Rendezvous Problem Research Articles

Using model checking to formally verify rendezvous algorithms for robots with lights in Euclidean space

The paper details the first successful attempt at using model checking techniques to verify the correctness of distributed algorithms for robots evolving in a continuous environment. The study focuses on the problem of rendezvous of two robots with lights.There exist many different rendezvous algorithms that aim at finding the minimal number of colors needed to solve rendezvous in various synchrony models (e.g., FSYNC, SSYNC, ASYNC). While these rendezvous algorithms are typically very simple, their analysis and proof of correctness tend to be extremely complex, tedious, and error-prone as impossibility results are based on subtle interactions between the activation schedules of the robots.The paper presents a generic verification model that can be concretely expressed in available software model-checkers. In particular, we explain the subtle design decisions that allow to keep the search space finite and tractable, as well as prove several important theorems that support them. As a sanity check, we use the model to verify several known rendezvous algorithms in six different models of synchrony. In each case, we find that the results obtained from the model checker are consistent with the results known in the literature. The model checker outputs a counter-example execution in every case that is known to fail.In the course of developing and proving the validity of the model, we identified several fundamental theorems, including the ability for a well chosen algorithm and ASYNC scheduler to produce an emerging property of memory in a system of oblivious mobile robots, and why it is not a problem when robots executing the gathering algorithms are equipped with lights.

Connectivity-preserving rendezvous of heterogeneous multi-agent systems under denial-of-Service attacks

This article investigates the rendezvous problem of heterogeneous multi-agent systems against Denial-of-Service attacks with preserving initial connectivity under a dynamic communication topology. The algorithm of resisting Denial-of-Service attack is first introduced to connectivity-preserving rendezvous problem of heterogeneous multi-agent systems. First of all, in order to observe the information of the leader, the distributed observers are designed to estimate the state of the leader with the communication network in the presence of Denial-of-Service attack by adaptive algorithm. Then, a switching system model is constructed by taking into account the influence of Denial-of-Service attacks. By means of the obtained model combined with a artificial potential function technique, the proposed distributed control algorithm allows all agents to accomplish rendezvous assignment with connectivity preservation as well as resisting Denial-of-Service attacks. Finally, detailed simulation validates the effectiveness of the proposed distributed observer and controller algorithm.

On the Effect of Symmetry Requirement for Rendezvous on the Complete Graph

We consider a classic rendezvous game in which two players try to meet each other on a set of n locations. In each round, every player visits one of the locations, and the game finishes when the players meet at the same location. The goal is to devise strategies for both players that minimize the expected waiting time till the rendezvous. In the asymmetric case, when the strategies of the players may differ, it is known that the optimum expected waiting time of [Formula: see text] is achieved by the wait-for-mommy pair of strategies, in which one of the players stays at one location for n rounds, while the other player searches through all the n locations in a random order. However, if we insist that the players are symmetric—they are expected to follow the same strategy—then the best known strategy, proposed by Anderson and Weber [Anderson EJ, Weber RR (1990) The rendezvous problem on discrete locations. J. Appl. Probab. 27(4):839–851], achieves an asymptotic expected waiting time of [Formula: see text]. We show that the symmetry requirement indeed implies that the expected waiting time needs to be asymptotically larger than in the asymmetric case. Precisely, we prove that for every [Formula: see text], if the players need to employ the same strategy, then the expected waiting time is at least [Formula: see text], where [Formula: see text]. We propose in addition a different proof for one our key lemmas, which relies on a result by Ahlswede and Katona [Ahlswede R, Katona GOH (1978) Graphs with maximal number of adjacent pairs of edges. Acta Mathematica Academiae Scientiarum Hungaricae 32(1–2):97–120]: the argument is slightly shorter and provides a constant larger than [Formula: see text], namely, [Formula: see text]. However, it requires that n be at least 16. Both approaches seem conceptually interesting to us. Funding: This work is a part of project TOTAL (Mi. Pilipczuk) that has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program [Grant 677651]. It is also partially funded by French ANR projects [Grant ANR-16-CE40-0023] (DESCARTES) and [Grant ANR-17-CE40-0015] (DISTANCIA).

