Iorfida E, Palmer P, Roberts RM (2016) Novel Approach on the Optimisation of Mid-Course Corrections Along Interplanetary Trajectories, Astrodynamics Network AstroNet-II, The Final Conference, Astrophysics and Space Science Proceedings 44 pp. 121-136
The primer vector theory, firstly proposed by Lawden, defines a set of necessary conditions to characterise whether an impulsive thrust trajectory is optimal with respect to propellant usage, within a two-body problem context. If the conditions are not satisfied, one or more potential intermediate impulses are performed along the transfer arc, in order to lower the overall cost. The method is based on the propagation of the state transition matrix and on the solution of a boundary value problem, which leads to a mathematical and computational complexity.In this paper, a different approach is introduced. It is based on a polar coordinates transformation of the primer vector which allows the decoupling between its in-plane and out-of-plane components. The out-of-plane component is solved analytically while for the in-plane ones a Hamiltonian approximation is made.The novel procedure reduces the mathematical complexity and the computational cost of Lawden?s problem and gives also a different perspective about the optimisation of a transfer trajectory
LAUTERBACH R, ROBERTS M (1992) HETEROCLINIC CYCLES IN DYNAMIC-SYSTEMS WITH BROKEN SPHERICAL-SYMMETRY, JOURNAL OF DIFFERENTIAL EQUATIONS 100 (1) pp. 22-48 ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS
Manoel M, Roberts M (2015) Gradient systems on coupled cell networks, NONLINEARITY 28 (10) pp. 3487-3509 IOP PUBLISHING LTD
For networks of coupled dynamical systems we characterize admissible functions,
that is, functions whose gradient is an admissible vector field. The schematic representation
of a gradient network dynamical system is of an undirected cell graph, and we
use tools from graph theory to deduce the general form of such functions, relating it to
the topological structure of the graph defining the network. The coupling of pairs of
dynamical systems cells is represented by edges of the graph, and from spectral graph
theory we detect the existence and nature of equilibria of the gradient system from the
critical points of the coupling function. In particular, we study fully synchronous and 2-
state patterns of equilibria on regular graphs.These are two special types of equilibrium
configurations for gradient networks. We also investigate equilibrium configurations of
-invariant admissible functions on a ring of cells.
Roberts RM, Dias MERD (1997) Bifurcations from relative equilibria of Hamiltonian systems, NONLINEARITY 10 (6) pp. 1719-1738 IOP PUBLISHING LTD
Montaldi J, Roberts M (2000) Note on semisymplectic actions of Lie groups, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE 330 (12) pp. 1079-1084 EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER
Roberts M, Schmah T, Stoica C (2006) Relative equilibria in systems with configuration space isotropy, JOURNAL OF GEOMETRY AND PHYSICS 56 (5) pp. 762-779 ELSEVIER SCIENCE BV
MONTALDI J, ROBERTS M, STEWART I (1990) EXISTENCE OF NONLINEAR NORMAL-MODES OF SYMMETRICAL HAMILTONIAN-SYSTEMS, NONLINEARITY 3 (3) pp. 695-730 IOP PUBLISHING LTD
Kristiansen KU, Palmer P, Roberts RM (2012) The Persistence of a Slow Manifold with Bifurcation, SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 11 (2) pp. 661-683 SIAM PUBLICATIONS
Horri NM, Palmer P, Roberts M (2011) Design and validation of inverse optimisation software for the attitude control of microsatellites, Acta Astronautica 69 (11-12) pp. 997-1006
Cushman R, Roberts M (2002) Poisson structures transverse to coadjoint orbits, BULLETIN DES SCIENCES MATHEMATIQUES 126 (7) PII S0007-4497(02)01118-1 pp. 525-534 GAUTHIER-VILLARS/EDITIONS ELSEVIER
Horri NM, Kristian Kristiansen, Palmer PL, Roberts RM (2012) Relative attitude dynamics and control for a satellite inspection mission, Acta Astronautica Volume 71, pp. 109-118
ROBERTS M (1985) ON THE GENERICITY OF SOME PROPERTIES OF EQUIVARIANT MAP GERMS, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 32 pp. 177-192 LONDON MATH SOC
Wulff C, Roberts M (2002) Hamiltonian systems near relative periodic orbits, SIAM Journal on Applied Dynamical Systems 1 (1)
We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties of the Hamiltonian system is described, and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.
