Pylinkage

Pylinkage is a Python library for building and optimizing planar linkages using Particle Swarm Optimization. It is still in beta, so don’t hesitate to post pull requests or issues for features you would like to see!

Table of Contents

• Installation

• Quick Start

• Complete Example

  • Joints Definition

  • Linkage Definition and Simulation

  • Visualization

  • Optimization

• Project Structure

• Architecture

  • Abstraction Levels

  • Key Design Principles

  • Performance

• Requirements

• Contributing

Installation

Using pip

This package is on PyPI as pylinkage, and can be installed using:

pip install pylinkage

Using uv (recommended for development)

If you’re using uv as your package manager:

uv add pylinkage

For development, clone the repository and run:

uv sync

Quick Start

Here’s a minimal example creating and visualizing a four-bar linkage:

import pylinkage as pl

# Main motor
crank = pl.Crank(0, 1, joint0=(0, 0), angle=.31, distance=1)
# Close the loop
pin = pl.Revolute(
    3, 2, joint0=crank, joint1=(3, 0),
    distance0=3, distance1=1
)

# Create the linkage and visualize
my_linkage = pl.Linkage(joints=(crank, pin))
pl.show_linkage(my_linkage)

The package also supports automatic optimization to achieve specific movements. See the Complete Example below for optimization details.

Complete example

Let’s start with a crank-rocker four-bar linkage, as a classic mechanism.

Joints definition

First, we define at least one crank because we want a kinematic simulation.

crank = pl.Crank(0, 1, joint0=(0, 0), angle=0.31, distance=1)

Here we are actually doing the following:

• 0, 1: x and y initial coordinates of the tail of the crank link.

• joint0: the position of the parent Joint to link with, here it is a fixed point in space. The pin will be created on the position of the parent, which is the head of the crank link.

• angle: the crank will rotate with this angle, in radians, at each iteration.

• distance: distance to keep constant between crank link tail and head.

Now we add a pin joint to close the kinematic loop.

pin = pl.Revolute(3, 2, joint0=crank, joint1=(3, 0), distance0=3, distance1=1)

In human language, here is what is happening:

• joint0, joint1: first and second Joints you want to link to, the order is not important.

• distance0, distance1: distance to keep constant between this joint and his two parents.

And here comes the trick: Why do we specify initial coordinates 3, 2? They even seem incompatible with distance to parents/parents’ positions!

• This explanation is simple: mathematically a pin joint the intersection of two circles. The intersection is often two points. To choose the starting point, we calculate both intersection (when possible), then we keep the intersection closer to the previous position as the solution.

Note: A linkage with a single motor and only one pin joint? Behind the curtain, many joints are created on the fly. When you define a Crank joint, it creates a motor and a pin joint on the crank’s link head. For a Revolute joint, it creates 3 pin joints: one on each of its parents’ positions, and one at its position, which forms a deformable triangle. This is why pylinkage code is so concise.

Linkage definition and simulation

Once your linkage is finished, you can either use the reload method of each Joint in a loop, or put everything in a Linkage that will handle this for you.

Linkage definition is simple:

my_linkage = pl.Linkage(joints=(crank, pin))

That’s all!

Now we want to simulate it and to get the locus of pin. Just use the step method of Linkage to make a complete rotation.

locus = my_linkage.step()

You can also specify the number of steps with the iteration argument, or subdivisions of each iteration withdt.

Let’s recap.

import pylinkage as pl

# Main motor
crank = pl.Crank(0, 1, joint0=(0, 0), angle=.31, distance=1)
# Close the loop
pin = pl.Revolute(
    3, 2, joint0=crank, joint1=(3, 0),
    distance0=3, distance1=1
)

my_linkage = pl.Linkage(joints=(crank, pin))

locus = my_linkage.step()

Visualization

First thing first, you made a cool linkage, but only you know what it is. Let’s add friendly names to joints, so the communication is simplified.

crank.name = "B"
pin.name = "C"
# Linkage can also have names
my_linkage.name = "Four-bar linkage"

Then you can view your linkage!

pl.show_linkage(my_linkage)

Last recap, rearranging names:

# Main motor
crank = pl.Crank(0, 1, joint0=(0, 0), angle=.31, distance=1, name="B")
# Close the loop
pin = pl.Revolute(
    3, 2, joint0=crank, joint1=(3, 0),
    distance0=3, distance1=1, name="C"
)

# Linkage definition
my_linkage = pl.Linkage(
    joints=(crank, pin),
    order=(crank, pin),
    name="My four-bar linkage"
)

# Visualization
pl.show_linkage(my_linkage)

Optimization

Now, we want automatic optimization of our linkage, using a certain criterion. Let’s find a four-bar linkage that makes a quarter of a circle. It is a common problem if you want to build a windshield wiper for instance.

Our objective function, often called the fitness function, is the following:

# Save initial state for later reset
init_pos = my_linkage.get_coords()
init_constraints = my_linkage.get_num_constraints()

@pl.kinematic_minimization
def fitness_func(loci, **_kwargs):
    """
    Return how fit the locus is to describe a quarter of circle.

    It is a minimization problem and the theoretic best score is 0.
    """
    # Locus of the Joint 'pin', last in linkage order
    tip_locus = tuple(x[-1] for x in loci)
    # We get the bounding box
    curr_bb = pl.bounding_box(tip_locus)
    # We set the reference bounding box, in order (min_y, max_x, max_y, min_x)
    ref_bb = (0, 5, 3, 0)
    # Our score is the square sum of the edge distances
    return sum((pos - ref_pos) ** 2 for pos, ref_pos in zip(curr_bb, ref_bb))

Please note that it is a minimization problem, with 0 as lower bound. On the first line, you notice a decorator; which plays a crucial role:

• The decorator arguments are (linkage, constraints), it can also receive init_pos

• It sets the linkage with the constraints.

• Then it verifies if the linkage can do a complete crank turn.

  • If it can, pass the arguments and the resulting loci (path of joints) to the decorated function.

  • If not, return the penalty. In a minimization problem the penalty will be float('inf').

• The decorated function should return the score of this linkage.

With this constraint, the best theoretic score is 0.0.

Let’s start with a candide optimization, the trial-and-error method.

Here it is a serial test of switches.

# Exhaustive optimization as an example ONLY
score, position, coord = pl.trials_and_errors_optimization(
    eval_func=fitness_func,
    linkage=my_linkage,
    divisions=25,
    n_results=1,
    order_relation=min,
)[0]

Here the problem is simple enough, so that method takes only a few seconds and returns 0.05.

However, with more complex linkages, you need something more robust and more efficient. Then we will use particle swarm optimization.

Here are the principles:

• The parameters are the geometric constraints (the dimensions) of the linkage.

• A dimension set (an n-uplet) is called a particle or an agent. Think of it like a bee.

• The particles move in an n-vectorial space. That is, if we have n geometric constraints, the particles move in an n-D space.

• Together, the particles form the swarm.

• Each time they move, their score is evaluated by our fitness function.

• They know their best score, and know the current score of their neighbor.

• Together they will try to find the extreme in the space. Here it is a minimum.

It is particularly relevant when the fitness function is not resource-greedy.

# Reset the linkage to initial state before optimization
my_linkage.set_num_constraints(init_constraints)
my_linkage.set_coords(init_pos)

# Generate bounds for the optimization search space
bounds = pl.generate_bounds(my_linkage.get_num_constraints())

score, position, coord = pl.particle_swarm_optimization(
    eval_func=fitness_func,
    linkage=my_linkage,
    bounds=bounds,
    order_relation=min,
)[0]

Here the result can vary, but it is rarely above 0.2.

So we made something that says it works, let’s verify it:

With a bit of imagination, you have a wonderful windshield wiper!

