Numerics
- State
- The minimum variables needed to continue a simulation: commonly pose and velocity, plus controller, sensor, cache, task, time, and RNG state.
- Derivative
- A rate of change. Velocity is the time derivative of position; acceleration is the derivative of velocity.
- Timestep (h or Δt)
- The simulated time advanced by one numerical update.
- Control step
- The interval between policy actions. It may contain several smaller physics substeps.
- Local truncation error
- Error introduced by one idealized integration step starting from exact state.
- Global error
- Accumulated difference after many steps, including propagated earlier errors.
- Stability
- Whether numerical error stays bounded under a chosen method, timestep, and system stiffness.
- Stiffness
- The presence of fast dynamics that force an explicit method to use a small timestep.
- Condition number
- Sensitivity of a linear-system solution to small changes in its inputs.
- Residual
- How badly a candidate solution still violates its equation or fixed-point condition.
Rigid bodies and coordinates
- Rigid body
- An ideal body whose internal distances do not change.
- Center of mass (COM)
- Mass-weighted mean position; translational dynamics can be represented there.
- Body frame
- Coordinates attached to a body and moving with it.
- World frame
- The fixed coordinate system shared by the scene.
- Pose
- Position plus orientation.
- Quaternion
- Four-number unit representation of 3D orientation; LavenderSim orders it (x, y, z, w).
- Angular velocity (ω)
- Instantaneous rotation axis multiplied by angular speed, expressed in a declared frame.
- Inertia tensor (I)
- A 3 × 3 matrix relating angular velocity and momentum about the COM.
- Generalized coordinate (q)
- A coordinate chosen to describe configuration, often one number per joint degree of freedom.
- Generalized velocity (q̇)
- Velocity coordinates associated with q; not always the literal derivative of every representation component.
- Degree of freedom (DoF)
- One independent direction of allowable configuration change.
- Kinematic tree
- A parent–child hierarchy of bodies and joints without closed loops.
- Forward kinematics
- Computing body or site poses from joint configuration.
- Jacobian (J)
- Linear map from a small state change or velocity to a task-space change or constraint velocity.
- Singularity
- A configuration where a Jacobian loses rank and some task motion or force direction becomes unavailable or ill-conditioned.
Dynamics and algorithms
- Momentum
- Mass times linear velocity; angular momentum additionally depends on inertia and rotation.
- Impulse
- Force integrated over time; it changes momentum immediately in an impulse solver.
- Generalized force (τ)
- Force expressed in generalized coordinates, such as joint torque.
- Mass matrix (M)
- Configuration-dependent map from generalized acceleration to generalized force.
- RNEA
- Recursive Newton–Euler Algorithm: computes inverse dynamics for an articulated tree.
- CRBA
- Composite Rigid Body Algorithm: constructs an articulated system's joint-space mass matrix.
- ABA
- Articulated Body Algorithm: computes forward dynamics in linear time for a tree.
- Armature
- Additional inertia reflected at an actuator or joint.
- Activation dynamics
- Internal actuator state that approaches the requested control over time.
Constraints, collision, and contact
- Constraint
- A restriction on configuration, velocity, or force, represented by equalities or inequalities.
- Holonomic constraint
- A configuration-level equation φ(q)=0.
- Lagrange multiplier (λ)
- A constraint-space force or impulse magnitude used to enforce a constraint.
- Baumgarte stabilization
- Velocity-level feedback proportional to position error.
- Position projection
- A correction that moves configuration toward a constraint manifold after integration.
- Broad phase
- Cheap rejection of shape pairs that cannot intersect.
- Narrow phase
- Detailed geometric query for a candidate pair.
- AABB
- Axis-aligned bounding box used for inexpensive overlap tests.
- Support mapping
- Function returning the farthest shape point in a direction; used by GJK.
- GJK
- Gilbert–Johnson–Keerthi algorithm for convex distance or intersection queries.
- EPA
- Expanding Polytope Algorithm for penetration normal and depth after a convex intersection.
- Contact manifold
- One or more points representing a contacting shape pair.
- Non-penetration
- Unilateral condition that contact may push bodies apart but not pull them together.
- Restitution
- Bounce parameter relating separating to approaching normal speed.
- Compliance
- Controlled softness: finite displacement under constraint load.
- Coulomb friction
- Tangential force bounded by a coefficient times normal force.
- Friction cone
- Set of tangential forces satisfying the Coulomb bound; often approximated by a pyramid.
- Complementarity
- Either a contact gap is open or its normal reaction is active, with their product zero in the ideal model.
- LCP
- Linear complementarity problem, a common linearized form for unilateral constraints.
Solvers, environments, and observability
- Gauss–Seidel
- Iterative linear solver that immediately reuses each newly updated unknown.
- PGS
- Projected Gauss–Seidel: a row update followed by projection onto legal bounds.
- Sequential impulses
- Matrix-free PGS applied directly to rigid-body velocities.
- Warm start
- Initialize a solve with impulses cached from a previous nearby state.
- Solver iteration
- One pass over the selected constraint rows; it is a budget, not an accuracy guarantee.
- Site
- Named pose attached to a body for targets, sensors, or telemetry without adding mass or collision.
- Proprioception
- Measurements of a robot's own motion, such as joint state and IMU output.
- Observation
- Numeric data exposed to an agent at one environment step.
- Telemetry
- Richer diagnostic state intended for inspection or visualization, not necessarily policy input.
- Domain randomization
- Seeded variation of scene or task parameters to broaden training experience.
- Dwell condition
- A success condition that must remain true for a specified duration.
- Snapshot
- Versioned serialized simulator and task state used for restore, replay, or branching.
- Backend parity
- A scoped test that native and WASM implementations expose equivalent semantics within a stated tolerance.