Laurent-Polz F, Montaldi J, Roberts M (2011) POINT VORTICES ON THE SPHERE: STABILITY OF SYMMETRIC RELATIVE EQUILIBRIA, JOURNAL OF GEOMETRIC MECHANICS 3 (4) pp. 439-486
Impey M, Roberts M, Stewart I (1996) Hidden symmetries and pattern formation in Lapwood convection, DYNAMICS AND STABILITY OF SYSTEMS 11 (3) pp. 155-192 CARFAX PUBL CO
Kozin IN, Roberts RM, Tennyson J (2000) Relative equilibria of D(2)H(+) and H(2)D(+), MOLECULAR PHYSICS 98 (5) pp. 295-307 TAYLOR & FRANCIS LTD
GOLUBITSKY M, ROBERTS M (1987) A CLASSIFICATION OF DEGENERATE HOPF BIFURCATIONS WITH O(2) SYMMETRY, JOURNAL OF DIFFERENTIAL EQUATIONS 69 (2) pp. 216-264 ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS
HAAF H, ROBERTS M, STEWART I (1992) A HOPF-BIFURCATION WITH SPHERICAL-SYMMETRY, ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK 43 (5) pp. 793-826 BIRKHAUSER VERLAG AG
Herrera-Sucarrat E, Palmer PL, Roberts RM (2014) Asteroid Observation and Landing Trajectories Using Invariant Manifolds, JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 37 (3) pp. 907-920 AMER INST AERONAUTICS ASTRONAUTICS
In this paper a study of the equilibrium points of a rotating non-spherical asteroid
is performed with special emphasis on the equilibria aligned with the longest axis of
the body. These equilibrium points have the same spectral behaviour as the collinear
Lagrange points of the Restricted Three Body Problem (RTBP), saddle-centres, and
therefore unstable and stable invariant manifolds can be computed. The invariant
manifolds of the equilibrium point or periodic orbits around it, which are fuel-free
trajectories, can approach the surface of the asteroid, orbit around it for diýerent
amounts of time, and even impact on it. This paper studies the dependence of the
existence of fuel-free trajectories to the surface of the asteroid from the equilibrium
point on the shape and rotation rate of the body. A possible manoeuvre to orbit
the asteroid to observe it and later achieve vertical landing is proposed. The theory
developed is then applied to asteroid 4660 Nereus, for which an approach, observation
phase and landing manoeuvre is designed.
