Nonlinear Relative Motion Research Articles

Nonlinear Observability for Relative Orbit Determination with Angles-Only Measurements

This paper presents nonlinear observability criteria for the relative orbital dynamics represented by the solutions of the two-body problem. It is assumed that a chief is on a circular orbit with a prescribed orbital radius, and it measures lines-of-sight toward a deputy only. A differential geometric method, based on the Lie derivatives, is used to derive sufficient conditions for observability of the orbital properties of the deputy. It is shown that under certain geometric conditions on the relative configuration between the chief and the deputy, the nonlinear relative motion is observable from angles-only measurements. The second part of this paper presents a quantitative measure of observability for the relative orbits, and it is formulated by generalizing the observability Gramian of linear dynamic systems. An extended Kalman filter is also developed to numerically illustrate the observability of nonlinear relative orbits with angles-only measurements and to show correspondence between the proposed observability measure and filtered solution accuracy.

Adaptive control for two-spacecraft electromagnetic formation keeping

Electromagnetic spacecraft formation flying is a new type of spacecraft formation using electromagnetic forces between spacecraft to control relative motion. Primary principles of the relative motion of a two-craft electromagnetic formation were discussed. Analytical solutions of control magnetic moments according to commanded control forces for the case of two-spacecraft electromagnetic formation were developed considering the energy consumption equilibrium. Nonlinear relative motion dynamic models for a two-spacecraft electromagnetic formation were derived based on polar coordinates. Parameter uncertainties of the equations because of the approximation of the electromagnetic model and unknown disturbed forces were analyzed. Nonlinear adaptive feedback control laws for formation keeping were then designed,and the closed-loop dynamic procedure of a two-spacecraft electromagnetic formation in a low earth circular orbit was simulated. Results indicate that the relative motion equations and the adaptive control laws are valid. The formation can effectively converge to desired states and the unknown parameters are estimated accurately. The simulations demonstrate that electromagnetic forces between spacecraft can be used to realize spacecraft formation keeping and the electromagnetic formation flying is feasible.

Probability analysis and parameter estimation for nonlinear relative wave motions on a semi-submersible using the method of LH-moments

Nonlinear relative wave motions and air gap performance greatly affect the operation and security of semi-submersibles in harsh environments. An appropriate air gap is essential to preventing potential impact loads, and accurate prediction for long-return-period events is of great concern. Given the random and nonlinear nature of relative wave motions, a reliable approach for their prediction is a challenging task that warrants further research. In this study, a modified three-parameter Rayleigh distribution model for nonlinear relative wave crests was derived utilising a quadratic transformation of incident waves. The method of LH-moments was used to estimate the unknown parameters from sample data, and the relationship between the parameters and LH-moments was derived. The sample data used for the probability analysis were obtained from a model test programme for a semi-submersible. The first-order parameter of the model was found to decrease with increasing η, whereas the second-order parameter was found to increase. A larger second-order parameter indicates higher nonlinearity of the relative wave motions for a 100-year-return sea state than for a 1-year-return sea state. In general, using LH-moments to estimate the parameters of the probability distribution results in better prediction of large relative wave crests with low probabilities of exceedance, because the estimation of the three parameters in the nonlinear Rayleigh model using LH-moments released the constraint on the linear term.

Optimal Elliptic Orbital Maneuver with Continuous Radial Thrust on the Chaser Using Generating Functions

AbstractThis paper solves the time-fixed optimal elliptic orbital maneuver problem with continuous radial thrust on the chaser based on the generating function technique. Firstly, by using the relative direction cosine matrix, the two-body nonlinear relative motion equations with a continuous radial acceleration on the chaser are obtained in the target’s local-vertical-local-horizontal frame. Secondly, considering that the relative range is small, a dimension of the system is weakly controllable by analyzing the controllability of the system, and then the problem is simplified into a two-dimensional optimal control problem with integral and terminal constraints. Thirdly, the optimal control problem is transformed into a two-point boundary value problem by using the Pontryagin’s minimum principle. Then a proper form of the generating function is proposed, and the differential equations associated with initial conditions are derived. Finally, the initial value of the adjoint variable is obtained by solving ...

