# Professor Mark Roberts

## Academic and research departments

Faculty of Engineering and Physical Sciences, Department of Mathematics.## Publications

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.

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.

One of the key research issues in wireless systems is how to improve the system capacity. MIMO has been proven as an effective method to achieve this. Previously, the focus of MIMO-OFDM research in High Performance Metropolitan Area Network (HIPERMAN) systems was on Space Time Coding (STC) and beamforming. Recently, Multi-User Detection (MUD) has emerged as a novel approach in MIMO-OFDM-based HIPERMAN systems. In this paper, we have proposed a new MAC design, which includes the new and flexible MAC frame structure and an efficient dynamic resource allocation algorithm, in order to accommodate the MUD techniques in uplink transmission in HIPERMAN systems. The performance of the new MAC design has been evaluated via simulation means. The simulation results show that the new MAC design based on MUD can significantly increase the system capacity.

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.

© 2015. Milling is a critical process for controlling the properties of the granules produced by roll compaction. In the current study, the positron emission particle tracking (PEPT) technique was used to examine the milling kinematics of roll-compacted ribbons at various milling speeds. Microcrystalline cellulose (MCC, Avicel PH-102) was used as the model feed material and a radioactive particle (tracer) was mixed with the MCC powder and roll-compacted to form sample ribbons. They were then milled using an oscillating mill at various speeds and the kinematics of the ribbons (trajectory, velocity, and occupancy) were quantitatively determined using PEPT. A close examination of the PEPT data reveals that, for milling MCC PH-102 ribbons using the oscillating mill considered in this study, the milling speed plays an important role: at low values, the milling process is dominated by cooperative motion of the ribbons with the blade (i.e. the speeds of the ribbons and the blade are similar, and the ribbons move along with the blade) and the ribbons are milled primarily by abrasion; as the speed increases the ribbons undergo more random motion involving collisions that results in an increase in ribbon breakage and hence an increase in the milling efficiency. It is shown that the PEPT technique is a useful technique for examining milling kinematics of roll-compacted ribbons.

Within the field of trajectory optimisation, Lawden developed the primer vector theory, which defines a set of necessary conditions to characterise whether a transfer trajectory, in the two-body problem context, is optimum with respect to propellant usage. If the conditions are not satisfied, a region of the transfer trajectory is identified in which one or more potential intermediate impulses are performed in order to lower the overall cost. The method is computationally complex owing to having to solve a boundary value problem. In this paper is presented a new propagator that reduces the mathematical complexity and the computational cost of the problem, in particular it exploits a separation between the in-plane and out-of-plane components of the primer vector along the transfer trajectory. Using this propagator, the optimality of the transfer arc has been investigated, varying the departure and arrival orbits. In particular, keeping fixed the transfer trajectory, the optimality has been extensively analysed varying both the initial and final positions on the orbit, together with the directions of the initial and final thrust impulses.

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 dierent 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.

Coronary artery disease (CAD) has a significant genetic contribution that is incompletely characterized. To complement genome-wide association (GWA) studies, we conducted a large and systematic candidate gene study of CAD susceptibility, including analysis of many uncommon and functional variants. We examined 49,094 genetic variants in ~2,100 genes of cardiovascular relevance, using a customised gene array in 15,596 CAD cases and 34,992 controls (11,202 cases and 30,733 controls of European descent; 4,394 cases and 4,259 controls of South Asian origin). We attempted to replicate putative novel associations in an additional 17,121 CAD cases and 40,473 controls. Potential mechanisms through which the novel variants could affect CAD risk were explored through association tests with vascular risk factors and gene expression. We confirmed associations of several previously known CAD susceptibility loci (eg, 9p21.3:p

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

For the case of impulsive thrust trajectories, Lawden’s primer vector theory gives a set of necessary conditions that determines if intermediate impulses have to be applied in order to obtain a fuel optimal trajectory. In this paper, a novel approach is presented in which, through the representation of the primer vector in polar coordinates, a separation of the in-plane and out-of-plane components occurs. This procedure gives a complete analytic solution for the out-of-plane component of the primer vector, which is shown to be independent of the semimajor axis of the transfer orbit. In the case where the initial and final thrusts are both perpendicular to the orbital plane, the optimality of the transfer arc is fully analyzed. The analytic correlations between the boundary conditions on the transfer orbit and the profile of the primer vector are derived. In particular, the novel approach allows the development of a simple procedure based on a graphical representation from which, given only the initial and final position vectors, the optimality of the transfer orbit can be determined.

During pharmaceutical powder compaction, temperature rise in the compressed powder can affect physiochemical properties of the powder, such as thermal degradation and change in crystallinity. Thus, it is of practical importance to understand the effect of process conditions and material properties on the thermal response of pharmaceutical formulations during compaction. The aim of this study was to examine the temperature rise of pharmaceutical powders during tableting, in particular, to explore how the temperature rise depends on material properties, compression speed and tablet shape. Three grades of microcrystalline cellulose (MCC) were considered: MCC Avicel PH 101, MCC Avicel PH 102 and MCC DG. These powders were compressed using a compaction simulator at various compaction speeds (10 - 500 mm/s). Flat faced, shallow convex and normal convex tablets were produced and temperature distributions on the surface of theses tablets upon ejection were examined using an infrared thermoviewer. It was found that an increase in the compaction speed led to an increase in the average surface temperature. A higher surface temperature was induced when the powder was compressed into a tablet with larger surface curvature. This was primarily due to the increasing degree of powder deformation (i.e. the volume reduction) and the effect of interparticule/wall friction.

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.

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.