TH1: Modeling from scratch

TH1: modeling from scratch

Weight: 25%. Suggested release: week 2. Estimated effort: 8–12 hours. Use the engine only as an independent check where requested, not as a substitute for derivation.

Required systems

Implement a scalar oscillator or pendulum, a free rigid body, and a planar two-link point-mass arm. The point-mass arm uses massless links and angles measured from downward vertical as in Lecture 6. Keep the model assumptions explicit.

Required work

1. Derive the state equations and units. For the arm, derive M and c from kinetic and potential energy and verify the unit-parameter mass matrix at the aligned pose.
2. Implement explicit Euler, semi-implicit Euler where applicable, and RK4. Compare fixed-final-time convergence on the oscillator against its analytic solution.
3. Integrate a free body's quaternion and angular velocity using the wxyz body-to-world convention. Compare normalized Euler and exponential attitude updates on constant angular velocity, then study torque-free dynamics.
4. Measure quaternion norm, energy, and world angular-momentum drift. Explain why one invariant alone is insufficient.
5. Design one independent check of the arm dynamics: symbolic differentiation, finite-difference energy/power balance, or an equivalent carefully matched engine model. Matching link length alone is insufficient.

Required evidence

Include timestep-refinement plots with labeled axes, an error table or machine-readable CSV, analytic checks, and at least one deliberately broken convention whose effect you diagnose. Restore the correct implementation in the submitted runnable default.

Rubric

• 30%: derivations, assumptions, and units.
• 25%: numerical methods and quaternion conventions.
• 25%: quantitative convergence and invariant checks.
• 10%: interpretation of a failure or mismatch.
• 10%: reproducibility and attribution.

Submit source, YAML configs, tests, results, and a report of at most six pages. Do not claim exact energy conservation for the normalized RK4 reference. Optional variational integration is enrichment rather than a core requirement.