FORCED VIBRATION OF TIP-MASSED CANTILEVER WITH NONLINEAR MAGNETIC INTERACTIONS

This paper predicts the nonlinear relative motion of a cantilever with a tip mass and magnets based on a distributed-parameter model. The Kelvin viscoelastic model is used to account for the damping in the cantilever. Under the harmonic base excitation, the governing equation is deduced via a coordinate transformation linked to its static equilibrium. To qualitatively validate those results captured from the approximate methods, the finite difference method and the multi-scale method are respectively employed to determine the natural frequency and the steady-state response of forced vibration. It is analytically demonstrated that those modes uninvolved in a certain resonance actually have no effect on the response amplitude of the stable steady-state motion. The effects of forcing amplitude, viscoelastic damping, and tip mass on the steady-state response are detailed via the amplitude–frequency response curves. For the first time, the current works theoretically illustrated the conversion from the hardening-type behavior to the softening-type versus the augmenting magnetic force, as well as the opposing effect of different tip masses on the first mode and the higher modes.

Analysis of the applicability of collision probability algorithms for nonlinear relative motion

In the calculation of the collision probability between space objects, the assumption of linear relative motion is generally adopted to simplify the problem because most encounters are at high relative velocity. Nevertheless, the assumption is no longer valid for encounters at extremely low velocities, and a new algorithm is urgently needed for computing collision probability for space objects having nonlinear relative motion. In this particular case, the direction associated with relative velocity is reintroduced for integration. The different integral limits would lead to the variations of probability and integral time. Moreover, the application scope of this new algorithm is also presented. Since the nonlinear effect is only significant in some certain situations, the new algorithm needs to be considered only in such certain situations. More specifically, when space objects in circular orbits encounter with a tiny inclined angle (the extreme situation), the new algorithm can derive much more accurate collision probability than the linear method, that is to say, the linearity assumption involved in general collision probability formulation is not adequate anymore. In addition, the deviation of the probability derived by the linear method (linear collision probability) from that derived by the nonlinear method (nonlinear collision probability) also weakly depends on the relative distance and combined covariance, and essentially depends on their ratio.

Composite disturbance attenuation based saturated control for maintenance of low Earth orbit (LEO) formations

Maintenance of high performance formation control is important for low Earth orbit (LEO) formation missions of small spacecraft. In this paper, a model of nonlinear relative motion dynamics is built, and then nonlinear and important perturbations affecting the formation configuration, such as J 2 and atmospheric drag, are analyzed as disturbances. Global navigation satellite system based relative positioning with nonlinear filtering is adopted to provide state information associated with the perturbations. By combining disturbance observer based control with H ∞ state feedback, a composite disturbance attenuation controller is proposed for maintenance of continuous and accurate formation. With consideration of precise control relying on micro thrusters, a composite disturbance attenuation based saturated controller is designed and its stability is proved. Finally, through numerical simulations, we demonstrate that control accuracy is improved after effectively avoiding perturbations and that stabilization can be satisfied using this method.

Three-Dimension $H_\infty$ Guidance Law and Capture Region Analysis

A new three-dimension nonlinear H∞ guidance law is proposed to intercept an unknown maneuvering target. By using the modified polar coordinate (MPC), nonlinear relative motion equations only contain three state variables. Then we apply the nonlinear H∞ control theory into system equations to get an analytic H∞ guidance law, which has the same expression as true proportional navigation (TPN). The proposed H∞ guidance law can ensure system L2 gain less than or equal to a positive constant. Some results of performance analysis including capture region, capture region with acceleration constraint, an integral expression of time-to-go, and miss distance are derived analytically. Particularly, the capture region with acceleration constraint is an important improvement upon the capture region of TPN and is significant for applications. Finally, some numerical examples illustrate the theories clearly, and some simulations are conducted to validate the academic results.

Spacecraft Attitude and Orbit Coupled Nonlinear Adaptive Synchronization Control

Precise dynamic model of spacecraft is essential for the space missions, to be completed successfully. Nevertheless, the independent orbit or attitude dynamic models can not meet high precision tasks. This paper developed a 6-DOF relative coupling dynamic model based upon the nonlinear relative motion dynamics equations and attitude kinematics equations described by MRP. Nonlinear synchronization control law was designed for the coupled nonlinear dynamic model, whose close-loop system was proved to be global asymptotic stable by Lyapunov direct method. Finallly, the simulation results illustrate that the nonlinear adaptive synchronization control algorithm can robustly drive the orbit and attitude errors to converge to zero.

A Research on the Collision Probability Calculation of Space Debris for Nonlinear Relative Motions†

The calculation of collision probability is the foundation of collision detection and avoidance maneuver for space objects. Now an assumption of linear relative motion is usually applied in the calculation of collision probability and then the complex 3-dimensional problem can be reduced to a 2-dimensional integral of probability density function over the area of circle. However, if the relative velocity value is very small, the term of linear relative motion is not valid. So it is necessary to consider the calculation of collision probability for nonlinear relative motions. The method used to calculate collision probability for nonlinear relative motion is studied, and test cases are designed to justify the validity of this method. It is applicable to collision probability problems involving relative velocity and error covariance varying with time. The results indicate that it is necessary to calculate collision probability with this nonlinear method under certain circumstances. For example, for elliptical relative motions in Satellite Formation Flying, when the relative velocity is below 100 m/s, the relative error between the linear method and the nonlinear method exceeds 5%; for the problem of conjunction analysis of two satellites with circular orbits, when the relative velocity is below 10 m/s, the relative error is also larger than 1%. Some significant conclusions are obtained for the collision detection system of our country.

Integrated System for Autonomous Proximity Operations and Docking

An integrated system composed of guidance, navigation and control (GNC) system for autonomous proximity operations and the docking of two spacecraft was developed. The position maneuvers were determined through the integration of the statedependent Riccati equation formulated from nonlinear relative motion dynamics and relative navigation using rendezvous laser vision (Lidar) and a vision sensor system. In the vision sensor system, a switch between sensors was made along the approach phase in order to provide continuously effective navigation. As an extension of the rendezvous laser vision system, an automated terminal guidance scheme based on the Clohessy-Wiltshire state transition matrix was used to formulate a “V-bar hopping approach” reference trajectory. A proximity operations strategy was then adapted from the approach strategy used with the automated transfer vehicle. The attitude maneuvers, determined from a linear quadratic Gaussian-type control including quaternion based attitude estimation using star trackers or a vision sensor system, provided precise attitude control and robustness under uncertainties in the moments of inertia and external disturbances. These functions were then integrated into an autonomous GNC system that can perform proximity operations and meet all conditions for successful docking. A sixdegree of freedom simulation was used to demonstrate the effectiveness of the integrated system.

Optimal Control for Proximity Operations and Docking

This paper proposes optimal control techniques for determining translational and rotational maneuvers that facilitate proximity operations and docking. Two candidate controllers that provide translational motion are compared. A state-dependent Riccati equation controller is formulated from nonlinear relative motion dynamics, and a linear quadratic tracking controller is formulated from linearized relative motion. A linear quadratic Gaussian controller using star trackers to provide quaternion measurements is designed for precision attitude maneuvering. The attitude maneuvers are evaluated for different final axis alignment geometries that depend on the approach distance. A six degrees-of-freedom simulation demonstrates that the controllers successfully perform proximity operations that meet the conditions for docking.

Effect of Kinematic Rotation-Translation Coupling on Relative Spacecraft Translational Dynamics

