Guidance, Navigation & Control (GNC)

Q1. (4 marks)

• A unit quaternion represents attitude: $q=(q_0,q_1,q_2,q_3)$ with scalar part $q_0$ and vector part $(q_1,q_2,q_3)$. (1)
• Constraint: $q_0^2+q_1^2+q_2^2+q_3^2=1$. (1)
• With $q=(0.5,0.5,0.5,c)$: $0.25+0.25+0.25+c^2=1\Rightarrow c^2=0.25$. (1)
• $c=0.5$ (positive root). (1) Why: attitude quaternions must lie on the unit 3-sphere to represent a pure rotation.

Q2. (5 marks)

• Gimbal lock is the loss of one rotational degree of freedom when two of the three Euler rotation axes align. (2)
• In 3-2-1 convention it occurs at pitch $\theta=\pm90^\circ$. (1)
• At this point the roll and yaw axes become parallel, so roll and yaw produce the same motion. (1)
• The DCM-to-Euler inversion becomes singular (division by $\cos\theta=0$), only $\phi\pm\psi$ is defined. (1)

Q3. (4 marks) $R_3(\psi)=\begin{pmatrix}\cos\psi & \sin\psi & 0\\ -\sin\psi & \cos\psi & 0\\ 0 & 0 & 1\end{pmatrix}$

• Correct 2×2 rotation block (2), correct sign convention (rotating reference→body, active-passive) (1), third row/column identity (1).

Q4. (5 marks)

• Kinematic equation: $\dot C = -[\omega\times]\,C$. (2) ($C$ = DCM inertial→body, $\omega$ = body angular velocity)
• Skew matrix: (3) $[\omega\times]=\begin{pmatrix}0 & -\omega_3 & \omega_2\\ \omega_3 & 0 & -\omega_1\\ -\omega_2 & \omega_1 & 0\end{pmatrix}$

Q5. (5 marks)

• $\dot x = Ax + Bu$, $\;y = Cx + Du$. (2)
• $A$ = system/dynamics matrix, $B$ = input matrix, $C$ = output matrix, $D$ = feedthrough matrix. (2)
• Stability determined by eigenvalues of $A$ (all with negative real part ⇒ stable). (1)

Q6. (6 marks)

• Proportional $K_p e$: acts on present error; increases speed of response, reduces (not eliminates) steady-state error. (2)
• Integral $K_i\int e\,dt$: acts on accumulated past error; eliminates steady-state error but can add lag/overshoot. (2)
• Derivative $K_d\dot e$: acts on rate of change (predictive); adds damping, reduces overshoot, sensitive to noise. (2)

Q7. (5 marks)

• Law: $a_c = N\,V_c\,\dot\lambda$ (commanded lateral acceleration). (2)
• $N$ = navigation constant, $V_c$ = closing velocity, $\dot\lambda$ = line-of-sight (LOS) rate. (2)
• Typical $N\in[3,5]$. (1)

Q8. (4 marks)

• Diagonal matrix ⇒ eigenvalues $\lambda_1=-2$, $\lambda_2=-5$. (2)
• Both have negative real part. (1)
• Therefore the system is (asymptotically) stable. (1)

Q9. (6 marks)

• State prediction: $\hat x_{k+1}^- = F\hat x_k^+$. (2)
• Covariance prediction: $P_{k+1}^- = F P_k^+ F^T + Q$. (2)
• Symbols: $F$ = state transition matrix, $P$ = state error covariance, $Q$ = process noise covariance, superscript $-$ = a-priori (predicted), $+$ = a-posteriori (updated). (2)

Q10. (6 marks)

• Pseudorange = measured distance from receiver to satellite computed as $c\times$(signal travel time) using receiver clock. (2)
• It differs from geometric range because of the unknown receiver clock bias (and atmospheric delays). (2)
• Minimum 4 satellites: 3 for the position coordinates $(x,y,z)$ plus 1 to solve for the receiver clock bias. (2)

[
  {"claim":"Unit quaternion c=0.5 for q=(0.5,0.5,0.5,c)","code":"c=sqrt(1-3*Rational(1,4)); result = (c==Rational(1,2))"},
  {"claim":"Eigenvalues of diag(-2,-5) are -2,-5 and stable","code":"A=Matrix([[-2,0],[0,-5]]); ev=list(A.eigenvals().keys()); result = (set(ev)=={-2,-5}) and all(re(e)<0 for e in ev)"},
  {"claim":"Skew matrix times own axis vector is zero (omega x omega =0)","code":"w1,w2,w3=symbols('w1 w2 w3'); S=Matrix([[0,-w3,w2],[w3,0,-w1],[-w2,w1,0]]); w=Matrix([w1,w2,w3]); result = simplify(S*w)==zeros(3,1)"},
  {"claim":"R3(psi) is orthogonal with determinant 1","code":"p=symbols('psi'); R=Matrix([[cos(p),sin(p),0],[-sin(p),cos(p),0],[0,0,1]]); result = simplify(R.det())==1 and simplify(R*R.T)==eye(3)"}
]