rparak / parametric_curves

An open-source library for interpolation functions of parametric curves (Bézier, B-Spline) useful for robotics applications.

An Open-Source Parametric Curves Library Useful for Robotics Applications

Requirements

Programming Language

Python

Import Libraries

More information can be found in the individual scripts (.py).

Supported on the following operating systems

Windows, Linux, macOS

Project Description

An open-source library for interpolation functions of parametric curves (Bézier, B-Spline) useful for robotics applications, such as path planning, etc. The library provides access to specific classes for working with two types of parametric curves: Bézier curves and B-Spline curves. The specific classes focus on the problem of interpolating both two-dimensional and three-dimensional curves from input control points.

The classes also include the methods to calculate the derivative of the individual curves, arc-length, bounding box, simplification of the curves, and much more. The B-Spline interpolation class contains a function to optimize control points using the least squares method.

Path (Bézier): ..\Collision_Detection\src\Interpolation\Bezier\Core.py
Path (B-Spline): ..\Collision_Detection\src\Interpolation\B_Spline\Core.py

In particular, the library focuses on solving the path planning problem of the industrial/collaborative robotic arms. But, as an open-source library, it can be used for other tasks, as creativity knows no limits.

The repository also contains a transformation library with the necessary project-related functions. See link below.

/rparak/Transformation

The library can be used within the Robot Operating System (ROS), Blender, PyBullet, Nvidia Isaac, or any program that allows Python as a programming language.

Bézier Curves

Bernstein Polynomials

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/Bezier
$ ../Evaluation/Bezier> python3 Bernstein_Polynomials.py

Demonstration of two-dimensional (2D) Curves

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/Bezier
$ ../Evaluation/Bezier> python3 test_2d_1.py

Demonstration of three-dimensional (3D) Curves

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/Bezier
$ ../Evaluation/Bezier> python3 test_3d_1.py

A simple program that describes how to work with the library can be found below. The whole program is located in the individual evaluation folder.

# System (Default)
import sys
# Numpy (Array computing) [pip3 install numpy]
import numpy as np
# Matplotlib (Visualization) [pip3 install matplotlib]
import matplotlib.pyplot as plt
# Custom Lib.:
#   ../Interpolation/Bezier/Core
import Interpolation.Bezier.Core as Bezier

"""
Description:
    Initialization of constants.
"""
# Bezier curve interpolation parameters.
#   'method': The name of the method to be used to interpolate the parametric curve.
#               method = 'Explicit' or 'Polynomial'.
#   N: The number of points to be generated in the interpolation function.
CONST_BEZIER_CURVE = {'method': 'Explicit', 'N': 100}
# Visibility of the bounding box:
#   'limitation': 'Control-Points' or 'Interpolated-Points'
CONST_BOUNDING_BOX = {'visibility': False, 'limitation': 'Control-Points'}

def main():
    """
    Description:
        A program to visualize a parametric two-dimensional Bézier curve of degree n.
    """

    # Input control points {P} in two-dimensional space.
    P = np.array([[1.00,  0.00], 
                  [2.00, -0.75], 
                  [3.00, -2.50], 
                  [3.75, -1.25], 
                  [4.00,  0.75], 
                  [5.00,  1.00]], dtype=np.float32)

    # Initialization of a specific class to work with Bézier curves.
    B_Cls = Bezier.Bezier_Cls(CONST_BEZIER_CURVE['method'], P, 
                              CONST_BEZIER_CURVE['N'])
    
    # Interpolation of parametric Bézier curve.
    B = B_Cls.Interpolate()

    # Obtain the arc length L(x) of the general parametric curve.
    L = B_Cls.Get_Arc_Length()

    # Create a figure.
    _, ax = plt.subplots()

    # Visualization of relevant structures.
    ax.plot(P[:, 0], P[:, 1], 'o--', color='#d0d0d0', linewidth=1.0, markersize = 8.0, 
            markeredgewidth = 4.0, markerfacecolor = '#ffffff', label='Control Points')
    ax.plot(B[:, 0], B[:, 1], '.-', color='#ffbf80', linewidth=1.5, markersize = 8.0, 
            markeredgewidth = 2.0, markerfacecolor = '#ffffff', label=f'Bézier Curve (N = {B_Cls.N}, L = {L:.03})')

    # Show the result.
    plt.show()

if __name__ == '__main__':
    sys.exit(main())

B-Spline Curves

Basic Functions

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/B_Spline
$ ../Evaluation/B_Spline> python3 Basic_Functions.py

Demonstration of two-dimensional (2D) Curves

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/B_Spline
$ ../Evaluation/B_Spline> python3 test_2d_1.py

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/B_Spline
$ ../Evaluation/B_Spline> python3 test_2d_2.py

Demonstration of three-dimensional (3D) Curves

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/B_Spline
$ ../Evaluation/B_Spline> python3 test_3d_1.py

$ /> cd Documents/GitHub/Parametric_Curves/Evaluation/B_Spline
$ ../Evaluation/B_Spline> python3 test_3d_2.py

A simple program that describes how to work with the library can be found below. The whole program is located in the individual evaluation folder.

# System (Default)
import sys
# Numpy (Array computing) [pip3 install numpy]
import numpy as np
# Matplotlib (Visualization) [pip3 install matplotlib]
import matplotlib.pyplot as plt
# Custom Lib.:
#   ../Interpolation/B_Spline/Core
import Interpolation.B_Spline.Core as B_Spline

"""
Description:
    Initialization of constants.
"""
# B-Spline interpolation parameters.
#   n: Degree of a polynomial.
#   N: The number of points to be generated in the interpolation function.
#   'method': The method to be used to select the parameters of the knot vector. 
#               method = 'Uniformly-Spaced', 'Chord-Length' or 'Centripetal'.
CONST_B_SPLINE = {'n': 3, 'N': 100, 'method': 'Chord-Length'}
# Visibility of the bounding box:
#   'limitation': 'Control-Points' or 'Interpolated-Points'
CONST_BOUNDING_BOX = {'visibility': False, 'limitation': 'Control-Points'}

def main():
    """
    Description:
        A program to visualize a parametric two-dimensional B-Spline curve of degree n.
    """

    # Input control points {P} in two-dimensional space.
    P = np.array([[1.00,  0.00], 
                  [2.00, -0.75], 
                  [3.00, -2.50], 
                  [3.75, -1.25], 
                  [4.00,  0.75], 
                  [5.00,  1.00]], dtype=np.float32)

    # Initialization of a specific class to work with B-Spline curves.
    S_Cls = B_Spline.B_Spline_Cls(CONST_B_SPLINE['n'], CONST_B_SPLINE['method'], P, 
                                  CONST_B_SPLINE['N'])
    
    # Interpolation of parametric B-Spline curve.
    S = S_Cls.Interpolate()

    # Obtain the arc length L(x) of the general parametric curve.
    L = S_Cls.Get_Arc_Length()

    # Create a figure.
    _, ax = plt.subplots()

    # Visualization of relevant structures.
    ax.plot(P[:, 0], P[:, 1], 'o--', color='#d0d0d0', linewidth=1.0, markersize = 8.0, 
            markeredgewidth = 4.0, markerfacecolor = '#ffffff', label='Control Points')
    ax.plot(S[:, 0], S[:, 1], '.-', color='#ffbf80', linewidth=1.5, markersize = 8.0, 
            markeredgewidth = 2.0, markerfacecolor = '#ffffff', label=f'B-Spline (n = {S_Cls.n}, N = {S_Cls.N}, L = {L:.03})')

    # Show the result.
    plt.show()

if __name__ == '__main__':
    sys.exit(main())

Blender

The library for parametric curves can also be used within the Blender software. See the instructions below for more information.

Bézier Curves

A description of how to run a program to visualize a parametric three-dimensional Bézier curve of degree n.

1. Open Bezier_Curve.blend from the Blender folder.
2. Copy and paste the script from the evaluation folder (../Bezier_Curve.py).
3. Run it and evaluate the results.

$ /> cd Documents/GitHub/Parametric_Curves/Blender/Interpolation
$ ../Collision_Detection/Blender> blender Bezier_Curve.blend

B-Spline Curves

A description of how to run a program to visualize a parametric three-dimensional B-Spline curve of degree n.

1. Open B_Spline.blend from the Blender folder.
2. Copy and paste the script from the evaluation folder (../B_Spline.py).
3. Run it and evaluate the results.

$ /> cd Documents/GitHub/Parametric_Curves/Blender/Interpolation
$ ../Collision_Detection/Blender> blender B_Spline.blend

YouTube

Contact Info

[email protected]

Citation (BibTex)

@misc{RomanParak_ParametricCurves,
  author = {Roman Parak},
  title = {An open-source parametric curves library useful for robotics applications},
  year = {2023},
  publisher = {GitHub},
  journal = {GitHub repository},
  howpublished = {\url{https://github.com/rparak/Parametric_Curves}}
}

License

MIT