Orbital Rendezvous Research Articles

Feedback Control for Directional Rendezvous Using Constant-Magnitude Low Thrust

Feedback control for close-range orbital rendezvous, which permits choosing the direction of the final approach, is beneficial for numerous space missions. However, developing such a feedback controller for satellites operating with constant-magnitude low thrust without introducing chattering is nontrivial due to the underactuated nature of the dynamics. This paper develops a feedback control law for rendezvous with a target on a nearly circular orbit, assuming that the chaser satellite utilizes constant-magnitude low thrust. Based on the Clohessy–Wiltshire (CW) model, a sliding surface is chosen on which the states approach the origin. A feedback law driving given initial states to the sliding surface is developed first. Close to the origin, a bang-bang type controller is then employed, which drives the state errors to zero in finite time while allowing to choose the final approach direction along the R-, V-, or H-bar. Solutions to the state trajectories are obtained in closed form, except for the case of the R-bar approach, which is proven to be finite-time stable. It is shown that the higher-order dynamics neglected in the CW equations can be handled through dynamic inversion combined with a hopping maneuver.

Fault-tolerant control for satellite autonomous rendezvous with quality characteristics and actuator uncertainties

An autonomous fault detection and fault-tolerant control solution combining Model Predictive Control (MPC) and Label Distributed Learning (LDL) is developed for the position tracking and attitude synchronization problems in autonomous satellite rendezvous and docking. The proposed algorithm can deal with center of mass (CM) variations and time-varying mass characteristics due to satellite fuel consumption, actuator saturation and fault, and external disturbances. Firstly, a six-degree-of-freedom (6-DOF) satellite translation-rotation coupling model was established to express the relative motion between the satellite and the target, taking the above factors into account. Subsequently, an on-board autonomous closed-loop fault-tolerant control algorithm using MPC is proposed, in which an LDL-based fault detection mechanism is embedded. The scheme specifies that propulsion faults are estimated by calculating the difference between the MPC's prediction of the satellite's future state and the measured change in the actual state. And the mapping matrix between the commanded force/torque and the actual force of the propulsion system is updated in real time to achieve autonomous fault detection and compensation on board the satellite. Among them, the LDL-based fault detection scheme is trained offline. It requires only simple matrix operations for on-orbit use, which is low-cost and suitable for resource-limited space environments. The additional creation of a time-varying disturbance observer allows the estimation and compensation of unmodelled errors and external disturbances. The stability of the proposed fault-tolerant control algorithm is rigorously demonstrated using Lyapunov analysis. Finally, numerical simulations show that the proposed method can successfully achieve autonomous satellite docking under different combinations of thruster faults.

On-ground validation of orbital GNC: Visual navigation assessment in robotic testbed facility

AbstractCubeSats have become versatile platforms for various space missions (e.g., on-orbit servicing and debris removal) owing to their low cost and flexibility. Many space tasks involve proximity operations that require precise guidance, navigation, and control (GNC) algorithms. Vision-based navigation is attracting interest for such operations. However, extreme lighting conditions in space challenge optical techniques. The on-ground validation of such navigation systems for orbital GNC becomes crucial to ensure their reliability during space operations. These systems undergo rigorous testing within their anticipated operational parameters, including the exploration of potential edge cases. The ability of GNC algorithms to function effectively under extreme space conditions that exceed anticipated scenarios is crucial, particularly in space missions where the scope of errors is negligible. This paper presents the ground validation of a GNC algorithm designed for autonomous satellite rendezvous by leveraging hardware-in-the-loop experiments. This study focuses on two key areas. First, the rationale underlying the augmentation of the robot workspace (six-degree-of-freedom UR10e robot + linear rail) is investigated to emulate relatively longer trajectories with complete position and orientation states. Second, the control algorithm is assessed in response to uncertain pose observations from a vision-based navigation system. The results indicate increased control costs with uncertain navigation and exemplify the importance of on-ground testing for system validation before launch, particularly in extreme cases that are typically difficult to assess using software-based testing.

