Global Trajectory Optimization Research Articles

GLIM: 3D range-inertial localization and mapping with GPU-accelerated scan matching factors

This article presents GLIM, a 3D range-inertial localization and mapping framework with GPU-accelerated scan matching factors. The odometry estimation module of GLIM employs a combination of fixed-lag smoothing and keyframe-based point cloud matching that makes it possible to deal with a few seconds of completely degenerated range data while efficiently reducing trajectory estimation drift. It also incorporates multi-camera visual feature constraints in a tightly coupled way to further improve the stability and accuracy. The global trajectory optimization module directly minimizes the registration errors between submaps over the entire map. This approach enables us to accurately constrain the relative pose between submaps with a small overlap. Although both the odometry estimation and global trajectory optimization algorithms require much more computation than existing methods, we show that they can be run in real-time due to the careful design of the registration error evaluation algorithm and the entire system to fully leverage GPU parallel processing.

GLIM: 3D range-inertial localization and mapping with GPU-accelerated scan matching factors

This article presents GLIM, a 3D range-inertial localization and mapping framework with GPU-accelerated scan matching factors. The odometry estimation module of GLIM employs a combination of fixed-lag smoothing and keyframe-based point cloud matching that makes it possible to deal with a few seconds of completely degenerated range data while efficiently reducing trajectory estimation drift. It also incorporates multi-camera visual feature constraints in a tightly coupled way to further improve the stability and accuracy. The global trajectory optimization module directly minimizes the registration errors between submaps over the entire map. This approach enables us to accurately constrain the relative pose between submaps with a small overlap. Although both the odometry estimation and global trajectory optimization algorithms require much more computation than existing methods, we show that they can be run in real-time due to the careful design of the registration error evaluation algorithm and the entire system to fully leverage GPU parallel processing.

An improved multi-operator differential evolution with two-phase migration strategy for numerical optimization

Over the last decade, the multi-operator differential evolution (MODE) has become one of the most popular research areas in the differential evolution (DE) family. Among these MODEs, an improved multi-operator differential evolution (IMODE) has achieved success as the winner in CEC2020. Despite its good performance, the random information sharing strategy may not efficiently handle the balance between exploration and exploitation for the complex numerical optimization. To address this limitation, we propose a two-phase migration strategy (TMS) to improve the performance of IMODE, called IMODE-TMS. In the first phase, the top-ranked individuals in each sub-population are retained for exploitation, and all the bottom-ranked individuals are assigned to three sub-populations to maintain diversity. Furthermore, the second phase plays a crucial role in eliminating stagnation. When a sub-population is identified as stagnant by the stagnation indicator, the optimal individual will migrate to that sub-population following the uni-directional ring structure. This process is mainly used to increase the off-trap capability on some problems with complex fitness landscapes. IMODE-TMS is tested on the CEC2020, CEC2021 and CEC2022 benchmark functions, and seven well-known complex global trajectory optimization problems (GTOP). The experimental results show that IMODE-TMS significantly outperforms not only IMODE but also other state-of-the-art comparison algorithms.

A Greedy-Strategy-Based Iterative Optimization Method for Articulated Vehicle Global Trajectory Optimization in Complex Environments

This paper considers the problem of trajectory planning for articulated vehicles in complex environments. We formulate this problem as an optimal control problem (OCP) and propose a greedy-strategy-based planner. This planner consists of three stages. In stage 1, an IAA* algorithm is proposed to identify the homotopy class. In stage 2, the collision-free tunnels are constructed along the guiding trajectory generated in stage 1 to simplify the intractable collision-avoidance constraints. In stage 3, a greedy-strategy-based iterative optimization (GSIO) framework is designed, which contributes to escaping from local optimums, making the optimization process more targeted, and converging to the global optimum solution quickly, especially in complex tasks. One feature of the proposed planner is that it is suitable for any type of articulated vehicle, and the proposed optimization framework can be used as an open framework to optimize any criterion that can be described explicitly by a polynomial. Furthermore, in the set simulation cases, our work shows significant competitiveness, under the premise of ensuring moderate CPU processing time, our algorithm achieves approximately a 40% performance improvement in optimization effects compared to selected comparative algorithms.

