Core Concepts¶

This section provides a high-level mathematical overview of how Axion formulates and solves the physics simulation problem. Understanding these concepts is essential for grasping the theoretical foundation underlying the simulator's robust and unified approach.

Mathematical Foundation¶

Axion's physics engine is built on a unified mathematical framework that treats all physical phenomena—articulated body dynamics, contact interactions, and joint constraints—as a single, coupled system of equations. This approach provides superior stability and accuracy compared to traditional methods that handle these phenomena separately.

Articulated Bodies¶

Axion represents articulated body systems using generalized coordinates and velocities that describe the system's configuration and motion. The dynamics are governed by:

\[ \mathbf{\tilde{M}}(\mathbf{q}) \Delta\mathbf{u} = \mathbf{f}_{\text{ext}} h + \mathbf{J}^T(\mathbf{q}) \boldsymbol{\lambda} \]

This captures how velocity changes result from external forces and constraint impulses. The meaning and derivation of these constraint impulses will be explained in Gauss's Principle of Least Constraint.

Mathematical Notation

For detailed definitions of all symbols (\(\mathbf{q}\), \(\mathbf{u}\), \(\mathbf{\tilde{M}}\), \(\mathbf{J}\), \(\boldsymbol{\lambda}\), etc.), see the Notation page.

Contact and Constraint Formulation¶

Physical interactions are mathematically encoded as constraints:

Joint Constraints (bilateral): Enforce exact geometric relationships between bodies
Contact Constraints (unilateral): Prevent body interpenetration
Friction Constraints: Model stick-slip behavior through complementarity conditions

These constraints create a system mixing equalities and inequalities, requiring specialized mathematical treatment to solve simultaneously.

Solution Approach¶

Axion's approach follows a four-step mathematical progression:

1. Constraint Formulation¶

First, we mathematically formulate how articulated bodies and their interactions are represented as constraint equations. This establishes the mathematical foundation for describing joints, contacts, and friction.

→ Next: Constraints Formulation

2. Optimization Principle¶

We apply Gauss's Principle of Least Constraint, which provides a principled way to determine how the system should evolve when subject to constraints. This principle frames constraint enforcement as an optimization problem.

→ Next: Gauss's Principle of Least Constraint

3. Nonlinear System¶

The optimization principle, combined with time discretization, leads to a large nonlinear system of equations that must be solved at each time step. This system encodes all physical laws simultaneously.

→ Next: Nonlinear System

4. Numerical Solution¶

Finally, we numerically solve this nonlinear system using a specialized Newton-type method designed to handle the non-smooth nature of contact and friction.

→ Next: Numerical Solution

Why This Unified Approach?¶

Traditional physics engines handle dynamics, contacts, and joints in separate phases, leading to:

Instability in tightly coupled systems
Drift and constraint violation accumulation
Artificial softness in joints and contacts

Axion's unified mathematical formulation addresses these issues by:

Solving everything simultaneously — no artificial sequencing
Position-level constraint enforcement — eliminates drift by design
Principled optimization framework — mathematically grounded decisions

This mathematical rigor enables stable simulation of complex scenarios like articulated robots making contact with the environment, which often challenge traditional approaches.