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Coordinate Systems and Euler Angles

Reference: FM04.pdf, Fig. 1 ("Kinematics Equation", "Navigation Equation"), Eq. (3)-(4). Code: src/simulator/frames.py.

Frames used in this lab

Frame Symbol (paper) Origin Axes Used for
Body-fixed F_B rocket C.G. x: forward along symmetry axis; y: right; z: down (completing right-hand triad) aerodynamic forces/moments, angular rates p,q,r
Local geodetic (NED) F_E launch point x: North; y: East; z: Down position, velocity in "range/altitude/drift" terms students actually plot
Earth-centered — Earth center ellipsoidal, rotating only used if include_earth_rotation=True (assumption d)

Body axes are chosen because the rocket's moments of inertia (Ixx, Iyy, Izz) are constant in that frame — a rigid body's mass distribution doesn't change relative to itself as it tumbles, but it would appear time-varying if you tried to write Euler's equations in an inertial frame. This is the standard justification in Etkin's Dynamics of Atmospheric Flight (paper reference [1]), the primary textbook source underlying this paper's equations.

Euler angles (3-2-1 sequence)

The paper uses the standard yaw→pitch→roll (ψ→θ→φ) 3-2-1 Euler sequence to describe body attitude relative to the geodetic frame:

  • φ (phi) — roll / bank angle, rotation about the final body x-axis.
  • θ (theta) — pitch / inclination angle, rotation about the intermediate y-axis.
  • ψ (psi) — yaw / azimuth angle, rotation about the geodetic z-axis (first rotation).

The direction cosine matrix L_BE (Eq. 4)

L_BE transforms a vector expressed in body axes into the geodetic frame: v_E = L_BE @ v_B. Its transpose, L_BE.T = L_EB, does the reverse. This single 3×3 matrix is used in two places in Fig. 1:

  1. Navigation equation — rotating body velocity [u v w] into [Ṅ Ė Ḋ] to update position.
  2. Earth-rotation term (Eq. 3) — rotating the Earth's angular velocity vector into body axes so it can be added to the relative body rates.

Navigation equation: flat vs. rotating/curved Earth

state_derivative's include_earth_rotation flag controls two independent pieces of physics, both disabled by default:

  • False (default) — flat, non-rotating Earth. Body rates [p q r] are used as-is, and position is simply [Ṅ Ė Ḋ] = L_BE @ [u v w]. This matches the paper's short-range 122mm case study exactly and is left byte-for-byte unchanged by the rotating-Earth work below.

  • True — full rotating/curved-Earth model:

    • Eq. (3) Coriolis term on body rates (earth_rotation_body_rates, unchanged from before).
    • A rotating/curved-Earth Navigation Equation for the geodetic-frame velocity [V_N V_E V_D] (now carried as three extra integrated states, growing the state vector from 12 to 15): 
      V_N_dot = (mu_dot + 2*omega)*sin(lambda)*V_E - lambda_dot*V_D + a_N
V_E_dot = -(mu_dot + 2*omega)*sin(lambda)*V_N - (mu_dot+2*omega)*cos(lambda)*V_D + a_E
V_D_dot = (mu_dot + 2*omega)*cos(lambda)*V_E + lambda_dot*V_N + a_D
lambda_dot = V_N / (R_meridian + H_G)
mu_dot     = V_E / ((R_normal + H_G) * cos(lambda))
      where lambda/mu are geodetic latitude/longitude, H_G is geodetic altitude, and a_N, a_E, a_D are the body-axis velocity rotated into the geodetic frame via L_BE (i.e. what the flat-Earth path calls vel_ned).

    Simplification, documented explicitly: the paper doesn't publish a closed-form rotating/curved-Earth Navigation Equation, only the flat-Earth form and the Eq. (3) body-rate term. This lab therefore uses the standard spherical-Earth approximation R_meridian = R_normal = R_EARTH (mean Earth radius, 6,378,137 m — WGS84 semi-major axis), recovering the current latitude from the North distance travelled as lambda = lambda_0 + N / R_EARTH. A full WGS84 ellipsoidal model (latitude-dependent R_meridian/R_normal) is not implemented — left as a possible extension exercise. See equations_of_motion.py for the code.

from src.simulator.frames import euler_to_LBE
L_BE = euler_to_LBE(phi, theta, psi)   # radians in, 3x3 matrix out

The kinematic equation (Euler angle rates)

Body angular rates [P Q R] are not the same thing as the rates of change of the Euler angles [φ̇ θ̇ ψ̇] — they are related by a matrix that itself depends on the current attitude:

⎡φ̇⎤   ⎡1  sinφ·tanθ   cosφ·tanθ⎤ ⎡P⎤
⎢θ̇⎥ = ⎢0  cosφ        -sinφ    ⎥ ⎢Q⎥
⎣ψ̇⎦   ⎣0  sinφ·secθ   cosφ·secθ⎦ ⎣R⎦

Gimbal lock: this matrix is singular at θ = ±90° (secθ → ∞). A rocket pitching to vertical would break this representation — the practical fix (beyond this teaching lab's scope) is a quaternion attitude representation, which has no singularities but is less intuitive to read. See Exercise 5 in assignments.md.

Aerodynamic angles: angle of attack and sideslip

Given the (relative-to-wind) body velocity components u_r, v_r, w_r:

α (alpha)  = atan2(w_r, u_r)             angle of attack
β (beta)   = asin(v_r / V)               sideslip angle
V          = sqrt(u_r² + v_r² + w_r²)    airspeed magnitude

These angles are what the aerodynamic coefficients respond to (see aerodynamic-model.md) — physically, α and β measure how far the oncoming relative wind is misaligned from the nose of the rocket in the pitch and yaw planes respectively.

Professor Notes

Ask students to derive the L_BE matrix from three elementary rotation matrices (R_z(ψ), R_y(θ), R_x(φ)) multiplied in the correct order — this is a classic "do it once by hand" exercise that makes the Fig. 1 block diagram click. See Exercise 1.

Student Exercises

See assignments.md Exercises 1 and 5.