Attitude is defined from an inertial coordinate system, where spacecraft attitude is measured as an angular deviation of the spacecraft body axes from the inertial coordinates. Attitude control deals with a spacecrafts rotation about its axes keeping pitch, yaw and roll angles within limits, whereas guidance and control deals with an objects position in geographical coordinates.

External and internal perturbing torques act upon the spacecraft.

These include:

- Gravity gradient – this is the difference in gravity between the top and bottom of the spacecraft and exerts a resultant torque force.
- Solar radiation pressure – electromagnetic radiation exerts a very weak pressure upon any surface exposed to it, it becomes more pronounced at higher orbits. Solar paddles.
- Internal mass distribution – if the fuel inside sloshes around then there will be an internally produced torque.
- Thermal snap – the side exposed to the sun can expand and flex the spacecraft.
- Aerodynamic pressure at low altitudes.

- Earth sensors (detect infrared emission differential at earth-space interface) – which measures the orientation of the spacecraft with respect to earth.
- Sun sensors (measures solar array orientation with respect ot the sun).
- Star sensors
- Magnetometer sense the orientation of the geomagnetic field and measure the spacecraft’s orientation – not as accurate as other methods. Likely to be used on a cubesat.

Receives the attitude errors and sends correction signals to torquing devices.

- Momentum/reaction wheels
- Thrusters
- Magnetic torquers
- Solar sailing
- Nutation dampers

The steps in setting ACS requirements are as follows:

- Summarise mission pointing direction requirements. This is usually defined by the payload as well as the mission design [often nadir (or Earth) pointing, inertial target or scanning].
- Summarise mission and payload pointing accuracy requirements. Pointing accuracy is a function of both knowledge and control accuracy.
- Knowledge accuracy is a function of the inherent accuracy of the sensors used and the attitude determination method used.
- Control accuracy is a function of the capability of the actuators and the accuracy of the control law.
- Typical pointing accuracy requirements
- Solar array: 4 to 10 deg.
- High-gain antenna: 0.1 to 0.5 deg.
- Optics, telescopes and cameras: 0.001 to 0.1 deg.

- Define the translation and rotational manoeuvring required for the mission including angles, durations, rates and frequency.
- Select the ACS type. The selection of system type is a major driver for the entire spacecraft design. Type selection requires knowledge of:
- Payload pointing requirements [usually the major diver].
- Control authority and allowable attitude error, for translational ∆V manoeuvres.
- Manoeuvre slew rates [angle to rotate/time], a driver if greater than about 0.5 deg/s.
- Approximate solar panel area. [If panel area is large compared to the spacecraft body area, three-axis stabilization is indicated].

- Quantify the internal and external disturbance torques
- Select and size the major hardware elements.
- Define the control law and attitude determination method
- Conduct trade studies to improve the requirements and selections.

- Nutation – Where the axis of rotation of an object changes orientation in a rocking fashion
- Steradians – describes a two-dimensional angular span in three-dimensional space. Similar concept to radians except applied to a sphere, i.e. a sphere is 4π steradians.

Spin Stabilised

- Requires thrusters.
- Uses gyroscopic stability, so that the spacecraft can maintain it’s attitude through simply spinning at a set angular velocity.

Dual Spin

- Requires thrusters.

Three-axis Stabilised

- Requires thrusters.
- The difference between three-axis and spin stabilization is that the entire spacecraft does not rotate. Three axis stabilization can be achieved through different methods.

Reaction Wheels

- Used in threes, one for each axis.
- Problems with static friction (starting from zero velocity can be difficult)
- This can be alleviated by using magnetic bearings (no physical contact with housing)
- If the attitude disturbance is too great, thrusters can be used to dump the momentum.
- Wheels spin in three different planes.

Momentum Wheels

- Similar idea to the spin-stabilization technique.
- Wheels aligned along pitch axis and give gyroscopic stability
- Coupling between inertial stiffness along separate axes can then be used to control attitude in all three senses.
- Wheels all spin in same plane.

A momentum-bias system uses a momentum wheel to provide inertial stiffness in two axes and control of wheel speed provides control in the third axis.

This system is particularly useful for a nadir pointing spacecraft using wheel speed to hold z axis on nadir.

The system is relatively simple and good for long-life missions. It offers good point in one axis (usually pitch) and poor accuracy in the wheel axes (usually yaw/roll).

Gravity-gradient stabilisation takes advantage of the tendency of a spacecraft to align its long axis with the gravity vector. For this stabilising technique to work, it is necessary that the gravity-gradient torques are greater than any disturbance torque; this criteria can usually be met in orbits lower than 1000 km. It is necessary for the moment of inertia about the x and y axis to be much greater than the moment of inertia about the z axis. Deployed booms have been used on the long axis to improve the inertial properties. Gravity-gradient stabilises the pitch and roll axes only and no the yaw axis. It is common practise to use a momentum wheel with its axis perpendicular to the orbit plane to provide stiffness in yaw.

This form of ADAC is most effective in LEO. The magnetic field of the Earth is calculated by the equation:

B=B_0 (R_E/r)^3 (1-3 sin^2L )^(1/2)

B_0=strength at surface

R_E=radius of the Earth

r=distance from centre of earth to this point

L=latitude from magnetic equator

Intensity decrease as the cube, and therefore becomes very feeble at GEO.

- Magnetic component acts as dipole to produce aligning torque for the satellite.
- Satellite makes two rotations per orbit period.
- Magnetic torquers, or torque rods are used.

A large reflective balloon would, when attached to a mass, stabilise the oscillation of a cubesat and point it in the direction of the sun. Some oscillation of the balloon around the mass may continue but the cubesat should be sun pointing. This method is passive, therefore there would be little control without other devices on board.

Hardware | Gravity Gradient | Momentum bias | Solar Radiation Pressure |
---|---|---|---|

Sensors | sun or horizon | Sun sensors, horizon sensors | sun sensors |

Control | control electronics | control electronics, programmable computer | control electronics |

Torquers | boom, mumentum wheel | momentum wheel | Reflective sail, momentum wheel |

Mechanism | None | antenna pointing, solar array pointing, slip ring | Sail Deployment |

Table 1: Possible Hardware implementation

Requirment | Gravity-gradient | Momentum bias | Solar radiation pressure |
---|---|---|---|

Nadir pointing | Yes | OK | No |

Geosynchronous | No | OK | OK |

Manoeuvring | No | Poor | No |

Pointing accuracy | 5 | 0.1 to 3 | N/A |

Table 2: Mission Suitability

A cubesat would be affected by the following torques, where torque is generated by the asymmetry of the spacecraft relative to the disturbance:

Solar Torque

The momentum change form a photon striking the cubesat’s surface will result in a force exerted upon that surface. The pressure produced is proportional to the projection of the surface area perpendicular to the sun and the solar intensity, as well as depending on whether the photon was absorbed, specularly reflected (mirror), or diffusely reflected.

Absorption

If solar radiation impinging on a surface its totally absorbed the force on the surface will be aligned with the sun vector and have a magnitude found as follows:

## Equation 1 F=P_s A cosα

Where, I_s = incident solar radiation (W/m^2), C = speed of light (m/s), P_s = solar pressure (N/m^2);

P_s=I_s/C=1376/(2.998x〖10〗^8 )=4.59x〖10〗^(-6) N/m^2

Specular Reflection

The force resulting from impingement on a specularly reflective surface is normal to the surface regardless of sun line and is an elastic collision with twice the magnitude of that of an absorbing surface.

## Equation 2 F=2P_s A cosα

Diffuse Reflection

A diffusely reflective surface can be considered to be absorption and reradiation uniformly distributed over a hemisphere. The absorption component is aligned with the sun vector with a magnitude given by Equation 1. The net force vector resulting from the reflected component is normal to the surface; all tangential components cancel. The magnitude of the diffuse component is two-thirds that of the absorption component.

Note that the extent to which solar energy is absorbed, diffusely reflected or specularly reflected is a surface property of the material upon which the sunlight falls.

