SADC-Exam2

What is the basic purpose of attitude determination?

Goal of attitude determination is to figure out the orientation (rotation) of an object, such as a satellite, aircraft, or spacecraft.

What does a DCM do in this course's convention?

It maps vector components from the fixed frame to the body frame: v_B = DCM_FB v_F.

What properties does a proper DCM have?

Its columns/rows are orthonormal, DCM^T DCM = I, and DCM^{-1} = DCM^T; its determinant is +1.

Why does one vector measurement give only 2 equations?

A sensor might measure the direction of a known vector (for example, the Earth's magnetic field or the Sun's direction).

A vector has 3 components, but because it is a unit vector (length = 1), those components are linked together:

x²+y²+z²=1

This means only 2 components can vary independently. Once you know two of them, the third is fixed.

Therefore:

1 vector measurement = only 2 independent equations

Why do we need 2 sensors?

We need 3 independent equations to find the 3 unknown orientation values.

But one vector measurement only gives 2 equations, which isn't enough.

So we use at least 2 vector measurements:

• Sensor 1 → 2 independent equations

• Sensor 2 → 2 independent equations

Total:

2+2=4

independent equations. We only need 3 equations to solve for the 3 unknowns, but now we have 4 equations.

That means: over-determined.

Why is static attitude determination usually an estimation problem rather than an exactly determined problem?

In the real world, sensors are not perfect.

For example:

• Sun sensor says orientation = 30.1°

• Magnetometer says orientation = 29.8°

The measurements don't agree exactly because of noise and errors.

Since we have extra equations, there is usually no single orientation that satisfies all measurements perfectly.

Instead, we find the orientation that best fits all the measurements. This process is called attitude estimation (often using methods like least squares, QUEST, or Kalman filtering).

What is the unit-quaternion constraint?

q^T q=1, or q1²+q2²+q3²+q4²=1

Why are quaternions useful for spacecraft attitude?

They avoid the singularity associated with Euler angles, require only four numbers with one constraint, and compose rotations efficiently through quaternion multiplication.

What is the quaternion inverse for a unit quaternion?

The inverse is the conjugate: if q = [q_vec; q4], then q^-1 = [-q_vec; q4].

What is the small-angle relationship between Euler-angle error and quaternion error?

For small rotation angle, the vector part is approximately half the angle-error vector: q_vec ≈ (1/2) theta_error, or theta_error ≈ 2 q_vec.

What is the main danger of Euler angles?

They can become singular at certain attitudes, commonly called gimbal lock, and their rates do not generally equal body angular-rate components.

What objective does the q-method minimize?

The weighted squared mismatch between measured body vectors and modeled fixed-frame vectors after applying the candidate attitude: J = 1/2 sum(w_k ||v_kB - DCM_FB v_kF||^2).

What is the key matrix equation in the q-method?

K q = lambda q, where K is a symmetric 4x4 matrix constructed from the weighted vector measurements.

Which eigenvector solves the q-method?

The normalized eigenvector associated with the largest eigenvalue of K

What are the main q-method intermediate quantities?

B = sum(w_k v_kB v_kF^T), S = B + B^T, sigma = trace(B), and Z formed from the antisymmetric part of B. Then K = [[S - sigma I, Z], [Z^T, sigma]].

Why is the q-method computationally expensive?

It solves the full 4x4 eigenvalue/eigenvector problem, although it is generally the most accurate of the listed static methods.

What does QUEST change relative to the q-method?

QUEST avoids solving the full eigenvalue problem exactly. It estimates the largest eigenvalue, approximately the sum of the measurement weights, and solves a smaller system.

What are Gibbs-Rodrigues parameters in the QUEST derivation

A three-parameter representation related to the quaternion vector part divided by the scalar part: approximately q_vec/q4.

What is the speed/accuracy ordering from the lectures?

q-method: slowest and most accurate; QUEST: slightly faster and slightly less accurate; ESOQ2: faster than QUEST with a different tradeoff; TRIAD: fastest and least accurate.

How does TRIAD construct the body-frame triad?

x_B = v1_B, y_B = (v1_B x v2_B)/||v1_B x v2_B||, and z_B = x_B x y_B.

How is the TRIAD DCM formed?

Build the same orthonormal triad in both frames, then DCM_FB = [x_B y_B z_B][x_F y_F z_F]^T.

When does TRIAD fail or become poorly conditioned?

When the two measured vectors are parallel or nearly parallel, because their cross product is zero or very small.

Why use gyroscopes instead of differentiating a static attitude measurement?

Static sensor noise is amplified by differentiation. Gyroscopes measure angular rate directly and avoid that noisy numerical derivative.