FirebotSLAM: Thermal SLAM to Increase Situational Awareness in Smoke-Filled Environments.

Operating in extreme environments is often challenging due to the lack of perceptual knowledge. During fire incidents in large buildings, the extreme levels of smoke can seriously impede a firefighter's vision, potentially leading to severe material damage and loss of life. To increase the safety of firefighters, research is conducted in collaboration with Dutch fire departments into the usability of Unmanned Ground Vehicles to increase situational awareness in hazardous environments. This paper proposes FirebotSLAM, the first algorithm capable of coherently computing a robot's odometry while creating a comprehensible 3D map solely using the information extracted from thermal images. The literature showed that the most challenging aspect of thermal Simultaneous Localization and Mapping (SLAM) is the extraction of robust features in thermal images. Therefore, a practical benchmark of feature extraction and description methods was performed on datasets recorded during a fire incident. The best-performing combination of extractor and descriptor is then implemented into a state-of-the-art visual SLAM algorithm. As a result, FirebotSLAM is the first thermal odometry algorithm able to perform global trajectory optimization by detecting loop closures. Finally, FirebotSLAM is the first thermal SLAM algorithm to be tested in a fiery environment to validate its applicability in an operational scenario.

Duty-cycle-aware low-thrust trajectory optimization using embedded homotopy

In space mission design, an important operational constraint is planned periodic coast arcs during which various mission-critical tasks such as navigation or ground communications are performed. These constraints, typically implemented as thruster duty cycles, can be characterized as discrete coast arcs enforced periodically during intervals of deterministic thrusting. A typical approach to incorporate duty cycles is by enforcing maximum thrust levels that are less than a nominal value. This work presents an embedded homotopy indirect optimization method for a smooth enforcement of discrete duty-cycle constraints for low-thrust trajectory optimization. The complexity of enforcing such discrete constraints is two-fold: (1) a precise event-detection strategy is needed to determine the beginning of deterministic thrust arcs, which must be successively updated along the trajectory and (2) a smooth modeling of the discrete coast events occurring on the order of 10s or even 100s depending on the duration of deterministic thrust arcs. Duty cycle constraints are enforced at the level of the Hamiltonian using a Composite Smooth Control (CSC) method, producing high-resolution optimal control solutions conscious of planned duty cycle coast arcs. Numerical solutions are presented for the heliocentric transfer phases of three space missions. The results are then validated against the Evolutionary Mission Trajectory Generator (EMTG), one of NASA’s global trajectory optimization tools used for interplanetary mission design and capable of the explicit enforcement of periodic duty cycle coasting. The proposed indirect method is suitable for obtaining high-resolution trajectory solutions with realistically enforced thruster duty cycle, which could then be used as high-quality initial guesses in flight-fidelity mission planning tools. The proposed smooth modeling of discrete coast arcs can also be customized to enforce periodic and non-periodic thrust shutdown constraints.

Newton’s Method for Global Free Flight Trajectory Optimization

Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton’s method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.

Reinforcement learning-based hybrid differential evolution for global optimization of interplanetary trajectory design

The global optimization and design of interplanetary trajectories is one of the most important tasks in deep space exploration. Its search space is often characterized by multi-constraints, extreme non-linearity and sensitivity to initial conditions. To cope with these difficulties, an improved reinforcement learning-based hybrid differential evolution (DE) named RL_HDE is proposed in this paper. In RL_HDE, a novel multi-mutation strategy LSHADE based on an adaptive Q-Learning framework is proposed for global exploration. To further balance the global exploration and local exploitation of RL_HDE, a new parameter adaptive strategy based on Q-Learning is designed to control two trigger parameters. The performance of RL_HDE is verified by the well-known Global Trajectory Optimization Problems (GTOP), which are developed by Advanced Concepts Team of European Space Agency (ESA-ACT). A comparison of RL_HDE with three sets of algorithms, including ten state-of-the-art hybrid evolutionary algorithms, four interplanetary trajectory design algorithms, and seven reinforcement learning-based hybridizations. Experimental statistical results demonstrate that RL_HDE outperforms other competitors in terms of convergence efficiency and accuracy. RL_HDE has better performance for solving complex interplanetary trajectory design problems such as Cassini2 and Messenger-full.

Global-local hierarchical path planning scheme for unmanned surface vehicles under dynamically unforeseen environments

Unmanned surface vehicles (USVs) should be allowed to respond to dynamic ocean environments and self-adjust their paths safely and efficiently. In this paper, considering the challenges posed by time-varying, partially unknown complex environments, a novel hierarchical motion planning framework is elaborately designed for a USV, which includes a global trajectory optimization and a local reactive collision avoidance strategy. By encapsulating the intricate nature of ocean environment, a global optimization path planning problem is developed to systematically strengthen the model's adaptability to the complex engineering problem. Incorporating adaptive elite selection and fuzzy probability set, an adaptive-elite GA with fuzzy inference (AEGAfi) is devised to fully exploit the underlying optimization problem, providing high-quality global paths. By applying virtual sensory vector onto the USV's sensing module, the COLREG-compliant local-reaction is achieved by governing feasible actions of USVs under dynamically unforeseen environments. Seamlessly bridged by the transition Clothoid path, the linkage between global optimization and dynamic-avoidance is strengthened by softening the replanning time restriction and maintaining path continuity. Eventually, the motion planning framework merits autonomous global-planning and local-reaction in an organically modular manner. Comprehensive simulations and comparisons in various ocean scenarios demonstrate the effectiveness and superiority of the proposed framework.

Optimal Steering of Matrix Product States and Quantum Many-Body Scars

Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of non-trivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states~(MPS). We compare counter-diabatic and leakage minimization approaches to the so-called local steering problem, that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS state in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counter-diabatic framework and use it to construct a Floquet model with quantum scars. We perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model. Finally we apply our leakage minimization approach to construct quantum scars in the periodically driven non-integrable Ising model.

Design of impulsive asteroid flybys and scheduling of time-minimal optimal control arcs for the construction of a Dyson ring (GTOC 11)

This paper describes the approach used by the team named the Eccentric Anomalies to obtain the sixth best solution to the problem of the eleventh Global Trajectory Optimization Competition, whose futuristic scenario of Dyson sphere building remained mathematically relevant for current space mission design and engineering in general. As usual for this recurring challenge, it involved large-scale combinatorics at a high level and a multitude of optimal control problems at a lower level. Furthermore it proposed additional layers of complexity by adding a strong scheduling component to the usual flyby sequencing, and by featuring both impulsive and continuous-thrust trajectories. The authors took advantage of modern theoretical techniques and open-source tools to put together a sequential process including analysis based on analytical trajectory models, tree searches using efficient data structures, global and local finite-dimensional optimization and multi-objective trade-offs. The optimal control part was both tackled with direct transcription as well as indirect shooting methods, and the mixed-integer scheduling reformulated as a bi-level optimization. From a programming point of view, the main framework was set in an interpreted language whilst using as much as possible dependencies written in compiled ones for speed.

Team theAntipodes: Solution methodology for GTOC11

This paper presents the solution approach developed by the team “theAntipodes” for the 11th Global Trajectory Optimization Competition (GTOC11). The approach consists of four main blocks: (1) mothership chain generation, (2) rendezvous table generation, (3) the dispatcher, and (4) the refinement. Blocks 1 and 3 are purely combinatorial optimization problems that select the asteroids to visit and allocate them to the Dyson ring stations. The rendezvous table generation involves interpolating time-optimal transfers to find all transfer opportunities between selected asteroids and the ring stations. The dispatcher uses the data stored in the table and allocates the asteroids to the Dyson ring stations in an optimal fashion. The refinement ensures each rendezvous trajectory meets the problem accuracy constraints, and introduces deep-space maneuvers to the mothership transfers. We provide the details of our solution that, with a score of 5992, was worth 3rd place.

Dyson sphere building: On the design of the GTOC11 problem and summary of the results

The National University of Defense Technology and the Xi’an Satellite Control Center organized the 11th edition of the Global Trajectory Optimization Competition (GTOC11) in 2021. The GTOC11 problem was created as a link between the ninth and tenth editions of the Global Trajectory Optimization Competition to bridge the gap between the planetary and galactic civilizations by introducing an intermediate stellar civilization scenario. The problem involves the construction of a Dyson sphere, a theoretical mega-structure that encircles a star with platforms orbiting in a tight formation to capture the maximum energy from it. Challenges in astrodynamics including the construction-orbit selection, combinatorial flyby of multiple asteroids, and mass distribution among stations were considered in the Dyson sphere design. A total of 94 teams registered for the competition, of which 25 teams provided solutions and passed automatic verification on the website. In this article, we describe the selection of the problem and its design process. In addition, an overview of the entire competition and an analysis of its results are presented.

