Mai Bando

Spacecraft formation flying dynamics and control using the geomagnetic lorentz force

The motion of a charged satellite subjected to the Earth’s magnetic field is considered. The Lorentz force, which acts on a charged particle when it is moving through a magnetic field, provides a new concept of propellantless electromagnetic propulsion. A dynamic model of a charged satellite, including the effect of the Lorentz force in the vicinity of a circular or an elliptic orbit, is derived and its application to formation flying is considered. Based on Hill–Clohessy–Wiltshire equations and Tschauner–Hempel equations, analytical approximations for the relative motion in Earth orbit are obtained. The analysis based on the linearized equations shows the controllability of the system by stepwise charge control. The sequential quadratic programming method is applied to solve the orbital transfer problem of the original nonlinear equations in which the analytical solutions cannot be obtained. A strategy to reduce the charge amount using sequential quadratic programming is also developed.

Periodic Orbits of Nonlinear Relative Dynamics and Satellite Formation

In this paper, leader―follower formation and reconfiguration problems based on the periodic orbits of the nonlinear relative dynamics along a circular orbit are considered. First, initial conditions of coplanar and noncoplanar relative orbits are characterized by the initial true anomaly, mean motion, semimajor axis, eccentricity, and inclination angle of the followers inertial orbit. Based on the property of null controllability with vanishing energy of the Hill―Clohessy―Wiltshire equations, L 1 suboptimal feedback controllers are designed via the linear quadratic regulator theory. Simulation results for three examples are given. A comparison with the reconfiguration problem based on the periodic orbits of the Hill―Clohessy―Wiltshire equations shows that replacement by nonlinear periodic orbits does not increase the L 1 -norm of the feedback control.

New Lambert Algorithm Using the Hamilton-Jacobi-Bellman Equation

T HE two-point boundary-value problem (TPBVP) for Keplerian motion,which is known asLambert’s problem, is a fundamental problem in space trajectory design. Lambert’s problemmay be stated as follows: given initial and final position vectors, determine the initial velocity which will allow a transfer in a specified flight time. The classical approach to Lambert’s problem is based on geometry of the radius vectors, which directly stems from Lambert’s theorem [1]. After the early works of Gauss [2], a number of numerical algorithms were proposedbyLancaster et al. [3],Battin [4], andother researchers [5–7]. However, these solution procedures cannot be applied to the perturbed problem, because the geometry of the dynamics changes. On theother hand, general TPBVPs are solvedbymeans of numerical methods such as the shooting method and the finite difference method. Lizia et al. [8] and Armellin and Topputo [9] developed an integration scheme for the solution of the TPBVP and astrodynamics applications concerning the computation of trajectories in thevicinity of the libration points in the restricted threeand four-body problem. Guibout and Scheeres [10] solved the TPBVP using the theory of canonical transformation. These approaches [8–10] rely on a power series expansion along a nominal trajectory. Hamilton’s principle [11] is an alternative formulation of the differential equations of motion of spacecraft, which states that the trajectory between two specified states at two specified times is an extremum of the action integral. Motivated by this observation, this paper attempts to show that a solution toLambert’sproblemisdirectly obtained by minimizing the action integral. This problem can be viewedasanoptimalcontrolproblembyreplacingkineticenergywith a quadratic performance index of the control input, so that the initial velocity is found as the optimal control. Then, the solution is given by the Hamilton–Jacobi–Bellman (HJB) equation. The approximation methods to the solutions to the HJB equation were studied by many authors based on series expansion techniques [12–14]. In [14], a closed-formsolutionof theHJBequation isobtainedbyexpandingthe value function as a power series in terms of the state and the constant Lagrange multipliers. Although higher-order approximations are possible to obtain by series expansion solutions, their computations are time-consuming and there is no guarantee that the resulting solution improve the performance. Another approach is through successive approximation, where the HJB equation is reduced to a sequence of the first-order linear partial differential equations [15– 17]. Mizuno and Fujimoto [18] showed that the HJB equation is effectively solved by the Galerkin spectral method with Chebyshev polynomials based on successive approximation. In this paper, the TPBVP of the Hamiltonian system is treated as an optimal control problemwhere the Lagrangian function plays a role as a performance index. Similar toMizuno and Fujimoto, our approach is based on the expansionof thevalue function in theChebyshevserieswithunknown coefficients, considering the computational advantages of the use of Chebyshev polynomials. The differential expressions that arise in the HJB equation are also expanded in Chebyshev series with the unknown coefficients. As a consequence, the algorithm is much simpler than the procedure based on series expansion, and higherorder approximations are possible to obtain for more complicated nonlinear dynamics. Our algorithm can provide a solution to the TPVBP using the spectral information about the gravitation potential function and is also applicable to the TPVBP under a higher-order perturbed potential function without any modification. The paper is organized as follows. Section II is the problem statement. Section III.A reviews the Hamilton principle and the motivation of our method, Sec. III.B introduces the HJB equation in optimal control theory, and Sec. III.C formulates the solution procedure. Section IV presents simulation results for the two-body problem and the circular restricted three-body problem.

