en:stabilization [ОрбиКрафт 3D]

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en:stabilization

## Lesson 04. Stabilization mode

Satellite stabilization mode means maintaining a zero angular velocity. This mode is necessary, for example, to obtain clear images or transfer them to a ground receiving point, when the data transmission time is long and the satellite antenna is not allowed to deviate from the ground receiving point. The theory described in this lesson is also suitable for maintaining any desired angular velocity (not only zero velocity), and for such tasks as tracking a moving object.

## How to implement stabilization mode

You can change the satellite’s angular velocity using reaction wheels, jet engines, electromagnetic coils, and gyrodyne engines. In this example we consider the control over the control moment using the reaction wheel. The action of this device is based on the Law of conservation of angular momentum. For example, when the reaction wheel engine spins in one direction, the spacecraft (SC), respectively, begins to rotate in the other direction. It happens under the action of the same unwinding moment, but directed in the opposite side in accordance with the Newton's Third Law. If, under the influence of external factors, the spacecraft begins to turn in a certain direction, it is enough to increase the rotation speed of the reaction wheel in the same direction. So, the unwanted rotation of the spacecraft will stop because the reaction wheel will “take” the rotational moment instead of the satellite. . The information about the angular velocity of the satellite will be received by use of angular velocity sensor. In this example, we consider how to calculate control commands for the reaction wheel from the indications of the angular velocity sensor and data on the speed of the reaction wheel. It is needed for the satellite to stabilize or maintain the required angular velocity

## Theory

### Analogies between translational and rotational motion

The analogue of the Law of conservation of momentum for rotational motion is the Law of conservation of angular momentum or the Law of conservation of kinetic momentum:

$\sum\limits_{i=1}^{n}{{{J}_{i}}\cdot {{\omega }_{i}}}=const \label{eq:1}$

In general, the rotational motion of a satellite is described by laws similar to thosefor translational motion. For example, for each parameter in the translational motion there is a similar parameter for the rotational motion:

Translational motion Analogy Rotational motion
Force $F\leftrightarrow M$ Momentum
Distance $S\leftrightarrow \alpha$ Angle
Speed $V\leftrightarrow\omega$ Angular velocity
Acceleration $a\leftrightarrow\epsilon$ Angular acceleration
Weight $m\leftrightarrow J$ Moment of inertia

The laws of motion also look similar.

Title of law Translational motion Rotational motion
Newton's second law $F=m\cdot a$ $M=J\cdot \epsilon$
kinetic energy $E=\frac{m\cdot {{V}^{2}}}{2}$ $E=\frac{J\cdot {{\omega}^{2}}}{2}$
law of momentum conservation $\sum\limits_{i=1}^{n}{{{m}_{i}}\cdot {{V }_{i}}}=const$ $\sum\limits_{i=1}^{n}{{{J}_{i}}\cdot {{\omega }_{i}}}=const$

### Derivation of ratio for required angular velocity of reaction wheel

Let us write the law of conservation of kinetic moment of the system ‘satellite + reaction wheel’ for the moments of time “1” и “2”:

${{J}_{s}}\cdot {{\omega }_{s1}}+{{J}_{m}}\cdot {{\omega }_{m1}}={{J}_{s}}\cdot {{\omega }_{s2}}+{{J}_{m}}\cdot {{\omega }_{m2}}$

The absolute speed of the reaction wheel, i.e. the reaction wheel speed in an inertial coordinate system (for example, associated with the Earth) is the sum of the satellite angular velocity and the angular velocity of the reaction wheel relative to the satellite, i.e. reaction wheel angular velocity:

${{\omega }_{mi}}={{\omega }_{si}}+{{{\omega }'}_{mi}}$

Please note: the reaction wheel can measure its own angular velocity relative to the satellite body or relative angular velocity.

Let usxpress the desired speed of the reaction wheel which must be set

${{J}_{s}}\cdot {{\omega }_{s1}}+{{J}_{m}}\cdot \left( {{\omega }_{s1}}+{{{{\omega }'}}_{m1}} \right)={{J}_{s}}\cdot {{\omega }_{s2}}+{{J}_{m}}\cdot \left( {{\omega }_{s2}}+{{{{\omega }'}}_{m2}} \right)$

$\left( {{J}_{s}}+{{J}_{m}} \right)\left( {{\omega }_{s1}}-{{\omega }_{s2}} \right)=-{{J}_{m}}({{\omega }_{m1}}-{{\omega }_{m2}})$

${{\omega }_{m2}}={{\omega }_{m1}}+\frac{{{J}_{s}}+{{J}_{m}}}{{{J}_{m}}}\left( {{\omega }_{s1}}-{{\omega }_{s2}} \right)$

Denote the relation $\frac{{{J}_{s}}+{{J}_{m}}}{{{J}_{m}}}$ as $k_d$.

Operation of the algorithm does not require the exact value of $\frac{{{J}_{s}}+{{J}_{m}}}{{{J}_{m}}}$ because the reaction wheel cannot instantly set the required angular velocity. Also, the precision of measurements is not absolute: the satellite’s angular velocity measured with an angular velocity sensor is not accurate, since measurements always contain measurement noise. Note that measurement of the angular velocity and command issuing to the reaction wheel occur with some minimum time step. All these limitations lead to the fact that $k_d$ should be experimentally selected. If it does not work we build detailed computer models which take into account all the above limitations. In our case, the coefficient $k_d$ will be selected experimentally.

${{\omega }_{m2}}={{\omega }_{m1}}+{{k}_{d}}\left( {{\omega }_{s1}}-{{\omega }_{s2}} \right)$

The angular velocity $\omega_{s2}$ at time “2” is the target angular velocity; we denote it by $\omega_{s\_goal}$. Thus, if the satellite is supposed to maintain the angular velocity $\omega_{s\_goal}$, then knowing the current angular velocity of the satellite and the current angular velocity of the reaction wheel, it is possible to calculate the desired velocity of the reaction wheel to maintain the “rotation with constant speed” mode:

${{\omega }_{m2}}={{\omega }_{m1}}+{{k}_{d}}\left( {{\omega }_{s1}}-{{\omega }_{{s\_goal}}} \right)$

Using the rotation mode with a constant speed, it is possible to make the satellite turn at any angle if the satellite is rotated at a constant speed for a certain time. Then the time that the satellite needs to rotate at a constant speed $\omega_{s\_goal}$ to turn to the required angle $\alpha$ is determined by dividing these values:

$t=\frac{\alpha}{\omega_{{s\_goal}}}$

If the satellite is required to be stabilized, then $\omega_{s\_goal}=0$ and the expression becomes simpler:

${{\omega }_{m2}}={{\omega }_{m1}}+{{k}_{d}}\cdot {{\omega }_{s1}}$

## Python implementation

### Sample of Python code using the formula:

# request for angular velocity sensor (AVS) and reaction wheel data
hyro_state, gx_raw, gy_raw, gz_raw = hyro_request_raw(hyr_num)
mtr_state, mtr_speed = motor_request_speed(mtr_num)

# conversion of angular velocity values in degrees/sec
gx_degs = gx_raw * 0.00875

# if AVS is set up with the z axis, then the angular velocity
# of the satellite coincides with the readings of the AVS along the z axis, otherwise
# it is necessary to change the sign: omega = - gz_degs
omega = gz_degs

mtr_new_speed = int(mtr_speed+ kd*(omega-omega_goal))

### Example of complete code of the satellite stabilization program in Python:

# Differential feedback coefficient.
# The coefficient is positive if the reaction wheel is located with the z axis up
# and AVS is also z-axis up.
# The coefficient is chosen experimentally, depending on the form
# and the mass of your  satellite.
kd = 200.0
# The time step of the algorithm, sec
time_step = 0.2
# Target satellite angular velocity, degrees/sec
# For stabilization mode is equal to 0.0 degrees/sec
omega_goal = 0.0
# reaction heel number
mtr_num = 1
# Maximum allowed reaction wheel speed, rpm
mtr_max_speed = 4000
# Number of AVS (angular velocity sensor)
hyr_num = 1

# Functions for determining the new reaction wheel speed.
# New reaction heel speed is made up of
# current reaction wheel speed and speed increments.
# Incrementing speed in proportion to angle error
# and error in angular velocity.
# mtr_speed - reaction wheel current angular speed, rpm
# omega - current satellite angular velocity, degrees/sec
# omega_goal - target angular velocity of the satellite, degrees/sec
# mtr_new_speed - required angular velocity of the reaction wheel, rpm
def motor_new_speed_PD(mtr_speed, omega, omega_goal):
mtr_new_speed = int(mtr_speed
+ kd*(omega-omega_goal)
)
if mtr_new_speed > mtr_max_speed:
mtr_new_speed = mtr_max_speed
elif mtr_new_speed < -mtr_max_speed:
mtr_new_speed = -mtr_max_speed
return mtr_new_speed

# The function includes all devices
# to be used in the main program.
def initialize_all():
print "Enable motor №", mtr_num
motor_turn_on(mtr_num)
sleep(1)
print "Enable angular velocity sensor №", hyr_num
hyro_turn_on(hyr_num)
sleep(1)

# The function disables all devices
# to be used in the main program.
def switch_off_all():
print "Finishing..."
print "Disable angular velocity sensor №", hyr_num
hyro_turn_off(hyr_num)
motor_set_speed(mtr_num, 0)
sleep (1)
motor_turn_off(mtr_num)
print "Finish program"

# The main function of the program in which  remaining functions are called up.
def control():
initialize_all()
# Initialize flywheel status
mtr_state = 0
# Initialize the status of the AVS
hyro_state = 0

for i in range(1000):
print "i = ", i

# Аngular speed sensor (AVS) and reaction wheel requests.
hyro_state, gx_raw, gy_raw, gz_raw = hyro_request_raw(hyr_num)
mtr_state, mtr_speed = motor_request_speed(mtr_num)

# Processing the readings of the angular velocity sensor (AVS),
# calculation of the satellite angular velocity.
# If the error code of the AVS is 0, i.e. there is no error
if not hyro_state:
gx_degs = gx_raw * 0.00875
gy_degs = gy_raw * 0.00875
gz_degs = gz_raw * 0.00875
# if AVS is set up with the z axis, then the angular velocity
# of the satellite coincides with the readings of the AVS along the z axis, otherwise
# it is necessary to change the sign: omega = - gz_degs
omega = gz_degs
print "gx_degs =", gx_degs, \
"gy_degs =", gy_degs, "gz_degs =", gz_degs
elif hyro_state == 1:
print "Fail because of access error, check the connection"
elif hyro_state == 2:
print "Fail because of interface error, check your code"

# Processing the reaction wheel and setting the target angular velocity.
if not mtr_state:	# if the error code is 0, i.e. no error
print "Motor_speed: ", mtr_speed
# setting of new reaction wheel speed
mtr_new_speed = motor_new_speed_PD(mtr_speed,omega,omega_goal)
motor_set_speed(mtr_num, mtr_new_speed)

sleep(time_step)

switch_off_all()

1. Change the program so that the satellite rotates at a constant speed.
3. Rewrite the program in C and get it working. 