2.4 Disturbance Torques
2.4.2 Gravity Gradient Torque
The gravity gradient disturbance torque on a satellite is caused by the gravitational force of Earth. This force is inversely proportional to the magnitude of the orbital radiuskRk cubed. It can be used to stabilise an elongated satellite passively with the longitudinal axis of the satellite aligned with the nadir vector.
The gravity gradient torque [20] can be determined by: NGG= 3µ kRk3 (ue× Iue) (2.4.7) = 3µ kRk3 (Izz− Iyy) A23A33 (Ixx− Izz) A13A33 (Iyy− Ixx) A13A23 (2.4.8)
where µ = 398600.5 km3/s2 is the gravitational constant of Earth, u
e= AO/B
h
0 0 1 iT
is the unit vector towards nadir in satellite body coordinates and I is the satellite’s moment of inertia tensor.
2.5
Summary
In order to describe the position and attitude of the satellite, three coordinate systems are needed. The satellite body coordinate frame is used to describe the position of the sensors and actuators in the satellite. All measurements are made with respect to this reference frame. The orbit reference frame is used to describe the attitude of the satellite. The position of the satellite can be described in terms of the earth centred inertial frame. The modelled vectors are produced with respect to this reference frame. The use of Euler angles and quaternions are discussed, as well as the different transformation matrices needed to convert vectors from one coordinate frame to another.
The satellite’s external structure was designed to suit aerodynamic control of the satellite. It consists of a 3U main bus, two aerodynamic roll control paddles and four aerodynamic feather antennae. The paddles are used for active roll control while the feathers form a passive pitch-yaw stabilisation system [8]. The 1U at the front of the main bus is dedicated to the attitude determination and control system. The sensors and actuators used comprise of a magnetometer, fine sun sensor, nadir sensor, three magnetic torque rods, the roll control paddles and three nano-reaction wheels.
The satellite’s change in attitude can be modelled by the kinematic and dynamic equa- tions [10]. The kinematic equation illustrates the change in the attitude of the satellite irrespective of the torques that caused the change. The dynamic equation models the effect of these external and control torques on the time-derivative of the angular momen- tum vector. Only two environmental disturbance torques are included in this project’s dynamic equation, namely the gravity gradient torque and the aerodynamic torque. In this project, however, the aerodynamic torque is incorporated as a control torque by choosing an appropriate structure for the satellite.
Chapter 3
Simulation Environment
In order to test the attitude determination methods of Chapter 4 and the control methods discussed in Chapter 5, a simulation environment is required. The final orbit for this project has not yet been decided on and the simulations are based on the orbit parameters of an existing satellite whose orbit corresponds with the orbital needs of this project. A simulation environment is set up in Matlab that consists of three control blocks, a satellite dynamic and kinematic model, an orbit propagator, as well as models of the space environment of the satellite and an attitude determination block. The final Simulink model and its various subsystems are discussed in this chapter.
3.1
Orbital Elements
The simulation orbit used in this project is the orbit of the SumbandilaSat microsatellite. The orbit information, shown in Table 3.1, is extracted from a set of two line elements (TLE) provided by the North American Aerospace Defence Command (NORAD) that monitors man-made objects in space [30].
Table 3.1: Simulation orbit information.
Parameter Value
Eccentricity 0.0002704 Inclination 97.2927◦ Semi-major axis 6 879.55 km Mean motion 15.215 rev/day
Period 5 678.7 seconds
The simulation orbit is a near circular sun-synchronous polar orbit with an eccentricity
of 0.0002704. The orbit is retrograde, meaning that the satellite moves from east to west, with an inclination of 97.29◦. The semi-major axis of the orbit is 6 879.55 km and relates to an orbital height of approximately 500 km. The satellite has a mean motion of 15.215 revolutions per day in this orbit with an orbit period of 5 678.7 seconds. For a 9 am/pm sun-synchronous orbit the eclipse time TE is calculated as 1 829.6 seconds which is 32.2%
of the orbit period. In Figure 3.1 the rising edge of the rectangular wave represents the transition from eclipse to the sunlit part of the orbit and the falling edge represents the transition back to eclipse. All simulations will experience the same eclipse and sunlit parts. 0 2000 4000 6000 8000 10000 12000 −1 −0.5 0 0.5 1 1.5 2 Time (s) Sunlit = 1, Eclipse = 0
Figure 3.1: Eclipse and sunlit parts of the simulation.
3.2
Simulation Environment
A simplified representation of the simulation environment as a control loop is shown in Figure 3.2. It consists of three control blocks, the satellite model and two blocks representing the sensors and estimation methods of the satellite. The simulation has a fixed time step of 1 second. The simulation time is chosen to best suit the specific control or estimation method being tested. The contents of the simulation blocks, the inputs they require and the outputs they provide are discussed in this section.
3.2.1
Plant
The plant in the control loop is represented by the satellite model block shown in Fig- ure 3.3. The satellite model is the implementation of the kinematic and dynamic equations of the satellite discussed in Section 2.3. It requires the principle moments of inertia of
Magnetic Control Reaction Wheel Control Paddle Control Satellite Model ADCS Models Estimation Control Plant
Sensors and Estimation
Attitude Command +
_
Figure 3.2: Simulation environment as a control loop.
the satellite, the generated aerodynamic, magnetic and wheel control torques, the angular momentum vector of the reaction wheels, the orbital rate and the radius of the satellite. The initial attitude and angular body rate vector of the satellite for the simulation can be set as an input to this block. The outputs of the satellite model are the current true attitude of the satellite expressed as the quaternion vector and the direction cosine matrix and the true ORC and ECI angular body rate vectors.
Satellite Model
Ixyz DCM RPYinitial Wboinitial hwheel Nm Nw Orbit Naero Wbo Q WbiFigure 3.3: Satellite Model.