A CCURATEmodeling of the differential translation and rotation between two spacecraft is essential for cooperative distributed space systems, spacecraft formation flying (SFF), rendezvous, and docking. High-fidelity relative motion modeling, as opposed to absolute motion modeling, is particularly important for autonomous missions [1]. Point-mass models for relative spacecraft translational motion have been extensively studied over the past 50 years, since Clohessy and Wiltshire (CW) presented a rendezvous model for a circular reference orbit and a spherical Earth [2]. Following the work of Clohessy and Wiltshire, variants on the point-mass model were developed, such as generalizations to elliptic reference orbits [3–5] and an oblate Earth [6,7]. The growing interest in SFF motivated the research of relative spacecraft motion modeling, yielding more accurate and complete equations and solutions for perturbed relative motion [8–10]. However, most of the works focused on point-mass, 3 degrees-offreedom (DOF) spacecraft. Obviously, performing a space mission that consists of several cooperative space vehicles requires modeling the relative rotational motion in addition to the relative translation, that is, 6-DOF models. Models for the relative motion of 6-DOF spacecraft have gained attention in the literature only in recent years. Among the first to suggest treating the spacecraft relative angular velocity in an SFF control problem were Pan and Kapila [11], who addressed the coupled translational and rotational dynamics of two spacecraft. By defining two body-fixed reference frames, one attached to the leader and the other attached to the follower, it was proposed [11] to use a two-part relative motion model: one that accounts for the relative translational dynamics of the body-fixed coordinate frame origins, and another that captures the relative attitude dynamics of the two body-fixed frames. A similar modeling approach was used for relative motion estimation [1]. In addition, tensorial equations of motion for a formation consisting ofN spacecraft, each modeled as a rigid body, were derived [12]. However, only the absolute equations of motion were developed [12]; a relative version of these equations was not given. Moreover, a clear mathematical relationship between the developed models and the traditional nonlinear point-mass relative motion and CW models was not provided. The coupling between the translational and rotational motion in the aforementioned models [1,11] was induced by gravity torques. The kinematic coupling, which is essentially a projection of the rotational motion about the center of mass (c.m.) onto the relative translational configuration space, was neglected. It is this kinematic coupling that the current paper is concerned with. In general, rigid-body dynamics can be represented as translation of the c.m. and rotation about the c.m. [13]. Thus, spacecraft relative motion must be composed by combining the relative translational and rotational dynamics of arbitrary points on the spacecraft. Whenever one of these points does not coincide with the spacecraft’s c.m., a kinematic coupling between the rotational and translational dynamics of these points is obtained. The purpose of this paper is to quantify the kinematic coupling effect and to show that this effect is key for high-precision modeling of tight SFF, rendezvous, and docking. This effect is also important in vision-based relative attitude and position control, where arbitrary feature points on a target vehicle are to be tracked. Given two rigidbody spacecraft, the model presented herein is formulated in a general manner that describes the motion between any two arbitrary points on the spacecraft. The relative translational motion is then generated by both the spacecraft orbitalmotion and the rotation about the c.m. In addition, this paper provides a CW-like approximation of the relative motion that includes the kinematic coupling. This new approximation is aimed at alleviating an apparent contradiction in linearized relative motion theories: to obtain linear equations of motion, the spacecraft are assumed to operate in close proximity. However, if the spacecraft are close to each other, then they can no longer be treated as point masses, because the spacecraft shape and size affects the relative translation between off-c.m. points. This effect is accentuated as the distances between spacecraft decrease. The remainder of this paper is organized as follows. First, a background on the relative position and attitude dynamics is given. Then, a new coupled relative spacecraft motion model is presented. The newly developed model is then examined in a simulation.

Satellite Formation Design and Optimal Stationkeeping Considering Nonlinearity and Eccentricity

