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Reduced UR3 Robot Arm Simulation (RRR)

This Python program simulates the trajectory of a reduce UR3 robot arm (RRR) in 3D space using the Denavit-Hartenberg parameters and motion laws for each segment.

Overview

The program is organized into a UR3 class, representing the UR3 robot arm. The class contains methods for setting parameters, reading parameters, and displaying trajectory data. The main functionality includes:

Initialization

def __init__(self, o0o1: float = 152, o1o2: float = 120, o2o3: float = 244, la: float = 296, lb: float = 92)
  • Parameters:
    • o0o1: Distance between joint 0 and joint 1.
    • o1o2: Distance between joint 1 and joint 2.
    • o2o3: Distance between joint 2 and joint 3.
    • la: Length of link A.
    • lb: Length of link B.

Setting Parameters

def set_param(self, param_dict, param_type="const")
  • Parameters:
    • param_dict: A dictionary containing parameter names and their values.
    • param_type: Type of parameters, either "const" for constant or "trajectory" for trajectory-related.

Reading Parameters

def read_param(self, param_dict, param_type)
  • Parameters:
    • param_dict: A dictionary containing parameter names and their values.
    • param_type: Type of parameters, either "const" for constant or "trajectory" for trajectory-related.

Inputting Parameters

def input_param(self, param_dict, param_type)
  • Parameters:
    • param_dict: A dictionary containing parameter names and their corresponding attribute names.
    • param_type: Type of parameters, either "const" for constant or "trajectory" for trajectory-related.

Displaying Data

def display_data(self, data: list, title: str, labels: list[str])
  • Parameters:
    • data: List of data to be plotted.
    • title: Title of the plot.
    • labels: List of labels for each data series.

Setting Constant Parameters

def set_const_param_to_ROM(self, o0o1: float = None, o1o2: float = None, o2o3: float = None, la: float = None, lb: float = None)
  • Parameters:
    • o0o1, o1o2, o2o3, la, lb: Constant parameters.

Reading Constant Parameters

def read_const_param(self)

Inputting Constant Parameters

def input_const_param_to_ROM(self)

Setting Trajectory Parameters

def set_trajectory_param_to_ROM(self, xA: float = None, yA: float = None, zA: float = None, xB: float = None, yB: float = None, zB: float = None, V1: float = None, V2: float = None)
  • Parameters:
    • xA, yA, zA: Initial position coordinates.
    • xB, yB, zB: Final position coordinates.
    • V1: Initial speed.
    • V2: Final speed.

Reading Trajectory Parameters

def read_trajectory_param(self)

Inputting Trajectory Parameters

def input_trajectory_param_to_ROM(self)

Trajectory Calculation

def traj(self, A: list[float] = None, B: list[float] = None, V1: float = None, V2: float = None)

  • Parameters:

    • A: Initial position coordinates.
    • B: Final position coordinates.
    • V1: Initial speed.
    • V2: Final speed.

  • Returns:

    • q0: Joint angles for each time step.
    • q1: Joint speeds for each time step.

Displaying $s(t)$, $\dot{s}(t)$, $\ddot{s}(t)$ 

def display_s(self)

Displaying $x(t)$, $\dot{x}(t)$, $\ddot{x}(t)$ 

def display_x(self)

Displaying $y(t)$, $\dot{y}(t)$, $\ddot{y}(t)$ 

def display_y(self)

Displaying $z(t)$ $\dot{z}(t)$ $\ddot{z}(t)$ 

def display_z(self)

Displaying end-effector speed

def display_tool_speed(self)

Displaying Operational Trajectory

def display_Xs(self)

Displaying Joint Positions and Velocities

def display_q_dotq(self, q, dotq)
  • Parameters:
    • q: List of joint angles for each time step.
    • dotq: List of joint velocities for each time step.

Foreward Kinematics (MGD)

def MGD(self, q: list[float])

  • Parameters:

    • q: Joint angles.

  • Returns:

    • X, Y, Z: Cartesian coordinates of the end-effector.

Inverse Kinematics (MGI)

def MGI(self, X: float, Y: float, Z: float)

  • Parameters:

    • X, Y, Z: Cartesian coordinates of the end-effector.

  • Returns:

    • all_q: List of possible joint angle solutions for the given Cartesian coordinates.

Jacobian Calculation

def Jacobian(self,q)

  • Parameters:

    • q: List of joint angles for a given time step.

  • Returns:

    • t: Jacobian matrix.

Motion Differential Inverse (MDI)

def MDI(self, q, dotX)

  • Parameters:

    • q: List of joint angles for a given time step.
    • dotX: List of Cartesian velocities for each time step.

  • Returns:

    • Joint velocities qdot calculated using the inverse Jacobian and transpose of Cartesian velocities.

Usage

  1. Initialization: Create an instance of the UR3 class.

    ur3_robot = UR3()

  2. Setting Parameters: Set constant and trajectory parameters using the appropriate methods.

    ur3_robot.set_const_param_to_ROM(o0o1=152, o1o2=120, o2o3=244, la=296, lb=92)
ur3_robot.set_trajectory_param_to_ROM(xA=0, yA=0, zA=0, xB=300, yB=100, zB=150, V1=50, V2=50)

  3. Trajectory Calculation: Calculate the trajectory using the traj method.

    q0, q1 = ur3_robot.traj()

  4. Displaying Trajectory: Visualize the trajectory using the display_Xs method.

    ur3_robot.display_Xs()

  5. Additional Displays: Check other trajectory-related data using methods like display_s, display_x, display_y, display_z, display_tool_speed, and display_q_dotq.

  6. Jacobian Calculation and Motion Differential Inverse (MDI): Calculate the Jacobian and use the MDI method as needed.

    jacobian = ur3_robot.Jacobian(q=q0[0])
q_dot = ur3_robot.MDI(q=q0[0], dotX=... )  # Provide Cartesian velocities as needed

Main example

main.py

import os
os.system("pip3 install numpy")
os.system("pip3 install matplotlib")

from UR3 import UR3
import numpy as np

# Create an instance of the UR3 robot and initialize the q parameters
robot = UR3()
qdeg = [np.pi/2, 0, 0]

# Modify the values of the robot constants via the terminal
robot.input_const_param_to_ROM()
# or
#robot.set_const_param_to_ROM(o0o1=152, o1o2=120, o2o3=244, la=296, lb=92)  # Modify the values of the robot constants easily - parameters are optional
robot.read_const_param()  # Read the values of the robot constants

# Enter the values of points (A, B), V1, and V2 via the terminal
robot.input_trajectory_param_to_ROM()
# or
#robot.set_trajectory_param_to_ROM(xA=200, yA=200, zA=100, xB=300, yB=350, zB=300, V1=1, V2=1.5)  # Enter the values of points (A, B), V1, and V2
robot.read_trajectory_param()  # Read the values of points (A, B), V1, and V2

# Test the traj(A, B, V1, V2) function - parameters are optional
q,dotq=robot.traj()

# Display the three curves of s(t), s*(t), s**(t) (with the display of the switching times)
robot.display_s()

# Display the three curves x(t), x*(t), x**(t) (with the display of the switching times)
robot.display_x()

# Display the three curves y(t), y*(t), y**(t) (with the display of the switching times)
robot.display_y()

# Display the three curves z(t), z*(t), z**(t) (with the display of the switching times)
robot.display_z()

# Display the curve of the end-effector speed
robot.display_endEffector_speed()

# Display the trajectory X(s) in the operational space (∈R³)
robot.display_Xs()

# Display the trajectories of the curves in q(t) q point(t) for each joint (with the display of the switching times)
robot.display_q_dotq(q,dotq)

Terminal

Input const parameters
o0o1 = 152
o0o1 set to 152.0
o1o2 = 120
o1o2 set to 120.0
o2o3 = 244
o2o3 set to 244.0
la = 296
la set to 296.0
lb = 92
lb set to 92.0

Read const parameters
o0o1    =       152.0
o1o2    =       120.0
o2o3    =       244.0
la      =       296.0
lb      =       92.0

Input trajectory parameters
xA = 200
xA set to {self.A[0]}
yA = 200
yA set to {self.A[1]}
zA = 100
zA set to {self.A[2]}
xB = 300
yB set to {self.B[0]}
yB = 350
yB set to {self.B[1]}
zB = 300
zB set to {self.B[2]}
v1 = 1
v1 set to {self.v1}
v2 = 1.5
v2 set to {self.v2}

Read trajectory parameters
A       =       [200.0, 200.0, 100.0]
B       =       [300.0, 350.0, 300.0]
v1      =       1.0
v2      =       1.5

Graph of the imposed trajectory checking the laws of evolution s(t), s*(t) and s**(t)
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]

Graph of the imposed trajectory checking the laws of evolution x(t), x*(t) et x**(t)
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]

Graph of the imposed trajectory checking the laws of evolution y(t), y*(t) et y**(t)
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]

Graph of the imposed trajectory checking the laws of evolution z(t), z*(t) et z**(t)
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]

Check speed profile of the tool against s*(t)
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]
switch_times    =        [76.93092581620719, 153.86185163241439, 230.79277744862156, 307.72370326482877]

Note

  • The program uses the Denavit-Hartenberg parameters to model the robot arm's geometry and kinematics.
  • The trajectory is calculated based on specified initial and final positions and speeds.
  • Inverse kinematics methods (MGI and MGD) are provided for converting between Cartesian coordinates and joint angles.

Feel free to explore and modify the program to suit your specific use case.