Periodic Orbits of Nonlinear Relative Dynamics Along an Eccentric Orbit

This paper is concerned with formation acquisition and reconfiguration problems with an eccentric reference orbit. As a first step, the characterization problem is considered for all initial conditions that constitute periodic solutions of the nonlinear equations of relative motion. Under the condition that the inertial orbits of the leader and a follower are coplanar, initial conditions of all periodic relative orbits are generated in terms of parameters of their orbits and their initial positions. Then the inertial orbit of the follower is rotated successively around the two axes of its perifocal reference system, and the initial conditions of all noncoplanar relative orbits are derived. Based on these periodic relative orbits, formation acquisition and reconfiguration problems by a feedback controller are formulated. The main performance index of a feedback controller is the total velocity change during the operation. Using the property of null controllability with vanishing energy of the Tschauner-Hempel equations, suboptimal controllers are designed via the differential Riccati equation of the linear regulator theory of periodic systems.

Active Formation Flying Along an Elliptic Orbit

T HE relative motion of a follower satellite with respect to the leader in a given circular orbit is described by autonomous nonlinear differential equations. The linearized equations around the null solution are known as Hill–Clohessy–Wiltshire (HCW) equations [1–3]. The HCWequations were used by many authors to study rendezvous problems (see [4] and references therein). The Tschauner–Hempel (TH) equations replace the HCW equations, when the orbit of the leader is eccentric [2,5]. Rendezvous problems along an eccentric orbit were studied in [4,5]. The HCW equations possess periodic solutions, which are useful as temporary orbits before mission and for proximity operations such as inspection and repair. The leader–follower formation and reconfiguration problems based on the periodic solutions were studied bymany authors [6–10]. The TH equations also have periodic solutions. They are characterized by Inalhan et al. [11], and the initialization procedure to periodic motion is given. Periodic solutions also follow from the transition matrix of the TH system given byYamanaka and Ankersen [12]. The effects of eccentricity on the shape and size of relative orbits are studied by Sengupta andVadali [13]. Periodic solutions of the TH equations are used for formation flying [14,15] because no control efforts are needed to maintain them. However, their period is fixed and is equal to that of the leader orbit, which would be inconvenient for a quick inspection of the leader. The shape of periodic solutions is irregular compared with that of the HCWequations, which would be undesirable for some missions. In this Note, active formation flying for the TH system is considered, in which the desired relative orbit of the follower is generated by an exosystem. This allows for flexibility of the shape and period of the reference orbit. Typical examples are elliptic relative orbits of the HCWequations with higher frequencies. Formation flying for the TH system with generated reference orbits has not been studied in the literature. To realize such a formation flying, the output regulation theory for linear periodic systems recently given by Ichikawa and Katayama [16] is employed. Output regulation covers tracking and disturbance rejection [17], but the tracking aspect of the theory is used. The regulator equation, which is necessary to achieve output regulation, is a differential equationwith periodic coefficients. The main contribution of this Note is to show that the regulator equation can be solved algebraically if the system is in the controllable canonical form and the observation matrix is of a special form. To achieve asymptotic tracking, stabilizing feedback controls are necessary. They are designed by the differential Riccati equation (DRE) of the linear quadratic regulator theory [18]. To show the effectiveness of this approach, two examples are given. In the first example, the orbit of the leader is assumed to be circular and the HCW equations are considered. The reference orbit of the follower for formation flying is circularwith arbitrary frequency. Starting from a periodic solution of the HCW equations, the follower satellite is asymptotically steered to the reference orbit. In this case, both the regulator equation and theRiccati equation become algebraic. TheL1 norm of the input ΔV and the settling time are given as functions of the weight parameter in the Riccati equation. The L1 norm of the input to maintain the reference orbit for one period is also calculated. These performance indices are further examined by varying the frequency of the reference orbit. The second example is concerned with the TH equations for the elliptic leader orbit. The reference relative orbit of the follower is circular, as in the first example. The regulator equation has periodic coefficients, but it is algebraically solved and explicitly given. The DRE is solved backward and the stabilizing periodic solution is obtained. TheL1 normof the input and the settling time are computed by varying eccentricity and frequency of the reference orbit.