Iorfida E, Palmer PL, Roberts M (2016) Geometric Approach to the Perpendicular Thrust Case for Trajectory Optimization, JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 39 (5) pp. 1059-1068 AMER INST AERONAUTICS ASTRONAUTICS
Asghar S, Palmer PL, Roberts M (2006) Exact steering law for pyramid-type four control moment gyro systems, Collection of Technical Papers - AIAA/AAS Astrodynamics Specialist Conference, 2006 2 pp. 1439-1448
An exact approach for gimbal steering based on generalised-inverse for a cluster of Control Moment Gyros (CMG) is presented, iteedback gains are calculated from analytical solutions of simplified model for desired closed-loop attitude dynamics of a satellite and corresponding angular momentum response of CMGs. It is highly desirable to be able to use full angular momentum workspace of CMG cluster for rapid slew manoeuvres. However, the troublesome internal elliptic singularities restrict the angular momentum workspace in most of the pseudo-inverse-based steering logics. Therefore, we propose a Generalised Inverse Steering Logic (GISL) different from Moore-Penrose inverse but exact unlike variants of Singularity Robust laws. The proposed method gives exact control while avoiding internal elliptic singularities and using full momentum capability of the CMG cluster. The important features of the proposed steering law are demonstrated by performing simulations. Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
ROBERTS M (1985) EQUIVARIANT MILNOR NUMBERS AND INVARIANT MORSE APPROXIMATIONS, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 31 pp. 487-500 LONDON MATH SOC
JANECZKO S, ROBERTS M (1991) CLASSIFICATION OF SYMMETRICAL CAUSTICS .1. SYMPLECTIC EQUIVALENCE, LECTURE NOTES IN MATHEMATICS 1463 pp. 193-219 SPRINGER VERLAG
ROBERTS M, STEWART I (1991) SINGULARITY THEORY AND ITS APPLICATIONS - WARWICK 1989 .2. SINGULARITIES, BIFURCATIONS AND DYNAMICS - PREFACE, 1463 pp. U1-U1 SPRINGER VERLAG
Horri NM, Palmer PL, Roberts MR (2009) Optimal Satellite Attitude Control: a Geometric Approach, 2009 IEEE AEROSPACE CONFERENCE, VOLS 1-7 pp. 2339-2349 IEEE
Patrick GW, Roberts RM (2000) The transversal relative equilibria of a Hamiltonian system with symmetry, NONLINEARITY 13 (6) pp. 2089-2105 IOP PUBLISHING LTD
MONTALDI JA, ROBERTS RM, STEWART IN (1988) PERIODIC-SOLUTIONS NEAR EQUILIBRIA OF SYMMETRIC HAMILTONIAN-SYSTEMS, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 325 (1584) pp. 237-293 ROYAL SOC
BRUCE JW, ROBERTS RM (1988) CRITICAL-POINTS OF FUNCTIONS ON ANALYTIC VARIETIES, TOPOLOGY 27 (1) pp. 57-90 PERGAMON-ELSEVIER SCIENCE LTD
Turconi A, Palmerr P, Roberts M (2014) Efficient modelling of small bodies gravitational potential for autonomous approach, Proceedings of the International Astronautical Congress, IAC 7 pp. 5144-5150
The the main challenge of proximity operations about asteroids is their weak and non-uniform gravitational attraction. On-board measurements of relative distance and velocity, from optical navigation and LIDAR, has been proven sufficient to control the terminal leg of landing and touch-and-go trajectories. However, a spacecraft in orbit about an asteroid needs also a model of the gravitational potential in order to achieve autonomous guidance and control. Because of spacecraft's limited computational resources, the on-board gravity model has to be governed by a small number of parameters and the calculation of the effects of each of its elements has to be straightforward. Reviewing the main characteristics of the dynamical environment about asteroids, we identify a class of approximate models which are not required to be globally accurate but rather to represent well those dynamical features that can be exploited for control purposes. In particular we are interested in modelling correctly the positions, energy levels and behaviour of the equilibrium points which are present around most asteroids in uniform rotation. In this paper we propose an optimisation strategy to derive simple models of asteroids from higher accuracy data. We analyse the performance of such optimisation and we give an example of their use in a control law which negotiates the spacecraft to the equilibrium point at the lowest energy level.
In this paper, different conservative models of tethered satellites are related mathematically, and it is established in what limit they may provide useful insight into the underlying dynamics. An infinite dimensional model is linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Moser's twist map theorem. Finally, numerical computations provide insight into the dynamics of the billiard model.
Lim C, Montaldi J, Roberts M (2001) Relative equilibria of point vortices on the sphere, PHYSICA D 148 (1-2) pp. 97-135 ELSEVIER SCIENCE BV
Asghar S, Palmer PL, Roberts M (2006) An exact steering law for twin control moment Gyro systems, European Space Agency, (Special Publication) ESA SP (606) pp. 407-414
In this paper we present an exact solution for the attitude control of a satellite with a twin Control Moment Gyros (CMG) system. A method is described to design feedback gains and/or size the CMGs such that the closed-loop response of the system with the exact steering law gives the fastest possible manoeuvre without hitting a singularity. Moreover, the effect of gimbal rate saturation on feedback control synthesis is also analysed and the steering law shown to give rise to a near-optimal control.