Singular optimal control for rendezvous problem for cooperative vehicle control

In this paper an unmanned aerial vehicle works cooperatively with a unmanned ground vehicle to aid in the automatic control of the ground vehicle. One challenge is when the aerial vehicle has to urgently rendezvous with the ground vehicle. An optimal control problem is formulated for such a case, the key contribution is the singular optimal control rendezvous problem formulation. Illustrative results are utilised to demonstrate this key formulation of singular optimal control solution as well as the optimal rendezvous trajectory analysis.

Optimal Multi-impulse Linear Rendezvous via Reinforcement Learning

A reinforcement learning-based approach is proposed to design the multi-impulse rendezvous trajectories in linear relative motions. For the relative motion in elliptical orbits, the relative state propagation is obtained directly from the state transition matrix. This rendezvous problem is constructed as a Markov decision process that reflects the fuel consumption, the transfer time, the relative state, and the dynamical model. An actor–critic algorithm is used to train policy for generating rendezvous maneuvers. The results of the numerical optimization (e.g., differential evolution) are adopted as the expert data set to accelerate the training process. By deploying a policy network, the multi-impulse rendezvous trajectories can be obtained on board. Moreover, the proposed approach is also applied to generate a feasible solution for many impulses (e.g., 20 impulses), which can be used as an initial value for further optimization. The numerical examples with random initial states show that the proposed method is much faster and has slightly worse performance indexes when compared with the evolutionary algorithm.

Approximate time-optimal low-thrust rendezvous solutions between circular orbits

Traditional planetocentric low-thrust solutions mainly focus on orbital rephasing or transfer maneuvers, while the rendezvous maneuver is more challenging to obtain and cannot be approximated by existing solutions in some mission scenarios. In this paper, approximate solutions to the time-optimal rendezvous problem between two close circular orbits are presented. First, the optimal control problem is established in terms of the true longitude by using the Sundman transformation, and a set of unified linearized equations of motion is derived to reduce the number of key parameters characterizing this problem. Then, we show that two key parameters can characterize the rendezvous problem, and that the rephasing (or transfer) corresponds to the case when one of the key parameters equals zero. Thus, a set of piecewise functions is developed to approximate the rephasing and transfer solutions. The rendezvous solutions are finally obtained by a traversal method and summarized in a set of contour maps. Numerical simulations demonstrate that the proposed solutions provide good initial guesses for solving the problem with nonlinear dynamics and accurately approximate the performance index for the preliminary mission design.

Indirect optimization for finite thrust orbit transfer and cooperative rendezvous using an initial guess generator

In this paper, the finite thrust nonplanar transfer and cooperative rendezvous are investigated. The indirect method is adopted in the optimization process. The energy-optimal to fuel-optimal continuation is performed by using the homotopy method. An initial guess generator (IGG) is designed based on a simplified dynamical model to overcome the difficulty in providing the initial guess. Some reasonable simplifications on the dynamical model are applied based on the near-circular near-coplanar transfer. The initial guess can be obtained by analytically solving a system of linear equations. Therefore, the IGG has high computational efficiency. Moreover, the terminal time and state are estimated by IGG to deal with the cooperative rendezvous problem. Numerical examples are presented and the validity of the “IGG-Homotopy” combined algorithm is verified.

Deterministic Collision-Resilient Channel Rendezvous: Theory and Algorithm

We formulate and investigate the problem of distributed channel rendezvous in collision-prone wireless networks. Existing researches on this topic are mainly devoted to designing channel hopping sequences, each pair of which can overlap on a common channel within bounded delay. However, this overlap-based canonical rendezvous design does not take into account channel collision, which may render existing rendezvous algorithms fail to achieve bounded delay in collision-prone environment. Motivated by this observation, we formulate and investigate the collision-aware channel rendezvous problem in a generic scenario, where a collision occurs if more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> packets overlap in time on a same channel. Our generic formulation allows to model both the baseline single packet reception model with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C=1$ </tex-math></inline-formula> and the more sophisticated multiple packet reception model with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C &gt; 1$ </tex-math></inline-formula> . We further abstract the collision-aware rendezvous problem as the problem of constructing a robust rendezvous system. We establish the theoretical limit of the problem, guided by which we design a collision-resilient distributed rendezvous algorithm with truly bounded rendezvous delay. We then demonstrate the performance of our rendezvous algorithm both analytically and numerically.

Оптимізація перельотів між низькими орбітами з суттєвою різницею довгот висхідних вузлів

At present, satellite systems, each comprising hundreds of satellites, are, and are to be, deployed in low orbits. In addition, existing satellite systems are replenished. There has appeared a trend towards the development of modular satellites, which will lead to the development of easy-to-maintain spacecraft consisting of many small structural modules with standardized interface mechanisms. To extend the life of all these systems and reduce their maintenance cost, it is advisable to develop a system for their maintenance. Despite the relatively large number of works on the rendezvous problem, this problem is considered in a somewhat simplified formulation, which is not sufficient for spacecraft servicing in low orbits. As a rule, the consideration is limited to coplanar rendezvous problems in an impulse formulation. In real conditions, rendezvous maneuvers in low orbits are nontrivial. As is known, the orbital parameters of low-orbit spacecraft may differ significantly: the difference in the longitude of ascending nodes (LAN) may reach tens and even hundreds of degrees. Because of this, the energy consumption for an orbital plane change becomes unacceptably high for modern service spacecraft. This energy consumption can be reduced by using the precession of the line of nodes due to the non-centrality of the Earth's gravitational field. A waiting maneuver of a service spacecraft in a well-chosen orbit makes it possible to eliminate the mismatch between the LANs of the service spacecraft’s parking and destination orbits, thus significantly reducing the orbital transfer energy consumption. However, the long wait time of the service spacecraft in its parking orbit significantly increases the total orbital transfer time. The aim of this article is to develop a mathematical model of bicriteria optimization of a transfer of a service spacecraft with a low constant thrust engine between low near-circular orbits with significantly different LANs. This problem is solved by averaging the service spacecraft’s dynamics equations over a fast parameter and using a genetic algorithm of global Pareto optimization. The novelty of the results obtained lies in a formulation of a bicriteria optimization problem and the development of a mathematical model for choosing an optimal service spacecraft parking orbit. The mathematical model developed may be used in planning service spacecraft transfers between low near-circular orbits with significantly different LANs.

Finite-Time Distributed Approximate Optimization Algorithms of Higher Order Multiagent Systems via Penalty-Function-Based Method

This article investigates the finite-time distributed approximate optimization problem of higher order multiagent systems, where the local cost functions are considered to be quadratic functions. This problem is solved via penalty-function-based method. First, by the penalty-function method, a global approximate cost function is constructed. Second, nonlinear distributed optimization algorithms are proposed for higher order multiagent systems by the tool of adding a power integrator technique. In the optimization algorithms design, the gradients of the approximate cost function are utilized. Under the proposed optimization algorithms, the agents approach the approximate optimal solution in finite time. Although there exist errors (they may be called approximation errors) between the approximate minimizers and the global accurate minimizer, the approximation errors can be regulated by penalty parameter and the relationship between the bound of the approximation errors and the penalty parameter is given explicitly. Furthermore, the proposed distributed approximate optimization algorithms are applied to the optimal rendezvous problem of wheeled multimobile robots, making the mobile robots achieve approximate optimization rendezvous in finite time. The effectiveness of the proposed distributed optimization algorithms and their applications to optimal rendezvous problem are validated by simulations.

Convex Optimization for Rendezvous and Proximity Operation via Birkhoff Pseudospectral Method

Rapid and accurate rendezvous and proximity operations for spacecraft are crucial to the success of most space missions. In this paper, a sequential convex programming method, combined with the first-order and second-order Birkhoff pseudospectral methods, is proposed for the autonomous rendezvous and proximity operations of spacecraft. The original nonlinear and nonconvex close-range rendezvous problem with thrust constraints and no-fly zone constraints is converted into its convex version by using the sequential convexification techniques; then, the Birkhoff pseudospectral method is used to transcribe the dynamic constraints into a series of linear algebraic equality constraints, in other words, a convex second-order conic programming problem with a relatively small condition number. Thus, the resulting problem can be accurately and efficiently solved by a convex solver. The simulation results indicate that the proposed methods, especially the second-order Birkhoff pseudospectral method, have obvious advantages over other methods in computational efficiency and sensitivity.

