Rendezvous Problem Research Articles

Coordination of optimal guidance law and adaptive radiated waveform for interception and rendezvous problems

The authors present an algorithm that allows an interceptor aircraft equipped with an airborne radar to meet another air target (the intercepted) by developing a guidance law and automatically adapting and optimising the transmitted waveform on a pulse-to-pulse basis. The algorithm uses a Kalman filter to predict the relative position and speed of the interceptor with respect to the target. The transmitted waveform is automatically selected based on its ambiguity function and accuracy properties along the approaching path. For each pulse, the interceptor predicts its position and velocity with respect to the target, takes a measurement of range and radial velocity and, with the Kalman filter, refines the relative range and range rate estimates. These are fed into a linear quadratic Gaussian controller that ensures the interceptor reaches the target automatically and successfully with minimum error and with the minimum guidance energy consumption.

Optimal short-range rendezvous using on–off constant thrust

A control parameter direct optimization method for the optimal short-range elliptic orbit rendezvous problem using on–off constant thrust is proposed. This method is based on analytical state propagation expressions for linear relative motion under on–off constant control acceleration. This avoids numerical integration for the orbit dynamic equations. Normalized optimization variables are introduced to replace switching times. Then this problem is transformed into a nonlinear programming problem with bound constraints on the optimization variables and terminal equality constraints. The along-track and cross-track thrusts are used for the in-plane and out-of-plane motions, respectively. Both the minimum-fuel and minimum-time solutions are obtained by using the Matlab function fmincon. Numerical examples are provided to verify the proposed method. The results indicate that the proposed method is accurate and fast.

A Minimum-Fuel Fixed-Time Low-Thrust Rendezvous Solved with the Switching Systems Theory

In this paper, a fuel optimal rendezvous problem is tackled in the Hill-Clohessy-Wiltshire framework with several operational constraints as bounds on the thrust, non linear non convex and disjunctive operational constraints (on-off profile of the thrusters, minimum elapsed time between two consecutive firings...). An indirect method and a decomposition technique have already been combined in order to solve this kind of optimal control problem with such constraints. Due to a great number of parameters to tune, satisfactory results are hard to obtain and are sensitive to the initial condition. Assuming that no singular arc exists, it can be shown that the optimal control exhibits a bang-bang structure whose optimal switching times are to be found. Noticing that a system with a bang-bang control profile can be considered as two subsystems switching from one with control on to with control off, and vice-versa, a technique coming from the switching systems theory is used in order to optimise the switching times.

Rendezvous of multiple nonholonomic unicycles-based on backstepping

ABSTRACTRendezvous problem of nonholonomic unicycles has been considered in this paper and nonlinear distributed controllers based on backstepping are constructed to guarantee the consensus of both position and direction simultaneously. Cascaded theory of nonlinear system is utilised to reduce the rendezvous problem of nonholonomic unicycles to consensus problem of moving direction and consensus problem of position, respectively. The uncontrollable challenge of position system after consensus of moving direction has been successfully removed through dilated coordinate transformation. Then, a modified backstepping methodology based on the coupling structure of position system is derived to guarantee the consensus of each unicycle's position. In the end, not only consensus of position but also consensus of direction has been realised for multiple nonholonomic unicycles using the controllers proposed in this paper. Simulation results using Matlab illustrate the effectiveness of the results of this paper.

Experimental Validation of Distributed Cooperative Control of Multiple Mobile Robots via Local Information Exchange

In this paper, we present the experimental testing results of distributed cooperative control algorithms for multiple mobile robots with limited sensing/communication capacity and kinematic constraints. Rendezvous and formation control problems are considered, respectively. To deal with the inherent kinematic constraints with robot model, the input/output linearization via feedback is used to convert the nonlinear robot model into a linear one, and then the distributed cooperative control algorithms are designed via local information exchange among robots. Extensive experiments using Quanser's QBot2 mobile robot platforms are conducted to validate the effectiveness of the proposed distributed cooperative control algorithms. Specifically, the robot's onboard Kinect vision sensor is applied to solve the localization problem, and the information exchange is done through an ad-hoc peer-to-peer wireless TCP/IP connection among neighboring robots. Collision avoidance problem is also addressed based on the utilization of fuzzy logic rules.