Roberts M, Wulff C, Lamb JSW (2002) Hamiltonian systems near relative equilibria, JOURNAL OF DIFFERENTIAL EQUATIONS 179 (2) pp. 562-604 ACADEMIC PRESS INC ELSEVIER SCIENCE
ROBERTS M (1986) CHARACTERIZATIONS OF FINITELY DETERMINED EQUIVARIANT MAP GERMS, MATHEMATISCHE ANNALEN 275 (4) pp. 583-597 SPRINGER VERLAG
This paper builds upon thework of Palmer and Imre exploring the relative motion
of satellites on neighbouring Keplerian orbits.We make use of a general geometrical setting
from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler
equations in an Earth centred inertial coordinate frame in terms of the relevant conserved
quantities: relative energy, relative angular momentum and the relative eccentricity vector.
The paper extends the work on relative satellite motion by providing solutions about any
elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum
case. The geometrical framework assists the design of complex formation flying trajectories.
This is demonstrated by the construction of a tetrahedral formation, described through the
relevant conserved quantities, for which the satellites are on highly eccentric orbits around
the Sun to visit the Kuiper belt.
ROBERTS M (1986) A NOTE ON COHERENT G-SHEAVES, MATHEMATISCHE ANNALEN 275 (4) pp. 573-582 SPRINGER
KRUPA M, ROBERTS M (1992) SYMMETRY-BREAKING AND SYMMETRY LOCKING IN EQUIVARIANT CIRCLE MAPS, PHYSICA D 57 (3-4) pp. 417-435 ELSEVIER SCIENCE BV
Kristiansen KU, Palmer PL, Roberts RM (2012) Numerical modelling of elastic space tethers, Celestial Mechanics and Dynamical Astronomy 113 (2) pp. 235-254
In this paper the importance of the ill-posedness of the classical, non-dissipative massive tether model on an orbiting tether system is studied numerically. The computations document that via the regularisation of bending resistance a more reliable numerical integrator can be produced. Furthermore, the numerical experiments of an orbiting tether system show that bending may introduce significant forces in some regions of phase space. Finally, numerical evidence for the existence of an almost invariant slow manifold of the singularly perturbed, regularised, non-dissipative massive tether model is provided. It is also shown that on the slow manifold the dynamics of the satellites are well-approximated by the finite dimensional slack-spring model. © 2012 Springer Science+Business Media B.V.
Iorfida E, Palmer PL, Roberts M (2016) A Hamiltonian approach to the planar optimization of mid-course corrections, CELESTIAL MECHANICS & DYNAMICAL ASTRONOMY 124 (4) pp. 367-383 SPRINGER
ASHWIN P, MOROZ I, ROBERTS M (1995) BIFURCATIONS OF STATIONARY, STANDING AND TRAVELING WAVES IN TRIPLY DIFFUSIVE CONVECTION, PHYSICA D 81 (4) pp. 374-397 ELSEVIER SCIENCE BV
Pihajoki P, Herrera-Sucarrat E, Palmer P, Roberts M (2012) L-3 DYNAMICS AND POINCARE MAPS IN THE RESTRICTED FULL THREE BODY PROBLEM, BALTIC ASTRONOMY 21 (3) pp. 271-297 INST THEORETICAL PHYSICS ASTRONOMY
Lamb JSW, Roberts M (1999) Reversible equivariant linear systems, JOURNAL OF DIFFERENTIAL EQUATIONS 159 (1) pp. 239-279 ACADEMIC PRESS INC
Kristiansen KU, Vereshchagin M, Gozdziewski K, Palmer PL, Roberts RM (2012) The two-body problem of a pseudo-rigid body and a rigid sphere, Celestial Mechanics and Dynamical Astronomy 112 (2) pp. 169-190 Springer
In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational and ?re-labelling? symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the notions of locally central and planar equilibria coincide. Finally, we show that Riemann?s theorem on pseudo-rigid bodies has an extension to this system for planar relative equilibria.