Zero-G Lab: A multi-purpose facility for emulating space operations

During orbital rendezvous, the spacecraft typically approach in the same orbital plane, and the phase of the orbit eventually aligns. Potential rendezvous and docking missions need to be emulated and tested in an on-ground facility for micro-gravity research prior to meeting the harsh conditions of space environment. For orbital docking, the velocity profile of the two spacecraft must be matched. The chaser is placed in a slightly lower orbit than the target. Since all these tasks are quite complex and the realization of space missions are very expensive, any space-related hardware or software’s performance must be tested in an on-ground facility providing zero gravity emulation before initiating its operation in space. This facility shall enable emulation conditions to mimic pseudo zero gravity. It is of critical importance to be equipped with all the necessary ”instruments and infrastructure” to test contact dynamics, guidance, navigation and control using robotic manipulators and/or floating platforms. The Zero-G Laboratory at the University of Luxembourg has been designed and built to emulate scenarios such as rendezvous, docking, capture and other interaction scenarios between separate spacecraft. It is equipped with relevant infrastructure including nearly space-representative lightning conditions, motion capture system, epoxy floor, mounted rails with robots, capability to integrate on-board computers and mount large mock-ups. These capabilities allow researchers to perform a wide variety of experiments for unique orbital scenarios. It gives a possibility to perform hybrid emulations with robots with integrated hardware adding pre-modeled software components. The entire facility can be commanded and operated in real-time and ensures the true nature of contact dynamics in space. The Zero-G Lab also brings great opportunities for companies/startups in the space industry to test their products before launching the space missions. The article provides a compilation of best practices, know-how and recommendations learned while constructing the facility. It is addressed to the research community to act as a guideline to construct a similar facility.

A novel on–off linear quadratic regulator control approach for satellite rendezvous

A novel on–off linear quadratic regulator control approach for satellite rendezvous

Simplified optimization model for low-thrust perturbed rendezvous between low-eccentricity orbits

Trajectory optimization of low-thrust perturbed orbit rendezvous is a crucial technology for space missions in low Earth orbits, which is difficult to solve due to its initial value sensitivity, especially when the transfer trajectory has many revolutions. This paper investigated the time-fixed perturbed orbit rendezvous between low-eccentricity orbits and proposed a priori quasi-optimal thrust strategy to simplify the problem into a parametric optimization problem, which significantly reduces the complexity. The optimal trajectory is divided into three stages including transfer to a certain intermediate orbit, thrust-off drifting and transfer from intermediate orbit to the target orbit. In the two transfer stages, the spacecraft is assumed to use a parametric law of thrust. Then, the optimization model can be then obtained using very few unknowns. Finally, a differential evolution algorithm is adopted to solve the simplified optimization model and an analytical correction process is proposed to eliminate the numerical errors. Simulation results and comparisons with previous methods proved this new method’s efficiency and high precision for low-eccentricity orbits. The method can be well applied to premilitary analysis and high-precision trajectory optimization of missions such as in-orbit service and active debris removal in low Earth orbits.

Autonomous Satellite Rendezvous and Proximity Operations with Time-Constrained Sub-Optimal Model Predictive Control

This paper presents a time-constrained model predictive control strategy for the 6 degree-of-freedom (6DOF) autonomous rendezvous and docking problem between a controllable “deputy” spacecraft and an uncontrollable “chief” spacecraft. The control strategy accounts for computational time constraints due to limited onboard processing speed. The translational dynamics model is derived from the Clohessy-Wiltshire equations and the angular dynamics are modeled on gas jet actuation about the deputy's center of mass. Simulation results are shown to achieve the docking configuration under computational time constraints by limiting the number of allowed algorithm iterations when computing each input. Specifically, we show that upwards of 90% of computations can be eliminated from a model predictive control implementation without significantly harming control performance.

