June 29, 2013

Attitude Determination and Control

We're going to talk now about the first of two parts of spacecraft Guidance, Navigation, and Control.  The motion of a spacecraft is defined by its position and velocity, and its attitude and rotation.  Here we're looking at the latter part, spacecraft attitude, how the vehicle is oriented in space.
Motion of a vehicle is said to have as many degrees of freedom as there are ways the vehicle can move.  In one dimension a vehicle has one degree of freedom: it can move backwards or forwards.  In two dimensions it has three degrees of freedom: it can move backwards and forwards, left and right, and it can also spin clockwise or anticlockwise.  In three dimensions it has six degrees of freedom: it can move in any of three dimensions, and it can rotate in any two of them.
We describe the motion of a vehicle in Cartesian coordinates.  In a Cartesian, or orthogonal, coordinate system there are three axes which point in directions ninety degrees apart, forming right angles with each other.  On a vehicle the x-axis points in the direction of motion, the y-axis points laterally (right and left), and the z-axis points perpendicular (ninety degrees) from these two (up and down).  A vehicle can move along any or all of these axes, and movement in any direction can be described as a combination of such moves.  It can also rotate around any of these axes.  Vehicle axes are always said to point out from the center of gravity, or balance point of the vehicle.

Motion along any axis is called translation.  Motion about any axis is called rotation.  Rotation about the x-axis is called roll, about the y-axis is called pitch, about the z-axis is called yaw.
We always describe translation as a movement of the center of gravity.  Rotation is described as the center of gravity remaining fixed.  Even if both are occurring at once we solve them separately.  Translation is caused by a force being applied to the vehicle. Rotation is caused by a torque.  Torque, or moment, is caused when a force is applied off from the center of mass.  When a force points in a straight line through the center of mass no moment results.  The shortest line connecting the center of mass, or axis of rotation, and the line of force is called the moment arm.  This line forms a right angle with the line of force.
Resulting moment is the product of the force and moment arm:
$\tau&space;=&space;R&space;\times&space;F$
Force and moment arm are always separated by a right angle, and the axis of rotation will always be at a right angle to the two.  Direction of the axis of rotation can be found using the right hand rule.  If the fingers point along the line of force and the palm is pointing towards the moment arm, then the upraised thumb of the right hand will point in the direction of the axis.
Generally we want a control system to produce either translation or rotation.  However a single off-center force will cause both.  So we use two off center forces.  The forces are of equal magnitude and pointing in opposite directions, so that there is no translation.  They are removed an equal distance, on opposite sides, from the center of mass so that they produce equal torques which twist it in the same direction.  Called a moment couple, this method eliminates translation and doubles torque, similar to the torque wrench shown below:
Forces cause acceleration, in the same way torques cause angular acceleration.  Force and acceleration are related by mass, torque and angular acceleration are related by moment of inertia.  Moment of inertia is a property of an object which includes not only its mass, but also how that mass is distributed about its center.  For most common, simple shapes the moment of inertia can be easily found.  Torque is a change in the angular momentum, just as force is a change in linear momentum:
$\tau&space;\equiv&space;{\Delta&space;H&space;\over&space;t}&space;\rightarrow&space;\tau&space;=&space;I&space;\alpha&space;\newline&space;F&space;\equiv&space;{\Delta&space;p&space;\over&space;t}&space;\rightarrow&space;F=ma$
Angular acceleration over time results in a change in angular velocity, which in turn results in a rotation.  Rotation is just a change in the angle the body makes.  The final angle is related to the initial angle, initial angular velocity, and the angular acceleration by time:
$\theta_f&space;=&space;\theta_i&space;+&space;\omega_i&space;+&space;\alpha&space;t^2$

A spacecraft in freefall is acted upon by a number of forces.  Environmental effects which alter a spacecraft's orientation are called disturbance torques.  Disturbance torques can be caused by a gravity gradient, atmospheric drag, magnetic fields, and radiation pressure.
Gravity acts more strongly the closer a body is to the center of the earth.  So a body is more strongly pulled down at its lower part than its upper part.  This will naturally tend to pull the heaviest part of an object down.  For a simple example consider a pendulum, whenever it is removed from the center it is always pulled back down.
Earth's atmosphere thins with increasing altitude, but even at low earth orbit it isn't completely negligible.  Any object trying to move through a fluid, no matter how thin, is subject to some drag force.  Since the atmosphere is thinner higher up, drag will slow the lower part of the spacecraft more than it does the upper part, causing it to pitch forward.
Light has momentum, carried in the form of photons.  Even though they have no mass, when photons collide with an object they impart some of their momentum to it.  A large surface facing the sun will have more photons collide with it, imparting more momentum, than one with a smaller area.  Naturally this effect would push those surfaces gently away from the sun, analogous to a sail exposed to a gentle wind.
Earth has a powerful magnetic field resulting from its iron core.  Spacecraft can develop their own electric field from poorly shielded electronics, particle radiation, or certain forms of propulsion.  When the two fields interact they result in a force.  Unless the entire spacecraft has an evenly distributed charge (unlikely) this means one part of the vehicle will be pushed or pulled by earth's magnetic field more than the rest.  Such an off center force results in a torque.

