A comparison between models of flexible spacecrafts1

Flexibility plays an important role in the design of space missions. Algorithms able to derive the dynamical equations for a generic chain of flexible and rigid bodies have been developed in the past decades so that accurate dynamic simulations of large multi-body chains are possible. On the other hand control devices are getting more and more sophisticated so that the guidance of satellite platforms equipped with last generation fly-wheels is a quite complex task. In order to develop control laws, restrictive hypotheses are commonly introduced to make the equation of motion as simple as possible: rigid body dynamic, small angular velocities, small deformations, symmetric appendages bending, reduced degrees of freedom and many other hypotheses lead to as many different mathematical models. Simple modeling leads to fast simulations and allows an easier design of attitude steering laws or vibration suppression controls. In this work a mathematical model describing the full non-linear dynamic of a flexible satellite platform equipped with a system of rigid fly-wheels is developed in an explicit form. The equations are than compared to those obtained by applying standard multi-body analysis. The advantages of having an explicit form of the equations in term of the sole state of the central platform (the state of the wheels is considered as a control) are particularly attractive for control design. The set of Ordinary Differential Equations written returns widely used model when restrictive hypotheses are introduced and has to be used in connection to a preliminary Finite Element Method analysis in order to evaluate the structural invariants. Introduction In the future space missions the flexibility of structural elements will certainly become an increasingly important issue. Large solar arrays are being considered for a number of classical and advanced mission concepts. Flexible booms have are already used in interplanetary missions (having RTGs, magnetometers or other payloads on their tip) and for stabilization purposes. Large space structures are continuously proposed in advanced concepts regarding the exploration strategy of our solar system 1Professor Chiara Valente died before she could see the final release of this work . We would like to dedicate our work in this paper to her, being a little thank to her invaluable support. (interplanetary gateways concepts) and beyond. As a consequence multi-body dynamic has attracted the attention of many researchers in the past years. Methods to obtain the set of decoupled non linear equations governing a generic holonomic system made of rigid and deformable bodies have been researched. Both assumed modes methods and finite element methods have been used to model the flexible degrees of freedom these systems. Newton-Euler approach, Lagrange approach and the more recent Kane approach have been used to write the final equations governing the dynamic of the holonomic system. Depending on the complexity of the problem these equations may result to be rather complicated and heavily coupled. Methods and algorithms for the formulation and solution of the equation of motion have been proposed during the last decade and are heavily based on a computer approach. The great interest that Kane method has recently arisen in many researcher is greatly due to the fact that it does not require any awkward method or trick to compute the component of the generalized forces, its whole procedure is quite systematic and therefore well designed for a straight forward computer implementation. Banerjee et al. [1] used a recursive approach and Kane’s method to derive the equation of motion of a chain of both flexible and rigid bodies interconnected with hinges. Some results on non linear elasticity behaviour are also available. Recent works by Bajodah et al. [2] and by Meghdari and Fahimi [3] showed how, by properly defining the generalized accelerations, the final system of equations of a generic multi-body system has to be uncoupled. On the other hand, in recent years, many works focused their attention on the design and test of control steering laws for satellites and vibration suppression of flexible modes. These designs require an explicit model representing the dynamical system to control. In all these works a somewhat incomplete model of a spacecraft had therefore to be considered. A satellite equipped with any system of fly-wheels is a good example of a multi-body system for which efficient control laws are extremely desirable. The platform may be considered flexible and the control devices rigid bodies constrained to the main platform. Such a multi-body system is particularly interesting as new fly wheels systems such as Control Moment Gyroscopes (CMG) and Variable Speed Control Moment Gyroscopes (VSCMG) are becoming more and more studied as actuators in aerospace applications. The influence of structural vibrations on the attitude dynamic will be more and more meaningful as these new control devices will appear on the modern satellites. In fact, since these new actuators are able to produce greater torques (and therefore fastyer manoeuvres), they increase the effects due to the nonlinear coupling between the flexible balance and the angular momentum balance. In such a situation it is important to develop an explicit model that can describe in a precise way these effects, keeping the formulation as easy as possible for control design purposes. Equations of motion In this section a set of equations able to describe the motion of a satellite platform equipped with flexible appendages and a cluster of Variable Speed Control Moment Gyroscopes is presented. The model applies also to Control Moment Gyroscopes and Reaction Wheels controlled satellites. The model is mainly taken from [4] and the set of equations is here presented again to correct some typos that affected the quoted paper. The equations of motion for the system under consideration are: