Guidance, Navigation & Control (GNC)

Level: 3 — Production (from-scratch derivations, code-from-memory, explain-out-loud) Time limit: 45 minutes Total marks: 60

Answer all questions. Show every derivation step. Where code is requested, pseudocode or Python-style is acceptable but must be logically complete.

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Question 1 — Quaternion kinematics from scratch (10 marks)

(a) State the unit quaternion constraint and give the quaternion kinematic equation $\dot{q}=\tfrac{1}{2}\,\Xi(q)\,\omega$, writing out the $4\times3$ matrix $\Xi(q)$ explicitly for $q=(q_0,q_1,q_2,q_3)$. (4)

(b) Derive the scalar equation $\dot{q}_0$ in terms of $q_{1,2,3}$ and $\omega=(\omega_x,\omega_y,\omega_z)$. (3)

(c) Show that this kinematics preserves the norm, i.e. $\frac{d}{dt}(q^\top q)=0$. (3)

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Question 2 — Proportional navigation derivation (10 marks)

(a) Starting from the line-of-sight (LOS) geometry, derive the PN guidance law $a_c = N\,V_c\,\dot\lambda$, defining every symbol. (5)

(b) A missile closes on a target with closing velocity $V_c = 800\ \text{m/s}$, LOS rate $\dot\lambda = 0.02\ \text{rad/s}$, and navigation constant $N=4$. Compute the commanded lateral acceleration and express it in units of $g$ ($g=9.81$). (3)

(c) Explain in one or two sentences why $N$ is chosen in the range $3$–$5$ in practice. (2)

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Question 3 — Kalman gain optimality (12 marks)

(a) Write the linear KF measurement update: posterior mean and covariance $P^+ = (I-KH)P^-(I-KH)^\top + KRK^\top$ (Joseph form). (3)

(b) By minimizing $\operatorname{tr}(P^+)$ with respect to $K$, derive the optimal Kalman gain $K = P^- H^\top (H P^- H^\top + R)^{-1}$. Show the differentiation step. (6)

(c) Substitute the optimal $K$ back to show the simplified $P^+=(I-KH)P^-$. (3)

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Question 4 — DCM / gimbal lock (10 marks)

(a) Construct the 3-2-1 (yaw-pitch-roll) DCM $C = R_1(\phi)R_2(\theta)R_3(\psi)$ symbolically (write the product's entries in terms of $s_\bullet,c_\bullet$). You may show the three elementary matrices and one representative product entry. (4)

(b) Give the standard formulas to extract $\phi,\theta,\psi$ from the DCM entries. (3)

(c) Explain mathematically why gimbal lock occurs at $\theta=\pm90^\circ$ — reference the specific DCM entries that become indeterminate. (3)

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Question 5 — LQR / stability (8 marks)

(a) State the continuous-time algebraic Riccati equation (CARE) whose solution $P$ gives the LQR gain $K=R^{-1}B^\top P$. (3)

(b) For the scalar system $\dot x = a x + b u$ with cost weights $q,r$, solve the scalar Riccati equation for $P>0$ and give the closed-loop pole. Use $a=0$, $b=1$, $q=1$, $r=1$ to get numeric values. (5)

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Question 6 — Code from memory: complementary filter (10 marks)

(a) Write pseudocode for a 1-axis complementary filter fusing gyro rate $\omega$ (integrated) and accelerometer-derived angle $\theta_{acc}$, with blend coefficient $\alpha$ and timestep $dt$. (5)

(b) Explain the frequency-domain interpretation: which sensor dominates at high frequency, which at low, and how $\alpha$ relates to the crossover time constant $\tau$. (3)

(c) State one advantage of a Kalman filter over the complementary filter for this task. (2)