Aerospace and Electronic Systems Magazine June 2017 - 41

Macˇ iulis and Buzas
time-dependent relative orientation between an inertial and a body
fixed satellite frame by solving a set of nonlinear differential dynamic and kinematic equations of motion in the form of [17]
x = f ( x , u, w , t )

(4)

where
T

x = q1 , q2 , q3 , q4 , ω1 , ω2 , ω3  is
the state vector;
u = Tctrl is the control torque;
w = Tdist is the disturbance torque.
The states are the quaternion representing the rotation and the
angular velocity of the principal body frame relative to the Earth
centered inertial frame. In the case of LituanicaSAT-2, the control
torque in nominal mode is the torque created by magnetic torquers
that are controlled by rate damping B-dot controller.
During simulations gravity gradient, solar pressure, magnetic
residual, and aerodynamic disturbance torques were taken into
account. A simplified gas-surface interaction model presented by
Wheeler and based on Schamberg's model has been chosen to calculate the aerodynamic pressure and shear acting on the differential area [18].

6 DOF ATTITUDE SIMULATION RESULTS
To check the attitude simulator accuracy a 3U CubeSat test model
was created with the same orbital, inertia, geometry, and damping parameters as the model developed and simulated by [11]. Rawashdeh et al.'s [11] simulator results have been compared with
observations obtained by NASA PAMS (Passive Aerodynamically
Stabilized Magnetically Damped Satellite) mission with good results [11], thus it was considered a reliable data for comparison.
For Rawashdeh et al.'s [11] test case, a 3U CubeSat with 30 cm

length fins deployed at 20 deg angle was simulated at 380 km altitude and 51.6 deg inclination orbit assuming an initial tumble
rate of 10 deg/s. The active damping solution was employed using
B-dot gain setting K = 18000.
Simulation results obtained by the attitude propagator have
shown that detumbling time was about 20 h-the same result as obtained by Rawashdeh. However the steady state tracking accuracy
of 2 deg was measured compared with 0.1 deg as stated by [11].
The difference in accuracy was quickly found to be due to the effects of the magnetic residual torque. In Rawashdeh's model, magnetic residual torque is not accounted for. Simulations performed
without considering magnetic residual torque revealed very similar
results obtained by [11] (see Figure 12). As seen from Figure 11,
aerodynamic torque at 380 km altitude is almost 10 times greater than magnetic residual torque produced by 0.01 A/m residual
dipole moment and about 50 times greater than gravity gradient
torque. The solar radiation torque is about 2 orders of magnitude
smaller than aerodynamic torque and thus was not shown on the
graph.
Nevertheless, these torques still tend to disturb the steady state
and thus increase velocity vector pointing error. LituanicaSAT-2
model parameters are summarized in Table 1. The modeling variables were fin deployment angle, initial tumble rates, and B-dot
gain. Under nominal conditions, 50 deg/s tip-off rates were considered. 0, 10, and 20 deg fin angle combinations have been simulated
assuming bang-bang type rate damping controller (see Figures 14,
15, 16). The results are summarized in Table 2. It can be observed
that steady state accuracy is dependent on fin angle but the detumble time is not. Fin angles of 10 and 20 deg ensure steady state
velocity vector pointing accuracy within 10 deg three-sigma value
as required by QB50 system requirements. It has been found that
the bang-bang type B-dot control law provides more stiffness to
the satellite and consequently decreases the steady state error. This

Figure 11.

Disturbance torques for 3U CubeSat with 30 cm fins deployed at 20°, 380 km altitude.

JUNE 2017

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine June 2017