RocketDynamicsLab

Assignments

Graduate-level (MSc/PhD) exercises built around FM04.pdf and this repository’s implementation. Each exercise names the files you’ll touch.

Exercise 1 — Derive the equations

Starting from F = ma and dH/dt = M (Newton’s second law, translational and rotational forms) in an inertial frame, derive:

(a) the Coriolis-coupling terms in Eq. (1) (equations.md) by expressing the time derivative of a vector in a rotating body frame as (dv/dt)_inertial = (dv/dt)_body + ω × v;

(b) Euler’s equations (the paper’s rotational-dynamics box) from dH/dt = M with H = Iω, showing where the gyroscopic ω×Iω terms come from;

(c) the L_BE direction cosine matrix (Eq. 4) as the product of three elementary rotation matrices in the 3-2-1 (yaw-pitch-roll) sequence.

Show your work; compare your derived matrix element-by-element against frames.euler_to_LBE().

Exercise 2 — Implement RK4 from scratch, then compare to solve_ivp

Without looking at integrators.py, implement your own rk4_step() function. Validate it against a problem with a known analytical solution (e.g. simple harmonic oscillator, or two-body Kepler orbit) before applying it to the 6-DOF system. Then:

• Run the full trajectory with your RK4 at dt = 0.01 s and with integrate_solve_ivp() (adaptive RK45) at tight tolerance.
• Plot impact range, drift, and time-of-flight for both.
• Quantify the discrepancy. Is it within the tolerance you’d expect given RK4’s fixed step vs. RK45’s adaptive error control?

Exercise 3 — Timestep sensitivity and numerical instability

Using examples/timestep_sensitivity.py as a starting point:

(a) Run the trajectory with forward Euler at dt ∈ {0.1, 0.05, 0.01, 0.005, 0.001, 0.0005} s. Plot impact range vs. dt. At what dt does the solution visibly diverge (blow up)? Explain why, referencing the gyroscopic coning frequency discussed in equations.md and numerical-methods.md.

(b) Repeat with RK4. At what dt does RK4 diverge? How much larger can RK4’s stable dt be than Euler’s, for comparable accuracy?

(c) Empirically estimate each method’s order of convergence: compute the impact-range error (vs. a very-fine-dt reference solution) at several dt values, plot log(error) vs. log(dt), and fit a line. Does the slope match the expected O(dt) for Euler and O(dt⁴) for RK4?

Exercise 4 — Variable-mass correction

Eq. (1) as published does not show an explicit ṁ-dependent correction term, yet the rocket loses ~30% of its mass during boost. Derive whether a -(ṁ/m)·V-type term should appear in a rigorous variable-mass Newton’s law (hint: consider whether the propellant exhaust’s momentum is already fully accounted for by the thrust force Tx). Implement both versions in equations_of_motion.py (behind a flag) and compare the resulting burn-out velocity against the paper’s reported 705 m/s at the burn-out point (Sec. 3.3).

Exercise 5 — Gimbal lock

Show analytically that the kinematic-equation matrix in coordinate-systems.md is singular at θ = ±90°. Construct a launch scenario (very steep elevation angle) that drives θ close to 90° and observe the numerical behavior of kinematic_rates() there. Research and briefly describe (2-3 paragraphs) how a quaternion attitude representation avoids this singularity, and what it would take to refactor this codebase to use one.

Exercise 6 — Model assumptions and Earth-rotation fidelity

Enable include_earth_rotation=True in state_derivative() and compare trajectories with and without it, at different launch latitudes. Given the rocket’s ~1-2 minute time of flight and tens-of-km range, is the Earth-rotation (Coriolis) effect significant here? Contrast with a long-range ballistic missile (~30 min flight, thousands of km) where this effect is well known to matter (e.g. WWII long-range artillery lore). Discuss which of the paper’s five modeling assumptions (mathematical-model.md) you would relax first if asked to build a higher-fidelity model, and why.

Exercise 7 — Reproduce and discuss the paper’s figures

Using the default ROCKET_122MM parameters at 45° elevation (the paper uses 50°), reproduce and compare against the paper:

• Fig. 2/3 — 3D trajectory and altitude vs. time.
• Fig. 4 — velocity magnitude vs. time.
• Fig. 5/6 — axial and normal acceleration vs. time.
• Fig. 7/8 — spin rate and pitch angle vs. time.
• Fig. 9 — aerodynamic angles (α, β, total AoA) vs. time.

For each, note qualitative agreement (shape, key transition points like burn-out) and explain any quantitative discrepancy, referencing the aerodynamic-coefficient caveat in aerodynamic-model.md (this lab’s coefficients are a representative reconstruction, not the paper’s exact Datcom output).

Exercise 8 — Full-parameter Monte Carlo dispersion

Extend monte_carlo_dispersion() (currently one-parameter-at-a-time) to perturb all ten DEFAULT_UNCERTAINTIES parameters simultaneously, drawing each from an appropriate distribution over its Table-2 range, for N=200-500 trajectories. Plot the resulting (range, drift) impact-point scatter. Fit an ellipse (or compute a covariance matrix) and estimate the 50% Circular/Elliptical Error Probable. Compare the combined dispersion to a naive sum of the individual one-at-a-time sensitivities from Exercise review of Figs. 10-21 — are they consistent? Why or why not?

Submission format

For each exercise: a short write-up (derivation/discussion, as applicable) plus a runnable script under examples/ or a notebook, and any figures saved to your own scratch directory (do not commit generated figures into assets/, which is reserved for hand-authored teaching diagrams).

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