An internal‐model‐based event‐triggered strategy for rendezvous with connectivity preservation problem of multi‐agent systems

AbstractIn this article, an internal‐model‐based event‐triggered control is proposed for the rendezvous problem of linear uncertain multi‐agent systems. Our approach is not only robust to parametric uncertainty and external disturbances, but also capable of preserving the connectivity of the position‐dependent time‐varying communication network. The event‐triggered mechanism is in a piecewise constant form and thus can be directly implemented in the digital platform. Technically, different from traditional internal model design, a class of distributed dynamic compensators are designed based on sampled relative position and velocity signals to convert the rendezvous problem into a stabilization with connectivity problem of a special augmented system, which is then solved by a piecewise constant stabilizer. Finally, it is proved that the rendezvous problem of the original system is solvable by stabilizing such augmented system and excluding Zeno behavior.

A greedy-based hybrid trajectory optimization method for the “Dyson Ring” building problem

The method used and results found by the team “ASTL-NUAA” from NUAA for the problem of GTOC 11 are presented in the paper. The problem is mainly decomposed into three parts: multiple targets flyby sequence optimization for the Mother Ships, station-building database construction problem of asteroid transfer arcs, and station-building mission planning. Therefore, a three-step optimization framework is proposed to solve the problem. After analyzing the data of all the asteroids and selecting according to the orbital elements and mass of the asteroids, an asteroid candidate set RS1 is chosen as candidate targets for the Mother Ships. A two-level optimization algorithm based on a genetic principle is applied to solve the two-impulse asteroid flyby problem and rapidly determine the asteroid sequence of each Mother Ship. Subsequently, a method of filtering input data for the time-optimal rendezvous problem by the optimal flight time and the corresponding optimal relative phase is proposed to construct a station-building database with less computation burden. The optimal relative phase is obtained by solving free final-phase continuous low-thrust time-optimal problem. The station-building database is composed of the results of continuous low-thrust time-optimal rendezvous problems. At last, a branch-and-bound algorithm based on the greedy strategy is adopted to rapidly get the result by maximizing the minimal mass among the stations. The final score of our solution is 4315.6663, which is obtained after the competition.

Distributed dynamic rendezvous control of the AUV-USV joint system with practical disturbance compensations using model predictive control

This paper addresses the distributed dynamic rendezvous control problem of the AUV-USV heterogeneous joint system in the presence of unknown external disturbances using nonlinear model predictive control. By utilizing the nonlinear model predictive control framework, the rendezvous problem can be solved without constructing explicit error dynamics of the heterogeneous system. The inputs and states constraints such as spatial and velocity constraints can also be considered simultaneously in an intuitive way. To facilitate distributed implementations, distributed optimization problems with incorporated moving suppression are then constructed to guarantee nominal stability. Considering unknown external disturbances, a practical disturbance compensating method is finally proposed, ensuring the satisfaction of state constraints. The nominal recursive feasibility and stability are proved in this paper. Compared with existing works, the proposed scheme improves the applicability and robustness against external disturbances for dynamic rendezvous between two heterogeneous robots. Simulations and comparisons are conducted to demonstrate the effectiveness and advantages of the proposed method.

Nonlinear rendezvous guidance law design for a receiver UAV based on the Lyapunov stability theory

The rendezvous problem in the autonomous aerial refueling (AAR) was researched for the receiver unmanned aerial vehicle (UAV). The three-dimensional guidance model was constructed for the UAV to pursue the virtual tanker. In order to guarantee the zero miss distance and the intercept angle constraints, a nonlinear rendezvous guidance law was designed for the UAV based on the Lyapunov stability theory. The guidance law generated the normal acceleration command and the lateral acceleration command. The velocity control system was designed by the sliding mode control method. It controled the UAV’s velocity to equal to that of the virtual tanker at the end of the rendezvous phase. The simulation results validate the effectiveness of the designed nonlinear guidance law and the velocity control system.

