# Analytical Mechanics of Space Systems, Fourth Edition

... The two-body problem STM is defined as, [4,27] ...

... In this section, Analytic Continuation technique has been implemented to compute 2 -6 perturbation accelerations and their higher order partial derivatives with respect to the initial states and implemented on the STM and the second order State Transition Tensors. The 2 -6 perturbation acceleration vectors are defined and simplified using Analytic Continuation as, [19,27], ...

... × 10 −6 , 5 = −0.15 × 10 −6 and 6 = 0.57 × 10 −6 [27]. ...

In this work, the Taylor series based integration approach, Analytic Continuation, has been implemented to compute Higher Order State Transition Tensors for the 2 − 6 gravity and drag perturbed Two-body problem. Analytic Continuation is an integration method applied to solve different fundamental problems of astrodynamics. In this method, two scalar quantities and are defined and differentiated to arbitrary order using the Leibniz product rule to obtain higher order time derivatives of the variables which are implemented in the Taylor series expansion of the solution. Previously, this method has been proved to be highly precise and computationally efficient in propagating Two-body trajectories with full spherical harmonics gravity and atmospheric drag perturbation. More recently, this method has been implemented in propagating gravity and drag perturbed State Transition Matrix with machine precision level of accuracy. An expansion of the procedure to compute 2 gravity perturbed Higher Order State Transition Tensors has also been presented in the subsequent work. In this paper, the method is further expanded to incorporate up to 6 gravity and drag perturbation in the Higher Order State Transition Tensors. Four types of orbits are considered for numerical simulations: LEO, MEO, GTO, and HEO. First, RMS error of the unperturbed STMs comparing to the closed form solution of Battin are presented. Then, the error in the symplectic nature of the gravity perturbed STM and State Transition Tensors are checked, showing double precision accuracy of the STMs and tensors. Finally, initial error of the states of the 2 − 6 gravity and drag-perturbed orbits are propagated using the computed perturbed Higher Order State Transition Tensors and compared against the results obtained using the perturbed State Transition Matrix, showing at least 3 to 4 digits of accuracy improvement while using Higher Order Tensors.

... Mean counterparts of orbital elements are indicated via a superscript bar (Keplerian elements E as E and nearly-nonsingular elements E ns as E ns , respectively). Mean elements are calculated from their osculating counterparts by means of a Brouwer-Lyddane contract transformation [15]. ...

... By specifying the relative orbit geometry in mean element space, the true relative spacecraft motion closely follows the prescribed relative orbit geometry [32]. Note that the relative obit description from Eq. (12) does not make any assumptions on how large the relative orbit is compared to the chief orbit radius, nor does it require the chief orbit to be circular [15]. ...

... As an example, the magnitude of the out-of-plane relative motion is a direct result of differences in the inclination and in the ascending nodes. Whereas differences in the inclination angle δi specify how much out-of-plane motion the relative orbit will have as the satellite crosses the northern-or southernmost regions, ascending node differences δΩ, however, indicates the out-of-plane motion as the satellite crosses the equatorial plane (at the ascending node) [15]. ...

Differential drag is a promising option to control the relative motion of distributed satellites in the Very Low Earth Orbit regime which are not equipped with dedicated thrusting devices. A major downside of the methodology, however, is that its control authority is (mainly) limited to the in-plane relative motion control. By additionally applying differential lift, however, all three translational degrees-of-freedom become controllable. In this article, we present a tool to flexibly plan optimal three-dimensional formation flight maneuvers via differential lift and drag. In the planning process, the most significant perturbing effects in this orbital regime, namely the J2 effect and atmospheric forces, are taken into account. Moreover, varying atmospheric densities as well as the co-rotation of the atmosphere are considered. Besides its flexible and high-fidelity nature, the major assets of the proposed methodology are that the in-and out-of-plane relative motion are controlled simultaneously via deviations in the yaw angles of the respective satellites and that the planned trajectory is optimal in a sense that the overall decay during the maneuver is minimized. Thereby, the remaining lifetime of the satellites is maximized and the practicability and sustainability of the methodology significantly increased. To the best of the authors knowledge, a tool with the given capabilities has not yet been presented in literature. The resulting trajectories for three fundamentally different relevant formation flight maneuvers are presented and discussed in detail in order to indicate the vast range of applicability of the tool.

... Denote the relative position from O t to O p as ρ = [ρ x , ρ y , ρ z ] T ∈ R 3 in F l . As a result, the relative translational dynamics can be described according to [33] ...

... Afterwards, the relative rotational dynamics can be described as follows [33]: ...

... The purpose of the relative position is to keep the pursuer in a safe position r d = [0, 5, 0] T [m], while the relative attitude σ = [0, 0, 0] T [rad] synchronously. The gravity gradient torque τ gg and J 2 perturbation force f J 2 are applied in the simulation [33], i.e., ...

This study presents an adaptive robust fault-tolerant control (FTC) method for spacecraft proximity operations, in the presence of external disturbances, actuator faults, and input saturation. Firstly, a coupled 6-degrees-of-freedom dynamics model is constructed to show the relative motion of the pursuer spacecraft to the target spacecraft. To deal with actuator faults and external disturbances, a basic robust FTC method is designed. Subsequently, an adaptive robust FTC approach is developed to address the negative effect from the input saturation. In particular, by incorporating a novel dead-zone model to represent the saturation nonlinearity, an adaptive technology is applied to compensate for the nondifferentiable integral term in the saturation model. According to Lyapunov stability theory, all the signals in the whole system are proved to be ultimately bounded, and the relative motion tracking errors can converge to arbitrarily small neighborhood around the origin by choosing the suitable parameters. Last but not least, comparative simulations are carried out to validate the superiority of the proposed control strategy.

... Using this recursive method the torques for each joint can be calculated using D-H paramters approach shown in Eq. (1) or PoE approach described by Eq. (13). To generate the torque and the angular positions, velocities and accelerations of each joint must be specified. ...

... While the spherical air-bearing can be kinematically described as a system of three coincident revolute joints each actuating about a different axis, describing the joints in the recursive algorithm is not as trivial as it sounds, resulting in singularities and yielding incorrect results. Instead, an analytical dynamic approach was taken to model the system, using the definition of angular momentum [13] to develop the equations of motion about the spherical air-bearing. ...

... To simplify the problem, this process can be conducted using one link at a time first, as seen in Figure 19, and then building up to seven links to better resemble the dynamics of the system. To correctly represent the inertia of each link about the spherical air-bearing the parallel axis theorem [13] was used. The angular momentum of the i th link about the spherical air-bearing in the body frame can now be determined to be: ...

This paper focuses on the development of a ground-based test-bed to analyze the complexities of contact dynamics between multibody systems in space. The test-bed consists of an air-bearing platform equipped with a 7 degrees-of-freedom (one degree per revolute joint) robotic arm which acts as the servicing satellite. The dynamics of the manipulator on the platform is modeled as an aid for the analysis and design of stabilizing control algorithms suited for autonomous on-orbit servicing missions. The dynamics are represented analytically using a recursive Newton-Euler multibody method with D-H parameters derived from the physical properties of the arm and platform. In addition, Product of Exponential (PoE) method is also employed to serve as a comparison with the D-H parameters approach. Finally, an independent numerical simulation created with the SimScape modeling environment is also presented as a means of verifying the accuracy of the recursive model and the PoE approach. The results from both models and SimScape are then validated through comparison with internal measurement data taken from the robotic arm itself.

... Given the importance stability plays in spacecraft trajectory design, many CLFD control laws are based on Lyapunov control theory. 14,15 The most well known and widely used is perhaps the Petropoulos Q-law. 9,10 Lots of recent work has looked to combine these Lyapunov control laws with global optimisers for Earth-Moon spiral transfers. ...

... Maddock and Vasile 23 extended the Q-law to handle both solar radiation pressure and 3 rd -body effects, as did the aforementioned Epenoy and Pérez-Palau. 17 Baresi et al. 24 derived a Lyapunov controller using relative orbital elements for orbit maintenance around the Martian moon Phobos in a similar fashion to [15]. Pontani et al. have explored Lyapunov stability in the presence of perturbations for the station-keeping and guidance problem. ...

... Lyapunov's second theorem states that for a systemŻ = f (Z), Z = X − X T , the equilibrium point X T is asymptotically stable if there exists a scalar Lyapunov function Q(Z) such that Q(0) = 0; it is positive-definite (Q(Z) > 0, ∀Z = 0); the derivative is negative-definite (Q(Z) < 0, ∀Z = 0); and lim |Z|→∞ , Q(Z) = ∞. 15 A very thorough discussion on the implications of this for trajectory design using nonlinear control can be found in [25]. For the Q-law a stable control is one that ensuresQ < 0 throughout the transfer. ...

Future missions to the Moon and beyond are likely to involve low-thrust propulsion technologies due to their propellant efficiency. However, these still present a difficult trajectory design problem. Lyapunov control laws can generate sub-optimal trajectories with minimal computational cost and are suitable for feasibility studies and as initial guesses for optimisation methods. In this work we combine Lyapunov control laws with state-dependent weights trained via reinforcement learning to design low-thrust transfers from GTO towards low-altitude Lunar orbits. The agent is able to explore third-body effects during training and learn to remain stable to perturbations during the different transfer phases. Three different approaches are investigated: backwards propagation, backwards propagation with freed geometry, and forwards propagation including rendezvous capability with the Lunar SOI. The last of these proves to be the most successful, coming within 6.6% of the optimal solution.

... Mean counterparts of orbital elements are indicated via a superscript bar (Keplerian elements E asĒ and nearly-nonsingular elements E ns asĒ ns , respectively). Mean elements are calculated from their oscu-lating counterparts by means of a Brouwer-Lyddane contract transformation [15]. Aerodynamic drag, a non-conservative perturbation force that is a result of the interchange of momentum between the Earth's atmosphere and the spacecraft surface, retards the motion of satellites orbiting in VLEO. ...

... By specifying the relative orbit geometry in mean element space, the true relative spacecraft motion closely follows the prescribed relative orbit geometry [32]. Note that the relative obit description from Eq. 12 does not make any assumptions on how large the relative orbit is compared to the chief orbit radius, nor does it require the chief orbit to be circular [15]. ...

... As an example, the magnitude of the out-of-plane relative motion is a direct result of differences in the inclination and in the ascending nodes. Whereas differences in the inclination angle δi specify how much out-ofplane motion the relative orbit will have as the satellite crosses the northern-or southernmost regions, ascending node differences δΩ, however, indicates the out-of-plane motion as the satellite crosses the equatorial plane (at the ascending node) [15]. ...

Differential drag is a promising option to control the relative motion of distributed satellites in the Very Low Earth Orbit regime which are not equipped with dedicated thrusting devices. A major downside of the methodology, however, is that its control authority is (mainly) limited to the in-plane relative motion control. By additionally applying differential lift, however, all three translational degrees-of-freedom become controllable. In this article, we present a tool to flexibly plan optimal three-dimensional formation flight maneuvers via differential lift and drag. In the planning process, the most significant perturbing effects in this orbital regime, namely the J2 effect and atmospheric forces, are taken into account. Moreover, varying atmospheric densities as well as the co-rotation of the atmosphere are considered. Besides its flexible and high-fidelity nature, the major assets of the proposed methodology are that the in-and out-of-plane relative motion are controlled simultaneously via deviations in the yaw angles of the respective satellites and that the planned trajectory is optimal in a sense that the overall decay during the maneuver is minimized. Thereby, the remaining lifetime of the satellites is maximized and the practicability and sustainability of the methodology significantly increased. To the best of the authors knowledge, a tool with the given capabilities has not yet been presented in literature. The resulting trajectories for three fundamentally different relevant formation flight maneuvers are presented and discussed in detail in order to indicate the vast range of applicability of the tool.

... As noted by Kozai (1959), the long-periodic perturbations of the first order come from the terms of the second order, and the singularity associated with the critical inclination only arises when such long-period effects are retained. Nevertheless, the distinction between secular and mean elements has often been muddled in the literature, and tabulated mean-to-osculating transformations (Gim and Alfriend 2003;Schaub and Junkins 2018) apply Brouwer's full periodic corrections. This approach will inevitably lead to significant errors near the critical inclination, but can be properly amended by neglecting the long-periodic terms (Breakwell and Vagners 1970). ...

... We establish a numerical averaging approach based on the fast-Fourier transform (Uphoff 1973;Ely 2015) and make detailed comparisons between our vectorial solution and the classical Brouwer-Lyddane (BL) theory. For the latter, we adapt the more streamlined formulas presented in Schaub and Junkins (2018) and Gim and Alfriend (2003). ...

... 2.2. For Brouwer-Lyddane, we have verified that the more streamlined formulas presented in Schaub and Junkins (2018) have been correctly transcribed according to their original sources, excepting the missing sin(2ω) factor in the long-period terms of M, ω, and Ω, which has only been noted in the recent erratum of the latest edition of this widely used monograph. 3 Furthermore, rather than using Lyddane's adhoc modification, Gim and Alfriend (2003) developed a new theory based on Brouwer's generating function that uses equinoctial elements. ...

We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

... The unconstrained equations of motion are here derived as a first step towards writing the constrained equations of motion. The Lagrangian for this buoy system can be written as the summation of three quantities [34]: ...

... and, the Lagrange equation for the distributed parameters is [34]: ...

... where the and are real scalars called the mass and stiffness matrix multipliers with units 1/sec and sec, respectively [40][41][42]. Combining the equations of motion from Eqs. (34), (35) and (73) yields the equation of motion of the flexible buoy: ...

In the recently introduced Variable-Shape heaving wave energy converters, the buoy changes its shape in response to changing incident waves actively. In this study, the dynamic model for a spherical Variable-Shape Wave Energy Converter is developed using the Lagrangian approach. The classical bending theory is used to write the stress-strain equations for the flexible body using Love's first approximation. The elastic spherical shell is assumed to have an axisymmetric vibration behavior. The Rayleigh-Ritz discretization method is adopted to find an approximate solution for the vibration model of the spherical shell. One-way Fluid-Structure Interaction simulations are performed using MATLAB to validate the developed dynamic model and to study the effect of using a flexible buoy in the wave energy converter on its trajectory and power production.

... However, it is computationally burdensome to include high-order terms of this gravitational perturbation in the EKF model. Two common models for this perturbation are the zonal harmonic model and the spherical harmonic model [8]. The zonal harmonic model assumes an axisymmetric gravity potential, only capturing latitude-dependent variations in the gravity field. ...

... The spherical harmonic model has a latitude-and longitude-dependent potential and therefore can more accurately map the planet's gravity field at the cost of computational resources. For EKF propagation, we only consider a second-order zonal harmonic expansion of the gravity field, also known as the J 2 perturbation [8]. With this, the equations of motion are given in [8] as ...

... For EKF propagation, we only consider a second-order zonal harmonic expansion of the gravity field, also known as the J 2 perturbation [8]. With this, the equations of motion are given in [8] as ...

Onboard orbit determination can be challenging in GPS-denied environments. For
these situations a navigation architecture is proposed and numerically investigated.
This architecture involves two satellites, a Chief satellite and a Deputy satellite, orbiting Earth. The Chief satellite is capable of obtaining precise attitude measurements and measuring the body-relative range and direction to the Deputy satellite. These measurements are then used with an Extended Kalman Filter (EKF)
equipped with a second-order zonal harmonic gravity model to estimate the inertial position and velocity of both satellites. The numerical investigation is comprised of multiple Monte Carlo analyses and the effects of several factors on the
steady-state accuracy such as filter tuning parameters and the six Keplerian orbital
elements are quantified to provide insight into mission capabilities. The best performance is achieved when the satellites are in a high-Earth orbit (HEO) and have
a separation varying from 100 km to 200 km. This scenario yields a minimum
3σ steady-state position estimation accuracy of 22 m per axis and minimum 3σ
steady-state velocity estimation accuracy of 0.17 cm/s per axis.

... The state transition matrix (STM) of the two-body problem propagates the error of the initial states over time by working as a sensitivity matrix of the current states to the initial states. The guidance, navigation and control of spacecraft are directly dependent on the accurate computation of the STM [8][9][10][11]. Several works studied the computation of the perturbed STM for various astrodynamics applications. An approximate Cartesian STM computation method with the incorporation of J2 perturbation has been studied [10]. ...

... The detailed derivation of the recursive formulas for higher order time derivative of J2 gravity and drag perturbation are shown herein. The J2 and drag perturbed acceleration vectors are given by, [8,48] ...

... The same approach is used to compute the STM from t0 to t k . The STM is divided into four sub-matrices that represent the sensitivity of the current states with respect to the initial position and velocity as [8], ...

In this work, a new method for space-based angles only orbit estimation is presented. The approach relies on the integration of the novel and highly accurate Analytic Continuation technique with a new measurement model for multiple observers for inertial orbit estimation. Analytic Continuation computes the perturbed orbit dynamics as well as the perturbed State Transition Matrix (STM) in the inertial frame. The new measurement model is developed for simultaneous measurements from a constellation of low-cost observers with monocular cameras producing angles-only measurements. Analytic Continuation and the new measurement model are integrated in an Extended Kalman Filter (EKF) framework where, Analytic Continuation propagates the perturbed dynamics and computes the perturbed STM and the error covariance, whereas the measurements are obtained via the new measurement model. Two case studies that comprise small and large constellations of observers are presented, along with cases of sparse measurements and a study of the computational efficiency of the present approach. Results show that, the new approach is capable of producing highly accurate and computationally efficient perturbed orbit estimation results versus classical EKF implementations.

... The reference axes are first rotated about theb 3 axis by the yaw angle ψ, then about theb 2 axis by the pitch angle θ, and finally about theb 1 axis by the roll angle φ. Thus, the standard yaw-pitch-roll (ψ, θ, φ) angles are the (3-2-1) set of Euler angles [33]. For a fixed frame B orbiting around another body at an angular velocity Ω, the kinematic differential equations can be written as in the following subsection. ...

... The relation between the angular velocities in a body frame B and the Euler rates (ψ,θ,φ) using the (3-2-1) set of Euler angles is [33,34]: ...

... Using the transport theorem, Euler's equation is expressed as [33]: ...

In this article, we propose a mathematical model using the port-Hamiltonian formalism for a satellite’s three-axis attitude system comprising fluid rings. Fluid rings are an alternative to reaction wheels used for the same purpose, since, for the same mass, they can exert a greater torque than a reaction wheel as the fluid can circulate the periphery of the satellite. The port-Hamiltonian representation lays the foundation for a posterior controller that is feasible, stable, and robust based on the interconnection of the system to energy shaping and/or damping injection components, and by adding energy routing controllers. The torques exerted by the fluid rings are modeled using linear regression analysis on the experimental data got from a prototype of a fluid ring. Since the dynamics of turbulent flows is complex, the torques obtained by the prototype lead to a simpler first approach, leaving its uncertainties to a controller. Thus, the attitude system model could be tested in a future prototype before considering a spatial environment.

... The DCM is the most conventional form of representing rotations and orientations between reference frame. Unfortunately it is highly redundant as only 3 parameters are required to represent rotation whereas 9 components of the matrix must be specified [145]. The use of DCM is justified by the simplicity in reference frame conversion, i.e. rotations of vectors, and rotation composition. ...

... Modified Rodriguez Parameters (MRP) are a recent development in orientation representation [146]. It is well suited for optimization and estimation as the representation is minimal and non-singular [145,147]. This implies that no algebraic constraints must be considered for optimization and estimation purposes. ...

... Moreover, it is important to note that MRPs are non-singular but for Φ = 2π which represents a complete rotation about the rotation axis. This singularity can be avoided by switching σ to its shadow set σ S [145]. The shadow set is defined as: ...

... Furthermore, the coupling between rotational and translational motion can cause additional perturbations. For example, the gravitational force acting on a spacecraft can consider orbit-attitude coupling that will perturb the spacecraft motion from the two-body problem with a point-mass spacecraft [1]. Extensions for oblate bodies using spherical harmonic potential models have been derived and utilized in the literature [2][3][4][5]. ...

... , where g and g are [1] In Eq. (9), denotes the gravitational parameter of the central body, B = T is the position vector of the center of mass of the spacecraft expressed in the B frame, and the unit vector ̂ B is defined via ̂ B = B ∕‖ ‖ . As mentioned before, it is assumed in Eq. (9) that the central body has a uniform gravitational field. ...

... The gravitational potential of a central body with a nonuniform gravitational field acting on a point mass is given in the celestial-body fixed frame F by [17], [27] where and are the spacecraft's latitutde and longitude with respect to the F frame, respectively. Also, in Eq. (10), the parameter R M denotes the mean radius of the body being orbited, P n,p denotes the normalized associated Legendre functions, and C n,p and S n,p are the normalized gravity coefficients obtained from experimental data [17,18] or analysis based on assumptions made on the celestial body [1,27]. The normalized associated Legendre functions P n,p , associated Legendre functions P n,p , and Legendre polynomial P n , are given, respectively, by [28], [27] where and (9a) ...

A novel representation of the gravitational force and gravity-gradient torque acting on a rigid-body spacecraft is presented. This formulation considers orbit-attitude coupling perturbations acting on the spacecraft when the gravitational field is modeled using spherical harmonics. Furthermore, the main contribution of this work is the generalization of these coupling terms to any degree and order of spherical harmonic representation. The magnitude of the accelerations are presented for point-mass and rigid-body spacecraft up to degree and order 256 for the case of a spacecraft in orbit around a central body. Numerical simulations are provided for spacecraft orbits near two different central bodies: The Moon and a “Bennu-like” object. The “Bennu-like” object is assumed to have the same size, mass, gravitational parameter, and angular velocity as those of the titular asteroid, but with spherical harmonic gravity constants of the Moon. It is shown that the magnitudes of the accelerations caused by the orbit-attitude coupling are orders of magnitude smaller than the accelerations derived from assuming a point-mass spacecraft. It is also observed that the difference in these orders of magnitudes has the tendency to decrease appreciably as the size of the celestial body decreases, suggesting the importance of consideration of higher order attitude terms in scenarios with large, low-mass spacecraft and/or in orbits around small celestial bodies with highly nonuniform gravitational fields. Therefore, the formalism in this study can be used as an accurate tool to quantify the effects of translational-rotational coupling in space operations

... This section introduces four models of relative motion as follows: the unperturbed nonlinear rela tive model [57], the HillClohessyWiltshire model [18], the exact J2 nonlinear relative model and the linearized J2 relative model [58]. ...

... The unperturbed nonlinear equations of motion relative to the orbital frame are given by [57]: ...

... The tethered satellite's motion in the triangle is described by a set of equations of motion relative to the orbital frame [57] taking into account tether forces. The dynamics is described for each satellite position vector r j = [x j , y j , z j ] T as: ...

