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Position Based Dynamics

Simulates motion by directly adjusting positions to satisfy constraints.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Simulation
Difficulty: Advanced
Family: Constraint Simulation
Tier: Extended
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 5
Code example: Not ready
Visualization: Ready
Aliases: PBD

Overview

Position Based Dynamics predicts positions, solves constraints by moving points, and then updates velocities from the corrected positions.

Core Concepts

simulated statephysical stateconstraint propagationconstraint simulationstate updatetime stepnumerical stability

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(pc)

Average

O(i c)

Worst

O(i c)

Space

O(p + c)

When to Use

Use Position Based Dynamics when you need to predict particle positions.
Use it for cloth simulation with inputs like particles.
Use it when approximate behavior over time is more useful than a closed-form answer.
Use it when the Constraint Simulation tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when the time step, grid size, or sample count cannot support stable results.
Avoid it when the required output is not corrected positions.

What It Does

Predicts particle positions
Projects positions to satisfy constraints

Use Cases

Cloth simulation
Soft bodies
Interactive physics

Inputs

Particles
Constraints
Time step

Outputs

Corrected positions
Updated velocities

Key Formulas (1)

Constraint correction

δ_{p i} = −stiffness · C(p) · w_i · grad_i / ∑(w_j · |grad_j|²)

delta_p_i = -stiffness * C(p) * w_i * grad_i / sum(w_j * |grad_j|^2)

Moves particles to reduce a position constraint violation.

Pseudocode

1.Predict positions from velocity
2.Apply external forces
3.Iteratively solve constraints by moving positions
4.Update velocity from position changes
5.Commit corrected positions

Canonical Example

Input: position = 0, velocity = 2, time step = 0.5
Output: next simulated state
Explanation: Position Based Dynamics: The algorithm advances the system by applying its update rule over a small step.

Implementation Notes

Keep the input contract explicit: Particles; Constraints; Time step.
Use this step as the implementation spine: Predict positions from velocity.
Then preserve the next state change: Apply external forces.
Record why candidates are rejected so contradictions are debuggable.
Track units and invariants such as mass, energy, or count where they matter.

Common Pitfalls

PBD favors stable, controllable behavior over strict physical accuracy.
Using a time step that is convenient for rendering but unstable for the simulated system.
Returning corrected positions without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show particles being pulled back into valid constrained positions.

Step 1 of 5: Predict positions from velocity

constraint-particles

Notes

Position Based Dynamics operates in O(i c) time and O(p + c) space within the Constraint Simulation family. Key governing relation: delta_p_i = -stiffness * C(p) * w_i * grad_i / sum(w_j * |grad_j|^2). Often compared with Verlet Integration and Smoothed Particle Hydrodynamics.