An Iterative Guidance and Navigation Algorithm for Orbit Rendezvous of Cooperating CubeSats

Modern space missions often require satellites to perform guidance, navigation, and control tasks autonomously. Despite their limited resources, small satellites are also involved in this trend, as in-orbit rendezvous and docking maneuvers and formation flying have become common requirements in their operational scenarios. A critical aspect of these tasks is that these algorithms are very much intertwined with each other, although they are often designed completely independently of one another. This paper describes the design and simulation of a guidance and relative navigation architecture for the rendezvous of two cooperating CubeSats. The integration of the two algorithms provides robustness to the solution, by simulating realistic levels of noise and uncertainty in the guidance law implementation. The proposed guidance law is derived based on the linearized equations of orbital motion, written in terms of spherical coordinates. The trajectory is iteratively corrected at a fixed time step, so that errors from the navigation and the initial orbital condition can be recovered. The navigation algorithm processes the bearing and range measurements from a camera and an intersatellite link through an unscented filter to provide the information required from the guidance law. A Monte Carlo campaign based on a 3-DOF simulation demonstrates the effectiveness of the proposed solution.

Neural Network-Based Approximation Model for Perturbed Orbit Rendezvous

An approximation of orbit rendezvous is usually used in the global optimization of multi-target rendezvous missions, which can greatly affect the efficiency of optimization process. A fast neural network-based surrogate model is proposed to approximate the optimal velocity increment of perturbed orbit rendezvous in low Earth orbits. According to a dynamic analysis, the initial and target orbits together with the flight time are transformed into a nine-dimensional normalized vector that is used as the input layer of the neural network. An existing approximation method is introduced to quickly generate the training data. In simulations, different numbers of layer nodes and hidden layers are tested to choose the best parameters. The proposed neural network model demonstrates high precision and high efficiency compared with previous approximation methods and neural network models. The mean relative error is less than 1%. Finally, a case of an optimization of a multi-target rendezvous mission is tested to prove the potential application of the neural network model.

Discrete-Variable-Thrust Guidance for Orbital Rendezvous Based on Feedback Linearization

This research is focused on the analysis, design, and numerical testing of a feedback guidance algorithm for autonomous (unmanned) close-range maneuvering of a chaser spacecraft, in the context of orbital rendezvous with a target vehicle. The relative dynamics of the two vehicles, placed in nearby low Earth orbits, is modeled using the nonlinear Battin–Giorgi equations of relative motion, with the inclusion of all the relevant perturbations, i.e. several harmonics of the geopotential, atmospheric drag, solar radiation pressure, and third body gravitational pull due to Moon and Sun. Unlike several former contributions in the scientific literature, this research considers the orbit perturbing actions on both vehicles, proving that this is crucial for a successful maneuver. Feedback linearization provides the theoretical foundation for the definition and development of a guidance algorithm that is capable of driving the chaser vehicle toward the target spacecraft. Moreover, discrete-variable thrust is considered, and an effective modulation scheme is proposed that includes adaptation of the control gains. Monte Carlo simulations demonstrate that the guidance technique at hand is effective and accurate in driving the chaser spacecraft toward the target vehicle, in the presence of orbit perturbations and unpredictable displacements from the nominal initial conditions.

Optimization of low-thrust multi-debris removal mission via an efficient approximation model of orbit rendezvous

An efficient iterative approximation model for low-thrust rendezvous in low-earth orbit is proposed to be applied in optimization of multi-debris removal missions. Based on an existing method for impulsive orbit rendezvous and a novel iteration process, the propellant cost and thrust time of a low-thrust rendezvous can be quickly evaluated. Moreover, another iterative method is also designed to help find the minimal transfer duration required for orbit rendezvous. Then, its application in the max-reward optimization of debris removal mission with a large-scale targets to be selected is studied. In the three-steps optimization including target selection, sequence optimization, and low-thrust trajectory optimization, the iterative approximation model is used to deal with the constraints and evaluate the objective functions more efficiently. The problem of the eighth Chinese trajectory optimization competition was used to test the performance, and a leading solution including 93 targets was found. Comparisons with previous results were made to prove the advantages of the method in this paper.

Orbital Rendezvous via a General Nonsingular Method

This paper develops a general method to execute orbital rendezvous in continuation of previous work, where a spacecraft’s orbit was represented using the angular momentum and eccentricity vectors of the orbit, along with the orbital energy. A discrete feedback control method using this orbit representation that can be used to autonomously maneuver to arbitrary target orbits is designed. Full rendezvous is accomplished in two distinct phases: orbit matching and a phasing maneuver, which match the five-degree-of-freedom orbit and along-track position, respectively. Once within a short distance of the target position, the spacecraft could enter a close-proximity mode if higher precision is required, such as for docking. The method is verified in simulations, and the usage is compared to two-impulse transfer and rendezvous and continuous low-thrust trajectories. Effects of perturbations are also examined quantitatively. The developed control method is shown to give similar expenditure to unbounded impulsive maneuvers while being applicable to arbitrary orbits, as it avoids singularities or linearizing the dynamics. It is usable by small spacecraft using infrequent, impulsive actuation, which may improve the feasibility and robustness of small spacecraft missions.

Application of a high-precision calibration technology for lunar liftoff and verification using Chang’E-5 mission

<p indent=0mm>Chang’E-5 explorer landed on the lunar surface on December 1, 2020, Beijing time, completed China’s first lunar surface sampling, lunar surface liftoff, and lunar orbital rendezvous and docking, and then returned to the Earth successfully. Accurate positioning of the lander and ascending vehicle assembly are the key to the success of lunar liftoff and as well as lunar orbital rendezvous and docking. Aiming at fulfilling the requirement of high-precision calibration of lunar liftoff of the lander and ascending vehicle assembly of Chang’E-5 mission, the method of using a “three-way ranging and Doppler system + VLBI interferometry system” to realize high precision of a landing position is proposed herein. Then, a high-precision determination algorithm for the lunar liftoff datum mark is provided. Using 20-min observables of Chang’E-5 mission after landing, the landing location of the lander and ascending vehicle assembly could be calculated rapidly, which was 43.0595°N and −51.9214°E, having a −2446-m elevation. Results showed that the difference between the positioning results obtained using the tracking data only <sc>20 min</sc> after landing and the lunar reconnaissance orbiter (LRO) image data provided by NASA was <sc>180 m.</sc> When only longitude and latitude were calculated and elevation was obtained from the lunar digital elevation model, the difference between the positioning result and LRO image data was<sc>97 m,</sc> which could be effectively used as the lunar surface liftoff datum mark. Using <sc>48-h</sc> observables after landing, the landing location of the lander and ascending vehicle assembly was 43.0594°N and −51.9167°E, having a −2550-m elevation, with a difference of only <sc>60 m</sc> compared with the results of LRO.

Design and implementation of a structural subsystem ofChang&amp;rsquo;e-5 probe

<p indent=0mm>The Chang’e-5 probe has achieved the first unmanned lunar surface sampling and return mission in China. The probe faced several technical challenges, including complex configuration, high lightweight and precision requirements, and strict environmental and load conditions. The design scheme of the whole structure and substructure of the probe is briefly introduced in this paper. Particularly, for key technical problems, such as the lunar surface sampling, lunar orbital rendezvous and docking, and returning lunar samples to the earth at high speed <sc>(11 km/s,</sc> close to the second cosmic velocity), this paper expounds on the research contents of the lightweight of the lander and ascender, high-precision composites structure design method, structure scheme of the takeoff platform with the integrated heat-resistant and load-bearing technology, design of the returner bond buffer to adapt to the impact of oblique landing, and labyrinthine thermal-seal design of the hatch door. The results of the mission show that the design of the structural subsystem of the Chang’e-5 probe is a complete success.

A Safety Prediction System for Lunar Orbit Rendezvous and Docking Mission

In view of the characteristics of the guidance, navigation and control (GNC) system of the lunar orbit rendezvous and docking (RVD), we design an auxiliary safety prediction system based on the human–machine collaboration framework. The system contains two parts, including the construction of the rendezvous and docking safety rule knowledge base by the use of machine learning methods, and the prediction of safety by the use of the base. First, in the ground semi-physical simulation test environment, feature extraction and matching are performed on the images taken by the navigation surveillance camera. Then, the matched features and the rendezvous and docking deviation are used to form training sample pairs, which are further used to construct the safety rule knowledge base by using the decision tree method. Finally, the safety rule knowledge base is used to predict the safety of the subsequent process of the rendezvous and docking based on the current images taken by the surveillance camera, and the probability of success is obtained. Semi-physical experiments on the ground show that the system can improve the level of intelligence in the flight control process and effectively assist ground flight controllers in data monitoring and mission decision-making.