So, how do we know where the spacecraft is and where it is pointing?  One method is to look for some fixed point.  In orbit there are three easy fixed points that we can choose from: the earth, the sun, or  the stars.  So we design sensors which detect the location of one or more of them.
Earth horizon sensors look for the infra red signal emitted by earth, so called earth-shine.  They come it two forms: static and scanning.  Static earth horizon sensors are composed of a ring of infrared scanners, when each scanner is receiving the same amount of light then earth is centered in the scanner.  Scanning sensors use a spinning prism with an IR sensor behind it.  As the prism spins is deflects light into the sensor at different angles, sweeping a cone of light into the scanner.  By noting when the earth appears in view and leaves view the spacecraft can tell where it is.
Sun sensors detect light from the sun.  A thin slit at the top of the sensor lets through a thin line of light, which is then allowed so pass through small slits cut at specific distances and passes down to photo-cells.  From the height of the sensor and the separation of the slits the angle that the sun makes with the sensor is found.  This tells the spacecraft how it is pointing relative to the sun.
Star sensors also rely on looking for the light from a star.  In this case a highly sensitive CCD surface is used (similar to that of a digital camera) which looks at the stars.  Star trackers follow a single star specifically and keep it in view, similar to sailors who used the north star to tell which way they were going.  Star cameras compare a wide view of the sky to a stored starmap, like looking at constellations in the sky.
With the exception of the star camera, each of these sensors is two dimensional, that is it can resolve two of the three pointing angles.  An earth sensor can only tell roll and pitch, but not yaw.  A sun sensors and star trackers can determine any two, depending on where they are positioned, but not a third.  So two or more sensors are needed for full determination if all three angles are significant.
Another fixed constant for an orbiting vehicle is the earth's magnetic field, as we mentioned before.  Magnetometers measure the strength of the earth's magnetic field in one direction.  A three axis magnetometer is simply three magnetometers each pointing in a different direction.  This allows the spacecraft to determine its position relative to earth's magnetic field lines, which run between the north and south magnetic poles, like a compass.  The measurement is also inherently two dimensional as the magnetometer cannot detect any spin occurring about the magnetic field lines.
Each of the systems mentioned so far uses some external reference, either a body in space or a magnetic field, to determine orientation.  However if the spacecraft is already pointing in the right direction then all we really need to know is how it varies from that orientation.  To measure that change we can use a gyroscope.
Recall above that we said torque is a change in angular momentum.  So without a torque a spinning object will continue spinning in the same way.  This is the principle of a mechanical gyroscope, it is a spinning mass allowed to swivel so that its top always points the same direction.
A spacecraft can use this to measure how its orientation has changed in two ways.  Strap down gyros keep the spinning mass fixed, and measure the torque that the restraint puts on the gyro.  This torque is proportional to the torque experienced by the spacecraft itself, and strain gauges attached can measure this.  Gimbaled gyros are allowed to swivel freely, and a measurement is taken of how far the top of it has shifted from the desired location.
Other forms of gyros exist besides mechanical ones.  Laser gyros measure the phase shift of light caused by rotation.  Two laser beams of identical frequency travel in opposite directions, by measure the slight change in frequency of these two beams caused by motion it is possible to determine the angular velocity of the spacecraft.  Ring laser gyros use mirrors to direct the beams while fiber optic gyros use fiber optics, though both work on the same principle.  FOG's are less expensive and lighter, RLG's are more accurate.

