pylinkage

A linkage builder written in Python. This package is made to create planar linkages and optimize them kinematically thanks to Particle Swarm Optimization. It is still an early work, so it should receive great changes in the future.

Installation

Using pip

This package is in the PyPi as pylinkage, and can be installed using: pip install pylinkage

It is the recommended way downloading it, since the release versions are synced.

Setting up Virtual Environment

We provide an environment.yml file for conda. Use conda env update --file environment.yml --name pylinkage-env to install the requirements in a separate environment.

Usage

As of today, the code is segmented in three parts:

• geometry.py that module handles geometric primitives, such as circle intersections, distance claculation. It works in Euclidian space only. Aside from dist and sqr_dist functions, you might not use it directly.
• linkage.py this module describes joints and linkages
  • Due to the geometric approach, joints (instances of Joint object) are defined without links.
  • The Linkage class that will make your code shorter.
• optimizer.py proposes three optimizations based on three techniques:
  • The "exhaustive" optimization (exhaustive_optimization function) is a dumb optimization method, consisting or trying sequencially all positions. It is here for demonstration purposes only, and you should not use it if you are looking for an efficient technique.
  • The built-in Particle Swarm Optimizer (PSO). I started with it, so it offers a large set of useful options for linkage optimization. However, it is here for legacy purposes, and is much short than the PySwarms module.
  • PSO using PySwarms. We provide a wrapper function to PySwarm from ljvmiranda921, that will progressively be extended.
• visualizer.py can make graphic illustrations of your linkage using matplotlib.
  • It is also used to visualise your n-dimensional swarm, which is not supported by PySwarms.

Requirements

Python 3, numpy for calculation, matplotlib for drawing, and standard libraries. You will also need PySwarms for the optimization since the built-in PSO is deprecated and will be removed soon.

Example

Let's start with a crank-rocker four-bar linkage, as classic of mechanics.

Joints definition

The first thing you need if to define points belonging to the frame:

import pylinkage.linkage as pl

frame_first = pl.Static(0, 0)
frame_second = pl.Static(3, 0)

• O, O: "x, y" position of the first point.
• 3, 0: same for the second point.

Then we have to define at least one crank because we do a kinematic simulation.

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

Here you need some explanations:

• 0, 1: x and y initial coordinates of the tail of the crank link.
• joint0: the parent Joint to link with. 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.Pivot(3, 2, joint0=crank, joint1=frame_second, distance0=3, distance1=1)

Here comes some help:

• 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 pain: Why do we specify initial coordinates 3, 2? Moreover, they 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 strating point, we calculate both intersection (when possible), then we keep the intersection closer to the previous position as the solution.

Wait! A linkage with a single motor and only one pin joint? That doesn't make sense! :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 Pivot joint, it creates 3 pin joints: one on the position of each of its parents, and one its position, forming a reshapable triangle. This is why pylinkage is so short to write.

Linkage definition and simulation

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

Linkage definition is simple:

my_linkage = pl.Linkage(joints=(frame_first, frame_second, 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 recape.

import pylinkage.linkage as pl

# Static points in space, belonging to the frame
frame_first = pl.Static(0, 0)
frame_second = pl.Static(3, 0)
# Main motor
crank = pl.Crank(0, 1, joint0=frame_first, angle=.31, distance=1)
# Close the loop
pin = pl.Pivot(3, 2, joint0=crank, joint1=frame_second, 
               distance0=3, distance1=1)

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

locus = my_linkage.step()

Visualization

Firsting 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.

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

Then you can view your linkage!

import pylinkage.visualizer as visu

visu.show_linkage(my_linkage)

Last recap, rearranging names:

import pylinkage.linkage as pl
import pylinkage.visualizer as visu

# Static points in space, belonging to the frame
frame_first = pl.Static(0, 0, name="A")
frame_second = pl.Static(3, 0, name="D")
# Main motor
crank = pl.Crank(0, 1, joint0=frame_first, angle=.31, distance=1, name="B")
# Close the loop
pin = pl.Pivot(3, 2, joint0=crank, joint1=frame_second,
               distance0=3, distance1=1, name="C")

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

# Visualization
visu.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 make a quarter of a circle. It is a common problem if you want to build a windscreen wiper for instance.

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

from pylinkage.geometry import bounding_box

# We save initial position because we don't want a completely different movement
init_pos = my_linkage.get_coords()

def fitness_func(linkage, params, *args):
    """
    Return how fit the locus is to describe a quarter of circle.

    It is a minisation problem and the theorical best score is 0.
    """
    linkage.set_coords(init_pos)
    linkage.set_num_constraints(params)
    try:
        points = 12
        n = linkage.get_rotation_period()
        # Complete revolution with 12 points
        tuple(tuple(i) for i in linkage.step(iterations=points + 1,
                                             dt=n/points))
        # Again with n points, and at least 12 iterations
        n = 96
        factor = int(points / n) + 1
        L = tuple(tuple(i) for i in linkage.step(
            iterations=n * factor, dt=1 / factor))
    except UnbuildableError:
        return -float('inf')
    else:
        # Locus of the Joint 'pin", mast in linkage order
        tip_locus = tuple(x[-1] for x in L)
        # We get the bounding box
        curr_bb = bounding_box(tip_locus)
        # We set the reference bounding box with frame_second as down-left
        # corner and size 2
        ref_bb = (frame_second.y, frame_second.x + 2,
                  frame_second.y + 2, frame_second.x)
        # Our score is the square sum of the edges distances
        return -sum((pos - ref_pos) ** 2
                    for pos, ref_pos in zip(curr_bb, ref_bb))

Please not that it is a minization problem, with 0 as lower bound.

We need to get the geometric constraints as the optimization parameters. ``constraints = tuple(my_linkage.get_num_constraints())```

With this constraints, score should be around -3.

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 = opti.exhaustive_optimization(
    eval_func=fitness_func,
    linkage=my_linkage,
    delta_dim=.1,
    n_results=1,
)[0]

Here the problem is simple enough, so that method typical return the maximal theorical value of 0.0.

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 constrints (the dimensions) of the linkage.
• A dimension set (a n-uplet) is called a particule or an agents. Think of it like a bee.
• The particles move in a n-vectorial space. That is, if we have n geometric constraints, the particules move in a n-D space.
• Together, the particules 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 they neighbours.
• Together they will try find the extremum in the space. Here it is a minimum.

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

Due to incompatibilities, we need a wrapper.

# A simple wrapper
def PSO_fitness_wrapper(constraints, *args):
    """A simple wrapper to make the fitness function compatible."""
    return fitness_func(my_linkage, constraints, *args)


# We reinitialize the linkage (an optimal linkage is not interesting)
my_linkage.set_num_constraints(constraints)
# As we do for initial positions
my_linkage.set_coords(init_pos)


import numpy as np

# Optimization is more efficient with a start space
bounds = (np.zeros(len(constraints)), np.ones(len(constraints)) * 5)

score = opti.particle_swarm_optimization(
    eval_func=PSO_fitness_wrapper,
    linkage=my_linkage,
    bounds=bounds
).swarm.best_cost

Here again the result should be 0.0.

So we made something that say it works, let's verify it:

With a bit of imagination you have a wonderful windscreen wiper!