Horri NM, Palmer PL, Roberts M (2009) Inverse optimal satellite attitude control from a geometric viewpoint, Proceedings of the 11th IASTED International Conference on Control and Applications, CA 2009 pp. 68-74
A geometric optimal control approach is presented, which circumvents the tedious task of numerically solving online the partial differential equations of the nonlinear global optimal control problem. In practice, operational satellite attitude control is based on standard non optimal control techniques because of the implementation complexity of nonlinear optimal control. An inverse optimal control approach is therefore proposed, based on phase space geometry. It has the advantages of low implementation complexity and low computational demand. The optimal control objective is to minimise a norm of the control torque subject to a rapidity constraint on the convergence rate of a Lyapunov function. The constraint is that the convergence rate has to exceed that of a given stabilising benchmark controller. Two benchmark controllers are considered: PD and maximum rate sliding mode to illustrate the technique. The proposed optimisation method significantly enhances the torque-rapidity trade-off compared to the benchmark controller. The only practical issue is the possibility of torque saturation from certain initial conditions, which is resolved through gain scheduling.
Iorfida E, Palmer PL, Roberts RM (2014) Optimisation modelling of mid-course corrections along interplanetary transfers, Proceedings of the International Astronautical Congress, IAC 7 pp. 5293-5301
The primer vector theory, firstly proposed by Lawden, defines a set of necessary conditions to characterise whether a transfer trajectory is optimum with respect to propellant usage, within a two-body problem context. If the conditions are not satisfied, one or more potential intermediate impulses are performed along the transfer trajectory, in order to lower the overall cost. The method is based on the propagation of the state transition matrix and on the solution of a boundary value problem, which leads to a mathematical and computational complexity. A novel propagator has been developed and it is based on the decoupling between the in-plane and out-of-plane components of the primer vector on the orbital plane. It reduces the mathematical complexity and the computational cost of the problem presented by Lawden. In this paper it is proved how the method is independent from the semi-major axis of the transfer orbit. A case that exploits the properties of the novel propagator is also presented. The optimality has been analysed keeping the transfer arc fixed, while the departure and arrival trajectories are varying. The search space is defined only by the boundary conditions on the transfer orbit and its eccentricity.
ROBERTS M (1984) EXTREMA OF LANDAU POLYNOMIALS, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 17 (13) pp. 2573-2580 IOP PUBLISHING LTD
Horri NM, Palmer PL, Roberts M (2011) Energy optimal spacecraft attitude control subject to convergence rate constraints, Control Engineering Practice 19 (11) pp. 1297-1314
ZAKALYUKIN VM, ROBERTS RM (1992) ON STABLE SINGULAR LAGRANGIAN VARIETIES, FUNCTIONAL ANALYSIS AND ITS APPLICATIONS 26 (3) pp. 174-178 PLENUM PUBL CORP
MONTALDI J, ROBERTS M, STEWART I (1990) STABILITY OF NONLINEAR NORMAL-MODES OF SYMMETRICAL HAMILTONIAN-SYSTEMS, NONLINEARITY 3 (3) pp. 731-772 IOP PUBLISHING LTD
Despite the theoretical advances in optimal control, satellite attitude control is still predominantly performed by standard controllers, such as PD laws, which are easier to implement. A switched controller is proposed, based on inverse optimal control theory, which circumvents the complex task of numerically solving online the Hamilton Jacobi Bellman (HJB) partial differential equation of the global nonlinear optimal control problem. The inverse optimization problem consists of minimizing the norm of the control torque subject to a constraint on the convergence rate of a parameterized Lyapunov function, under the effect of the benchmark controller, which is chosen to be a PD law without loss of generality. The controller is then modified by gain scheduling to achieve a tradeoff enhancement compared to the benchmark controller, while maintaining torque saturation limits. The extent to which performance can be enhanced is shown to be dependent on the controller parameters. A controller tuning analysis shows how a design settling time limit can be achieved, within the problem?s constraints on the maximum torque and the total integrated torque. The proposed optimization approach is globally stabilizing and presents low implementation complexity, which is highly desirable given the limited resources onboard small satellites.