This research has been conducted as a part of the trade-off analysis for a tentative ionospheric mission, whose objective is to conduct multipoint ionospheric plasma measurements, distinguishing temporal and spatial variations of the measured parameters. It has been shown that for the experiment in question a formation of four CubeSats is required, which is deployed in a near polar low-Earth orbit and is maintained so as to keep intersatellite distances at 500 to 1000 meters. Furthermore, the four spacecraft should form a tetrahedron, whose shape must be as close to a regular tetrahedron as possible at least near the poles but preferably for as long a part of the orbit as possible.
Prior research shows that maintaining a regular tetrahedral formation of free-flying spacecraft even for the specially designed relative orbits optimized through the analysis of Hill-Clohessy-Wiltshire equations, requires frequent or continuous orbit corrections, which imposes additional constraints on CubeSats design and shortens the mission's lifetime. It turns out that adding space tethers into the system to connect some of the four satellites, on one hand, does not essentially change the system’s complexity, while on the other hand, allows significantly improving the characteristic formation quality. As a result, two alternatives of tethered tetrahedral satellite formations have been designed.
The first designed formation consists of a tethered triangle spinning out of the orbital plane and a free-flying spacecraft. It is shown that a proper choice of the spin rate of the tethered triangle causes it to change its orientation due to gravity-gradient torque and track the free-flying satellite that changes its periodic inertial position due to J2. Furthermore, the design of J2-invariant relative orbits is utilized to minimize the undesired drift. The proposed passive control strategy results in producing high-quality tetrahedra. The second formation, on the other hand, consists of a gravity-gradient stabilized tether and two free-flying spacecraft. It is shown that proper choice of initial conditions for the two connected spacecraft ensures gravity-gradient stabilization of the subsystem and prevents secular drift over time. This leads to a substantial decrease in the complexity of searching proper initial conditions for the four-spacecraft configuration by utilizing the linearized HCW equations. This setup with two of the four satellites tethered allows for high-quality tetrahedra.

... whereû i ∈ E 3 are the orthonormal local vertical, local horizontal (LVLH) basis vectors of the orbiting body, µ is the gravitational parameter of the central body, r is the radial distance from the central body, h is the specific orbital angular momentum magnitude, θ = ω + ν is the argument of latitude, and ν is defined momentarily. The sixth COE is related to the phase angle and is typically chosen as one of the following: true anomaly, νν = 1 he h 2 µ cos νû 1 − ( h 2 µ + r) sin νû 2 ·⃗ a + h r 2 mean anomaly, MṀ = b ahe ( h 2 µ cos ν − 2re)û 1 − ( h 2 µ + r) sin νû 2 ·⃗ a + n eccentric anomaly, EĖ = 1 nae (cos ν − e)û 1 − (1 + r a ) sin νû 2 ·⃗ a + a r n (2) where θ is simply the sum of ω and ν , b is the semi-minor axis, and n is the mean motion. † Unlike the first five COEs, the above equations do not reduce to zero for the Kepler problem (⃗ a = ⃗ 0 ). ...

... Unlike the COEs, the governing equations for the above elements are free of the singularities at e = 0 and i = 0 . 2,4,9 The term equinoctial elements, with no further descriptors, usually refers to the elements in the second column above. ‡ The elements in the third column above are typically referred to as the modified equinoctial elements (MEEs) and have seen significant use in the decades since their introduction. ...

The origin and geometric interpretation of the equinoctial elements is explained with a connection to orthogonal rotations and attitude dynamics in Euclidean 3-space. An identification is made between the equinoctial elements and classical Rodrigues parameters. A new set of equinoctial elements are then developed using the modified Rodrigues parameters.

... From a series of novel works in the existing literature [16,20,31,45,48], it is evident how the Gaussian approximate solutions interact with the space of orientations through the various attitude coordinate systems [57,59,61,63]. A very fundamental one, being presented in [19], expresses the motion using the Euler angles. ...

... Lastly, the control u ∈ U ⊆ R n 2 is considered as known input torques and G 1 2 is the square root of G ∈ R n 2 ×n 2 . The coefficients f 1 and f 2 express the kinematics and the dynamics of the physical motion respectively, as indicated by the Euler's equations of motion [57]. The process X comprises the orientation and angular rate respectively, where n 1 depends on the chosen coordinate system map. ...

This paper conveys attitude and rate estimation without rate sensors by performing a critical comparison, validated by extensive simulations. The two dominant approaches to facilitate attitude estimation are based on stochastic and set-membership reasoning. The first one mostly utilizes the commonly known Gaussian-approximate filters, namely the EKF and UKF. Although more conservative, the latter seems to be more promising as it considers the inherent geometric characteristics of the underline compact state space and accounts -- from first principles -- for large model errors. We address the set-theoretic approach from a control point of view, and we show that it can overcome reported deficiencies of the Bayesian architectures related to this problem, leading to coordinate-free optimal filters. Lastly, as an example, we derive a modified predictive filter on the tangent bundle of the special orthogonal group $\mathbb{TSO}(3)$.

... Equation (26) gives the relation for (¯ ,¯ ) and (¯ ,¯ ) which result from the point transformation (¯ ) given by equation (20). Equations (21) and (25) give the physical meaning for the non-minimal variables¯ and¯ . It will later be shown in section III.B.3 that the multiplier is a constant which may be set to zero. ...

... As an example, this section will consider a body in orbit around an oblate central body and include the 2 perturbation terms. The potential for the 2 perturbation is given in terms of inertial cartesian coordinates, = ( 1 , 2 , 3 ) , as follows [21]: ...

... All together, to switch from the equatorial frame to the rotating ecliptic frame, two rotations are needed. Then, the angular momentum vector can be mapped between the two frames so that it can be described in terms of the Earth equatorial frame using the transport theorem [95] in Equation 6.51. ...

... The vector of the frame that makes it a right handed coordinate system ist =ẑ ×d. The averaged Equations A.6 and A.7 are transformed into a rotating frame using the transport theorem, e.g.,ė =ė r +ḟẑ ×ê [95]. Then the solution can be found as Equations 2.14 and 2.15 in the paper. ...

This research approaches the problem of debris mitigation at high altitudes by leveraging naturally occurring perturbations. These perturbations include effects due to solar radiation pressure and effects due to third body gravitation. Solar radiation pressure can be used for a variety of high altitude orbits beyond where atmospheric effects dominate. For third body effects, they impact the work discussed in medium Earth orbit where luni-solar resonances affect the stability of the region. These instabilities cause trajectories to increase in eccentricity on the order of decades to centuries.
This research is broken up into four main goals. The first goal studies the averaging tools used in this research. Doubly averaged solutions provide rapid computation power for studying orbits over long time-spans but can lead to a degradation of the solution. This goal characterizes the uncertainties of this model in the unstable regime they are used in. The second and third goals relate to the instability of medium Earth orbit. The second goal studies the graveyard orbit approach, placing satellites in a disposal orbit at their end-of-life, and the long-term behavior of debris in these orbits. The third goal deciphers whether it is feasible or not to target these regions of instability for an atmospheric reentry to depopulate the orbits. The final goal involves using solar sailing for end-of-life debris mitigation at high altitudes. Similar to how satellites in low Earth orbit use drag sails to depopulate the orbit, the solar sail could be deployed at end-of-life to change the orbit and achieve an atmospheric reentry for high altitude orbits.

... Equation (26) gives the relation for (¯ ,¯ ) and (¯ ,¯ ) which result from the point transformation (¯ ) given by equation (20). Equations (21) and (25) give the physical meaning for the non-minimal variables¯ and¯ . It will later be shown in section III.B.3 that the multiplier is a constant which may be set to zero. ...

... As an example, this section will consider a body in orbit around an oblate central body and include the 2 perturbation terms. The potential for the 2 perturbation is given in terms of inertial cartesian coordinates, = ( 1 , 2 , 3 ) , as follows [21]: ...

