Simulation of Spacecraft Attitude and Orbit Dynamics

In this paper, the simulation model of satellite attitude and orbit dynamics is discussed. The satellite attitude model has been represented in term of a quaternion and a ordinary differential equation is used to describe the satellite orbital motion. The different actuators and sensors have been modeled with suitable faults and failures. The simulation model enables us to consider the satellite motion under different environmental perturbations (for example aerodynamic drag, external celestial body etc.) and failure in actuators and sensors. The simulation model is utilized in the development of attitude and orbit control algorithms or fault detection, isolation and recovery (FDIR) technologies. Simulation results are also given. INTRODUCTION During the last decades modeling, simulation, and wider computational science and engineering have become more and more important tools in the research and development projects. The design phase has to be reduced in time and cost when the use of new ideas and tools becomes possible. This is also the trend in space application in which the real tests are not possible or at least they are expensive. New demands on the aerospace and control engineering have become up and they have to be able to answer to requirements. Spacecraft simulators or simulators in general, are software tools that can be used by researchers, engineers, students or everybody to analyze and assess system operations, behaviors, and to answer to the questions regarding phenomenon or product. The simulations are essential tools in the mission and spacecraft control design. For example, the scientific missions are unique and the instrumentation of a spacecraft is designed only for this specific mission. There are not any ready-to-use platforms that can be used. Hence, it is not possible to verify the operation of control algorithms and strategies in real process but the simulation environments can be used. There are plenty of companies that offer their simulation services to the research institutes and space companies. MODEL STRUCTURE AND MATHEMATICS Model Structure The spacecraft simulation model is organized like any actual control loops (See Figures 1.). The interfaces of the components are defined and modeled in such a way that the simulation model would be as modular as possible. Modifications to the simulation model are easy to do and one part of the model can be easily replaced with another. The model is initialized and controlled from the coordination level. This means that the model parameters and possible faults and failures in the FDIR simulation case are defined. Figures 1. Spacecraft simulation model structure. Coordinate Systems Three different coordinate systems are defined in the simulator: 1. Inertial Coordinate System (ICS), 2. Orbit Coordinate System (OCS), and 3. Body Coordinate System (BCS). The inertial coordinate system is usually defined such that the center of mass of the Earth (cm) acts as origin and the direction of the axes are fixed to the solar system. This kind of coordinate system is not exactly inertial but it is enough for all engineering purposes (Sidi 1997). The Z-axis of the ICS is the rotation axis of the Earth in a positive direction and the X-Y plane is the equatorial plane of the Earth, which is perpendicular to the Earth’s rotation axis. The vernal equinox vector Υ is selected to be the X-axis of the ICS. Finally, the Y-axis has been chosen in such a way that the ICS is righthanded orthogonal coordinate system. Orbit coordinate system is also a right-handed orthogonal coordinate system with origin in the center of the satellite mass. The Z-axis is pointing towards the center of the Earth; X-axis to the direction of satellite perpendicular to the Z-axis, and Y-axis completes the coordinate system such that it is right-handed and orthogonal. The third coordinate system, which has been fixed to the moving and rotating spacecraft, defines satellite orientation. Rotation The attitude transformation in space can be executed by using various different aspects. In the simulation model, the quaternion technique is used. The main feature of quaternions is that they provide a convenient product rule for successive rotations and they have simple form of kinematics (Wertz 1978, Wis ́niewski 1996). The basic definition of the quaternion is a consequence of the property of the direction cosine matrix A that it has at least one eigenvalue of unity. This means that there is an eigenvector e (Euler axis) that is unchanged in every rotation. The quaternion is defined as a vector (1) where qi∈R, i, j and k satisfy the Hamilton’s rule (2) and where the length of the quaternion is unity. (Sidi 1997) 4 1 2 3 q q q q = + + + q i j k (1) 2 2 2 1 = = = = − = − = = − = = − = i j k ijk ij ji k jk kj i ki ik j (2) When the Euler axis e of the rotation is known the connection between quaternion and the rotation Euler axis is