F ORMATION flying has received much attention in recent years because of the possible advantages of replacing a single, complex satellite with a cluster of smaller ones. Flying a formation of satellites offers improved flexibility and redundancy and the ability to construct much large virtual sensors than can be flown on a single, monolithic satellite. Several missions have identified formation flying as an enabling technology for increasing the science return and reducing the total mission costs [1,2]. A critical problem in the formation-flying application is the formation design. The purpose of formation design is twofold. First, various flying missions have different requirements for formation design, such as the Aperture Synthesis Radar mission, the Earth Observer-1 mission, the Laser Interferometer Space Antenna mission, etc. The primary purpose of formation design is searching the proper formation array that fulfills the formation mission requirement. The scientific or other need of the mission is the main constraint for the formation design and can be defined as the mission constraint for the formation design. Second, formation-flying satellites operate on orbits with long time spans, in general, from several months to several years, and frequent thruster firings to keeping formation will consume so much propellant and cannot be accepted bymany consideringmissions. And so, it is more important to design nature periodic relative motion orbits and avoid the secular drift. This can be defined as the orbit constraint for formation design. The problem of satellite formation design has been studied by many researchers and many advances have been made. Sabol et al. [3] used the Hill’s equations to design formation and proposed four special formations for various formation-flying missions. Hill’s equations are the constant coefficient differential equations and have simple analytical solutions with an in-track secular term in the solutions. The drift can be avoided by a proper initialization. These simple analytical expressions permit the use of intuitive methods to design formations. However, Hill’s equations can only design valid formations for circular reference orbits and linearized relative motion. Carter et al. [4–6] studied the relative motion of two vehicles in nearby elliptical orbits and presented an analytical solution under assumptions of linearization and without perturbations, and developed an initialization procedure to obtain initial conditions for the formation satellites period relative motion at reference orbit perigee. But, the solutions of Carter’s have a definite integral and are complex in form and cannot be conveniently applied in formation design. Using orbital element differences and Cartesian coordinates, respectively, Lane and Axelrad [7] and Xing et al. [8] derived the simple periodic analytic solutions in eccentric orbits under assumptions of linearization. Theseworks investigated the linearized problem of relative motion of formation flying. Considering nonlinearity and eccentricity, Gurfil [9] and Xing et al. [10,11] independently proposed generalized periodic relative motion conditions (GPRMC) for formation flying on arbitrary Keplerian elliptic orbits. This paper focuses on the formation design for nonlinear relative motion and eccentric reference orbits under GPRMC. Because there are some linearized errors for formation design usingHill’s equations and solutions of Xing’s [8], a new method is proposed to design formation considering nonlinearity and eccentricity which use GPRMC to correct the initial conditions derived from Hill’s equations or Xing’s solutions and get the periodic relative motion orbits for formation flying. This approach is attractive for two reasons. First, the process of formation design uses the GPRMC and removes the secular drift of relative motion orbit designed by Hill’s equations or Xing’s solutions and avoids frequent thruster firings to keep formation. More important, the approach adequately uses the previous research results which were derived from Hill’s equations and could use the intuitive methods to design formations for various formation-flying missions. It is assumed that the formation designed by Hill’s equations fulfilled the requirement of the reality formation-flying mission. Considering nonlinearity and eccentricity, the designed formations were not perfectly consistent with Hill’s. The remainder of this paper developed an optimal impulse formation-keeping maneuver based on the orbit constraint (GPRMC) and the mission constraint (Hill’s solutions). We used the orbit constraint and the mission constraint to keep periodic relativemotion and fulfill the formation-flyingmission requirement, respectively.

Relative Motion and the Geometry of Formations in Keplerian Elliptic Orbits with Arbitrary Eccentricity

The well-known Hill-Clohessy-Wiltshire equations that are used for the design of formation flight relative orbits are based on a circular reference orbit Classical solutions such as the projected circular or general circular relative orbit are no longer valid in the presence of eccentricity. This paper studies the effects of eccentricity on relative motion for Keplerian orbits. A new linear condition for bounded motion in relative position coordinates is derived that is valid for arbitrary eccentricities and epoch of the reference orbit. It is shown that the solutions to the Tschauner-Hempel equations that are used for rendezvous in elliptic orbits are directly related to the description of relative motion using small orbital element differences. A meaningful geometric parameterization for relative motion near a Keplerian elliptic orbit of arbitrary eccentricity is also developed. The eccentricity-induced effects are studied and exploited to obtain desired shapes of the relative orbit. Equations relating these parameters to initial conditions and differential classical and nonsingular elements are also derived. This parameterization is very useful for the analysis of more complicated models, such as the nonlinear relative motion problem.

Cyclic Spacecraft Formations: Relative Motion Control Using Line-of-Sight Measurements Only

In this paper, we develop a new nonlinear relative spacecraft motion control law based on line-of-sight measurements only. Each spacecraft tracks its neighboring spacecraft to produce a line-af-sight vector measurement, and the last spacecraft tracks again the first spacecraft to create a cyclic formation. We show that cyclic leaderless formations controlled by line-of-sight measurements only are stable in the sense of energy matching. The initial semimajor axis errors relative to a reference semimajor axis all tend to the same value, which is the centroid of the initial semimajor axis errors. The stable behavior is achieved by using both full line-of-sight feedback (range and bearing) and partial line-of-sight feedback (bearing only). We also investigate the case of leader formations, where a designated leader does not track any other spacecraft. In this case, the semimajor axes of all follower spacecraft will converge to the initial semimajor axis of the leader. We also discuss an intermediate case, in which only part of the spacecraft forms a cycle. In this case, the cyclic formation members will determine the semimajor axes of all the other, noncyclic, spacecraft. We use elementary graph theory to describe the resulting formation topologies and conclude the development with an illustrating example.