View Video Presentation: https://doi.org/10.2514/6.2022-2458.vid The Hamiltonian formalism in over-parameterized, regularized, and redundant phase spaces is developed and applied to the two-body problem. A class of canonical and point transformations from minimal to non-minimal coordinates is derived that provides an explicit identification of the new momenta and the corresponding non-minimal coordinates. The resulting non-minimal phase space with four generalized coordinates is found to be equivalent to the radial unit vector and reciprocal of the radial distance. It is shown that these coordinates, along with their conjugate momenta, directly define the local vertical local horizontal basis vectors, the radial distance and radial velocity, and the angular momentum magnitude. The extended phase space is then used to transform the independent variable and obtain canonical equations of motion which are shown to be equivalent to a perturbed four-dimensional harmonic oscillator. An analytical solution and state transition matrix for the unperturbed (Kepler) problem is presented in closed form. Lastly, the perturbed equations are verified numerically using the J_2 perturbation as an example.

... It computes the propagation of error of the initial states over time. Computing the STM is ubiquitous to spaceflight dynamics, navigation and control [8,9,10,11]. Hence, derivation of STM consolidating gravity and drag perturbation can improve its practical application. ...

... We would refer the reader to [17] for the detailed derivation of J 3 − J 6 perturbation. The J 2 and drag perturbed acceleration vectors are given by, [8,34] ...

In this work, the Analytic Continuation method is studied for orbit propagation and State Transition Matrix(STM) computation, with J 2 − J 6 gravity and atmospheric drag perturbations, and combined with a new measurement model for inertial orbit estimation using low-cost observation satellites via extended Kalman filter. The Analytic Continuation is a Taylor series based semi-analytic method applied to solve fundamental problems in Astrodynamics. Previously, this method has been used to derive the STM incorporating J 2 − J 6 gravity and atmospheric drag perturbations. The test results show that the Analytic Continuation method is capable of computing the perturbed STM with machine precision level of accuracy regardless of the orbit type. The new measurement model revolves around a network of low-cost observation satellites that only require a monocular camera to provide angles-only line-of-sight measurements. Previously, using an extended Kalman filter, this measurement model has been implemented to unperturbed and J 2 gravity perturbed two distinct cases: first, a limited number of observation nodes were used to analyze the effect of each node's instantaneous observability on the estimation results and demonstrate the ease of removal and addition of nodes to the filter, and second, a constellation of observation satellites is used to track a single target. The results of each case show that this method is capable of producing accurate orbit estimation in the inertial frame that converges within a short period of time. When combining these two methods, by propagating the perturbed dynamics via the Analytic Continuation method within the extended Kalman filter using the new measurement model, the orbit estimation is improved by being able to incorporate J 2 − J 6 gravity and atmospheric drag perturbations. Results show that this approach is capable of producing highly accurate perturbed orbit estimation in the inertial frame. Due to simplicity and adaptability of this approach, it can easily be expanded to include higher order State Transition Tensors within a higher order extended Kalman filter to improve the orbit estimation.

... The vectors r i and r c are coordinatized in the camera frame of reference, while the projection displacements ρ i are described in the body-fixed frame attached to the moving object (aircraft). The time derivative of Eq. (5) is written using the transport theorem [30] as ...

Interferometric Vision-Based Navigation (iVisNav) is a novel optoelectronic sensor for autonomous proximity operations. iVisNav employs laser emitting structured beacons and precisely characterizes six degrees of freedom relative motion rates by measuring changes in the phase of the transmitted laser pulses. iVisNav's embedded package must efficiently process high frequency dynamics for robust sensing and estimation. A new embedded system for least squares-based rate estimation is developed in this paper. The resulting system is capable of interfacing with the photonics and implement the estimation algorithm in a field-programmable gate array. The embedded package is shown to be a hardware/software co-design handling estimation procedure using finite precision arithmetic for high-speed computation. The accuracy of the finite precision FPGA hardware design is compared with the floating-point software evaluation of the algorithm on MATLAB to benchmark its performance and statistical consistency with the error measures. Implementation results demonstrate the utility of FPGA computing capabilities for high-speed proximity navigation using iVisNav.

... The steps to transform the GPS coordinates logged by the multirotor flight controller are shown in Figure 6. The conversion from WGS84 to ENU as derived in [17] is shown in equations (20) and (21): ...

View Video Presentation: https://doi.org/10.2514/6.2022-0498.vid Onboard detection and tracking capability are integral to the sensing component in collision avoidance systems needed to safely operate autonomous urban air taxis and small Unmanned Aircraft Systems. Ground-based validation of detection and tracking systems is an important milestone towards the end goal of real-time collision avoidance using onboard sensors and algorithms. In this work, we evaluate three Extended Kalman Filter (EKF) based fusion trackers with radar and vision detection inputs, and compare them with baseline trackers for each sensor type. Performance is assessed using field collected data of ground to air test flights with the sensors co-located on a stable platform with an instrumented multirotor acting as the intruder performing a waypoint pattern at a distance of 1.1 km to 0.3 km to simulate a head-on collision geometry. Fusing an image-based morphological detector with a radar detector using an EKF covered 74% of the ground truth position updates logged by the flight controller on the multirotor within an error of 50 meters, after accounting for alignment offsets, while covering 15% more ground truth updates relative to radar only. Removing timestamps when the intruder aircraft is occluded by trees and only considering timestamps where the radar has an update, the EKF image-based morphological detector combined with the radar detector covered 90% of the ground truth position updates within an error of 50 meters and 97% within an error of 100 meters.

... The effect of this CAM is modelled following the framework presented in [1], composed of three main blocks: 1. Characterization of the orbit modification due to the CAM, expressed in terms of Keplerian elements. 2. Determination of the deviation at the TCA, using linearized relative motion equations to map changes in Keplerian elements at TCA into changes in relative position and velocity [3]. 3. Post-processing operations, for analysis and optimization purposes. ...

The continuous evolution of low-thrust propulsion technologies, both in performance and the range of platforms that can equip them, provides increasing advantages in propellant efficiency and enables new kinds of missions. However, their smaller control authority compared to impulsive thrusters makes it more challenging to react to unforeseen situations such as collision avoidance activities. Whereas impulsive propulsion allows for efficient Collision Avoidance Manoeuvres (CAMs) performed just a few orbits before the predicted close approach, low thrust CAMs can require a longer acting time. Moreover, low-thrust CAM models are more computationally costly to perform parametric analyses and optimisations to inform the decision-making process. To tackle some of these issues, we propose analytical and semi-analytical models for low-thrust CAMs, based on averaging techniques and with focus on computational efficiency. They are part of the latest iteration of the Manoeuvre Intelligence for Space Safety (MISS) software tool for CAM design. The main goal in this work is to improve the characterisation of the phasing change at the predicted close approach, as this is the leading contribution to collision probability reduction. To this end, the fully analytical model for constant, tangential low-thrust CAMs introduced in previous works is updated to use a differential time law in eccentric anomaly including first-order terms in thrust acceleration, replacing the previous time law derived from Kepler's equation. Numerical tests show that the new approach significantly improves the accuracy with no additional model complexity, except for quasi-circular orbits. For this case, the zeroth-order time law with a correction for the displacement of the apse line deals better with the singularity of Gauss equations at zero eccentricity. The computational efficiency of the model is leveraged to perform sensitivity analyses for representative test cases, highlight their qualitative characteristics. Finally, two future improvements and applications are introduced. First, the inclusion of normal thrust acceleration components is treated. Although an analytical result can be obtained, its complexity prevents from using it for efficient CAM computation in its current form. Then, the synergies with machine learning techniques for achieving on-board CAM autonomy are briefly discussed.

... This parametrization of the space of rotations is not global, in the sense that it is not able to represent rotations of magnitude θ = ±π. Similarly, the modified Rodrigues rotation vector [18], i.e., tan( θ /4) θ/ θ , is singular at θ = ±2π. On the contrary, the representation of the associated rotation matrix R(α) is free from trigonometric functions and the composition of two successive rotations α (1) and α (2) is determined by the following simple expression ...

We develop an explicit, second-order, variational time integrator for full body dynamics that preserves the momenta of the continuous dynamics, such as linear and angular momenta, and exhibits near-conservation of total energy over exponentially long times. In order to achieve these properties, we parametrize the space of rotations using exponential local coordinates represented by a rescaled form of the Rodrigues rotation vector and we systematically derive the time integrator from a discrete Lagrangian function that yields discrete Euler-Lagrange equations amenable to explicit, closed-form solutions. By restricting attention to spherical bodies and Lagrangian functions with a quadratic kinetic energy and potential energies that solely depend on positions and attitudes, we show that the discrete Lagrangian map exhibits the same mathematical structure, up to terms of second order, of explicit Newmark or velocity Verlet algorithms, both known to be variational time integrators. These preserving properties, together with linear convergence of trajectories and quadratic convergence of total energy, are born out by two examples, namely the dynamics on $\mathrm{SO}(3)$ of a three-dimensional pendulum, and the nonlinear dynamics on $\mathrm{SE}(3)^n$ that results from the impact of a particle-binder torus against a rigid wall.

... The rotational dynamics of the debris are given by [18,Chapter 4] ...

The Electrostatic Tractor (ET) concept utilizes attractive Coulomb forces to relocate retired satellites from Geostationary Earth Orbit (GEO) to a graveyard orbit several hundred kilometers above GEO without any physical contact. Prior research investigated the charged relative motion control performance of the ET for two spherical spacecraft, and how electric potential uncertainty affects the control stability. This work utilizes the Multi-Sphere Method (MSM) to consider general three-dimensional spacecraft shapes, and investigates how the attitude of the debris and electric potential uncertainty affect the control effort and reorbit time. The results show that the reorbit time is minimized if protruding structures of the debris, such as solar panels, are directed toward the servicing satellite. The control effort, on the other hand, is only marginally affected by the debris attitude. Elec-trostatic torques generally cause the debris to tumble, and Monte Carlo simulations show that the rotation of the debris averages out the effects of debris attitude on control effort and reorbit time. However, the sensitivity of the controller to electric potential estimation errors is not entirely eliminated by a tumbling debris.

... The rotational (or attitude) dynamics of a rigid body is covered extensively by Junkins and Schaub (2009), and only the relevant equations are reproduced here. Euler parameters are a type of unit quaternion which are a non-singular attitude representation, making them very convenient for numerical integration and simulation. ...

Near Earth Object (NEO) 2020 SO is believed to be a Centaur rocket
booster from the mid 1960's that was temporarily recaptured by the Earth. 2020 SO entered Earth's Hill sphere in November 2020, with close approaches in December 2020 and February 2021, where it became bright enough (approximately 14 V magnitude) to be observed by Raven-class (< 1 m) telescopes. In this paper, 2020 SO's spin state and reflective properties are estimated using data collected from multiple telescope sites around the world during both close approaches. The
95% Highest Posterior Density (HPD) region and Maximum A Posteriori (MAP) spin state and reflective properties of 2020 SO are estimated using Bayes' theorem via Markov Chain Monte Carlo (MCMC) sampling of a predictive light curve simulation that is based on an anisotropic Phong reflection model. We estimate ten parameters at the start of an observation epoch: attitude quaternion (4), angular
velocity vector (3), and di usive/specular reflectivity parameters (3). Using a Fourier fitting and least squares minimization technique we find a joint-estimated period of 9.328 +/- 0.275 s at a 2sigma confidence level in the light curves of 2020 SO that further provides support for it being an artificial object as the current most rapidly rotating known asteroid is 2017 QG18 with a period over 1.3 times slower. The method of light curve inversion employed in this paper can be applied directly to other NEOs given photometric observations with a high enough temporal density and knowledge of some approximate physical properties of the object.

... Furthermore, [33,34] developed a simplified approach for determining the minimum-fuel solution based on the minimum-fuel free-time solution by computing and comparing two candidates at most. In addition, the shooting technique [21], the homotopic approach [29], and differential algebra [2] are used to solve the multiple-revolution perturbed Lambert's problem. ...

NASA's Psyche mission will launch in August 2022 and begin a journey of 3.6 years to the metallic asteroid: Psyche, where it will orbits and examine this unique body. This paper presents an alternative opportunity of the Psyche mission as well as the return opportunity to the Earth. It uses Mars's gravity assists to rendezvous with and orbits to the largest metal asteroid in the solar system. The spacecraft orbits around Psyche for approximately 1710 solar days, then starts its return journey. In the outer layer of the proposed methodology, the differential evolution algorithm is used to find the optimal launch, flyby and arrival date. In the inner layer, Lambert's algorithm is used for finding the feasible and optimal space trajectories solution. Considering gravity assists, before the gravity assists impulse, an optimal thrust impulse has been calculated at periapsis of the fly-by planet that gives the maximum $ \Delta\nu_2 $ to the spacecraft.

... The illustration of the unit sphere S 3 embedded in the space R 4 is inspired byFig. 3.10in[63]. ...

This paper presents a novel Lagrangian approach to attitude tracking for rigid spacecraft using unit quaternions, where the motion equations of a spacecraft are described by a four degrees of freedom Lagrangian dynamics subject to a holonomic constraint imposed by the norm of a unit quaternion. The basic energy-conservation property as well as some additional useful properties of the Lagrangian dynamics are explored, enabling to develop quaternion-based attitude tracking controllers by taking full advantage of a broad class of tracking control designs for mechanical systems based on energy-shaping methodology. Global tracking of a desired attitude on the unit sphere is achieved by designing control laws that render the tracking error on the four-dimensional Euclidean space to converge to the origin. The topological constraints for globally exponentially tracking by a quaternion-based continuous controller and singularities in controller designs based on any three-parameter representation of the attitude are then avoided. Using this approach, a full-state feedback controller is first developed, and then several important issues, such as robustness to noise in quaternion measurements, unknown on-orbit torque disturbances, uncertainty in the inertial matrix, and lack of angular-velocity measurements are addressed progressively, by designing a hybrid state-feedback controller, an adaptive hybrid state-feedback controller, and an adaptive hybrid attitude-feedback controller. Global asymptotic stability is established for each controller. Simulations are included to illustrate the theoretical results.

... Since the debris is uncontrolled, its attitude generally changes during the reorbit process as a result of the electrostatic torque. The rotational dynamics of the debris are given by [19,Chapter 4] [ ...

The electrostatic tractor concept is an active debris removal method that has been proposed to remove retired satellites from Geostationary Earth Orbit without physical contact, using electrostatic forces. These forces are generated by charging the servicing satellite and the debris with an electron gun that is attached to the ser-vicer. Prior work investigated the effects of debris attitude on performance factors such as reorbit time and control effort. Uncertainty in the electric potential of the debris was also considered. This work extends the analysis of the electrostatic tractor performance by considering additional sources of uncertainty, such as uncertainty in mass properties, charge model errors, and electric potential uncertainty of the servicing satellite. The results suggest that errors in the estimated electric potential have the most significant impact on the reorbit performance.

... 6 The celestial mechanics formulations and equations, solutions to the Keplerian two-body problem, are drawn from well established material. 7,8 To demonstrate the concept, first consider the inertial Earth frame to be (x e , y e , z e ) and a satellite's frame to be (x o , y o , z o ), which for now coincides with the inertial Earth frame. In the PoE formulation it can be described as follows: ...

The Product of Exponentials (PoE) formula is a mathematical tool that is used extensively in robotics. The virtue of using the exponential mapping, Lie Algebra and screw theory is that it allows an elegant and concise way of describing the orientation and position of a body with respect to another body in a multi-body system. Although the PoE formula is mainly used in robotics, this work aims to demonstrate the utility of the PoE formula as an alternative method for defining and drawing orbits given an orbital elements set. The work also explores the first derivative of the adapted PoE formula in the framework of orbital mechanics, which allows obtaining the state of the satellite (position and velocity) from the orbital elements set using the developed formulation.

... According to Euler's rotation theorem, in three-dimensional space, any displacement of a rigid body that keeps at least one of its points fixed can be described by a rotation at angle α around a fixed axis e x , e y , e z , and the axis e x , e y , e z should pass through these fixed points. One can find a full characterization of the quaternion as a function of α and e x , e y , e z in [27]. ...