Figure 2: Specular and Diffuse Reflection

Solar Torque

The solar torque acting on the spacecraft is the sum of all the forces on all elemental surfaces times the radius from the centroid of the surface to the spacecraft centre of mass. In Equation 3 the reflectivity is represented by q where q=1 is a specular reflector and q=0 is an absorber.

## Equation 3 T_s=PAL(1+q)

Where,

T_s = solar torque on the spacecraft caused by a surface (Nm)

A = area of surface projected to sun line normal (m^2)

L = Distance from the centroid of the surface to the centre of mass of spacecraft, m

Q = reflectance factor between 0 and 1, unitless. Spacecraft bodies tend to be reflectors; a q of 0.5 is representative; solar panels tend to be absorbers, a q of 0.3 is representative

Magnetic Torques

The torque of any magnetic field on a current-carrying coil is:

## Equation 4 T=NIAB sinθ

Where,

T_ = magnetic torque (Nm)

N = number of loops of the coil

I = current in the coil (A)

A= coil area (m^2)

B= Earth’s (planet’s) magnetic field (T)

θ = angle between the magnetic field lines and the perpendicular coil

To calculate a planet’s magnetic field strength at a certain point:

## Equation 5 B=(B_0 〖r_0〗^3)/r^3 〖(3 〖sin〗^2L=1)〗^(1/2)

Where,

B = Earth’s (planet’s) magnetic field at any attitude or lattitude (T)

B_0= Earth’s (planet’s) magnetic field at sea level (T), approximately 3x〖10〗^(-5)T

r_ = spacecraft orbital radius (m)

r_0 = Earth surface radius (6,378,000 m)

L= latitude in magnetosphere (deg), where geographic latitude is an adequate approximation

Gravity-Gradient Torques

Gravity-gradient torques arise from the fact that the lower extremities of the spacecraft are subjected to exponentially higher gravity forces than the upper extremities. The gravity-gradient torque is found from:

## Equation 6 T_g=(3µ)/r^3 |I_z-I_y |θ

Where,

T_g = Gravity-gradient torque (Nm)

µ= Gravitational parameter for Earth(389,600.4 km^3/s^2)

r_ = radius from spacecraft centre of mass to central body centre if mass (km)

I_z = moment of inertia about the z axis

I_y = moment of inertia about the y axis

NOTE: for maximum torque use the lowest value of I_z or I_y

θ = angle between spacecraft z axis and nadir vector (rad)

Aerodynamic Drag

Aerodynamic drag is a source of torque as well as velocity reduction. Drag force can be estimated as follows:

## Equation 7 D=1/2 ρV^2 C_d A

Where,

D = Drag force (N), aligned with the velocity vector and opposite in sign

ρ = atmospheric density (kg/m^2)

V = spacecraft velocity (km)

I_z = moment of inertia about the z axis

I_y = moment of inertia about the y axis

NOTE: for maximum torque use the lowest value of I_z or I_y

θ = angle between spacecraft z axis and nadir vector (rad)

The torque is then found from:

## Equation 8 T=DL

Where,

T = Torque (Nm)

D= Drag (N)

L = the distance between the centre of pressure and the c.g.

Define Control Modes

The initial motion of the cubesat will be tumbling randomly after launch. There will have to be a sequence of coil activation to bring this motion under control.

External torques will have to be accounted for and different modes of stabilization used.

Sensor Configuration

In order to correctly understand the attitude of the cubesat at any point in time, an array of sensors must be used.

MEMS Gyroscope

Most MEMS Gyroscopes are rate gyros, this means that the output voltage changes proportionally to the angular rate of change of orientation. This signal can be integrated and the absolute angle of orientation found at a point.

CUTE-I used 4 gyroscopes for redundancy – being such sensitive devices they are prone to malfunction or environmental effects.

Multi-axis Accelerometer

These are used to detect the centrifugal acceleration of the cubesat. This data will then be compared to the gyroscope’s data. Note that there were also four separate accelerometers used in the past mission, this stresses the need for redundancy and error-checking when determining the attitude of the cubesat.