Autonomous Optimal Trajectory Planning for Orbital Rendezvous, Satellite Inspection, and Final Approach Based on Convex Optimization

Autonomous Optimal Trajectory Planning for Orbital Rendezvous, Satellite Inspection, and Final Approach Based on Convex Optimization

Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits. Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris. This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories, considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow. Firstly, a set of mean-element orbital dynamic equations of a chaser (spacecraft) and a target (debris) are derived by using the orbital averaging technique, and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem. Subsequently, the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle, and a practical optimization procedure is adopted to find the specific initial costate variables, wherein the necessary conditions of the optimal solutions are all satisfied. Afterwards, the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics. A simple correction strategy about the initialization of the mean elements, specifically the differential mean true longitude, is suggested, which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements. Finally, numerical examples are presented, and specifically, the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems, demonstrating the effectiveness of the optimization method proposed in this article.

A Modular MIMO Millimeter-Wave Imaging Radar System for Space Applications and Its Components

This article presents the design and prototyping of components for a modular multiple-input-multiple-output (MIMO) millimeter-wave radar for space applications. A single radar panel consists of 8 transmitters (TX) and 8 receivers (RX), which can be placed several times on the satellite to realize application-specific radar apertures and hence different cross-range resolutions. The radar chirp signals are generated by SiGe:C BiCMOS direct-digital-synthesizers (DDS) in the frequency range of 1 to 10.5GHz with a chirp repetition rate of 30dB gain with noise figure values of 9dB, respectively. The MIMO radar utilizes patch antenna arrays on organic multilayer printed circuit boards (PCB) with 18dBi gain and 18∘ half power beamwidth (HPBW). Generation of power supply and control signals, analog-to-digital conversion (ADC), and radar signal processing are provided centrally to each panel. The radar supports detection and tracking of satellites in distances up to 1000m and image generation up to 20m, which is required to support orbital maneuvers like satellite rendezvous and docking for non-cooperative satellites.

Linear Cotangential Transfers and Safe Orbits for Elliptic Orbit Rendezvous

This paper presents the theory for linear cotangential transfers and safe orbits for elliptic orbit rendezvous. Expressions for the transfer angle and the required ’s are derived. Singularities in the algorithm can occur if the two orbits intersect. Alternative maneuvers for such singular cases are developed. The linear cotangential transfer algorithm is compared with the nonlinear cotangential transfer and the algorithm is found to be very similar. The development of the linear cotangential transfer leads to a new set of relative orbital elements that are well suited for defining safe trajectories. The characteristics of safe trajectories are discussed and a linear safety checking algorithm is developed. Finally, the combination of the cotangential transfers and safe orbits is used to define safe rendezvous trajectories for elliptical orbit rendezvous.

Minimum Bounds on Multispacecraft Optimal Cooperative Rendezvous

Time- and orientation-free optimal transfers are used to investigate optimal rendezvous orbits for cooperative systems of two and more spacecraft. The problem of finding total optimal cooperative rendezvous orbits is formulated and solved as a nonlinear programming problem using analytic expressions for optimal orbit transfer costs. Each spacecraft has propulsive abilities, and optimal rendezvous orbits are found for planar (match semimajor axis and eccentricity) and three-dimensional (match semimajor axis, eccentricity, and inclination) scenarios. The costs found for this type of rendezvous are lower-bound costs for finite-time full rendezvous where all spacecraft would have matching sets of all six orbit elements. This lower-bound cost, which is realizable for full rendezvous given infinite time and secular perturbations, is shown to be also realizable for finite-time full rendezvous if certain initial conditions can be met. Finally, application of the cooperative rendezvous problem to constellation deployment is briefly discussed.