Reachable set estimation for spacecraft relative motion based on bang-bang principle

For spacecraft formation flight,the information of relative motion reachable set is very important,which can be used to predict the operating boundary of adjacent spacecraft and thus to ensure the safety of spacecraft operation.In this paper,we aim at developing a numerical method to approximate the reachable set for spacecraft relative motion.In particular,we focus on the quality of the approximation and the computational cost.Based on the bang-bang control principle,a polyhedral approximation algorithm is proposed to compute the reachable set of a relative motion spacecraft system.An inner approximation and an outer approximation of the reachable set for the system can be obtained.We prove that the approximation quality measured in Hausdorff distance can be guaranteed.The method is easy to implement and has low computational cost.Finally,the effectiveness of the algorithm is demonstrated by experimental simulation.

Attitude control of spacecraft during propulsion of swing thruster

As for orbit transfer vehicle (OTV) with multiple satellites/payloads carried,the release of each payload will bring serious change to the mass center of OTV and the thrust produced by the swing thruster will form a rather large disturbance to the attitude of OTV. Steering the nozzle to track the estimated center of mass (ECM) of OTV can reduce but not remove the disturbance due to the difference between the ECM and the practical mass center (PCM) of OTV. The practical propelling direction will change with the internal motion during the propulsion process and attitude control system should be enabled to guarantee that the propelling direction is collinear with the command. Since the structural parameters have changed,which is due to internal motion and fuel consumption,the dynamic model have to be formulated to determine these time-varying parameters and the required attitude of OTV should be determined as well. Modulating attitude quaternion results in quasi Euler angles. Based on the resulting quasi Euler angles,a novel attitude switching control law is introduced to control the variable-mass OTV. Simulation results show that,even in the case of structural asymmetry,control torque matrix asymmetry,attitude disturbance and strong coupling between the channels,the attitude of OTV can be controlled perfectly,and the proposed attitude control law is effective for the variable-mass OTV with swing thruster.

Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

Relative dynamics estimation of non-cooperative spacecraft with unknown orbit elements and inertial tensor

The state estimation for relative motion with respect to non-cooperative spacecraft in ren- dezvous and docking （RVD） is a challenging problem. In this paper, a completely non-cooperative case is considered, which means that both orbit elements and inertial tensor of target spacecraft are unknown. By formulating the equations of relative translational dynamics in the orbital plane of chaser spacecraft, the issue of unknown orbit elements is solved. And for the problem for unknown inertial tensor, we propose a novel robust estimator named interaction cubature Kalman filter （InCKF） to handle it. The novel filter consists of multiple concurrent CKFs interlacing with a max- imum a posteriori （MAP） estimator. The initial estimations provided by the multiple CKFs are used in a Bayesian framework to form description of posteriori probability about inertial tensor and the MAP estimator is applied to giving the optimal estimation. By exploiting special property of spherical-radial （SR） rule, a novel method with respect to approximating the likelihood probability of inertial tensor is presented. In addition, the issue about vision sensor＇s location inconformity with center mass of chaser spacecraft is also considered. The performance of this filter is demonstrated by the estimation problem of RVD at the final phase. And the simulation results show that the perfor- mance of InCKF is better than that of extended Kalman filter （EKF） and the estimation accuracy of oose and attitude is relatively high even in the comoletely non-coooerative case.

Close relative motion on distant retrograde orbits

Distant Retrograde Orbits(DROs)in the Earth-Moon system have great potential to support varieties of missions due to the favorable stability and orbital positions.Thus,the close relative motion on DROs should be analyzed to design formations to assist or extend the DRO missions.However,as the reference DROs are obtained through numerical methods,the close relative motions on DROs are non-analytical,which severely limits the design of relative trajectories.In this paper,a novel approach is proposed to construct the analytical solution of bounded close relative motion on DROs.The linear dynamics of relative motion on DRO is established at first.The preliminary forms of the general solutions are obtained based on the Floquet theory.And the general solutions are classified as different modes depending on their periodic components.A new parameterization is applied to each mode,which allows us to explore the geometries of quasi-periodic modes in detail.In each mode,the solutions are integrated as a uniform expression and their periodic components are expanded as truncated Fourier series.In this way,the analytical bounded relative motion on DRO is obtained.Based on the analytical expression,the characteristics of different modes are comprehensively analyzed.The natural periodic mode is always located on the single side of the target spacecraft on DRO and is appropriate to be the parking orbits of the rendezvous and docking.On the basis of quasi-periodic modes,quasi-elliptical fly-around relative trajectories are designed with the assistance of only two impulses per period.The fly-around formation can support observations to targets on DRO from multiple viewing angles.And the fly-around formation is validated in a more practical ephemeris model.

A simple method to design non-collision relative orbits for close spacecraft formation flying

A set of linearized relative motion equations of spacecraft flying on unperturbed elliptical orbits are specialized for particular cases, where the leader orbit is circular or equatorial. Based on these extended equations, we are able to analyze the relative motion regulation between a pair of spacecraft flying on arbitrary unperturbed orbits with the same semi-major axis in close formation. Given the initial orbital elements of the leader, this paper presents a simple way to design initial relative orbital elements of close spacecraft with the same semi-major axis, thus preventing collision under non-perturbed conditions. Considering the mean influence of J_2 perturbation, namely secular J_2 perturbation, we derive the mean derivatives of orbital element differences, and then expand them to first order. Thus the first order expansion of orbital element differences can be added to the relative motion equations for further analysis. For a pair of spacecraft that will never collide under non-perturbed situations, we present a simple method to determine whether a collision will occur when J_2 perturbation is considered. Examples are given to prove the validity of the extended relative motion equations and to illustrate how the methods presented can be used. The simple method for designing initial relative orbital elements proposed here could be helpful to the preliminary design of the relative orbital elements between spacecraft in a close formation, when collision avoidance is necessary.

Secure motion control of micro-spacecraft using semi-homomorphic encryption

This paper studies the secure motion control problem for micro-spacecraft systems.A novel semi-homomorphic encrypted control framework,consisting of a logarithmic quantizer,two uniform quantizers,and an encrypted control law based on the Paillier cryptosystem is developed.More specifically,a logarithmic quantizer is adopted as a digitizer to convert the continuous relative motion information to digital signals.Two uniform quantizers with different quantization sensitivities are designed to encode the control gain matrix and digitized motion information to integer values.Then,we develop an encrypted state-feedback control law based on the Paillier cryptosystem,which allows the controller to compute the control input using only encrypted data.Using the Lyapunov stability theory and the homomorphic property of the Paillier cryptosystem,we prove that all signals in the closed-loop system are uniformly ultimately bounded.Different from the traditional motion control laws of spacecraft,the proposed encrypted control framework ensures the security of the exchanged data over the communication network of the spacecraft,even when communication channels are eavesdropped by malicious adversaries.Finally,we verify the effectiveness of the proposed encrypted control framework using numerical simulations.