GTOC 11: Results found at Beijing Institute of Technology and China Academy of Space Technology

This paper presents the crucial method for the results found at Beijing Institute of Technology and China Academy of Space Technology in the 11th Global Trajectory Optimization Competition (GTOC-11). To build 12 solar-power stations placed in the “Dyson Ring” orbit around the Sun, GTOC-11 focuses on the flyby trajectory design, low-thrust trajectory optimization, and target assignment problem. The mission with time and state constraints causes many difficulties in the design and optimization, which is performed in the following four steps. The first step is to design the orbital elements of the “Dyson Ring” by the grid search method. An approximate model is utilized to evaluate the minimum transfer time for each candidate at first, and the accurate solutions are obtained by the indirect method. The second step involves the Mother Ship trajectories design. A large quantity of earth-to-asteroid and asteroid-to-asteroid subsequences are generated by parallel beam search. The complete trajectories of Mother Ship are designed by connecting the subsequences using beam search. The third step involves the station assignment problem, which is required to design the asteroid transfer device activation epoch and the target station assignment. Based on the results above, the last step involves the rendezvous trajectories optimization to obtain the accurate thrust direction profile. Finally, the submitted and post-competition results are reported.

GTOC-11: Results from the University of Alabama

In this work, Team ASRL’s solution approach for the 11th Global Trajectory Optimization Competition (GTOC-11) is described. This problem tasked the competing teams with constructing a futuristic Dyson Ring utilizing materials acquired from the asteroid belt. In total, 10 motherships would depart from Earth in the year 2121 and visit as many asteroids as possible. After visiting each asteroid, a low-thrust propulsion module would transfer the material down to the desired final Dyson stations. The final approach utilized a deterministic tree search that involved alternating between fixed time of flight Lambert searches and solutions to the full-fidelity optimal control problem. Once a single tour had been constructed, transfer trajectories were computed for each asteroid to as many of the building stations as possible. After computing a pool of thousands of these completed legs, a bin packing algorithm was used to determine the highest scoring combination of 10 solutions. This search process was implemented in Python using the soon-to-be-released trajectory optimization tool, ASSET. Ultimately, the team finished 5th with a score of 5525.38.

Global trajectory optimization, pathfinding, and scheduling for a multi-flyby, multi-spacecraft mission

The UT Austin team presents their methodology and result for GTOC11, constructing a hypothetical Dyson ring using asteroids encountered by 10 motherships departing from Earth. A pathfinding algorithm for the mothership is designed using a fast lookup table and a robust Lambert solver. Sequencing of unique mothership itineraries is performed by approximating the mass delivered to the ring via a linear data-based estimator trained on simulated asteroid-to-ring trajectories. A detailed derivation for the necessary boundary conditions of the constant-thrust-acceleration indirect optimization problem is presented, with key insights about the magnitudes of the co-states. An indirect boundary value solver finds feasible asteroid-to-ring trajectories for each of the 12 phasing locations in a high-performance parallel computing environment. A genetic algorithm informs the sequencing for the ring station arrival order that gives the highest performance index value. UT Austin’s final submitted solution placed 4th in the competition, with a performance index value of 5885.5, 235 asteroids collected, and a minimum build-station mass of 1.1328×1015 kg.

GTOC11: Methods and results from the team of Harbin Institute of Technology

This paper presents the methods and the results submitted by the team of Harbin Institute of Technology in the 11th Global Trajectory Optimization Competition (GTOC11). The mission of GTOC11 is to design the Mother Ship trajectories to execute asteroid flybys selected from over 83000 candidates, and then transfer these asteroids to 12 stations to build the “Dyson Ring” while maximizing the total performance index. The building mission is completed by five steps: “Dyson Ring” orbit selection, database creation, Mother Ship trajectory design, building station construction strategy, and asteroid transfer trajectory optimization. The database creation is mainly used to fast estimate the transfer time for the asteroid low-thrust trajectory, and it plays an essential role in the whole mission. Finally, the building of the “Dyson Ring” is realized within the given period. The final index of the submitted result is 5208.3463, which ranked seventh in GTOC11.

