2R Robotic Arm Simulator (Python)

AIM:  To write a program to simulate the forward kinematics of a 2R Robotic Arm in Python.

OBJECTIVE :

1. To solve and simulate the forward kinematics of a 2R Robotic Arm.
2. To create the animation of working of a 2R Robotic Arm.

INTRODUCTION: A robotic arm is a robot manipulator, usually programmable, with similar functions to a human arm. The links of such a manipulator are connected by joints allowing either rotational motion (such as in an articulated robot) or translational (linear) displacement. The links of the manipulator can be considered to form a kinematic chain. The use of industrial robots is increasing in areas such as food, consumer goods, wood, plastics, and electronics, but is still mostly concentrated in the automotive industry.

                                                (i)

In the image shown above, the robotic arm consists of 4 components,

• The Base, which is clamped to the surface and all other links are joined through the hinge joint.
• Link 1 (l1) making $\theta 1$ with horizontal.
• Link 2 (l2) making $\theta 2$ with horizontal.
• The device attached to the manipulator which interacts with its environment to perform tasks is called the end-effector.

SOLUTION STEPS : 

1. For our program, we have assumed the length of link 1 and link 2 to 1 m and 0.5 m.
2. In order to simulate the robotic arm, it is necessary to vary $\theta 1\text{and}\theta 2$ to obtain the desired animation and it cannot be possible if we manually enter the respective data every time. So we have created an array from $0°-90°$ with 10 evenly spaced points and stored $\theta 1\text{and}\theta 2$ as ${\theta }_{Start}\text{and}{\theta }_{end}$ between them and a loop counter is initiated from 1.
3. The coordinates of the base, i.e., is assumed to be at origin $(0,0)$ and the initial value of $\theta 1\text{and}\theta 2$ array is null. Then a for loop is executed with a range of 0 to 10 and each value of the loop is temporarily stored in a separate variable.
4. Another for loop is executed which stores each value obtained from the previous one and dumps into the trigonometric relation for simulating different values of angles and the loop counter increases by one for each iteration. 
5. For plotting graph 'plt.plot' the command is used with 'plt.figure' and 'plt.lim' to set the axis and connector in the desired way.  
6. The coordinates of endpoints of link 1 (l1) can be found by using trigonometry as,

   $x_1=l1\cdot \cos \left(\theta 1\right)$   and  $y_1=l1\cdot \sin \left(\theta 1\right)$

            Similarly, the coordinates of endpoint of link 2 (L2) can also be found from trigonometry as,

                          $x_2=x_1+l2\cdot \cos \left(\theta 2\right)$  and  $y_2=y_1+l2\cdot \sin \left(\theta 2\right)$

BODY OF THE CONTENT :

Python Code -

# A program to create simulation of 2R robotic arm manipulator
import math
import matplotlib.pyplot as plt 
# Inputs 
l1 = 1
l2 = 0.5
x0 = 0
y0 = 0
n_theta = 10
theta_start = 0
theta_end = math.pi/2
counter  = 1

theta_1 = []
theta_2 = []

for i in range(0,n_theta):
	tmp = theta_start + i*(theta_end - theta_start)/(n_theta - 1)
	theta_1.append(tmp)
	theta_2.append(tmp)

 
for T1 in theta_1:
	for T2 in theta_2:
		# Link 1 & Link 2 connector
		x1 = l1*math.cos(T1)
		y1 = l1*math.sin(T1)
		# Coordinates of manipulator
		x2 = x1 + l2*math.cos(T2)
		y2 = y1 + l2*math.sin(T2)
		filename = str(counter) + '.png'
		counter = counter + 1
		plt.figure()
		plt.plot([x0,x1],[y0,y1])
		plt.plot([x1,x2],[y1,y2])
		plt.xlim([0,3])
		plt.ylim([0,3])
		plt.savefig(filename)

RESULTS :

• Upon executing the given code, it will generate 100 plots as the program was inputted to have 10 values between $0°-90°$ and due to iterating it two times for two different angles. The following plot is obtained as one of the output: