Symbolic rigid body models

Before numerical integration, write down the state, parameters, invariants, and update equations for a rigid body. Symbolic work exposes missing assumptions before they become hard-to-find bugs.

Think like a programmer

Start with a typed model interface: body-frame inertia, world-frame pose, linear momentum, angular momentum, and a force/torque function. The equations are executable documentation for the update loop.

Model checklist

Inputs: Pose, mass, inertia tensor, momentum, force, torque, and time step.
State: World-frame linear and angular momentum.
Rule: Integrate momentum, derive velocity, then integrate pose.
Output: Next valid rigid-body state.
Check: Dimensions agree and declared invariants are monitored.

\[\dot{\mathbf p}=\mathbf F\qquad \dot{\mathbf L}=\boldsymbol\tau\qquad \mathbf v=\frac{\mathbf p}{m}\]

Try this experiment

Prediction: Writing torque as a function makes it easy to replace gravity with a motor or drag model.

Sketch a data structure for a body and separate it from a force callback. Which fields are parameters and which evolve each step?

Where this model breaks

Coordinate frames matter. Mixing body-frame and world-frame vectors is a common source of correct-looking but incorrect 3D rotation code.

Summary

Make the model explicit before coding it: name frames, store momentum, and derive velocities. Then numerical integration has a clear contract to test.

Glossary

Inertia tensor: matrix relating angular velocity and angular momentum in 3D.
Body frame: coordinates attached to an object.
World frame: shared scene coordinates.

Self-check

Why store momentum rather than only velocity?
Which frame must each vector declare?
What should a force callback return?

Sources

D. Baraff, Rigid Body Simulation, SIGGRAPH course notes.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Symbolic Rigid Body Models, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Rolling, Angular Momentum, and Gyroscopes, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Symbolic Rigid Body Models as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Symbolic Rigid Body Models into a test

Specify rigid-body state, frames, forces, and update equations before implementing an integrator.

Name the inputs and units that the rolling, angular momentum, and gyroscopes model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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