GTOC 11: Results from Tsinghua University and Shanghai Institute of Satellite Engineering

In this paper, the methods and results from Tsinghua University and Shanghai Institute of Satellite Engineering for the 11th Global Trajectory Optimization Competition (GTOC11) are presented. To deal with the complicated “Dyson Sphere” building problem, a three-stage procedure is conducted. First, the pre-analysis is performed to reduce search space. It is found that two-impulse maneuvers between asteroid flybys are near-optimal, the semi-major axis of the “Dyson Ring” should be better at 1.0–1.5 AU, and the larger arrival mass asteroids tend to be selected. Second, the globally optimal trajectory design problem is further divided into two sub-problems, the mothership trajectory design and the asteroid assignment to the “Dyson Ring” power stations. For the first problem, beam search is used to obtain numerous single mothership trajectories based on a pre-constructed flyby trajectory database of 3–8 asteroids. The overall trajectories and asteroids visited are obtained by selecting 10 mothership trajectories with a genetic algorithm. For the second problem, we build a database of optimal rendezvous times for all the 83,453 asteroids at different phase angles to reach power stations of different radii and phase angles, then a greedy algorithm is proposed to obtain the asteroid arrival schedule based on all the asteroids visited by motherships. Finally, local optimization of asteroid sequence and flyby epochs is conducted. The activation time adjustment in combination with indirect continuous-thrust trajectory optimization is used based on the global optimization result. In the final submission, motherships fly by 388 asteroids, and the minimum mass of twelve power stations reaches 94% of the theoretical upper bound, which is defined using the minimum-time orbital transfers with free initial and target phases.

Asteroid Flyby Cycler Trajectory Design Using Deep Neural Networks

Asteroid exploration has been attracting more attention in recent years. Nevertheless, we have just visited tens of asteroids, whereas we have discovered more than 1 million bodies. As our current observation and knowledge should be biased, it is essential to explore multiple asteroids directly to better understand the remains of planetary building materials. One of the mission design solutions is utilizing asteroid flyby cycler trajectories with multiple Earth gravity assists. An asteroid flyby cycler trajectory design problem is a subclass of global trajectory optimization problems with multiple flybys, involving a trajectory optimization problem for a given flyby sequence and a combinatorial optimization problem to decide the sequence of the flybys. As the number of flyby bodies grows, the computation time of this optimization problem expands maliciously. This paper presents a new method to design asteroid flyby cycler trajectories utilizing a surrogate model constructed by deep neural networks approximating trajectory optimization results. Because one of the bottlenecks of machine learning approaches is the heavy computation time to generate massive trajectory databases, we propose an efficient database generation strategy by introducing pseudo-asteroids satisfying the Karush–Kuhn–Tucker conditions. The numerical result applied to Japan Aerospace Exploration Agency’s mission shows that the proposed method is practically applicable to space mission design and can significantly reduce the computational time for searching asteroid flyby sequences.

A differential evolution algorithm with the guided movement for population and its application to interplanetary transfer trajectory design

The extremely sensitive and highly non-linear search space of interplanetary transfer trajectory design brings about big challenges on global optimization. As a representative, the current known best solution (fuel consumption) of the global trajectory optimization problem designed by the European space agency is very hard to be found. To deal with this difficulty, a powerful differential evolution with the guided movement for population, named G_DE is proposed in this paper. G_DE employs a two-stage evolutionary process, which concentrates on learning global structure in the earlier process, and tends to self-adaptively probe the structures of numerous local spaces at a later stage. During these two stages, four guidance strategies related to the learning from the population distribution are proposed. The experimental test results show that G_DE can find the current known best solutions of Cassini1 and Sagas directly. For the newly proposed GTOP-X problem, G_DE has found state-of-the-art solutions for four encounter sequences, and using five gravity assists, G_DE found encounter sequences with fuel consumption as low as 1.5676 km/s. The experimental test on 10D CEC2017 problems proves G_DE has a comparable performance with recently proposed differential evolution variant.