What is the physical principle behind a gyroscope?

A spinning rotor carries angular momentum. Changing the rotor's orientation requires a torque and produces gyroscopic reaction behavior.

What is the basic angular-momentum equation?

M = (dH/dt)_G + omega x H, with the derivative taken in the specified frame.

What does a three-axis gyro provide?

Measurements of the spacecraft angular-rate vector along three approximately orthogonal gyro axes

What is the difference between attitude determination and attitude propagation with a gyro?

Determination estimates the current attitude from observations. Gyro propagation integrates measured angular rate forward in time, but accumulates bias and drift unless corrected.

How is the orbit frame oriented?

Its x-axis points radially outward, its y-axis points opposite the orbital angular-momentum direction, and its z-axis completes the right-handed frame in the direction of motion convention used in the lecture.

What is the circular-orbit mean motion?

n = sqrt(mu/r^3).

What is the orbit-frame angular velocity relative to the fixed frame for the course convention?

It is along the negative orbit-frame y-axis with magnitude n: omega_O/F^O = [0, -n, 0]^T.

How are angular rates between frames related?

Angular velocities add: omega_B/F = omega_O/F + omega_B/O, with all vectors expressed in the same frame before adding.

What approximation is made in the small-angle simulation model?

Products of small attitude errors are neglected: theta_i theta_j ≈ 0 for small angles.

What does a quaternion-to-Euler-angle block do in the simulation?

It converts the integrated quaternion attitude into selected Euler angles for plotting and interpretation; it does not replace the quaternion state propagation.

What creates gravity-gradient torque?

Different spacecraft mass elements experience slightly different gravitational forces because they are at different distances from the central body. The distributed forces produce a net torque about the center of mass

What is the compact gravity-gradient torque equation used in the lectures?

T_g = (3 mu/r^3) (r x I r) when r is expressed in the body frame and I is the body inertia matrix.

Why does a point-mass spacecraft have no gravity-gradient torque?

A point mass has no spatial extent, so all of its mass experiences the same gravitational acceleration through its center of mass; there is no force gradient across the body.

What disturbance torques were listed in the lecture?

Aerodynamic, gravity-gradient, magnetic, solar-radiation, and micrometeor-impact torques.

What is Euler's rotational equation for a rigid spacecraft?

I dot(omega) + omega x (I omega) = T in the body frame, assuming constant body inertia.

What is the purpose of linearizing the gravity-gradient equations?

To study small perturbations about a nominal orbit-frame attitude and determine stability using a linear system rather than the full nonlinear equations.

What is the pitch stability condition from the linearized equation?

The pitch equation has the form ddot(theta2) + 3 k2 n^2 theta2 = 0; oscillatory stability requires k2 > 0, which means I1 > I3

What are the three nondimensional inertia parameters?

k1 = (I2-I3)/I1, k2 = (I1-I3)/I2, and k3 = (I2-I1)/I3.

What is the main stability conclusion for all three axes?

The all-axis stable regions satisfy the lecture's inequalities k1 > k3, k1 k3 > 0, 1 + 3k1 + k1 k3 > 0, and (1 + 3k1 + k1 k3)^2 - 16 k1 k3 > 0.

What are the two named stable regions in the k1-k3 (Smelt) plane?

The first-quadrant Lagrange region and the third-quadrant DeBra-Delp (D2) region.

What does a positive real part of an eigenvalue mean for the linearized attitude dynamics?

The perturbation contains a growing exponential and the equilibrium is unstable. Purely imaginary roots indicate undamped oscillation; negative real parts indicate decay.

How does a CMG create spacecraft torque?

It changes the direction of stored rotor angular momentum by gimbaling the rotor. The required change in angular momentum produces an equal-and-opposite spacecraft torque.

What is the single-CMG torque relationship used in the lecture?

The torque contribution is proportional to stored spin momentum, gimbal rate, and the transformed gimbal-axis direction; the sign follows from angular-momentum conservation.

What is the multi-CMG matrix equation?

T_CMG = C(theta_G) omega_G, where C is a 3-by-n configuration matrix and omega_G contains the n gimbal rates.

What is the minimum-norm solution for CMG gimbal rates?

omega_G = C^T (C C^T)^(-1) T_CMG, or the equivalent pseudoinverse form when the matrix has the needed rank.

Why does the CMG configuration matrix change with time?

The gimbal angles change the orientation of each CMG axis, so the torque directions and therefore the columns of C(theta_G) change.

What is a pyramid CMG array useful for?

It provides torque authority about all three spacecraft axes with redundancy, allowing a desired three-axis torque to be produced from four CMGs when the configuration is nonsingular.

What is a CMG singularity?