Optimal Pulse Strategies for Relative Orbit Transfer Along a Circular Orbit

This paper is concernedwith a fuel-optimal relative orbit transfer problem forHill–Clohessy–Wiltshire equations. The input is of the pulse type and the performance index is the total velocity change required for a transfer. For the in-plane motion, the existence of optimal three-pulse strategies is shown and the minimum total velocity change is explicitly given under the condition that the difference of orbit size is relatively larger than that of drift velocity. For the out-of-plane motion, the existence of optimal single-pulse strategies is shown. The orbit transfer problem covers formation reconfiguration problem as a special case. Midpoints of the optimal pulses coincide with those points at which the velocity along the radial direction is zero. Using this information and null controllability with vanishing energy of the Hill–Clohessy–Wiltshire equations, suboptimal feedback controllers are designed. Simulation results with optimal open-loop strategies and feedback controls are given for two formation reconfiguration problems.

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.

Spacecraft Relative Dynamics under the Influence of Geomagnetic Lorentz Force

The motion of a charged satellite subjected to the Earth’s magnetic field is considered. Lorentz force, which acts on charged particle when it is moving through the magnetic field, provides a new concepts of propellant-less electromagnetic propulsion. We derive a dynamical model of a charged spacecraft including the effect of Lorentz force in the vicinity of circular and elliptic orbit and consider its application to formation flight. Based on Hill-Clohessy-Wiltshire equations and Tschauner-Hempel equations, analytical approximations for the relative motion in Earth orbit are obtained. The analysis based on linearized equations in the equatorial case show that the stability of the periodic solutions. Numerical simulations revealed that the periodic solutions which are useful for the formation are obtained for both circular and elliptic reference orbits.The control strategy for the propellantless rendezvous and reconfiguration are developed.

Near-Earth Asteroid Flyby Survey Mission Using Solar Sailing Technology

The purpose of this paper is to investigate the possibility of on asteroid (NEA) survey mission enabled by advanced solar sailing technology. The study is focused not on the solar sail spacecraft itself but on its orbital dynamics to realize the missions. A novel NEA flyby survey mission with a lightweight solar sail spacecraft to increase the accessibility to NEA flybys located in the vicinity of the Earth’s orbit is proposed. A numerical study suggests that our approach increases the opportunities in proximity to NEAs, which have eccentric and inclined orbits.

In-Plane Motion Control of Hill–Clohessy–Wiltshire Equations by Single Input

T HE linearized equations of the relative motion for a circular leader orbit are known as Hill–Clohessy–Wiltshire (HCW) equations [1–4]. They are useful for rendezvous and docking [5–9]. In the HCW equations, the in-plane and out-of-plane motions are decoupled, and the latter is a simple sinusoidal motion. The in-plane orbit is in general a drifting ellipse and is characterized by four parameters representing the size, the drift velocity, the deviation of the center of the ellipse from the leader, and the position of the follower in the relative orbit. When the drift parameter is zero, the orbit is an ellipse, and it is useful for passive formation flying [4,10–15] because no energy is required to keep the follower in such an orbit. For rendezvous problems, impulsive maneuvers are often employed, and the minimum-fuel problems with fixed time and/or fixed end conditions have been extensively studied [5–8,16–18]. The optimal solutions are obtained by solving two point boundary problems. In the case of a circular reference orbit, a fixed time problemwas considered by Prussing [5,6], and two-, three-, and fourimpulse solutions were derived. The two-impulse problem with free time and fixed end conditions was studied by Jezewski and Donaldson [7]. Under the fixed time and fixed end conditions, necessary and sufficient conditions for the optimal solution were obtained by Carter [8] and Carter and Brient [16,17] for circular and elliptic orbits. For rendezvous and orbit transfer problems, the number of impulses needed is known, which is at most equal to the dimension of the state space [17,19–21]. Thus, at most four impulses are necessary for an optimal planar rendezvous. Impulsive maneuvers were also used for formation flying [15,22], and in particular, a linear quadratic regulator (LQR) [11–13] was employed for formation keeping [10,23]. Impulsive maneuvers were also employed for the initialization problem inwhich the initial conditions for the nonlinear equations of relative motion are adjusted to those corresponding to periodic solutions [22,24]. A single impulse was sufficient for the initialization of relative motion between Keplerian elliptic orbits, and the impulse with minimum velocity change was derived [24]. The parameterization of elliptic relative orbits was also used by Campbell [14], and the minimum-time problem and the minimum-fuel problemwith a fixed final time by continuous controls were considered. Sampled-data control, which keeps control inputs constant between sampling times, was used in [12]. The HCWequations and the Tschauner–Hempel equations for an elliptic leader orbit are null controllable with vanishing energy (NCVE) [25,26]. This is a property that any state of the system can be steered to the origin with an arbitrarily small amount of control energy in the L2 (square integral) sense [27]. This property guarantees that the L2 norm of the stabilizing feedback controller designed by the LQR theory converges to zero as the penalty matrix on the state decreases to zero. However, the total velocity change required for the transition to the origin is represented by the L1 norm of the feedback. Shibata and Ichikawa [25] showed numerically that the L1 norm also decreases as the penalty matrix decreases. They considered asymptotic formation flying and designed suboptimal feedback controllers through the algebraic and differential Riccati equations. To find the infimum of the L1 norm, a new minimum-fuel problem by impulse inputs has been formulated by Ichimura and Ichikawa [28] in which the number of impulses and impulse times are free, the final position on the reference orbit is free, and the performance index is the total velocity change ΔV required for the transfer. To solve this open-time problem, a new approach was employed. For the in-plane motion, two lower bounds of ΔV were obtained as functions of the difference of the size parameters and drift velocities.Under the condition that the first lower bound is larger than the second, optimal three-impulse strategies that attain the lower bound were constructed. This condition is always satisfied if the drift parameters are zero, namely, in the case of formation reconfiguration. In the optimal strategy, the three impulses are applied when the crosstrack velocity is zero and the impulse direction is along track. Based on this information, the orbit transfer problem by feedback was also considered in [28]. Suboptimal controllers were designed by the discrete-time LQR theory. Theminimum-fuel problem using a crosstrack impulse was also discussed. The minimum ΔV in this case doubles. For the out-of-plane motion, the relative orbit is described by two parameters representing the size and the phase, and optimal single-impulse strategies were obtained. The impulse of an optimal strategy is applied when the follower crosses the orbit plane of the leader. Jifuku et al. [29] replaced the impulse by a pulse and considered the formation-flying problem of [28]. Using the same approach, two lower bounds of ΔV were derived for the in-plane motion, and optimal three-pulse strategies were constructed. The midpoints of the three pulse intervals coincide with the optimal impulse times, and the pulse direction is also along track. The optimal impulse strategies are recovered by letting the pulse width go to zero. Asymptotic formation flying by feedback is also considered in [29]. To design feedback controllers, the HCW system was discretized with a sampling time ofT∕2. This is because the interval between two consecutive pulses in their optimal open-loop strategy is T∕2 or 2k 1 T∕2. For the out-of-plane motion, optimal single-pulse strategies are given. Motivated by the fact that the optimal inputs in [28] and [29] for the in-planemotion are along track, this Note studies formation flying for the in-plane motion with a single input. First, continuous-time feedback controllers are considered, and whether the along-track controller remains better than the two-input controller is examined. TheL1 norm of the controller is computed as a function of theweight parameter of the LQR theory, and whether it approaches to the minimum ΔV of the pulse control is also examined. The HCW systemwith cross-track input is not controllable, but the reconfiguration by cross-track impulse is possible, as shown in [28]. To see why Received 7 December 2011; revision received 21 August 2012; accepted for publication 21 August 2012; published online 26 February 2013. Copyright