Project Structure

The recommended workflow for using pylinkage is:

1. Define joints: Use joint types from pylinkage.joints (Crank, Revolute, etc.)

2. Create linkage: Combine joints in a Linkage from pylinkage.linkage

3. Simulate (optional): Run linkage.step() to compute motion

4. Optimize: Use pylinkage.optimization functions to find optimal dimensions

5. Visualize: Display results with pylinkage.visualizer

Architecture

Pylinkage is organized into distinct abstraction layers, from low-level numerics to high-level user APIs. This design ensures separation of concerns and enables both ease of use and high performance.

Abstraction Levels

Level 0: Geometry (pure numba math primitives)
├── geometry/core.py       → cyl_to_cart, get_nearest_point, dist
└── geometry/secants.py    → circle_intersect, circle_line_intersection
         ↑ Single source of mathematical truth

Level 1: Solver (numba-compiled Assur group solvers)
└── solver/joints.py
    ├── solve_crank()      → Driver rotation
    ├── solve_revolute()   → RRR dyad (circle-circle intersection)
    ├── solve_linear()     → RRP dyad (circle-line intersection)
    └── solve_fixed()      → Deterministic polar constraint
         ↑ Single source of solving logic (used by all higher layers)

Level 2: Hypergraph (abstract mathematical structures)
└── hypergraph/
    ├── core.py            → Node, Edge, Hyperedge
    ├── graph.py           → HypergraphLinkage
    └── components.py      → Component, Port (reusable subgraphs)
         ↑ Foundational graph theory (no dependencies on assur)

Level 3: Assur (kinematic theory built on hypergraph)
└── assur/
    ├── graph.py           → LinkageGraph, Node, Edge
    ├── groups.py          → DyadRRR, DyadRRP (delegate to solver)
    └── decomposition.py   → Assur group decomposition algorithm
         ↑ Formal kinematic analysis

Level 4: User API (thin wrappers for ease of use)
├── joints/
│   ├── crank.py           → Crank (calls solve_crank)
│   ├── revolute.py        → Revolute (calls solve_revolute)
│   ├── linear.py          → Linear (calls solve_linear)
│   └── fixed.py           → Fixed (calls solve_fixed)
└── linkage/
    └── linkage.py         → Linkage class (orchestrates simulation)
         ↑ Validation + delegation to solver

Level 5: Applications
├── optimization/          → PSO, grid search algorithms
├── visualizer/            → Matplotlib, Plotly, SVG backends
└── bridge/                → Linkage ↔ SolverData conversion

Key Design Principles

1. Single Source of Truth: The solver/joints.py module contains the canonical implementations of all solving algorithms. Joint classes and Assur groups delegate to these functions.

2. Hypergraph as Foundation: The hypergraph module provides abstract mathematical structures. The assur module builds kinematic theory on top of it, not the other way around.

3. Pure Numerics in Solver: The solver module uses only numpy and numba with no Python object dependencies. This enables maximum performance for optimization loops.

4. Thin User API: Joint classes (Crank, Revolute, etc.) are thin wrappers that handle validation and user-friendly errors, then delegate to solver functions.

5. Assur Group Theory: The package is built on Assur group decomposition, a formal approach to analyzing planar linkages:

   • RRR dyad (3 revolute joints): Solved via circle-circle intersection

   • RRP dyad (2 revolute + 1 prismatic): Solved via circle-line intersection

Performance

The solver provides two simulation methods:

• linkage.step(): Python-based, ~300-400k steps/sec

• linkage.step_fast(): Numba-compiled, ~1.5-2.5M steps/sec (4-7x faster)

Use step_fast() for optimization loops where performance matters.

Requirements

Python 3, numpy for calculation, matplotlib for drawing, and standard libraries. You will also need PySwarms for the Particle Swarm Optimization.

Contributing

Pylinkage is open to contribution. I may consider any pull request, but I ask you to respect the CODE_OF_CONDUCT.md and follow the guidelines as defined in CONTRIBUTING.md.