Level 1 — RecognitionGuidance, Navigation & Control (GNC)

Guidance, Navigation & Control (GNC)

⏱ 20 minutes40 marksprintable — key stays hidden on paper

Chapter: 3.5 Guidance, Navigation & Control Level: 1 — Recognition (MCQ, matching, true/false with justification) Time limit: 20 minutes Total marks: 40

Section A — Multiple Choice (1 mark each; 16 marks)

Choose the single best answer.

Q1. An accelerometer in free-fall (no thrust, only gravity) measures a specific force of:

(a) $g$ downward
(b) $g$ upward
(c) zero
(d) $2g$

Q2. The 3-2-1 Euler angle sequence rotates by angles in the order:

(a) roll, pitch, yaw
(b) yaw, pitch, roll
(c) pitch, yaw, roll
(d) yaw, roll, pitch

Q3. The unit quaternion constraint is:

(a) $q_0 + q_1 + q_2 + q_3 = 1$
(b) $q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1$
(c) $q_0^2 - q_1^2 - q_2^2 - q_3^2 = 1$
(d) $q_1^2 + q_2^2 + q_3^2 = 1$

Q4. Gimbal lock in a 3-2-1 Euler sequence occurs at pitch angle:

(a) $\theta = 0$
(b) $\theta = 45^\circ$
(c) $\theta = \pm 90^\circ$
(d) $\theta = 180^\circ$

Q5. The DCM kinematics equation is:

(a) $\dot{C} = [\omega\times]C$
(b) $\dot{C} = -[\omega\times]C$
(c) $\dot{C} = C[\omega\times]$
(d) $\dot{C} = -C^T[\omega\times]$

Q6. A continuous-time linear system $\dot{x}=Ax$ is asymptotically stable if and only if:

(a) all eigenvalues of $A$ have positive real part
(b) all eigenvalues of $A$ have negative real part
(c) $A$ is symmetric
(d) $\det(A)=0$

Q7. Proportional navigation guidance commands acceleration proportional to:

(a) range to target
(b) $N\,V_c\,\dot\lambda$ (LOS rotation rate)
(c) closing velocity only
(d) the target's absolute acceleration

Q8. The Kalman gain is chosen to:

(a) maximize the trace of the covariance
(b) minimize the trace of the posterior error covariance
(c) make the innovation zero
(d) set the process noise to zero

Q9. A PID controller output is:

(a) $K_p e + K_i \int e\,dt + K_d \dot{e}$
(b) $K_p e + K_i \dot{e} + K_d \int e\,dt$
(c) $K_p \int e\,dt$
(d) $K_p \dot{e}$

Q10. The controllability matrix for $(A,B)$ with $A\in\mathbb{R}^{n\times n}$ is:

(a) $[C\ CA\ \cdots\ CA^{n-1}]^T$
(b) $[B\ AB\ A^2B\ \cdots\ A^{n-1}B]$
(c) $[A\ B]$
(d) $A^{-1}B$

Q11. Gyroscope measurement error typically includes:

(a) bias and noise
(b) only quantization
(c) gravity coupling only
(d) no error (ideal)

Q12. GPS position is found from at least how many satellite pseudoranges (for 3D + clock)?

(a) 2
(b) 3
(c) 4
(d) 6

Q13. In the Nyquist stability criterion, stability is assessed by counting encirclements of the point:

(a) $0$
(b) $+1$
(c) $-1$
(d) $+\infty$

Q14. The LQR optimal gain requires solving the:

(a) Lyapunov equation
(b) algebraic Riccati equation
(c) Bellman ODE only
(d) characteristic polynomial of $B$

Q15. Modified Rodrigues parameters are attractive because they are:

(a) always exactly the Euler angles
(b) compact and singularity-free over a large range
(c) a 4-parameter set with a constraint
(d) only defined for small rotations

Q16. A control moment gyroscope (CMG) provides high torque but suffers from:

(a) zero-crossing friction
(b) gimbal singularities
(c) plume impingement
(d) pseudorange bias

Section B — Matching (1 mark each; 12 marks)

Match each item in Column X to the correct description in Column Y. Write pairs (e.g., M1–?).

Column X		Column Y
M1. Complementary filter		A. Two vector measurements → attitude
M2. TRIAD method		B. Optimal linear estimator, predict/update
M3. Kalman filter		C. Simple frequency-domain sensor blend
M4. EKF		D. Sigma points, no Jacobians
M5. UKF		E. Linearization via Jacobians
M6. IMU		F. Accelerometer + gyroscope package

Column X		Column Y
M7. Root locus		P. Magnitude & phase vs frequency
M8. Bode plot		Q. Poles migrate as gain varies
M9. Observability matrix		R. Distributes torque among thrusters/wheels
M10. Pole placement		S. Ackermann's formula
M11. RCS thruster selection		T. Stack of $C, CA, \dots, CA^{n-1}$
M12. Reaction wheel		U. Momentum storage, zero-crossing issue

Section C — True/False WITH one-line justification (2 marks each: 1 T/F + 1 justification; 12 marks)

Q17. "Quaternions avoid gimbal lock entirely." — True/False? Justify.

Q18. "The gravity vector coincides with the specific force measured by an accelerometer on a table." — True/False? Justify.

Q19. "Adding more, well-spread GPS satellites reduces the geometric dilution of precision (GDOP)." — True/False? Justify.

Q20. "A system can be stabilized by full-state feedback pole placement only if it is controllable." — True/False? Justify.

Q21. "In a 3-2-1 DCM, the total rotation matrix is $R_1(\phi)R_2(\theta)R_3(\psi)$ applied to a vector expressed in inertial frame to get body frame." — True/False? Justify.

Q22. "The LQG controller can be designed by separately designing the LQR and the Kalman filter (separation principle)." — True/False? Justify.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (c) zero. In free-fall, the only force is gravity; an accelerometer measures non-gravitational specific force $f = a - g$ . In free-fall $a=g$ , so $f=0$ . (This is why accelerometers can't sense gravity directly.)

Q2 — (b) yaw, pitch, roll. The "3-2-1" refers to rotation axes in order: 3rd axis (yaw ψ), then 2nd (pitch θ), then 1st (roll φ).

Q3 — (b) $q_0^2+q_1^2+q_2^2+q_3^2=1$ . A unit quaternion has norm 1 to represent a pure rotation.

Q4 — (c) $\theta=\pm90^\circ$ . At pitch $=\pm90^\circ$ the roll and yaw axes align → one DOF lost → gimbal lock.

Q5 — (b) $\dot{C}=-[\omega\times]C$ . Standard body-to-inertial DCM kinematics with body rates.

Q6 — (b) negative real parts. Hurwitz condition: all eigenvalues in open left half-plane ⇒ modes $e^{\lambda t}\to 0$ .

Q7 — (b) $N V_c \dot\lambda$ . PN commands lateral acceleration proportional to closing velocity times line-of-sight rate.

Q8 — (b) minimize trace of posterior covariance. Kalman gain minimizes expected squared error = trace of $P^+$ .

Q9 — (a) $K_p e + K_i\int e\,dt + K_d\dot e$ . Definition of PID.

Q10 — (b) $[B\ AB\ \dots\ A^{n-1}B]$ . Controllability matrix; full rank $n$ ⇒ controllable.

Q11 — (a) bias and noise. Gyro model: $\tilde\omega = \omega + b + n$ .

Q12 — (c) 4. Three coordinates plus receiver clock bias = 4 unknowns → 4 pseudoranges.

Q13 — (c) $-1$ . Nyquist counts encirclements of $-1+j0$ .

Q14 — (b) algebraic Riccati equation. $K=R^{-1}B^TP$ , with $P$ from the ARE.

Q15 — (b) compact and singularity-free over a large range. MRPs are a 3-parameter set with singularity only at $360^\circ$ .

Q16 — (b) gimbal singularities. CMGs give high torque amplification but can hit singular gimbal configurations.

Section B (1 mark each)

M1–C, M2–A, M3–B, M4–E, M5–D, M6–F, M7–Q, M8–P, M9–T, M10–S, M11–R, M12–U.

Rationale: complementary filter = simple frequency blend; TRIAD uses two vectors; KF is optimal predict/update; EKF linearizes (Jacobians); UKF uses sigma points; IMU = accel+gyro; root locus tracks poles vs gain; Bode = magnitude/phase; observability matrix stacks $C, CA,\dots$ ; pole placement via Ackermann; RCS selection distributes torque; reaction wheels store momentum with zero-crossing friction.

Section C (1 mark T/F + 1 mark justification)

Q17 — TRUE. Quaternions have no coordinate singularity; the 4-parameter representation with unit constraint parametrizes SO(3) globally (double cover), so no gimbal lock.

Q18 — FALSE. On a table $a=0$ , so specific force $f=-g$ (points opposite gravity, magnitude $g$ upward as reaction). The accelerometer reads the support force, not the gravity vector's direction—so it is anti-parallel to $g$ , not coincident.

Q19 — TRUE. More, geometrically well-distributed satellites make the geometry matrix better conditioned, lowering GDOP (better dilution). (Poorly spread satellites still give high GDOP.)

Q20 — TRUE. Arbitrary pole placement via state feedback is possible iff $(A,B)$ is controllable; controllability is the condition for full eigenvalue assignment. (Stabilizability suffices if uncontrollable modes are already stable, but full placement needs controllability.)

Q21 — TRUE. The 3-2-1 DCM from inertial to body is $C_b^i{}^{-1}=R_1(\phi)R_2(\theta)R_3(\psi)$ ; applied to an inertial vector it yields body-frame components. (Order of matrix multiplication = reverse order of physical rotation, consistent with 3-2-1.)

Q22 — TRUE. The separation principle guarantees LQR (state feedback) and Kalman filter (estimator) can be designed independently; combined LQG is optimal and the closed-loop eigenvalues are the union of both designs.

[
  {"claim":"Free-fall specific force is zero: f = a - g with a=g",
   "code":"a=g=9.81; f=a-g; result=(f==0)"},
  {"claim":"Unit quaternion norm-squared equals 1",
   "code":"q0,q1,q2,q3=Rational(1,2),Rational(1,2),Rational(1,2),Rational(1,2); result=(q0**2+q1**2+q2**2+q3**2==1)"},
  {"claim":"GPS 3D+clock requires 4 unknowns hence 4 satellites",
   "code":"unknowns=3+1; result=(unknowns==4)"},
  {"claim":"Hurwitz: eigenvalues of stable A=[[0,1],[-2,-3]] have negative real parts",
   "code":"A=Matrix([[0,1],[-2,-3]]); ev=list(A.eigenvals().keys()); result=all(re(e)<0 for e in ev)"},
  {"claim":"Controllability matrix of A=[[0,1],[0,0]],B=[[0],[1]] has full rank 2",
   "code":"A=Matrix([[0,1],[0,0]]); B=Matrix([[0],[1]]); Cm=B.row_join(A*B); result=(Cm.rank()==2)"}
]