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Control ExamSprint Handbook
Question 1. Which matrix property guarantees that a Direction Cosine Matrix (DCM) preserves vector lengths during rotation? A) Symmetry B) Orthogonality C) Positive definiteness D) Skew‑symmetry Answer: B Explanation: An orthogonal matrix satisfies (C^T C = I), ensuring that the Euclidean norm of any vector is unchanged after transformation. Question 2. In a 3‑ 2 ‑1 Euler angle sequence, which axis is rotated first? A) X‑axis B) Y‑axis C) Z‑axis D) None of the above Answer: C Explanation: The 3‑ 2 ‑1 sequence rotates about the inertial Z‑axis (3), then the intermediate Y‑axis (2), and finally the body‑fixed X‑axis (1). Question 3. What is the primary advantage of using quaternions over Euler angles for spacecraft attitude representation? A) Smaller memory footprint B) No singularities such as gimbal lock C) Easier to visualize D) Direct torque computation Answer: B Explanation: Quaternions provide a globally non‑singular representation, avoiding the gimbal‑lock problem inherent in Euler angles.
Control ExamSprint Handbook
Question 4. The eigenaxis of a rotation is defined as: A) The axis about which the rotation angle is zero B) The axis that remains invariant under the rotation C) The axis with the maximum moment of inertia D) The axis aligned with the magnetic field Answer: B Explanation: The eigenaxis (or Euler axis) is the line that does not move during the rotation; all points rotate about it. Question 5. Modified Rodrigues Parameters (MRPs) become singular when the rotation angle equals: A) 90° B) 180° C) 360° D) 45° Answer: B Explanation: MRPs use a stereographic projection that becomes undefined at a rotation of 180°, where the denominator vanishes. Question 6. Which term in Euler’s rotational equation (\dot{\mathbf{H}} + \boldsymbol{\omega} \times \mathbf{H} = \mathbf{M}) represents the effect of rotating reference axes? A) (\dot{\mathbf{H}}) B) (\boldsymbol{\omega} \times \mathbf{H}) C) (\mathbf{M}) D) None of the above Answer: B Explanation: The cross‑product term accounts for the change of angular momentum due to the rotating body frame.
Control ExamSprint Handbook
Question 10. Which environmental disturbance is most significant for a spacecraft in a sun‑synchronous low‑Earth orbit? A) Magnetic torque B) Solar radiation pressure C) Aerodynamic drag D) Gravity‑gradient torque Answer: C Explanation: At low altitudes, residual atmospheric particles produce drag that dominates the disturbance budget. Question 11. The residual magnetic dipole of a spacecraft interacts with Earth’s magnetic field to produce: A) Aerodynamic torque B) Gravity‑gradient torque C) Magnetic torque D) Solar radiation torque Answer: C Explanation: The torque is given by (\mathbf{M}_m = \mathbf{m} \times \mathbf{B}), where (\mathbf{m}) is the spacecraft dipole and (\mathbf{B}) the geomagnetic field. Question 12. In the Wahba problem, the loss function minimized is: A) Sum of squared differences between measured and predicted sensor readings B) Weighted sum of squared errors between body‑frame and reference vectors after rotation C) Integral of torque over time D) Determinant of the attitude matrix Answer: B Explanation: Wahba’s loss function (J = \frac{1}{2}\sum_i a_i | \mathbf{v}_i^{\mathrm{b}} - C \mathbf{v}_i^{\mathrm{r}} |^2) seeks the optimal rotation matrix (C).
Control ExamSprint Handbook
Question 13. The TRIAD algorithm requires how many independent vector observations to uniquely determine attitude? A) One B) Two C) Three D) Four Answer: B Explanation: Two non‑collinear vectors provide enough information to construct the orthonormal basis needed for TRIAD. Question 14. The QUEST algorithm solves Wahba’s problem by finding the eigenvector associated with: A) The smallest eigenvalue of the attitude profile matrix B) The largest eigenvalue of the attitude profile matrix C) The zero eigenvalue of the DCM D) None of the above Answer: B Explanation: QUEST computes the optimal quaternion as the eigenvector corresponding to the maximum eigenvalue of the Davenport K‑matrix. Question 15. In an Extended Kalman Filter (EKF) for attitude estimation, the state vector typically includes: A) Position only B) Quaternion and gyroscope bias C) Only angular rates D) Magnetic field measurements only Answer: B Explanation: The EKF state often comprises the attitude quaternion (or MRPs) and gyro bias to correct integration drift.
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A) Producing a magnetic dipole that interacts with Earth’s magnetic field B) Heating the spacecraft structure C) Using solar pressure D) Emitting photons Answer: A Explanation: The torque is (\mathbf{M} = \mathbf{m} \times \mathbf{B}), where (\mathbf{m}) is the magnetic moment generated by the coil. Question 20. In a PD controller for small‑angle attitude control, the “D” term primarily provides: A) Steady‑state error elimination B) Damping of oscillations C) Increased gain at low frequencies D) Integral action Answer: B Explanation: The derivative term adds damping, reducing overshoot and settling time. Question 21. Lyapunov‑based nonlinear control ensures stability by: A) Maximizing the control gain B) Designing a Lyapunov function that is always decreasing along system trajectories C) Using only proportional feedback D) Ignoring disturbances Answer: B Explanation: If a positive‑definite Lyapunov function has a negative‑definite derivative, the closed‑loop system is stable. Question 22. Momentum desaturation of reaction wheels is commonly performed using: A) Thrusters only B) Magnetorquers, thrusters, or a combination
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C) Solar panels D) Gyroscopes Answer: B Explanation: Desaturation transfers excess wheel momentum to the environment via magnetic torques or propulsive thrust. Question 23. The term “pointing budget” refers to: A) The total mass allocated for sensors B) The error budget allocated to jitter, drift, and settling time for a mission’s attitude requirement C) The power consumption of actuators D) The orbital fuel budget Answer: B Explanation: A pointing budget quantifies allowable errors from various sources to meet overall attitude performance. Question 24. In flexible‑body dynamics, the coupling between attitude control and structural vibrations is known as: A) Gravity‑gradient coupling B) Control‑Structure Interaction (CSI) C) Magnetorquer coupling D) Solar pressure coupling Answer: B Explanation: CSI describes how actuator commands can excite flexible modes and how those modes affect attitude. Question 25. For a nadir‑pointing satellite, the primary attitude reference is: A) Inertial stars B) Earth’s center direction (local vertical) C) Sun vector
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Explanation: Sun sensors output the Sun’s direction relative to the sensor’s body axes. Question 29. The “bias” term commonly estimated in gyro‑based attitude integration represents: A) Scale factor error B) Constant offset in measured angular rate C) Temperature drift only D) Random noise variance Answer: B Explanation: Gyro bias is a slowly varying constant error that must be estimated and subtracted to avoid drift. Question 30. In the Kalman filter, the process noise covariance matrix Q primarily models: A) Measurement errors B) Uncertainty in the system dynamics (e.g., torque disturbances) C) Sensor bias D) Computational rounding errors Answer: B Explanation: Q captures the uncertainty associated with the prediction step, including unmodeled torques and model inaccuracies. Question 31. The DCM element (C_{12}) represents the cosine of the angle between which pair of axes? A) Body X‑axis and Inertial Y‑axis B) Body Y‑axis and Inertial X‑axis C) Body Z‑axis and Inertial X‑axis D) Body X‑axis and Inertial Z‑axis Answer: A Explanation: In a DCM, element (i,j) is the cosine of the angle between body axis i and inertial axis j; thus (C_{12}) is body‑X vs inertial‑Y.
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Question 32. The parallel‑axis theorem is used to: A) Compute the magnetic dipole moment B) Shift an inertia tensor from the center of mass to another point C) Determine solar pressure torque D) Calculate aerodynamic drag Answer: B Explanation: The theorem adds (m d^2) to the inertia about a parallel axis displaced by distance (d). Question 33. Which attitude representation is most compact while still avoiding singularities for rotations up to 180°? A) Euler angles B) Direction Cosine Matrix C) Quaternion D) Gibbs vector Answer: C Explanation: Quaternions use four parameters, are unit‑norm constrained, and have no singularities up to 360°. Question 34. The term “shadow set” in MRPs refers to: A) The set of MRPs representing the same attitude after a 360° rotation B) The alternative parameter set obtained by stereographic projection when the primary set approaches singularity C) The error covariance matrix D) The set of unobservable states Answer: B Explanation: When MRPs approach the singularity at 180°, the shadow set provides an alternate representation avoiding the singularity.
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Question 38. In the EKF, the measurement update step primarily serves to: A) Predict the next state using dynamics B) Incorporate sensor data to correct the predicted state and reduce uncertainty C) Increase process noise D) Compute the DCM directly Answer: B Explanation: The update fuses new measurements with the prediction, yielding an improved state estimate. Question 39. The term “gyro drift” is synonymous with: A) Random walk noise B) Bias instability C) Scale factor error D) Quantization error Answer: B Explanation: Drift refers to the slow change in gyro bias over time, often modeled as a random walk. Question 40. For a spacecraft in a Sun‑synchronous orbit, the dominant solar radiation pressure torque direction is: A) Along the orbital angular momentum vector B) Along the Sun‑spacecraft line C) Perpendicular to the orbital plane D) Along the Earth‑pointing axis Answer: B Explanation: SRP acts along the Sun‑spacecraft vector; the torque depends on the offset between center of pressure and center of mass. Question 41. The “principal axes” of inertia are defined such that: A) The inertia tensor is diagonal in that coordinate system
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B) They are always aligned with the spacecraft’s geometric axes C) They minimize aerodynamic drag D) They maximize magnetic torque Answer: A Explanation: In the principal‑axis frame, products of inertia vanish, leaving only the diagonal moments (I_1, I_2, I_3). Question 42. In a dual‑spin satellite, the rotor is typically spun at a high speed to provide: A) Aerodynamic stability B) Gyroscopic stiffness that resists external torques C) Magnetic alignment D) Power generation Answer: B Explanation: The high‑speed rotor creates a large angular momentum vector, giving gyroscopic stability. Question 43. The “energy‑momentum diagram” for a rigid body illustrates: A) The relationship between kinetic energy and angular momentum magnitude for torque‑free motion B) The spacecraft’s orbital energy C) The power consumption of actuators D) The magnetic field strength vs. altitude Answer: A Explanation: It plots kinetic energy versus angular momentum, showing possible rotational states and stability. Question 44. A “gimbal lock” occurs in Euler angle representation when: A) Two rotation axes become aligned, causing loss of one degree of freedom B) The spacecraft exceeds 90° roll C) The quaternion norm deviates from unity
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Explanation: By biasing the rotor spin, the spacecraft gains gyroscopic stability about the bias axis. Question 48. The term “Euler parameters” is another name for: A) Euler angles B) Quaternions C) Direction cosine matrix elements D) Gibbs vectors Answer: B Explanation: Euler parameters are the four components of a unit quaternion. Question 49. In a star tracker, the “centroiding” algorithm is used to: A) Compute the spacecraft’s magnetic field B) Determine the precise pixel location of star images for attitude extraction C) Measure solar flux D) Estimate atmospheric drag Answer: B Explanation: Centroiding finds sub‑pixel star positions to compare with a star catalog. Question 50. The “B‑dot” control law for magnetic torquers uses which quantity? A) The time derivative of the magnetic field vector measured in the body frame B) The magnetic field magnitude only C) The spacecraft’s angular velocity D) The solar radiation pressure coefficient Answer: A Explanation: B‑dot control commands a magnetic moment proportional to (-\dot{\mathbf{B}}) to produce a damping torque.
Control ExamSprint Handbook
Question 51. Which of the following is a necessary condition for the quaternion multiplication to be associative? A) All quaternions must be unit‑norm B) Quaternion multiplication is always associative regardless of norm C) The quaternions must be pure vectors D) The scalar parts must be zero Answer: B Explanation: Quaternion multiplication is inherently associative; unit‑norm is required only for attitude representation. Question 52. The “angular rate bias” in a gyro is typically modeled as: A) A deterministic constant that never changes B) A random walk process in the filter state vector C) Pure white noise D) A sinusoidal variation with orbital period Answer: B Explanation: Bias drift is commonly modeled as a random walk, allowing the filter to estimate and track it over time. Question 53. The “Schmidt orthogonalization” process is used in attitude determination to: A) Convert measured vectors into an orthonormal basis for TRIAD B) Compute the DCM directly from sensor data C) Reduce magnetic interference D) Linearize the EKF equations Answer: A Explanation: Schmidt (or Gram‑Schmidt) orthogonalization creates orthonormal vectors from measured data for TRIAD. Question 54. In the context of control moment gyros, “singular configuration” occurs when:
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C) (V = \mathbf{\omega}^T \mathbf{I} \mathbf{\omega}) D) (V = \det(C)) Answer: B Explanation: (V = 1 - q_0) is positive‑definite and zero when the error quaternion aligns (i.e., (q_0 = 1 )). Question 58. In a low‑Earth orbit, the dominant source of atmospheric drag is: A) Molecular nitrogen at 300 km altitude B) Solar wind particles C) Charged particles trapped in the Van Allen belts D) Micrometeoroids Answer: A Explanation: At LEO altitudes, residual neutral atmosphere (mainly N2 and O2) provides the primary drag force. Question 59. The “B‑dot” control law inherently provides: A) Energy generation B) Passive damping of attitude motion without explicit knowledge of inertia C) High‑gain torque amplification D) Magnetic field amplification Answer: B Explanation: By commanding a magnetic moment opposite to (\dot{\mathbf{B}}), the resulting torque damps rotational motion regardless of inertia. Question 60. In the EKF, the “innovation” is defined as: A) The difference between predicted and measured sensor outputs B) The process noise covariance C) The state transition matrix D) The quaternion norm error
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Answer: A Explanation: Innovation (or residual) quantifies the mismatch between prediction and measurement, driving the update. Question 61. The “gravity‑gradient torque” is zero when: A) The spacecraft’s principal axes are aligned with the local vertical and horizontal directions B) The spacecraft is in geostationary orbit C) The inertia tensor is isotropic (sphere) D) Both A and C Answer: D Explanation: If the body is a sphere (isotropic inertia) or if the principal axes align with the gravity gradient, the torque vanishes. Question 62. The “reaction wheel momentum storage” is limited primarily by: A) Magnetic field strength B) Wheel material strength and motor speed capability C) Solar panel area D) Atmospheric density Answer: B Explanation: Maximum stored angular momentum depends on wheel mass, radius, and maximum safe rotational speed. Question 63. The “magnetic dipole moment” generated by a magnetorquer coil is proportional to: A) The square of the coil current B) The coil area times the current (NI) C) The orbital altitude D) The solar flux Answer: B