Collision Probability for Larger Bodies Having Nonlinear Relative Motion

d = cross-sectional radius of torus dz = subvolume height h = distance above x–y plane of torus P = collision probability PR = collision probability rate R = radius of torus r = polar coordinate in symmetrized encounter plane t = time Vz = in-track velocity in symmetrized encounter frame x, y, z = coordinates used in probability density function = contour integration parameter = three-dimensional probability density function = standard deviation in symmetrized encounter frame x;y;z = standard deviations of relative position uncertainty for each axis in diagonal frame

Second-Order Equations for Rendezvous in a Circular Orbit

Introduction T HIS Note develops an approximate analytical solution of the two-body orbital boundary-value problem, known as Lambert’s problem, using a time-explicit analytical solution of the relative motion problem. Explicitly stated, the problem is to determine the initial velocity of a satellite, given two position vectors and a time of flight between them. Although the classical origin of the problem related to the determination of planetary orbits, present-day applications are in the area of intercept and rendezvous guidance, where the initial velocity of a maneuvering spacecraft for which its position will coincide with a target position at a specified time must be determined. A short history of Lambert’s problem is given in Ref. 1. Numerical solutions of the problem have been developed by Battin and Vaughan1,2 and Gooding3 for example; however, for small separation distances, such as those encountered in many intercept and rendezvous problems, analytical solutions may be determined using a relative motion formulation. A closed-form solution of Lambert’s problem has been developed previously by Clohessy and Wiltshire in Ref. 4 and has been applied to a variety of intercept and rendezvous problems, as in Refs. 5–10, for example. This solution results from a linear approximation of the equations of motion, and thus the validity range is limited. A higher-order approximation of the equations of motion may extend the range of applicability and lead to an increased level of accuracy. There have been several attempts at developing relative motion equations that take nonlinear dynamics into account.11−14 The solutions presented in Refs. 11–14 result from a straightforward expansion, a method that constructs a truncated Taylor series representation of the exact solution. Anthony and Sasaki,12 Kechichian,13 and Kelly14 show how the nonlinear relative motion equations may be used to solve Lambert’s problem. While these solutions are more accurate than the solutions resulting from the linearized model, the validity region is limited to short periods of time due to the presence of secular terms. For a fixed time, solutions derived from the straightforward expansion will converge to the value of the exact solution as the number of terms in the expansion grows. For a fixed number of terms, however, the accuracy is diminished as time increases and the secular terms begin to dominate the solution. Karlgaard and Lutze15 used a multiple-scale perturbation method to determine a solution of the relative motion problem that included the effects of second-order terms kept in the expansion of the Keplerian equations of motion about a circular reference orbit. The multiple-scale approach allowed elimination of the secular terms by determination of the appropriate differential equations that

Application of symmetrized covariances in space confliction prediction

Space vehicle conflict prediction involving launch vehicles, space vehicles and electromagnetic radiation beams make use of position error covariance matrices of the respective objects. Such quantitative conflict predictions are needed to assess risk and help determine if a launch hold, space vehicle maneuver, or beam operational window closure is advisable. Computing the probability of conflict is greatly simplified by transforming respective positions, velocities and vehicle shapes to a coordinate frame in which the combined position error covariance is symmetric. Use of this symmetrized frame permits treatment of asymmetric vehicle shape, non-linear relative motion and time varying error covariance. Symmetrization also improves computation efficiency, which is especially beneficial in maneuver optimization to reduce conflict below a maneuver threshold, where large numbers of iterations are required.

Satellite Collision Probability for Nonlinear Relative Motion

A method for calculating the collision probability between two space vehicles when the relative motion is nonlinear is developed using contour integration methodology. The method involves transforming the problem to a scaled frame in which the error covariance matrix is symmetric in three dimensions. This enables the calculation of probability increments as a function of time throughout the encounter. Thus, changes in space vehicle position, velocity, and error covariance matrices throughout the encounter can be included in the formulation. This method is applicable to low-velocity space vehicle encounters that involve nonlinear relative motion. A software tool was created and exercised with both hypothetical and actual satellite data to demonstrate the method. Results differed from those of a Monte Carlo simulation by only 2%. Only 6% error resulted for a stress case in which the exact solution was known.