If the spacecraft is disturbed from its correct pointing then it must be returned.  Attitude control systems come in two forms: passive and active.  Passive control systems are open loop, they keep a spacecraft in position and can return it to equilibrium.  Active systems are closed loop and, in addition to maintaining position, they can reorient the spacecraft.  Passive control systems have the advantage of being lighter, simpler, and less expensive, however active control is capable of higher accuracy.
We've already talked about how the heaviest part of a spacecraft is naturally pulled down.  It's possible to design the spacecraft to take advantage of this, using gravity gradient stabilization.  The idea is fairly simple, one end of the spacecraft is meant to point towards earth.  So the other end has a long boom with some mass at the end.  Usually the end of the boom contains some sensor package or instrumentation that should be kept distant from the primary spacecraft to avoid interference.
In space there is little friction to resist movement or rotation.  However we can create friction using dampers.  The common design is to have a ball contained in a tube of fluid.  When the vehicle rotates the ball wants to stay put because of its own inertia.  Friction results initially from the stationary fluid interacting with the moving wall and then from the stationary ball with the moving fluid.  Over time this dissipates the angular momentum and the vehicle returns to a stop.
Another passive system is the yo-yo design.  Yo-yo's are deployed at the start of a mission to reduce rotation.  Two masses are released tied to two cables attached to opposite sides of the spacecraft.  As the vehicle spins the masses move further out, increasing the moment of inertia, and thus slowing the spacecraft.  The length of line is chosen so that when cable runs out the vehicle is spinning at the correct rate.  While inexpensive and light yo-yo's have limited use, being helpful only at mission outset, and send out fast moving masses to create a potential hazard to other spacecraft.
As already observed a spinning object will tend to maintain the same spin and pointing.  When a rotating body is acted on by a torque it tends to resist changing its orientation.  If the body already has high angular momentum then the relative change caused is less.  The behavior of a spinning object subjected to a torque is called precession.  Spacecraft can use precession as a form of passive attitude control.  Disturbance torques result in less change in rotation effect if the spacecraft already has some angular momentum.  Spin stabilization can be used with the vehicle spinning about a central axis, though when the entire vehicle spins it is difficult to keep instruments pointed correctly.  This can be avoided by using a dual spin system, where the central part of the spacecraft  acts as a fixed axis while the outer part spins.  Systems with one spinning section and one despun section are called dual spin.
Instead of having a large portion of the spacecraft spinning a mass inside the vehicle can be set to spin.  Momentum wheels are free rotating masses with high angular momentum, which resists a change in the momentum of the spacecraft.  The momentum of the wheel can also be altered to change the rotation of the spacecraft itself.  Passive momentum wheels spin with constant angular momentum and provide stability; active momentum wheels have variable momentum.
Momentum wheels can provide stiffness and some control, but can't be used to induce spin except about one axis.  Reaction wheels are sets of three, four, or more wheels.  Initially these have no angular momentum.  As the wheels spin the spacecraft naturally begins spinning in the opposite direction.  Reaction wheels can be used to reorient the spacecraft or counteract disturbance torques.  The wheels are aligned so that they can alter any combination of roll, pitch, and yaw.
Instead of 3+ wheels another method of achieving the same result is a single gimballed wheel.  Control momentum gyros have a single rapidly spinning mass that can be reoriented by motors to cause any spin desired.  They are capable of being much more responsive and causing faster rotation than reaction wheels, as the single wheel can have much larger mass than when there must be several wheels.
Over time these devices will build up momentum in one direction or the other.  Eventually this momentum would exceed what they were designed to tolerate.  When this happens the spacecraft has to slow down the wheels to avoid damaging them.  Momentum dumping would completely alter the spacecraft's orientation, so another means of active control is needed to offset the torque caused by slowing the momentum wheels.
One of the most basic forms of active attitude control is the use of attitude thrusters, small rockets.  As mentioned before when a body is subjected to forces off the center line it experiences a moment.  When two forces of equal magnitude and opposite direction are applied on opposite sides of the center line they produce pure rotation with no translation.  Such a technique is a simple way of rotating and orienting a vehicle.  Thrusters, of course, remain useful only so long as their propellant lasts.  On their own they are useful for short duration missions, or missions requiring a number of complex maneuvers, but not for long-term attitude maintenance.
Earlier we mentioned that if the spacecraft develops a magnetic field it will interact with Earth's magnetic field producing a torque.  This can be done deliberately through the use of magnetic torquers.  The system is simply a coil of wire running around either a bar or the spacecraft body itself.  When a current runs through the wire it produces a magnetic field.
For many small satellites and missions with low accuracy requirements passive stability is all that's needed.  However many mission with very high accuracy requirements can't rely on these methods alone.  Active control allows, through feedback, much higher accuracy at the cost of additional weight and complexity.
Low earth orbiting satellites frequently use spin stabilization alone, or combined with thrusters or magnetic torquers.  Above about 600km in altitude the remaining atmosphere is thin enough that it becomes practical to use gravity gradient stabilization.  At geostationary orbit, however, the gravity gradient isn't strong enough to provide stability and only thrusters or wheels are useful.  Momentum wheels provide moderate accuracy and are inexpensive, but for high accuracy reaction wheels or control momentum gyros are necessary.