Quadratic Hamiltonians on non-Euclidean spaces of arbitrary constant curvature

This paper derives explicit solutions for Riemannian and sub-Riemannian curves on non-Euclidean spaces of arbitrary constant cross-sectional curvature. The problem is formulated in the context of an optimal control problem on a 3-D Lie group and an application of Pontryagin’s maximum principle of optimal control leads to the appropriate quadratic Hamiltonian. It is shown that the regular extremals defining the necessary conditions for Riemannian and sub-Riemannian curves can each be expressed as the classical simple pendulum. The regular extremal curves are solved analytically in terms of Jacobi elliptic functions and their projection onto the underlying base space of arbitrary curvature are explicitly derived in terms of Jacobi elliptic functions and an elliptic integral

Nonlinearly stable equilibria in the Sun-Jupiter-Trojan-Spacecraft four body problem

The Trojan asteroids have been highlighted as a main target for future discovery missions, which will enable key questions about the formation of our Solar system to be answered. Programs like the Japanese Jupiter and Trojan Asteroids Exploration Programme are already testing technology demonstrators like the IKAROS spacecraft to enable future interplanetary missions to Jupiter and the Trojans. In this paper an analytic analysis of the stability of the Low thrust Sun Jupiter Asteroid Spacecraft system, is presented, from a Hamiltonian point of view. Setting the three primaries in the stable Lagrangian equilateral triangle configuration, eight natural (i.e. with zero thrust) equilibrium points are identified, four of which are close to the asteroid, two stable and two unstable, when considering as primaries the Sun and any other two bodies of the Solar System. Artificial equilibria, which can be seen as low thrust perturbations of the natural ones, are then taken into account with the aim of identifying their linearly stable subset. The Lyapunov stability of these marginally stable points is then analysed using basic KAM (Kolmogorov Arnold Moser) theory and Arnold’s stability theorem. In order to apply such theorem an iterative procedure to reduce the Hamiltonian into Birkhoff’s Normal Form is applied up to fourth order, explicitly defining, at each step, the generating function of a symplectic transformation. Despite the complexity of this process, Normal Forms are a fundamental, necessary step for any application of KAM theory; such theory, transforming a non-integrable system into a sum of perturbed integrable ones, enables the computation of a high order analytical approximation of the system dynamics, plus an estimation of the discrepancy from the initial model. As an application of KAM theory, a proof of the nonlinear stability for the low thrust generated equilibrium points under non resonant conditions is found using Arnold’s stability theorem. Results show that Lyapunov stability is guaranteed along the linearly stable domain with the exception of a set of points with zero measure where the conditions to apply Arnold‘s theorem are not satisfied

Analytic perturbative theories in highly inhomogeneous gravitational fields

Orbital motion about irregular bodies is highly nonlinear due to inhomogeneities in the gravitational field. Classical theories of motion close to spheroidal bodies cannot be applied as for inhomogeneous bodies the Keplerian forces do not provide a good approximation of the system dynamics. In this paper a closed form, analytical method for developing the motion of a spacecraft around small bodies is presented, for the so called fast rotating case, which generalize previous results to second order, arbitrary degree, gravitational fields. Through the application of two different Lie transformations, suitable changes of coordinates are found, which reduce the initial non integrable Hamiltonian of the system into an integrable one plus a negligible, perturbative remainder of higher degree. In addition, an explicit analytical formulation for the relegated, first and second order, arbitrary degree Hamiltonian for relatively high altitude motion in any inhomogeneous gravitational field is derived in closed-form. Applications of this algorithm include a method for determining initial conditions for frozen orbits around any irregular body by simply prescribing the desired inclination and eccentricity of the orbit. This method essentially reduces the problem of computing frozen orbits to a problem of solving a 2-D algebraic equation. Results are shown for the asteroid 433-Eros

The geometry of optimal control problems on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E3 , the spheres S3 and the hyperboloids H3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated

Time-delayed autosynchronous swarm control

In this paper a general Morse potential model of self-propelling particles is considered in the presence of a time-delayed term and a spring potential. It is shown that the emergent swarm behavior is dependent on the delay term and weights of the time-delayed function which can be set to induce a stationary swarm, a rotating swarm with uniform translation and a rotating swarm with a stationary center-of-mass. An analysis of the mean field equations shows that without a spring potential the motion of the center-of-mass is determined explicitly by a multi-valued function. For a non-zero spring potential the swarm converges to a vortex formation about a stationary center-of-mass, except at discrete bifurcation points where the center-of-mass will periodically trace an ellipse. The analytical results defining the behavior of the center-of-mass are shown to correspond with the numerical swarm simulations

Path planning for simple wheeled robots : sub-Riemannian and elastic curves on SE(2)

This paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryagin’s maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals and the expression for the position reduced to quadratures. Reachable sets are defined in the arbitrary speed case and a numerical plot of the time-limited reachable sets presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment

Attitude motion planning for a spin stabilised disk sail

While solar sails are capable of providing continuous low thrust propulsion the size and flexibility of the sail structure poses difficulties to their attitude control. Rapid slewing of the sail can cause excitation of structural modes, resulting in flexing and oscillation of the sail film and a subsequent loss of performance and decrease in controllability. Disk shaped solar sails are particularly flexible as they have no supporting structure and so these spacecraft must be spun around their major axis to stiffen the sail membrane via the centrifugal force. In addition to stiffening the structure this spin stabilisation also provides gyroscopic stiffness to disturbances, aiding the spacecraft in maintaining its desired attitude. A method is applied which generates smooth reference motions between arbitrary orientations for a spin-stabilised disk sail. The method minimises the sum square of the body rates of the spacecraft, therefore ensuring that the generated attitude slews are slow and smooth, while the spin stabilisation provides gyroscopic stiffness to disturbances. An application of Pontryagin’s maximum principle yields an optimal Hamiltonian which is completely solvable in closed form. The resulting analytical expressions are a function of several free parameters enabling parametric optimisation to be used to provide reference motions which match prescribed boundary conditions on the initial and final configurations. The generated reference motions are utilised in the repointing of a 70m radius spin-stabilised disk solar sail in a heliocentric orbit, with the aim of assessing the feasibility of the motion planning method in terms of the control torques required to track the motions

Passive orbit control for space-based geo-engineering

In this Note we consider using solar sail propulsion to stabilize a spacecraft about an artificial libration point. It has been demonstrated that the constant acceleration from a solar sail can be used to generate artificial libration points in the Earth-Sun three-body problem. This is achieved by directing the thrust due to the sail such that it adds to the centripetal and gravitational forces. These libration points have the potential for future space physics and Earth observation missions. Of particular interest is the possibility of placing solar reflectors at the L1 artificial libration point to offset natural and human driven climate change. One engineering challenge that presents itself is that these artificial libration points are highly unstable and require active control for station-keeping. Previous work has shown that it is possible to stabilize a solar sail about artificial libration points using variations in both pitch and yaw angles. However, in a practical sense, solar sails are large structures and active control of the sail's attitude is a challenging engineering problem. Passive stabilization of such reflectors is to be investigated here to reduce the complexity of space-based geo-engineering schemes

Rigid body trajectories in different 6D spaces

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately, since the influence of the moments of inertia on the trajectories tend to zero as the scaling factor increases. The semi-direct product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry

An application of adaptive fault-tolerant control to nano-spacecraft

Since nano-spacecraft are small, low cost and do not undergo the same rigor of testing as conventional spacecraft, they have a greater risk of failure. In this paper we address the problem of attitude control of a nano-spacecraft that experiences different types of faults. Based on the traditional quaternion feedback control method, an adaptive fault-tolerant control method is developed, which can ensure that the control system still operates when the actuator fault happens. This paper derives the fault-tolerant control logic under both actuator gain fault mode and actuator deviation fault mode. Taking the parameters of the UKube-1 in the simulation model, a comparison between a traditional spacecraft control method and the adaptive fault-tolerant control method in the presence of a fault is undertaken. It is shown that the proposed controller copes with faults and is able to complete an effective attitude control manoeuver in the presence of a fault