Modeling and simulation of the attitude dynamics and control of a multi-module satellite

In literature there are several methods that can be uses by an engineer charged with the responsibility for developing an attitude control system for a multi-body space vehicle.

. Discrete coordinate methods
They involve few restrictions or approximations, and in some cases are as general as Newton's law for the simulation of the dynamic response of a collection of interconnected rigid bodies. The limitations of these methods stem primarily from the difficulty of creating the required mathematical model of a real vehicle without exceeding the practical limits imposed on computation. A strong point is their generality, and a dramatic simplification when the application is restricted to a vehicle composed of n rigid bodies interconnected at n-1 point contacts so as to form a tree structure topologically. In this case Newton-Euler equations allow very easy calculation. Among these methods there are

o Augmented-body method

It turns out to be useful and allows an immediate physical interpretation of some terms which come out from the equations.

This method was developed by Margulies and Hooker. They have observed that when a system of n point-connected rigid bodies is assembled in a topological tree, certain inertia-like terms naturally appear in combination in the individual equations of motion of each of the rigid bodies in the set. These combinations admit of physical interpretation as the inertia dyadic (or tensors or matrices) of abstractions called the augmented bodies. Briefly, the ith augmented body consists of the ith body of the set together with certain particles (point masses) attached to each of the joints of that body. The point mass attached to a given joint of the ith body equals the total mass of all of the connected bodies located "outboard" of the joint. The mass center of the augmented body is called the connection barycenter (or simply barycenter). The inertia dyadic of the augmented body with respect to the corresponding barycenter is the term that appears in the equations.

o Nested-body method

With this method, one has to write vector equations of motion in turn for n different subsets of bodies, including a final set of rotational and translational equations for the composite vehicle. The idea of isolating in sequence such subsets of rigid bodies in an n-body system (called nested bodies) seems to have both advantages and disadvantages. Terms related to concept of augmented body and connection barycenter (which are helpful to physical interpretation) do not appear with this approach. In compensation, the nested-body derivation facilitates the elimination of internal constraint forces and torques.

o Generalized-forces method

Most of them are within the framework of Lagrange's equation. For this reason, there are restrictions to holonomic systems, and it is not possible the use of redundant, singularity-free attitude variables.

. Hybrid-coordinate methods

They may be applied only when some portions of the vehicle (flexible appendages) undergo deformations that may reasonably be assumed to remain "small", thereby permitting the transformation to modal coordinates for vehicle appendages. The key feature of this approach, as opposed to the discrete-coordinate methods, is the possibility of truncating the matrix of modal coordinates.

. Vehicle normal-coordinate methods

They involve transformations of all kinematics coordinates of the simulation, and not merely the appendage deformation coordinates. They are accordingly more limited, and may require more complex coordinate transformations than the hybrid-coordinate methods would involve. However, in the simplest cases, they afford the most efficient simulation, since the then permit the most severe coordinate truncation. This is due to independence of these normal-mode coordinates, which permits the independent calculation of their participation in the vehicle motion. A such method is the traditional approach to the vibration analysis of elastic systems.
Hence, a preliminary approach to the design of an attitude control system of the configuration we want to investigate may employ Discrete-Coordinate methods, because of their generality. In addition to that, we want to investigate a system with a small number of bodies, and therefore we do not care computation limits. Augmented-body method seems the more suitable for the case analyzed, since we are concerned with a topological tree with n point-connected rigid bodies, and it will allow us to give a physical description to the terms appearing in the equations. We will see this method in detail and we will achieve equations related to pitch dynamics, which results decoupled from roll and yaw dynamics when assuming small rotation angles.