A hands-on simulator textbook

Tutorials on simulation.

Start with one falling point mass. Finish by tracing an articulated, contact-rich robot through the complete native timestep. Every chapter pairs physical intuition with tested NumPy code, LavenderSim experiments, and the actual engine implementation.

Begin Chapter 1 →Browse lab code

12 chaptersFrom integration to the full simulation loop.

Build, then measureEvery important claim becomes a runnable numerical experiment.

Honest internalsReference algorithms and real LavenderSim behavior are labeled separately.

The map before the territory

Three implementations may appear in one lesson.

Knowing which system an equation describes is part of understanding simulation.

Label	Meaning
Reference lab	A small NumPy implementation designed to make an algorithm readable.
LavenderSim	Behavior verified against the current Python API and compact C++ engine.
MuJoCo comparison	A conceptual comparison grounded in MuJoCo's official computation documentation—not a compatibility claim.
Future feature	Instrumentation or an algorithm proposed by the curriculum but not yet part of the engine.

The key architectural contrast: MuJoCo primarily uses generalized coordinates for articulated dynamics. LavenderSim currently stores Cartesian rigid-body states and maintains joints with numerical constraints and sequential impulses. We will learn both models without pretending they are the same.

Keep beside the workbench

Reference shelves.

Recover a definition, locate an equation, trace current engine code, or continue into primary sources.

GlossaryPlain-language definitions for numerics, mechanics, contact, solvers, and RL.Equation referenceSymbols, units, assumptions, physical meaning, and chapter links.Engine mapFollow an action from Python through C++, diagnostics, WASM, and the browser.Further readingAn ordered shelf of primary texts and official simulator documentation.

Part I

Foundations

Turn continuous equations into stable, inspectable state updates.

available

Numerical foundations

Timesteps, integrators, stability, stiffness, and energy drift.

available

Linear algebra for mechanics

Frames, transforms, inertia, Jacobians, and sparse structure.

available

3D rigid-body mechanics

Momentum, torque, inertia tensors, quaternions, and angular velocity.

Part II

Articulated systems

Move from isolated bodies to robot coordinates and recursive dynamics.

available

Generalized coordinates

Degrees of freedom, kinematic trees, forward kinematics, and singularities.

available

Lagrangian mechanics

Energy, generalized forces, mass matrices, and gravity terms—selectively.

available

Rigid-body algorithms

Readable RNEA, CRBA, and ABA implementations for a planar chain.

Part III

Interaction

Understand the geometry and numerical constraints behind joints and contact.

available

Constraints

Jacobians, multipliers, stabilization, projection, joints, and limits.

available

Collision detection

Broad phase, AABBs, support mappings, GJK, EPA, and manifolds.

available

Contact mechanics

Non-penetration, impulses, restitution, compliance, and caching.

available

Friction

Coulomb limits, sticking, sliding, cones, rolling, and torsional resistance.

available

Constraint solvers

Direct solves, PGS, sequential impulses, warm starts, and residuals.

Part IV

The whole engine

Put control, collision, solving, integration, sensing, and RL into one timestep.

available

Simulation architecture

Trace, profile, replay, and compare the complete native and WASM pipeline.

How to study

Read less. Predict, run, and explain more.

01 / PREDICT

Write down what should happen

Direction, scale, conserved quantity, and likely failure mode are more useful than a vague guess.

02 / RUN

Change one thing

Keep the scene and initial state fixed while sweeping timestep, stiffness, friction, or solver iterations.

03 / EXPLAIN

Use the telemetry

A plot should support a causal explanation, not merely show that two curves differ.