Guidance, navigation and control

Draft

Three distinct questions, often conflated. Navigation is where the vehicle is. Guidance is where it should go. Control is how to make it go there. For most spacecraft the dominant problem is attitude rather than position, because pointing is what the payload cares about.

Attitude determination

No single sensor is sufficient. Each has a failure regime, and attitude determination is the business of combining several.

Sensor	Accuracy	Provides	Limitation
Star tracker	1 to 10 arcsec	Full three-axis attitude	Blinded by Sun, Earth or Moon in the field of view; degrades under slew rate
Sun sensor	0.1 to 1°	One vector	Useless in eclipse
Earth horizon sensor	0.05 to 0.5°	Two axes	Needs a well-defined limb; atmospheric variability
Magnetometer	0.5 to 3°	One vector	Only in low Earth orbit; field model errors
Gyroscope	Drift-limited	Rate, integrated to attitude	Drifts without absolute reference; must be updated

A star tracker gives full attitude from one measurement by matching an observed star pattern against an onboard catalogue. It is the most accurate instrument on most spacecraft and the most easily blinded, which is why gyros accompany it: gyros carry attitude through the blinding, star trackers correct the gyro drift when they can see. Neither works alone.

Two vector measurements determine attitude completely, and the classical solutions to that problem, TRIAD and QUEST, solve Wahba’s least-squares formulation. Operationally these are superseded by an extended Kalman filter that fuses all available measurements with the gyro-propagated state and simultaneously estimates gyro bias. For quaternion states the multiplicative EKF is standard, because it maintains the unit-norm constraint that an additive filter violates.

Attitude representations

Representation	Parameters	Singularity	Use
Euler angles	3	Gimbal lock at 90°	Human-readable display only
Direction cosine matrix	9	None	Coordinate transformation
Quaternion	4	None	Onboard state and propagation
Modified Rodrigues parameters	3	At 360° rotation	Control law derivation

Quaternions dominate flight software for three reasons: no singularity anywhere on the sphere, composition of rotations requires only multiplication and no trigonometric functions, and interpolation between attitudes is smooth and well-defined. The cost is that they are not intuitive to read and that every quaternion has an equivalent negative representing the same physical attitude, which must be handled consistently or a controller will occasionally command the long way round a rotation.

Euler angles survive in ground displays because humans can read roll, pitch and yaw. They should not survive in the onboard state.

Actuators

Actuator	Torque	Momentum storage	Consumes	Note
Reaction wheel	0.01 to 1 N·m	0.5 to 100 N·m·s	Power only	The workhorse; four in a pyramid for redundancy
Control moment gyro	10 to 1000 N·m	Large	Power only	Torque amplification; used for agile imaging and stations
Magnetorquer	~10⁻⁴ N·m	None	Power only	Free momentum dumping, low Earth orbit only
Thruster	0.1 to 100 N·m	None	Propellant	High authority, finite total impulse

Reaction wheels work by exchanging angular momentum with the spacecraft: to rotate the vehicle one way, spin a wheel the other. Total system momentum is unchanged, which is exactly the limitation. The wheel can only absorb momentum until it reaches its speed limit, at which point it saturates and provides no further torque.

Control moment gyros instead gimbal a constantly spinning rotor, so the output torque is the cross product of the rotor momentum and the gimbal rate. This amplifies torque by orders of magnitude for the same stored momentum, which is why the ISS and agile imaging satellites use them. The price is a control law that must avoid gimbal-lock singularities, geometric configurations in which the available gimbal rates cannot produce torque in the required direction. CMG steering laws are substantially harder than wheel control.

Disturbance torques

Wheels saturate because external torques act secularly. For a representative low Earth orbit spacecraft at 400 km with 10 m² of area and a 0.1 m offset between centre of pressure and centre of mass:

Source	Magnitude	Scaling
Gravity gradient	1.9 × 10⁻⁴ N·m	$\propto 1/r^3$ , and with inertia asymmetry
Aerodynamic	1.7 × 10⁻⁴ N·m	With atmospheric density; dominant in low orbits
Solar radiation pressure	5.9 × 10⁻⁶ N·m	Independent of altitude; dominant in GEO
Residual magnetic dipole	3.0 × 10⁻⁶ N·m	$\propto 1/r^3$ ; low Earth orbit only

Gravity gradient and aerodynamic torque dominate in LEO; solar radiation pressure dominates in GEO, where the other two have fallen away. That crossover determines the momentum management strategy for the entire mission.

Gravity gradient torque arises because the near side of an extended body is pulled harder than the far side, producing a torque that tends to align the minimum-inertia axis with the local vertical. It can be exploited: a passively gravity-gradient-stabilised satellite with a long boom needs no attitude control at all, at the cost of a few degrees of pointing accuracy.

Momentum management

Secular disturbance torque accumulates into the wheels continuously, so momentum must be removed, or desaturated, before they reach their limits.

Magnetorquers: Generate torque against the geomagnetic field as m cross B. Free, requiring only power, but limited to low Earth orbit where the field is strong enough and unusable where it is not.
Thrusters: Work anywhere, including GEO and interplanetary. Consume propellant, which makes momentum management a line in the delta-v budget.
Solar pressure trim tabs: Adjustable surfaces that bias solar radiation pressure torque to cancel the secular accumulation. Used on some GEO satellites to eliminate propellant use for desaturation.
Design mitigation: Balancing the vehicle so the centre of pressure sits near the centre of mass reduces the disturbance at source, and is cheaper than any of the above.

For a GEO satellite with no usable magnetic field, desaturation consumes propellant over a fifteen-year mission, and it must be budgeted alongside station keeping. See Station keeping.

The pointing budget

Pointing performance is three separate requirements that are routinely conflated, and getting them confused produces either an unbuildable specification or an unusable spacecraft.

Term	Definition	Set by
Accuracy, or control error	How close the actual attitude is to the commanded one	Sensors, actuators and the control law together
Knowledge	How well the actual attitude is known after the fact	Sensors and the estimator only
Stability, or jitter	How much the attitude moves over a short interval	Structural modes, wheel imbalance, mechanism vibration

The distinction matters because for many payloads knowledge is worth more than accuracy. An imaging satellite that points 100 arcseconds off but knows where it pointed to 2 arcseconds can geolocate its pixels correctly; the opposite cannot. Knowledge can be improved after the fact by reprocessing on the ground; accuracy cannot.

Jitter is a separate problem again and usually a structural one. Reaction wheel static and dynamic imbalance excites structural modes, and the resulting motion at tens to hundreds of hertz smears a long exposure. Mitigations are isolation mounts, wheel speed avoidance zones that keep excitation away from structural resonances, and in extreme cases turning the wheels off during observation and drifting.

References

Wertz, J. R. (ed.) Spacecraft Attitude Determination and Control, Kluwer, 1978. Still the standard reference.
Markley, F. L. and Crassidis, J. L. Fundamentals of Spacecraft Attitude Determination and Control, Springer, 2014.
Sidi, M. J. Spacecraft Dynamics and Control: A Practical Engineering Approach, Cambridge University Press, 1997.
Wahba, G. “A Least Squares Estimate of Satellite Attitude”, SIAM Review, 7:409, 1965.
Wie, B. Space Vehicle Dynamics and Control, 2nd ed., AIAA, 2008, on CMG steering and singularity avoidance.