Wokes S, Palmer P, Roberts M (2008) Classification of two-dimensional fixed-sun-angle solar sail trajectories, JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 31 (5) pp. 1249-1258 AMER INST AERONAUT ASTRONAUT
JANECZKO S, ROBERTS M (1993) CLASSIFICATION OF SYMMETRICAL CAUSTICS .2. CAUSTIC EQUIVALENCE, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 48 pp. 178-192 LONDON MATH SOC
Herrera-Sucarrat E, Palmer PL, Roberts RM (2013) Modeling the gravitational potential of a nonspherical asteroid, Journal of Guidance, Control, and Dynamics 36 (3) pp. 790-798 American Institute of Aeronautics and Astronautics
In this paper a simple and very general approximation of the gravitational potential for a nonspherical body is presented. The gravitational potential is expanded using spherical harmonics and spherical Bessel functions, and it satisfies Laplace's equation outside the circumscribing sphere and Poisson's equation inside the circumscribing sphere. Therefore, trajectories can be integrated near the surface of the asteroid, as well as far away from it. This paper focuses on the construction of a simple expansion of the gravitational potential that preserves the critical nonlinear dynamical behavior of other gravitational models for a nonspherical asteroid that are more complex and computationally more demanding. Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Kozin IN, Roberts RM, Tennyson J (1999) Symmetry and structure of rotating H-3(+), JOURNAL OF CHEMICAL PHYSICS 111 (1) pp. 140-150 AMER INST PHYSICS
Maintaining missions in proximity of small bodies involves extensive orbit determination and ground station time due to the current ground-in-the-loop approach. The prospect of having multiple concurrent missions around different targets requires the development of concepts and capabilities for autonomous proximity operations. Developments in on-board navigation by landmark maps paved the way for autonomous guidance at asteroids. The missing elements for achieving this goal are gravity models, simple enough to be easily used by the spacecraft to steer itself around the asteroid, and guidance laws that rely on such inherently simple models. In this research, we identify a class of models that can represent some characteristics of the dynamical environment around small bodies with sufficient accuracy to enable autonomous guidance. We found that sets of three point masses are suitable to represent the rotational equilibrium points generated by the balance of gravity and centrifugal acceleration in the body-fixed frame. The equilibrium point at the lowest Jacobi energy can be viewed as the energy-gateway to the surface. Information of the location and energy of this point can then be used by a control law to comply with a condition of stability against impact for orbital trajectories. In this thesis, we show an optimisation process for the derivation of three-point mass models from higher order ones and compare the profile of the Zero-Velocity curves between the two models. We define an autonomous guidance law for achieving body fixed hovering in proximity of the asteroid while ensuring that no impact will occur with the small body during the manoeuvre. Finally, we discuss the performance of this approach by comparing it with another autonomous guidance law present in literature and we suggest possible future developments.
This thesis investigates novel analytical models for a fixed-angle solar sail in a heliocentric three-dimensional orbit. The models presented here build on previous work with the hodograph transformation and adds a kinematic representation of the out-of-plane components. Rotational symmetry is used to both reduce the solution space and enable an analytic model of the inclination, longitude of ascending node and true latitude to be derived. Orbits transfers are shown to be analytically solvable using this model and are also presented here. The inclination is then shown to exhibit two distinct short term behaviours which are described as either converging or diverging. A region in the two-dimensional phase space was then computed that defined the global short term inclination evolution through the intersection of the converging and diverging behaviours. Finally an analytical asymptotic analysis is performed on the orbital angles and the inclination is shown to have an unexpected oscillation. The time period of this and the existence of equilibrium points are also demonstrated.