The attitude tracking synchronization control of an orbit-predetermined leader–follower spacecraft swarm for the space moving target is discussed in this paper. The information exchange between all spacecraft is assumed to be discrete in time and on the undirected connected graph. Moreover, due to the demand for saving communication resources, wireless interference has been utilized, which allows all the neighbors of a spacecraft to access the same channel frequency spectrum simultaneously. Then the backstepping control algorithm is designed to let the spacecraft (β, A)-practically stably synchronize their states and track a time-varying trajectory in the presence of unknown fading channels. Finally, simulation is provided to verify that using the proposed control scheme, the attitude tracking synchronization can be achieved with high precision.

... As for control schemes designed for a six-DOF spatial rigid body, those conventional control schemes based on local motion parameterization for limited motion ranges cannot be applied to rigid body systems with large range of motion [18]. In order to model a six-DOF spatial rigid body in a geometric framework, the configuration (position and orientation) of a rigid body is represented globally on a six-dimensional special Euclidean space SE(3) in a coordinate-free manner. ...

This work proposes a robust terminal sliding mode control scheme on Lie group space SE(3) for Gough–Stewart flight simulator motion systems with payload uncertainty. A complete dynamic model with geometric mechanical structures and a computer dynamic model built in the MATLAB/Simulink package are briefly presented. The robust control strategy on the Lie group SE(3) is applied at the workspace level to counteract the effects of imperfect compensation due to model simplification and payload uncertainty in flight simulator application. With exponential coordinates for configuration error and adjoint operator on Lie algebra se(3), the robust control strategy is designed to guarantee almost global finite-time convergence over state space through the Lyapunov stability theory. Finally, a describing function and a step acceleration response to characterize the performance of a flight simulator motion base are employed to compare the robustness performance of the proposed controller on SE(3) with the conventional terminal sliding mode controller on Cartesian space. The comparison experimental results verify that the proposed controller on SE(3) provides better robustness than the conventional controller on Cartesian space, which means higher bandwidth in two degrees of freedom and faster response with smaller tracking error in six degrees of freedom.

... This is a reasonable assumption since the mission duration in this case is short [16]. The force due to gravity gradient is given as [17] f SB gg = − 3µ ...

In this work, we propose a novel controller based on a simple adaptive controller methodology and model predictive control (MPC) to generate and track trajectories of a spacecraft in the vicinity of asteroids. The control formulation is based on using adaptive control as a feedback controller and MPC as a feed-forward controller. The spacecraft system model, asteroid shape and inertia are assumed to be unknown, with the exception of the estimated total mass and angular velocity of the asteroid. The MPC is used to generate feed-forward trajectories and control input using only the mass and angular velocity of the asteroid combined with obstacle avoidance constraints. However, since the control input from MPC is calculated using only an approximated model of the asteroid, it fails to control the spacecraft in the presence of disturbances due to the asteroid’s irregular gravitational field. Hence, we propose an adaptive controller in conjunction with MPC to handle unknown disturbances. The numerical results presented in this work show that the novel control system is able to handle unknown disturbances while generating and tracking sub-optimal trajectories better than adaptive control or MPC solely.

... where the gravity perturbation is modeled as J 2 (Earth oblateness) [22,15] ...

Presented within this work is a method to perform initial orbit determination (IOD) utilizing observations from multiple space-based observers. As a demonstration of the present approach, angles-only, and angles and angle-rates measurements are studied for the IOD problem. The angles-only measurements are used to define orthogonal geometric planes whose intersection represents the direct line of sight between each observer and the object of interest (target). The geometric planar equations are differentiated with respect to time to describe the position and velocity using the angles-only and angle-rates measurements. With multiple observers, the solution to the planar equations and their time derivatives results in the determination of position and velocity of the target. The present initial orbit determination method is highly accurate with position percent errors on the order of 10 −4. Results demonstrate that the present approach does not require any knowledge of the target dynamics. Hence, it can be readily applied to perform accurate IOD for a variety of orbits in different dynamical environments, as demonstrated with both Earth orbit IOD and Cislunar IOD.

... The application of the numerical integration method to solve ordinary differential equations was published by [8]. The numerical solution of SLDE was published by [9,11]. Matrix notation of the quaternion vector space was analysed by [1,7]. ...

The design an optimal numerical method for solving a system of ordinary differential equations simultaneously is described in this paper. System of differential equations was represented by a system of linear ordinary differential equations of Euler's parameters called quaternions. The components of angular velocity were obtained by the experimental way. The angular velocity of the centre of gravity was determined from sensors of acceleration located in the plane of the centre of gravity of the machine. The used numerical method for solving was a fourth-order Runge-Kutta method. The stability of solving was based on the orthogonality of a direct cosine matrix. The numerical process was controlled on every step in numerical integration. The algorithm was designed in the C# programming language.

... where ω i mean the rotation angular velocity of the target and chaser, [J ] i (i = t, c) donate the inertia tensor of the target and chaser; T Bt is the Eddy induced torque of the magnetic field on the target; T c is the control torque of the thruster on the chaser; T Bc is the Eddy induced torque of the magnetic field on the chaser. As both of the chaser and target have arbitrary, large rotation motions, the four Euler parameters (quaternions) which offer a redundant, non-singular attitude description [13] are adopted to define the rotation motion of target and chaser. And , c) represents the posture quaternion from the ECI frame to the target body reference frame or the chaser body reference frame. ...

Due to the increasing risk of space tumbling targets for spacecraft
and astronauts, de-tumbling technology of spacecraft become more and more important and various de-tumbling methods have been proposed. This paper mainly studies the fast and safe de-tumbling of space tumbling target. Considering that the required time of de-tumbling via previous methods is too long, this paper first takes the maximum de-tumbling torque as the objective function and solves the optimal trajectory in real time. Then, the MPC algorithm is used to
track the trajectory under the constraints of the safe area to ensure a fast and safe de-tumbling. The numerical simulation of large failure satellite verify that the method proposed in this paper is very effective on reducing de-tumbling time. However, it consumes huge control power. The controller will continue to be optimized in the future to reduce the consumption of control power while ensuring rapid de-tumbling.

In this article, we investigate the role of Lyapunov functions in evaluating nonlinear–nonquadratic cost functionals for Itô-type nonlinear stochastic difference equations. Specifically, it is shown that the cost functional can be evaluated in closed-form as long as the cost functional is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability in probability. This result is then used to analyze discrete-time linear as well as nonlinear stochastic dynamical systems with polynomial and multilinear cost functionals. Furthermore, a stochastic optimal control framework is developed by exploiting connections between stochastic Lyapunov theory and stochastic Bellman theory. In particular, we show that asymptotic and geometric stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic stability and optimality.

As interest in the use and launching of spacecraft for communications, earth observation, scientific experiment and navigation purposes increases and manned missions to the Moon and Mars intensify, there is need for the design of efficient and high-fidelity relative motion dynamics to reduce spacecraft collisions and increase return on investment. The main aim of this work is to develop new approximate solution of spacecraft relative motion in elliptical orbit via power series method. Advantage of this method is that it does not involve evaluating complex integral I as employed for developing approximate solutions of linearized Tschauner–Hempel equations. Cauchy product, used for the discrete convolution of power series, is employed for the development of power series solutions of the approximated nonlinear spacecraft relative motion. Application of Cauchy criterion shows that the new solutions are convergent making them useful for spacecraft formation flying, proximity and rendezvous mission analysis.

This paper introduces a new guidance algorithm for discrete-event drag-modulated aerocapture, inspired by the concepts of variation-of-parameters and osculating orbits. The algorithm is implemented and tested in a high-fidelity simulation developed by NASA's Jet Propulsion Laboratory for the case of an Earth flight demonstration of aerocapture using a rigid drag skirt. The algorithm takes the spacecraft navigation-estimated position and velocity and computes a jettison time for the drag skirt, and the performance of the new algorithm is compared to a numerical predictor-corrector (NPC) approach.

