Tschauner-Hempel Research Articles

Pattern control for large-scale spacecraft swarms in elliptic orbits via density fields

Space swarms, enabled by the miniaturization of spacecraft, have the potential capability to lower costs, increase efficiencies, and broaden the horizons of space missions. The formation control problem of large-scale spacecraft swarms flying around an elliptic orbit is considered. The objective is to drive the entire formation to produce a specified spatial pattern. The relative motion between agents becomes complicated as the number of agents increases. Hence, a density-based method is adopted, which concerns the density evolution of the entire swarm instead of the trajectories of individuals. The density-based method manipulates the density evolution with Partial Differential Equations (PDEs). This density-based control in this work has two aspects, global pattern control of the whole swarm and local collision-avoidance between nearby agents. The global behavior of the swarm is driven via designing velocity fields. For each spacecraft, the Q-guidance steering law is adopted to track the desired velocity with accelerations in a distributed manner. However, the final stable velocity field is required to be zero in the classical density-based approach, which appears as an obstacle from the viewpoint of astrodynamics since the periodic relative motion is always time-varying. To solve this issue, a novel transformation is constructed based on the periodic solutions of Tschauner-Hempel (TH) equations. The relative motion in Cartesian coordinates is then transformed into a new coordinate system, which permits zero-velocity in a stable configuration. The local behavior of the swarm, such as achieving collision avoidance, is achieved via a carefully-designed local density estimation algorithm. Numerical simulations are provided to demonstrate the performance of this approach.

Bounded relative motions near Keplerian orbits in spherical coordinates

The problem of relative motion for arbitrary Keplerian orbits is modeled in a curvilinear coordinate system. A set of spherical coordinates rather than the classical Cartesian coordinates is used. The techniques such as linearization and normalization are performed to simplify the relative equations of motion as a linear form that has a similar structure as the Tschauner-Hempel (T-H) equations but expressed in spherical coordinates. The analytical solution in spherical coordinates is obtained by using the same technique as shown in the classical T-H equations. Based on this solution, bounded relative motions and the boundedness conditions are recognized for elliptical, parabolic and hyperbolic orbits, respectively. The geometry of bounded relative motions near parabolic and hyperbolic orbits are revealed, which has not been reported in literatures yet. It shows a strong connection with the “leader-follower” formation near elliptical orbits. Numerical simulation demonstrates the improved accuracy of the newly-developed solution and shows the existence of bounded relative trajectories for unbounded reference orbits.

Application of Floquet theory to formation flying in elliptic orbits via a Hamiltonian structure-preserving control

In this study, an improved Hamiltonian structure-preserving controller is applied for formation flying in elliptic orbits. The Lyapunov-Floquet theory is applied to the Tschauner-Hempel (TH) equation in order to generate its transformed form with two zero eigenvalues and two pairs of imaginary eigenvalues. Then, an improved Hamiltonian structure-preserving (HSP) control is introduced to the transformed system to eliminate the instability of double zeros. All bounded trajectories can be generated by basic solutions, which effectively reveal the following performances of the controller, such as the relationship between the frequencies of basic motions and control gains are detected and the wide selection of gains for stability. Local and global optimizations on gains are discussed based on different performances of the controller. One focuses on the sensitivity of output with respect to input of controller and the other aims at a minimal fuel consumption for the maximum likelihood. The ergodic relationship between the fuel consumption and initial condition is established for the minimal consumption. To avoid in-orbit collision, a collision-free strategy based on the artificial potential function is proposed to enhance the proposed controller, which make sure a minimum safety distance for the formation.

Global fuel consumption optimization of an open-time terminal rendezvous and docking with large-eccentricity elliptic-orbit by the method of interval analysis

By defining two open-time impulse points, the optimization of a two-impulse, open-time terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit is proposed in this paper. The purpose of optimization is to minimize the velocity increment for a terminal elliptic-reference-orbit rendezvous and docking. Current methods for solving this type of optimization problem include for example genetic algorithms and gradient based optimization. Unlike these methods, interval methods can guarantee that the globally best solution is found for a given parameterization of the input. The non-linear Tschauner- Hempel(TH) equations of the state transitions for a terminal elliptic target orbit are transformed form time domain to target orbital true anomaly domain. Their homogenous solutions and approximate state transition matrix for the control with a short true anomaly interval can be used to avoid interval integration. The interval branch and bound optimization algorithm is introduced for solving the presented rendezvous and docking optimization problem and optimizing two open-time impulse points and thruster pulse amplitudes, which systematically eliminates parts of the control and open-time input spaces that do not satisfy the path and final time state constraints. Several numerical examples are undertaken to validate the interval optimization algorithm. The results indicate that the sufficiently narrow spaces containing the global optimization solution for the open-time two-impulse terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit can be obtained by the interval algorithm (IA). Combining the gradient-based method, the global optimization solution for the discontinuous nonconvex optimization problem in the specifically remained search space can be found. Interval analysis is shown to be a useful tool and preponderant in the discontinuous nonconvex optimization problem of the terminal rendezvous and docking than GA.

