Self-Assessment Quiz: Spacecraft Attitude Dynamics

Twenty questions on attitude versus orbit, rotation representations, torques, sensors, actuators, and stabilization. Answer each before opening the key. Aim for 16 or more. Recall the §14.2 conventions: passive DCM (with $+\sin\theta$ upper-right in $R_z$) and Hamilton, scalar-first quaternions.

Question 1

A spacecraft's attitude describes its:

A) position in orbit B) orientation (which way it points) C) altitude above the Earth D) orbital energy

Question 2

Attitude is driven by __, whereas orbit is driven by ____.

A) torque; force B) force; torque C) energy; momentum D) thrust; gravity

Question 3

A rigid spacecraft has how many total degrees of freedom, and how many are attitude?

A) 3 total, 3 attitude B) 6 total, 3 attitude C) 6 total, 6 attitude D) 9 total, 3 attitude

Question 4

Gimbal lock in a 3–2–1 Euler-angle set occurs when the pitch angle reaches:

A) $0^\circ$ B) $45^\circ$ C) $90^\circ$ D) $180^\circ$

Question 5

The main reason spacecraft flight software represents attitude with quaternions rather than Euler angles is that quaternions:

A) are easier for humans to read B) have no singularity (no gimbal lock) C) use fewer than three numbers D) do not need a reference frame

Question 6

A unit quaternion for a rotation of angle $\theta$ about axis $\hat{\mathbf{e}}$ has scalar part:

A) $\sin\theta$ B) $\cos\theta$ C) $\cos(\theta/2)$ D) $\theta/2$

Question 7

Euler's rotation theorem states that any orientation can be reached by:

A) exactly three rotations about three axes B) a single rotation about a single fixed axis C) an infinite series of small rotations D) two rotations about perpendicular axes

Question 8

Which environmental torque dominates at geostationary altitude?

A) aerodynamic B) gravity-gradient C) magnetic D) solar radiation pressure

Question 9

Gravity-gradient torque scales with orbital radius as:

A) $1/r$ B) $1/r^2$ C) $1/r^3$ D) independent of $r$

Question 10

A star tracker determines attitude by:

A) measuring the spacecraft's turn rate B) sensing Earth's magnetic field C) imaging the star field and matching it to a catalog D) measuring the direction to the Sun

Question 11

A single Sun-direction measurement determines how many of the three attitude degrees of freedom?

A) 0 B) 2 C) 3 D) all six

Question 12

A rate gyroscope measures:

A) absolute orientation B) angular velocity (turn rate) C) the direction to the Sun D) magnetic field strength

Question 13

A reaction wheel turns the spacecraft by:

A) expelling propellant B) pushing against Earth's magnetic field C) spinning a flywheel the opposite way (conservation of angular momentum) D) using aerodynamic surfaces

Question 14

A control moment gyroscope (CMG) produces torque by:

A) spinning a wheel up from rest B) tilting the gimbal of a constantly spinning wheel to redirect its momentum C) firing thrusters D) changing the spacecraft's orbit

Question 15

Momentum dumping is required because:

A) reaction wheels leak propellant B) internal actuators can only redistribute momentum, so an external torque must remove accumulated momentum C) star trackers get blinded D) gyros drift

Question 16 (True/False, justify)

"A spacecraft in orbit is weightless, so no torques act on it and it will hold its orientation on its own." True or false? Justify in one sentence.

Question 17 (True/False, justify)

"Because they carry no propellant, reaction wheels can hold a spacecraft against a steady one-directional disturbance torque indefinitely." True or false? Explain.

Question 18 (True/False, justify)

"A spinning spacecraft is stable only if it spins about its axis of maximum moment of inertia." True or false? Name the mission that demonstrated this.

Question 19 (Short answer)

A pointing budget has independent error terms of 30, 40, and 120 arcsec. Compute the root-sum-square total, and state which term you should reduce first and why.

Question 20 (Short answer)

In one or two sentences, explain the difference between pointing accuracy and pointing knowledge, and name the sensor that most directly improves knowledge.

Answer Key

Q Ans Note
1 B Attitude is orientation; position/altitude/energy are orbit quantities.
2 A Torque changes orientation; force changes the center-of-mass motion.
3 B Six DOF: 3 translational (orbit) + 3 rotational (attitude).
4 C At pitch $90^\circ$ two axes align; only $(\phi-\psi)$ survives, and the rate equations divide by $\cos\theta \to 0$.
5 B Quaternions have no singular orientation; they are also compact and numerically robust — but the defining win is no gimbal lock.
6 C $q_0 = \cos(\theta/2)$; the vector part is $\hat{\mathbf{e}}\sin(\theta/2)$.
7 B Euler's rotation theorem — the basis of the quaternion.
8 D Solar-radiation-pressure torque is nearly altitude-independent; the others fall off with distance from Earth.
9 C $M_{gg} \propto 3\mu/r^3 = 3n^2$.
10 C It matches an imaged star pattern to an onboard catalog for full 3-axis attitude.
11 B A single direction fixes 2 DOF; rotation about the Sun line is unresolved (need a second vector).
12 B Rate gyros measure angular velocity; you integrate (and it drifts) to get orientation.
13 C Wheel spin-up reacts on the body by conservation of angular momentum — no propellant.
14 B Redirecting a large stored momentum gives torque amplification.
15 B $\dot{\mathbf{H}}_{\text{total}} = \mathbf{M}_{\text{external}}$: only an external torque removes total momentum.
16 False Free fall removes weight, not torque; gravity-gradient, solar, magnetic, and aerodynamic torques act unopposed and accumulate.
17 False A wheel is internal; a steady disturbance piles momentum into it until it saturates. Only an external torque (dumping) removes it.
18 True Energy dissipation drives a real body to spin about the max-inertia axis; Explorer 1 flat-spun after being set to spin about its long (min-inertia) axis.
19 — $\sqrt{30^2+40^2+120^2} = \sqrt{900+1600+14400} = \sqrt{16900} = 130\ \text{arcsec}$; reduce the 120 term first — squaring makes the largest term dominate.
20 — Accuracy = how close actual pointing is to commanded (a control error); knowledge = how well you know where you point (a determination error). The star tracker most directly improves knowledge.

Topics to review by question

Questions Topic Section
1, 2, 3, 16 Attitude vs. orbit; degrees of freedom §14.1
4, 5, 6, 7 Euler angles, gimbal lock, quaternions, Euler's theorem §14.2
8, 9 Environmental torques and their scaling §14.3
10, 11, 12 Attitude-determination sensors §14.4
13, 14, 15, 17 Actuators and momentum dumping §14.5
18 Spin stability and the major-axis rule §14.6
19, 20 Pointing budgets; accuracy vs. knowledge §14.6