions possible. Among them, multiple asteroid missions have not been the subject of much study. The design of such missions requires visiting a high number of targets. The mathematical problem is of large dimensionality, and to guarantee high accuracy, an efficient solution method should be proposed. This paper presents a possible formulation of the problem.An indirectmethod is used to formulate a multiple asteroid mission problem, which includes specific control constraints necessary for effective control and operations when approaching an asteroid for flyby. No a priori knowledge of the control structure is required. Special care is given to the Jacobian structure formaximum efficiency. The algorithm is implemented with parallelization for computational speed. Overall, the formulation demonstrates robustness in solving highdimensional optimal control problems using an indirect approach, which would mostly be impossible with direct methodswithout significantly impacting the solution accuracy. Interplanetary trajectories are presented as examples of the method. For instance, the method is applied to the problem of the fourth edition of the Global Trajectory Optimisation Competition, and it optimizes a mission scenario with as many as 42 asteroid flybys.

Low-thrust trajectory optimization for multiple target bodies tour mission

Spacecraft science missions to planets or asteroids have historically visited only one or several celestial bodies per mission. The research goal of this paper is to create a trajectory design algorithm that generates trajectory allowing a spacecraft to visit a significant number of asteroids during a single mission. For the problem of global trajectory optimization, even with recent advances in low-thrust trajectory optimization, a full enumeration of this problem is not possible. This work presents an algorithm to traverse the searching space in a practical fashion and generate solutions. The flight sequence is determined in ballistic scenario, and a differential evolution method is used with constructing a three-impulse transfer problem, then the local optimization is implemented with low-thrust propulsion on the basis of the solutions of impulsive trajectories. The proposed method enables trajectory design for multiple asteroids tour by using available low thrust propulsion technology within fuel and time duration constraints.