State transition matrix for long-distance formation with J2 in eccentric orbits

The relative state transition of spacecraft formation flying is considered in this paper. In eccentric orbits, the Tschauner–Hempel (TH) equations are used to express the relative motion between a deputy spacecraft and a chief spacecraft. Perturbation forces are not considered in the TH equations, and the relative distance between two spacecraft is limited to a short range. In this paper, the effects of the J2 perturbation forces are considered, and the case of relatively long distance between two spacecraft is focused. A state transition matrix applicable to such cases is derived. The state transition matrix is expressed by adding some compensating terms to the state transition matrix of the TH equations. The usefulness of the proposed state transition matrix relative to the state transition matrix of the TH equations is shown through numerical simulations from the viewpoint of position error.

A Simple Time Domain Collocation Method to Precisely Search for the Periodic Orbits of Satellite Relative Motion

A numerical approach for obtaining periodic orbits of satellite relative motion is proposed, based on using the time domain collocation (TDC) method to search for the periodic solutions of an exactJ2nonlinear relative model. The initial conditions for periodic relative orbits of the Clohessy-Wiltshire (C-W) equations or Tschauner-Hempel (T-H) equations can be refined with this approach to generate nearly bounded orbits. With these orbits, a method based on the least-squares principle is then proposed to generate projected closed orbit (PCO), which is a reference for the relative motion control. Numerical simulations reveal that the presented TDC searching scheme is effective and simple, and the projected closed orbit is very fuel saving.

Guidance and control for satellite proximity operations

Purpose – In this paper, the development of relative guidance and control algorithms for proximity operations to satellite in elliptical orbit are presented. The paper aims to discuss these issues. Design/methodology/approach – The process of autonomous proximity is divided into three phases: proximity manoeuvre, flyaround manoeuvre, and hovering manoeuvre. The characteristics of the three phases are analyzed. Different guidance algorithms are based on using the analytical closed-form solution of the Tschauner-Hempel (TH) equations that is completely explicit in time. Lastly, the linear quadratic regulators control algorithm based on the linearized TH equations is developed to minimize the initial state errors in the last phase. Findings – This paper defines three phases in the satellite proximity operations and develops the guidance and control algorithms. Then, the relative guidance and control algorithms are illustrated through different numerical examples. And the results demonstrate the effectiveness...

Graphical Generation of Periodic Orbits of Tschauner-Hempel Equations

T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.

Digital LQR control scheme to maintain the separation distance of the NASA benchmark tetrahedron constellation

In previous papers, the authors developed an active control scheme based on the linear quadratic regulator (LQR) to maintain the separation distance constraints of the NASA benchmark tetrahedron constellation. Two different active control schemes were used to maintain the separation distance constraints, but one control scheme applied to every eccentricity of the NASA benchmark problem. Through this paper, the digital LQR strategy will be developed with this active control scheme to maintain the separation distance conditions of the tetrahedron constellation. The motion of a pair of satellites in an elliptical orbit about the Earth is defined by the linearized Tschauner–Hempel (TH) equations. These equations will be defined discretely in the true anomaly angle for the development of the digital control scheme. In summary, the digital LQR approach will be implemented and studied to determine how this active control scheme maintains the separation distance constraints for different specific sizes of the NASA benchmark tetrahedron constellation.

Solar Pressure Effects for a Constellation in a Highly Elliptical Orbit

I NHIGHLY elliptical orbits (HEO), the solar pressure can perturb themotion of a constellation because of the longer exposure of the satellites to the sun [1]. To study its effects on the controller, the Tschauner–Hempel (TH) equations are augmented to include the effects of the solar pressure force. These TH equations are also expressed to include the effects of the perturbation due to the oblateness of the Earth (or J2 perturbation) [2]. To formulate this problem, theCarter–Humi (CH) approach [3] is used inwhich theTH equations are expressed in terms of the eccentricity and the true anomaly angle. These terms are important for the development of the control scheme [4–6]. The TH equations contain nonlinear terms due to the J2 perturbation [2] that can be considered in the controller. Capo-Lugo and Bainum [2] developed a hierarchical control scheme, in discrete format, to maintain the separation distance constraints for the nonlinear TH equations. This controller is based on the linear quadratic regulator and compensates for these nonlinear effects. The solar pressure force is only dependent on the position of the satellite but can affect the control effort. In conclusion, this paper shows how the solar pressure effects can cause variations in the control effort for the correction of the separation distance constraints for a pair of satellites within a constellation.

Rotational Motion Control by Feedback with Minimum L1-Norm

Rotational Motion Control by Feedback with Minimum L1-Norm

Active Control Schemes to Satisfy Separation Distance Constraints

T HE proposed NASA benchmark tetrahedron constellation is a complex formation to be maintained at every apogee point. In previous papers, the authors developed a strategy and determined the initial conditions to maintain the tetrahedron formation [1,2]. With this strategy, the constellation maintains the separation distance constraints for a limited number of orbits, depending on the specific size of the NASA benchmark problem [3]. After a pair of satellites violates the separation distance constraints [1,2], a control strategy is needed to maintain the separation distance conditions [3]. The NASA benchmark problem [3] establishes that the nominal separation distance between any pair of satellites within the constellation is 10 km at the apogee point, and, at any other point in the orbit, the separation distance cannot be less than 1 km. To explain the dynamics of a pair of satellites in an elliptical orbit, the linearized Tschauner–Hempel (TH) equations are used. In the TH equations, the coefficient term that varieswith the true anomaly angle will affect the control scheme that is used to maintain the separation distance between a pair of satellites. The control strategy is based on the linear quadratic regulator (LQR) which was used by Bainum, Strong, and Tan (BST) [4–6]. These authors used the LQR control scheme to maintain the separation between a pair of satellites in an along-track formation (string of pearls), and the varying coefficient was adapted in a piecewisemanner. Instead, Carter–Humi (CH) [7,8] used a different approach in which the cost function is defined in terms of the true anomaly angle, and this control scheme will be adapted here to the LQR strategy. These two control schemes will be compared to determine the best response from the BST and CH control approaches. After the constellation is corrected, the Satellite Tool Kit (STK) [9] software is used to propagate the constellation motion and determine when the constellation will again violate the separation distance constraints. Hence, the main objective of this engineering note is to compare and apply these two techniques for the first time to the NASA Benchmark Tetrahedron Constellation to correct the violation of the separation distance constraints [3]. II. Definition of the Specific Sizes (or Phases)

Review of station keeping strategies for elliptically orbiting constellations in along-track formation

Three techniques for station keeping an orbiting constellation of satellites in an elliptical orbit are developed: (1) based on an application of the linearized Tschauner–Hempel (TH) equations for the motion of a daughter satellite relative to a reference (mother) satellite together with the linear quadratic regulator (LQR) control strategy which can be used in a piecewise adaptive manner; (2) since the mathematical model is inherently nonlinear and time varying, a control law based on a non-linear Lyapunov function is applied to daughter satellites’ osculating orbital elements; (3) by carefully selecting relative orbital design parameters so that the relative secular drifts due to the non-spherical Earth’s perturbation in the longitude of the ascending node, the argument of perigee and mean anomaly could vanish or be constrained to a desired value.

Utilization of the influences of gravitational and light fields of the sun for orientation of spacecraft: O. V. Gurko and L. I. Slabkii: (pp. 34–45)

Three techniques for station keeping an orbiting constellation of satellites in an elliptical orbit are developed: (1) based on an application of the linearized Tschauner–Hempel (TH) equations for the motion of a daughter satellite relative to a reference (mother) satellite together with the linear quadratic regulator (LQR) control strategy which can be used in a piecewise adaptive manner; (2) since the mathematical model is inherently nonlinear and time varying, a control law based on a non-linear Lyapunov function is applied to daughter satellites’ osculating orbital elements; (3) by carefully selecting relative orbital design parameters so that the relative secular drifts due to the non-spherical Earth’s perturbation in the longitude of the ascending node, the argument of perigee and mean anomaly could vanish or be constrained to a desired value.