A configuration where the columns of C lose sufficient rank, so some desired torque directions cannot be produced or require excessively large gimbal rates

What is the standard second-order form?

ddot(x) + 2 zeta omega_n dot(x) + omega_n^2 x = input.

What are the natural frequency and damping ratio in the mass-spring-damper model?

omega_n = sqrt(k/m) and zeta = c/(2 sqrt(km)) = c/(2 m omega_n).

What are the four damping cases?

Underdamped: 0 < zeta < 1; critically damped: zeta = 1; overdamped: zeta > 1; undamped: zeta = 0.

What are the roots of an underdamped second-order system?

r = -zeta omega_n +/- i omega_d, where omega_d = omega_n sqrt(1-zeta^2).

What does the real part of the pole control?

The exponential decay or growth rate. For stable underdamped motion, the envelope decays as e^(-zeta omega_n t).

What does the imaginary part of the pole control?

The oscillation frequency, specifically the damped natural frequency omega_d.

What is the approximate 2% settling-time formula?

t_s ≈ 4/(zeta omega_n).

What is the percent overshoot formula for an underdamped second-order step response?

%OS = 100 exp[-zeta pi/sqrt(1-zeta^2)] if percent is requested; the fraction is exp[-zeta pi/sqrt(1-zeta^2)].

What is the time to first peak?

t_p = pi/omega_d.

How does increasing damping ratio affect overshoot?

It decreases overshoot. Increasing omega_n generally makes the response faster, while increasing zeta suppresses oscillation.

What are the three PID terms?

Proportional responds to present error, integral accumulates past error, and derivative responds to the rate of error change.

What does proportional control mainly do?

It creates restoring control effort proportional to attitude error. It generally speeds the response and reduces error, but a constant disturbance can leave steady-state error.

What does derivative control mainly do?

It adds damping by opposing angular-rate or error-rate motion. It reduces overshoot but is sensitive to measurement noise.

What does integral control mainly do?

It removes steady-state error and rejects constant disturbances by continuing to build control effort while error persists.

What is a major practical problem with integral control?

Integral windup: the integrator can continue accumulating when the actuator saturates, causing a large overshoot or slow recovery.

What is the typical PD control law in the lecture's sign convention?

T = -k_p theta_error - k_d dot(theta) for each controlled axis.

How do PD gains map to second-order parameters for a single rotational axis?

With inertia I, k_d/I = 2 zeta omega_n and k_p/I = omega_n^2.

How do you design gains from settling time and overshoot?

Use t_s ≈ 4/(zeta omega_n) to obtain zeta omega_n, use the overshoot equation to obtain zeta, calculate omega_n, then set k_d = 2 I zeta omega_n and k_p = I omega_n^2.

What is the attitude error definition in the 1-D Simulink example?

theta_error = theta_observed - theta_target.

What is the quaternion attitude-error definition in the example?

q_error = q_target^{-1} otimes q_observed, using the course's multiplication/order convention

Why can a quaternion error be converted to a small-angle error by multiplying its vector part by two?

For small rotations, the quaternion vector part is approximately half the rotation-angle error vector.

What is the control vector form used in the PID example?

T = -K x, where x contains the integrated attitude error, attitude error, and angular-rate error; K = [K_i K_p K_d].

Why is tuning usually iterative even after calculating gains analytically?

The analytical design is based on an idealized linear model. Coupling, nonlinearities, actuator limits, disturbances, sensor noise, and implementation details can change the actual response, so simulation and retuning are required.

What is the tuning workflow from the lecture?

Choose desired settling time, choose desired overshoot, solve for gains, simulate, check the requirements, and tune the gains as necessary.

Given zeta = 0.7 and omega_n = 3 rad/s, estimate the 2% settling time.

t_s ≈ 4/(0.7*3) = 1.90 s.

Given zeta = 0.6 and omega_n = 4 rad/s, find the damped natural frequency.

omega_d = 4 sqrt(1-0.6^2) = 3.2 rad/s.

Given inertia I = 3 kg m^2, zeta omega_n = 2, find k_d.

k_d = 2 I zeta omega_n = 2(3)(2) = 12 N m s/rad.

Given I = 3 kg m^2 and omega_n = 2.9 rad/s, find k_p.

k_p = I omega_n^2 = 3(2.9)^2 ≈ 25.2 N m/rad.

Given 5% overshoot, what damping ratio should you expect approximately?

Use zeta = -ln(0.05)/sqrt(pi^2 + ln^2(0.05)) ≈ 0.69

What is the first check before trusting a calculated control gain?

Confirm units, sign convention, axis/frame convention, and whether the gain was derived for torque, angular acceleration, angle, or quaternion error.