A class of globally stabilizing feedback controllers for the orbital rendezvous problem

SummaryThe development of feedback control systems for autonomous orbital rendezvous is a key technological challenge for next‐generation space missions. This paper presents a new class of control laws for the orbital rendezvous problem. The controllers belonging to this class are guaranteed to globally asymptotically stabilize the relative dynamics of two satellites in circular or elliptic orbits. The proposed design procedure builds on control techniques for nonlinear systems in cascade form, by exploiting the geometric properties of the orbital element description of the satellite motion. A numerical simulation of a formation flying mission demonstrates the effectiveness of this approach for long‐range and low‐thrust rendezvous operations. Copyright © 2017 John Wiley & Sons, Ltd.

Generalized hierarchical block circulant structure of multi-agent systems

In this paper, we consider solving the rendezvous problem in multi-agent systems using factor circulant strategy and generalized hierarchical block circulant strategy. First, we provide a centralized control strategy designed under factor circulant interaction to enable the agents to rendezvous at a point dictated by the beacon. Second, we introduce a hierarchical block circulant structure with the aim of solving the rendezvous problem in a distributed manner and improve the rate of convergence of multiple agents to a point. Simulation results demonstrates the key theoretical results.

Asymptotic behaviour in the robot rendezvous problem

This paper presents a natural extension of the results obtained by Feintuch and Francis in (2012a,b) concerning the so-called robot rendezvous problem. In particular, we revisit a known necessary and sufficient condition for convergence of the solution in terms of Cesàro convergence of the translates Skx0, k≥0, of the sequence x0 of initial positions under the right-shift operator S, thus shedding new light on questions left open in Feintuch and Francis (2012a,b). We then present a new proof showing that a certain stronger ergodic condition on x0 ensures that the corresponding solution converges to its limit at the optimal rate O(t−1/2) as t→∞. After considering a natural two-sided variant of the robot rendezvous problem already studied in Feintuch and Francis (2012a) and in particular proving a new quantified result in this case, we conclude by relating the robot rendezvous problem to a more realistic model of vehicle platoons.

Distributed rendezvous and tracking for multiple unicycles with heterogeneous input disturbances

SummaryThis paper addresses the distributed rendezvous and tracking problems for multiple unicycles in the presence of unknown heterogeneous disturbances. Disturbance compensators are designed to attenuate the effect of unknown disturbances, which rely on a state estimator constructed to estimate the disturbance‐free states of the controlled agents. LaSalle's invariance principle and Barbalat's lemma are employed, respectively, to prove the convergence of the systems. For the rendezvous problem, all agents will rendezvous at a pre‐specified point eventually under the effect of input disturbances. For the tracking problem, all follower agents will track the leader agent and then move with the leader at the same linear and angular velocities under the effect of input disturbances. The potential function approach is employed in constructing the control laws that are capable of avoiding inter‐agent collisions. With this method, the tracking error can be made upper bounded while all the agents keep a ‘safe’ distance from each other. Simulation examples are finally presented to verify the effectiveness of the designed controllers. Copyright © 2017 John Wiley & Sons, Ltd.

Rendezvous with connectivity preservation for multi-robot systems with an unknown leader

ABSTRACTThis paper studies the leader-following rendezvous problem with connectivity preservation for multi-agent systems composed of uncertain multi-robot systems subject to external disturbances and an unknown leader, both of which are generated by a so-called exosystem with parametric uncertainty. By combining internal model design, potential function technique and adaptive control, two distributed control strategies are proposed to maintain the connectivity of the communication network, to achieve the asymptotic tracking of all the followers to the output of the unknown leader system, as well as to reject unknown external disturbances. It is also worth to mention that the uncertain parameters in the multi-robot systems and exosystem are further allowed to belong to unknown and unbounded sets when applying the second fully distributed control law containing a dynamic gain inspired by high-gain adaptive control or self-tuning regulator.

Hierarchical nearly cyclic pursuit for consensus in large‐scale multi‐agent systems

The authors solve the rendezvous problem of multi-agent systems using nearly cyclic pursuit (NCP) and hierarchical NCP (HNCP). First, the control law designed under the NCP strategy enables agents to converge at a point dictated by a beacon. Second, they elevate the NCP strategy into the generalised L-layer HNCP, so that a large-scale system under the NCP can be divided into small groups in the hierarchical structure, leading to increasing its convergence rate compared with the original NCP. Finally, they prove that the HNCP strategy with fewer communication links achieves the same convergence rate as the hierarchical cyclic pursuit. They provide simulation results to demonstrate their method.

Discrete optimal control for Birkhoffian systems and its application to rendezvous and docking of spacecrafts

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

Asymptotics for Infinite Systems of Differential Equations

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of $C_0$-semigroups to obtain a characterisation, in terms of convergence of certain Ces\`aro averages, of those initial values which lead to convergent solutions. Moreover, we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition. These results rely on a detailed spectral analysis of the operator describing the system, which is made possible by certain structural assumptions on the operator. The resulting class of systems is sufficiently broad to cover a number of important applications, including in particular both the so-called robot rendezvous problem and an important class of platoon systems arising in control theory. Our method leads to new results in both cases.

Deterministic Rendezvous with Detection Using Beeps

Two mobile agents, starting at arbitrary, possibly different times from arbitrary nodes of an unknown network, have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. Agents have different labels which are positive integers. Each agent knows its own label, but not the label of the other agent. In traditional formulations of the rendezvous problem, meeting is accomplished when the agents get to the same node in the same round. We want to achieve a more demanding goal, called rendezvous with detection: agents must become aware that the meeting is accomplished, simultaneously declare this and stop. This awareness depends on how an agent can communicate to the other agent its presence at a node. We use two variations of the arguably weakest model of communication, called the beeping model, introduced in [8]. In each round an agent can either listen or beep. In the local beeping model, an agent hears a beep in a round if it listens in this round and if the other agent is at the same node and beeps. In the global beeping model, an agent hears a loud beep in a round if it listens in this round and if the other agent is at the same node and beeps, and it hears a soft beep in a round if it listens in this round and if the other agent is at some other node and beeps. We first present a deterministic algorithm of rendezvous with detection working, even for the local beeping model, in an arbitrary unknown network in time polynomial in the size of the network and in the length of the smaller label (i.e., in the logarithm of this label). However, in this algorithm, agents spend a lot of energy: the number of moves that an agent must make, is proportional to the time of rendezvous. It is thus natural to ask if bounded-energy agents, i.e., agents that can make at most c moves, for some integer c, can always achieve rendezvous with detection as well. This is impossible for some networks of unbounded size. Hence we rephrase the question: Can bounded-energy agents always achieve rendezvous with detection in bounded-size networks? We prove that the answer to this question is positive, even in the local beeping model but, perhaps surprisingly, this ability comes at a steep price of time: the meeting time of bounded-energy agents is exponentially larger than that of unrestricted agents. By contrast, we show an algorithm for rendezvous with detection in the global beeping model that works for bounded-energy agents (in bounded-size networks) as fast as for unrestricted agents.

Shape-based analytic safe trajectory design for spacecraft equipped with low-thrust engines

To rapidly design three dimensional low-thrust safe trajectories for spacecraft orbit transfer and rendezvous problems, an analytic shape-based approach is proposed. Unlike some existing analytic methods, in which a fixed shape of the trajectory is assumed, the proposed trajectory model in this paper is based on the initial and target orbit elements, through which the fitting ability for the trajectory shape is effectively improved. Additionally, the trajectory safety is guaranteed by using the shaping function. Moreover, the proposed approach can avoid computationally intensive numerical integration of the motion equations. The low-thrust trajectories designed by the proposed method can be also used to provide initial guess for more-accurate numerical trajectory optimization. Extensive numerical simulations are presented to show the effectiveness of the proposed approach.

Adaptive Rendezvous for Heterogeneous Channel Environments in Cognitive Radio Networks

Rendezvous is the fundamental challenge in cognitive radio networks to find each other on a specific channel and establish a communication link. The primary focus of this paper is to design an algorithm for blind rendezvous, i.e., a rendezvous without a coordinator or common control channel. Channels to jump are assumed to be homogeneous in prior work; however, in the real world, channels exhibit different conditions. Therefore, we propose an adaptive jumping pattern such that a superior channel is considered more frequently based on its noise level and users have a greater possibility to rendezvous on such a channel. Furthermore, in this paper, we investigate the rendezvous problem with multiple radio interfaces. When multiple radio interfaces are available for users, the possibility of rendezvous increases; however, the number of channels to jump for each interface may not be the same as other users’ interfaces. We propose an adaptive rendezvous algorithm that can function for multiple interfaces and different sizes of channel lists and an adaptive jumping pattern for different channel conditions. We prove that the proposed algorithm provides guaranteed rendezvous and derive the maximum time-to-rendezvous (TTR), which is also verified via simulation results. Further, our algorithm outperforms jump-and-stay in terms of TTR while enabling users to rendezvous on a better channel.

Codes, lower bounds, and phase transitions in the symmetric rendezvous problem

ABSTRACTIn the rendezvous problem, two parties with different labelings of the vertices of a complete graph are trying to meet at some vertex at the same time. It is well‐known that if the parties have predetermined roles, then the strategy where one of them waits at one vertex, while the other visits all n vertices in random order is optimal, taking at most n steps and averaging about . Anderson and Weber (J. Appl. Prob. 1990, pp. 839–851) considered the symmetric rendezvous problem, where both parties must use the same randomized strategy. They analyzed strategies where the parties repeatedly play the optimal asymmetric strategy, determining their role independently each time by a biased coin‐flip. By tuning the bias, Anderson and Weber achieved an expected meeting time of about , which they conjectured to be asymptotically optimal.We change perspective slightly: instead of minimizing the expected meeting time, we seek to maximize the probability of meeting within a specified time T. The Anderson‐Weber strategy, which fails with constant probability when , is not asymptotically optimal for large T in this setting. Specifically, we exhibit a symmetric strategy that succeeds with probability in steps. This is tight: for any , any symmetric strategy with fails with constant probability. Our strategy uses a new combinatorial object that we dub a “rendezvous code,” which may be of independent interest.When , we show that the probability of meeting within T steps is indeed asymptotically maximized by the Anderson‐Weber strategy. Our results imply new lower bounds, showing that the best symmetric strategy takes at least steps in expectation. We also present some partial results for the symmetric rendezvous problem on other vertex‐transitive graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 742–765, 2016

A new geometric guidance approach to spacecraft near-distance rendezvous problem

For the near-distance rendezvous problem of spacecraft, the classical guidance methods, such as R-bar and V-bar, are restricted by the initial relative positions and velocities and require a long period of time to rendezvous. In this paper, a new geometric guidance approach based on continuous low-thrust, which avoids the restrictions on initial states and decreases rendezvous time dramatically and has the advantages of robustness, is presented. First, based on the classical differential geometric curve theory, the dynamics equation of near-distance rendezvous is presented and a derivative term of spacecraft velocity is introduced, by which the relative velocity can be adjusted. Then, a spacecraft guidance curvature law is derived and one navigation ratio and two velocity feedback control gains are provided based on the analysis of motion in the normal and tangential directions of line-of-sight. Moreover, a method to improve the geometric guidance by using a modified linear-quadratic regulator, in which the time and the cost for rendezvous may be optimized by adjusting a ‘decay factor’, is proposed. Finally, numerical simulations about the selection of guidance parameters, about the comparison with the glideslope method, and about the guidance convergence with measurement errors, which prove the validity of the geometric approach for the rendezvous guidance problem of spacecraft, are presented.

Efficient Asynchronous Channel Hopping Design for Cognitive Radio Networks

In a cognitive radio network (CRN), a necessary condition for nodes to communicate with each other is that they have a rendezvous (switching to the same channel at the same time). Most of the existing rendezvous-guaranteed schemes have some undesired requirements, such as demanding role preassignment, a common control channel (CCC), or a synchronous environment. In this paper, we first define the complete rendezvous problem and then propose a staggered channel hopping scheme (SCH) to solve the complete rendezvous problem. The SCH utilizes the Chinese remainder theorem (CRT) to enable a node to have a rendezvous with any of its neighbors. We have proved the correctness of SCH through the CRT. We have also analyzed the performance of the SCH under some sample scenarios. Simulation results verify the superiority of the SCH in terms of time-to-rendezvous (TTR), standard deviation of TTR, and maximum TTR (MTTR).

Tight Lower Bounds for Channel Hopping Schemes in Cognitive Radio Networks

In this paper, we consider the two-user multichannel rendezvous problem in a cognitive radio network (CRN) and derive tight lower bounds for maximum time-to-rendezvous (MTTR) and maximum conditional time-to-rendezvous (MCTTR) of various channel hopping (CH) schemes under a channel loading constraint. In the symmetric and synchronous setting, we propose a novel Cycle-Adjustable Channel Hopping (CACH) scheme to achieve the MTTR lower bound (when the channel loading is bounded above by $1/u$ with $u$ being a prime power). Thus, the MTTR lower bound is tight and the CACH scheme is optimal in minimizing MTTR among all the symmetric and synchronous CH schemes under the same channel loading constraint. In the asymmetric setting, we show that the classical wait-for-mommy strategy can be used to achieve the MCTTR lower bound, and thus it is optimal. In the symmetric and asynchronous setting, we also show a hierarchical construction of an asynchronous CH sequence by using two smaller asynchronous CH sequences. To further understand the effect of channel loading to the other performance metrics in a CRN, we perform various computer simulations for various CH schemes. Our simulation results show that the average time-to-rendezvous of CACH is independent of the total number of channels, and it is also robust to the disturbance of primary users.