We consider the problem of rendezvous, proximity operations, and docking of an autonomous spacecraft. The problem can be conveniently divided into three phases: (1) rendezvous phase; (2) docking phase; and (3) docked phase. On each phase the task to perform is different, and requires a different control algorithm. Angle and range measurements are available for the entire mission, but constraints and tasks to perform are different depending on the phase. Due to the different constraints, available measurements, and tasks to perform on each phase, we study this problem using a hybrid systems approach, in which the system has different modes of operation for which a suitable controller is to be designed. Following this approach, we characterize the family of individual controllers and the required properties they should induce to the closed-loop system to solve the problem within each phase of operation. Furthermore, we propose a supervisory algorithm that robustly coordinates the individual controllers so as to provide a solution to the problem. In addition, we present specific controller designs that appropriately solve the control problems for individual phases and validate them numerically.

In this chapter, we consider practical time-synchronized attitude control of the on-orbit service spacecraft under disturbances and input saturation, where transient performance (e.g., settling time, overshoot, trajectory), steady-state error, and fuel consumption are likewise evaluated herein. This chapter expands and enriches previous ratio persistence, defines a modified version of ratio restriction (slightly different from Definition 2.12 introduced in Chap. 2) and an extended time-synchronized stability concept—practical time-synchronized stability. These new techniques enable a control system to inherit most of the merits from standard ratio persistence and time-synchronized stability, but allow the closed-loop error to be within a small neighborhood of the origin rather than to be exactly at the equilibrium point. Practical time-synchronized controllers with command filter and low-pass filter for the spacecraft are designed, and the input saturation is considered. The effectiveness and advantages of the controllers proposed are verified by strict theoretical proof and sufficient comparative simulation. From a number of fair comparisons, the energy consumption is about 25–35% less than that of the state of the art controllers, e.g., finite-time control. This further approves the declared merit of saving energy consumption of (practical) time-synchronized control.

Nonlinear optimal control, a dynamic approach and one of the applications of calculus of variations, is widely used in aerospace applications because of the performance criticality required. In this paper, a new nonlinear optimal controller is proposed for time-varying nonlinear dynamics of spacecraft formation flying with periodic coefficients using State-Dependent Riccati Equation (SDRE) technique. Over a more classic LQR control approach, the SDRE control approach has advantages of better tracking response, robustness, and ability to capture time-varying nonlinear characteristics of the system. First, time-varying cubic approximation model of spacecraft relative motion is developed from the original spacecraft relative motion. Second, four different linear-like State-Dependent Coefficient (SDC) parameterized structures, carefully selected to ensure optimal relative motion trajectory tracking of the formation flying, are developed for the approximated nonlinear system dynamics using SDRE. Finally, the efficiency of the new nonlinear SDRE controller and its optimal performance are validated using numerical analysis.

This work presents a new method for space-based angles-only orbit estimation. The approach relies on the integration of a novel and highly accurate Analytic Continuation technique with a new measurement model for multiple observers for inertial orbit estimation. Analytic Continuation computes the perturbed orbit dynamics, as well as the perturbed state transition matrix (STM), in the inertial frame. A new measurement model is developed for simultaneous measurements using a constellation of low-cost observers with monocular cameras for angles-only measurements. Analytic Continuation and the new measurement model are integrated in an Extended Kalman Filter (EKF) framework, where the Analytic Continuation method is used to propagate the perturbed dynamics and compute the perturbed STM and error covariance, with the measurements obtained via the new measurement model. Two case studies comprising small and large constellations of observers are presented, along with cases of sparse measurements and a study of the computational efficiency of the proposed approach. The results show that the new approach is capable of producing highly accurate and computationally efficient perturbed orbit estimation results compared with classical EKF implementations.

This paper presents an Intelligent Decision Support System (IDSS) that can automatically assess the suitable robust deflection strategies to respond to an asteroid impact scenario. The input to the IDSS is the warning time, the orbital parameters and mass of the asteroid and the corresponding uncertainties. The output is the deflection strategies that are more likely to offer a successful deflection. Both aleatory and epistemic uncertainties on ephemerides and physical properties of the asteroid are considered. The training data set is produced by generating thousands of virtual impactors, sampled from the current distribution of Near Earth Objects (NEO). For each virtual impactor we perform a robust optimisation, under mixed aleatory/epistemic uncertainties, of the deflection scenario with different deflection strategies. The robust performance indices is considered by the deflection effectiveness, which is quantified by Probability of Collision post deflection. The IDSS is based on a combination Dempster-Shafer theory of evidence and a Random Forest classifier that is trained on the data set of virtual impactors and deflection scenarios. Five deflection strategies are modelled and included in the IDSS: Nuclear Explosion, Kinetic Impactor, Laser Ablation, Gravity Tractor and Ion Beam Shepherd. Simulation results suggest that the proposed decision support system can quickly provide robust decisions on which deflection strategies are to be chosen to respond to a NEO impact scenario. Once trained the IDSS does not require re-running expensive simulations to make decisions on which deflection strategies are to be used and is, therefore, suitable for the rapid pre-screening or reassessment of deflection options.

The ultimate need to design and develop high-fidelity dynamical models for future space missions is necessitated by the continuous and enormous interest in spacecraft formation flying, rendezvous and spacecraft proximity operations. In this paper, to obtain high-fidelity dynamics, higher-order relative motion model is developed via nonlinear mapping of orbit element differences and Hill coordinates. First, second-order variation of parameter technique of calculus of variations is applied to the direction cosine matrix (DCM), which maps vector components in inertial frame to vector components in Deputy Hill frame, and deputy spacecraft inertial position and velocity vectors in Deputy Hill frame. Second, after series of transformations and elimination of higher-order terms greater than quadratic terms, new, nonlinearly mapped radial, along-track and cross-track relative motion position and velocity equations are obtained. Using the new equation of motion, nonlinear state space model is developed. The new equations, validated via numerical simulations, are amenable for the analysis of spacecraft relative motion, formation flying, rendezvous and proximity operations in both circular and elliptical orbits.

The study of inverted pendulum configurations has attracted the attention of researchers during many decades. One of the main reasons is that inverted-pendulum models have the feature of approximating the dynamics of many real-world mechanisms. Therefore, this paper presents the detailed dynamic modeling and control of a novel spherical pendulum with a variable speed control moment gyroscope. The dynamic model is obtained from the generic 3D pendulum, and the necessary assumptions to model the spherical pendulum are conducted in order to avoid singularities. Furthermore, a proportional-derivative nonlinear controller based on Lyapunov theory is designed to use favorably the features of the variable speed control moment gyroscope to control the spherical pendulum combining the gyroscopic torque and the torque provided by the reaction wheel. The proposed dynamic model and nonlinear controller are evaluated through numerical simulations for two different scenarios, driving the pendulum to a sequence of attitude commands including the upright position and tracking a desired trajectory. The results have shown that the proposed model is nonsingular and that the control law has provided adequate rates controlling the pendulum in both scenarios.

The ground-track adjustment problems for overflying one, two, and three ground targets with a single coplanar impulse are studied by considering the J2-perturbed model. For the single-target case, the minimum-fuel solutions are obtained by the Newton-Raphson iterations. For the two-target case, when the impulse position is fixed, the analytical approximate solutions for the impulse vector are derived; then the minimum-fuel solution is obtained by considering free impulse position. For the three-target case, the original three-dimensional problem is transformed into solving a single-variable equation only of impulse position. In addition, for free impulse position, the short-period term of semimajor axis is considered, and it is numerically proven to effectively reduce the overflight error. Several numerical examples are provided to verify the effectiveness of the proposed method for solving the ground-track adjustment problem for multiple targets.

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