FERROMOTION · RUST → WASM

The equation of motion

Any rigid mechanical system obeys M(q)q̈ + C(q,q̇)q̇ + G(q) = τ. How much of that did the network in the last lesson know?

See the physics a black box has never heard of: the equation of motion M(q)q̈ + C(q,q̇)q̇ + G(q) = τ, computed live by ferromotion as a double pendulum swings. This structured knowledge is the prior every method from here on injects into a learner.

01Challenge

The MLP in the last lesson knew nothing about the world. Here is what it was missing. Any rigid mechanical system (a pendulum, an arm, a humanoid) obeys one equation: M(q)q̈ + C(q,q̇)q̇ + G(q) = τ. M(q) is the mass/inertia matrix (how hard each joint is to accelerate, and how joints are coupled); C(q,q̇)q̇ are the Coriolis/centrifugal terms; G(q) is gravity; τ is the applied torque. The Bench integrates a real double pendulum with ferromotion-core: every frame it builds M(q), the gravity bias, and solves M q̈ = τ − bias. Watch it swing (chaotically (this is the famous double pendulum), and watch M(q) and G(q) change as the configuration changes. That dependence on q) the coupling, the geometry, is knowledge no amount of data teaches a black box for free.

Predict firstAny rigid mechanical system obeys M(q)q̈ + C(q,q̇)q̇ + G(q) = τ. How much of that did the network in the last lesson know?

02Model

There are three equivalent ways to write a system's dynamics: Newton–Euler (sum forces and moments), Lagrangian (d/dt ∂L/∂q̇ − ∂L/∂q = τ from the energy L = T − V), and Hamiltonian (from H = T + V in position/momentum coordinates). For a rigid multibody system they all reduce to the manipulator equation M(q)q̈ + C(q,q̇)q̇ + G(q) = τ. M(q) is symmetric positive-definite by construction, it is the joint-space inertia. Forward dynamics, what you need to simulate, inverts it: q̈ = M(q)⁻¹(τ − C q̇ − G). ferromotion computes M and the C+G bias with the recursive Newton–Euler algorithm and solves via Cholesky. The crucial thing for learning: M and G are functions of q. The inertia felt at a joint depends on how the whole arm is folded; gravity's pull depends on posture. A physics prior hands a learner this structure; a black box must rediscover it from data, and usually cannot.

03Guided practice

Step A (watch): the pendulum releases from a raised pose and swings under gravity, no motor torque, τ = 0, so this is pure M q̈ = −bias. Step B (read M): the mass matrix M(q) shown below changes as the arm folds and unfolds; M(0,0), the effective inertia of the shoulder joint, is largest when the arm is extended and smallest when it is tucked. Step C (read G): the gravity vector G(q) swings between positive and negative as the links pass through horizontal, gravity's torque depends on posture. Step D (kick it): add energy with Kick and watch the chaos; the energy readout barely drifts because ferromotion integrates it symplectically (the subject of the next lesson).

04Feedback

PASS WHEN PASS once you have watched the real equation of motion integrate the pendulum and seen M(q), G(q) change with configuration.

If your run fails, check:

The pendulum plays automatically, let it swing and watch the M(q) and G(q) readouts update.
Press Kick to inject energy and see the chaotic double-pendulum motion.

05Retrieve & space

In M(q)q̈ + C(q,q̇)q̇ + G(q) = τ, what does each term mean, and which one makes it a coupled, nonlinear system?
Why do M and G depend on q, and why is that dependence exactly what a black-box network struggles to learn from limited data?
Newton, Lagrange, and Hamilton give three routes to the dynamics, what do they share for a rigid multibody system?

06Mastery & project

You watched ferromotion compute the true equation of motion in real time, M(q), G(q), and the forward dynamics that solve for q̈. This is the prior. The next lessons inject it into a learner three different ways: through the data, through the loss, and through the architecture.

Grounds every later module. The learned Lagrangian and Hamiltonian nets reconstruct this same M and energy from data; the PINN penalizes violations of this equation; the differentiable controller backpropagates through it.

GO FURTHER · THE LIVE PLAYGROUND

ferromotion dynamics: RNEA, mass matrix, forward dynamics ↗ · the same recursive Newton–Euler dynamics you just drove, in pure Rust.

← The universal approximator Energy, and why integrators drift →

FERROMOTION · RUST → WASM

The equation of motion

Live ferromotion-wasm: a double pendulum integrated by the real forward dynamics. Watch M(q)q̈+Cq̇+G=τ play out and read M(q), G(q) as it swings.

You read

the readout: what the Rust crate computes, recomputed as you change things

You change

the bench controls, and with them the assumption the measurement rests on

Fixed

the ferromotion crate doing the computing, and the check that decides pass