Optimal low-thrust linearized elliptic orbit rendezvous considering the communication window

The optimal low-thrust linearized elliptic orbit rendezvous problems are studied for both continuous thrust and piecewise constant thrust considering the communication window constraint. Based on the analytical state propagation expressions for linear relative motion under on–off constant thrust, the analytical energy-optimal solution using piecewise constant thrust is derived by the Lagrange multiplier method for given switching time points. This solution is further extended to obtain the analytical energy-optimal solution using continuous thrust. For the piecewise constant thrust with free switching time points, the fuel-optimal problem is formulated as a nonlinear programming problem, which is solved by a modified differential evolution algorithm. In addition, the derived analytical energy-optimal solution for given switching time points is used to reduce the number of optimization variables, and it is numerically proven to effectively reduce the optimization time. Several examples are presented to show the effectiveness of the proposed method for the low-thrust orbit rendezvous problem considering the communication window constraint.

Asymptotic behavior of infinite systems determined by polynomial shift operators

AbstractThis paper studies the asymptotic behavior of infinite dimensional systems determined by polynomial shift operators. The results from ergodic theory and C0‐semigroups are used to obtain a characterization of those initial states which lead to convergent solutions. Estimates on the rate of convergence for solutions are obtained under a stronger ergodic condition on corresponding initial states. The results can be used to solve the robot rendezvous problem and investigate platoon systems.

Fast Neighborhood Rendezvous

In the rendezvous problem, two computing entities (called \emph{agents}) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous \emph{neighborhood rendezvous problem}, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in $O(\Delta)$ rounds ($\Delta$ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in $o(\Delta)$ rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of $O(\sqrt{n})$ rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $\tilde{O}(\sqrt{n\Delta/\delta} + n/\delta)$ rounds for graphs of the minimum degree larger than $\sqrt{n}$, where $n$ is the number of vertices in the graph, and $\delta$ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is $O(n)$, and achieves $\tilde{O}\left( \frac{n}{\sqrt{\delta}} \right)$-round time complexity without using whiteboards. These algorithms attain $o(\Delta)$-round complexity in the case of $\delta = {\omega}(\sqrt{n} \log n)$ and $\delta = \omega(n^{2/3} \log^{4/3} n)$ respectively.

A Symbolic Programming Approach to the Rendezvous Search Problem

In this paper we solve the rendezvous problem on the line with markers that can be dropped at chosen times when the initial distance D between the players is known. In the case of one marker, the M_1 game, the marker is held by player II at the start of the game and, once dropped and found by player I, indicates in which direction player I must move. In the case of two markers, the M_2 game, each player holds one and the dropping times may differ. There is uncertainty regarding the problem initial configuration, and the goal is to minimize the expected rendezvous time that we call the rendezvous value (of the game) denoted R_1 and R_2 for the M_1 and M_2 games respectively. We present an algorithm that computes exactly the rendezvous value of the M_1 game as a function of the dropping time z, i.e. zmapsto R_1(z). Then we show that the function R_1(z) is locally an affine function and we compute the parameters of the local representations of R_1(z). Finally, the rendezvous value of the game R_1=min_z R_1(z) and the optimal dropping times can be determined with the expression of R_1(z). The same proceeding can be extended to apply to the problem M_2. Symbolic execution of programs is a classical technique of program testing in computer science, see King [1] for the pioneering work. In this work we adapt the symbolic execution technique to solve an optimization problem. To our knowledge this is the first time that this is attempted, in particular to deal with rendezvous problems.

Sample data game strategy for active rendezvous with disturbance rejection

In this paper, we investigate rendezvous problem between two spacecraft operating under J2 perturbations and unknown control errors. As is well known, the main challenge is how to produce optimal actions against disturbance and limited measurements. Toward this, we propose a worst-case Nash strategy for the underlying problem, where each player is devoted to minimizing its own cost function while the disturbance works toward maximizing its influence on each player. In addition, a sampled-data game strategy is introduced to alleviate the seeking process of the Nash equilibrium (i.e., NE), and we proved rigorously that intermittently available measurements are sufficient to ensure the existence of the NE. Simulation results verified effectiveness of the proposed scheme.