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Robotics Submission

The document details an end-term project report on a 2-link planar robotic arm with two degrees of freedom, focusing on tracing an 'L' shape using forward kinematics and encoder feedback. It includes descriptions of the robot's structure, Denavit-Hartenberg parameters, homogeneous transformation matrices, Jacobian matrix, and both forward and inverse kinematics equations. Additionally, it discusses singularity conditions and isotropic points for optimal manipulator performance.

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terabhavitha
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62 views3 pages

Robotics Submission

The document details an end-term project report on a 2-link planar robotic arm with two degrees of freedom, focusing on tracing an 'L' shape using forward kinematics and encoder feedback. It includes descriptions of the robot's structure, Denavit-Hartenberg parameters, homogeneous transformation matrices, Jacobian matrix, and both forward and inverse kinematics equations. Additionally, it discusses singularity conditions and isotropic points for optimal manipulator performance.

Uploaded by

terabhavitha
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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End Term Project Report

2-Link Planar Robotic Arm


Tracing ‘L’ using Forward Kinematics in 2R-Planar SCM with encoder
feedback
Team 08

a) Robot Description
The robot is a 2-link planar robotic arm with 2 Degrees of Freedom (DoFs). It consists
of two revolute joints:

• Joint 1: Revolute joint at the base (DC Motor with Encoder)

• Joint 2: Revolute joint at the elbow (Servo Motor)

• Link 1 Length: L1 = 19.9 cm

• Link 2 Length: L2 = 15 cm

1
Joint θ (variable) d a α
1 θ1 0 L1 0
2 θ2 0 L2 0

Table 1: Denavit-Hartenberg Parameters

b) DH Parameters
c) Homogeneous Transformation Matrices
From Frame 0 to 1: 0 T1
 
cos θ1 − sin θ1 0 L1 cos θ1
0
 sin θ1 cos θ1 0 L1 sin θ1 
T1 = 
 0

0 1 0 
0 0 0 1

From Frame 1 to 2: 1 T2
 
cos θ2 − sin θ2 0 L2 cos θ2
1
 sin θ2 cos θ2 0 L2 sin θ2 
T2 = 
 0

0 1 0 
0 0 0 1

From Frame 0 to 2: 0 T2 = 0 T1 · 1 T2
 
cos(θ1 + θ2 ) − sin(θ1 + θ2 ) 0 x
0
 sin(θ1 + θ2 ) cos(θ1 + θ2 ) 0 y
T2 =  
 0 0 1 0
0 0 0 1
Where,
x = L1 cos θ1 + L2 cos(θ1 + θ2 )
y = L1 sin θ1 + L2 sin(θ1 + θ2 )

d) Jacobian Matrix J(θ)


 
−L1 sin θ1 − L2 sin(θ1 + θ2 ) −L2 sin(θ1 + θ2 )
J(θ) =
L1 cos θ1 + L2 cos(θ1 + θ2 ) L2 cos(θ1 + θ2 )

e) Forward Kinematics
   
x L1 cos θ1 + L2 cos(θ1 + θ2 )
=
y L1 sin θ1 + L2 sin(θ1 + θ2 )
Velocity relationship:

Ẋ = J(θ) · θ̇

2
f ) Inverse Kinematics
Step 1: Compute θ2
x2 + y 2 − L21 − L22
cos θ2 = ⇒ θ2 = cos−1 (·)
2L1 L2

Step 2: Compute θ1
y  
−1 −1 L2 sin θ2
θ1 = tan − tan
x L1 + L2 cos θ2

g) Singularity Condition
det(J) = L1 L2 sin(θ2 )
Singularities occur when:

sin(θ2 ) = 0 ⇒ θ2 = 0◦ , 180◦

h) Isotropic Points
A manipulator is isotropic when the Jacobian has equal singular values, i.e., well-conditioned.
One such condition is:

L1 = L2 and θ2 = 90◦
This results in uniform manipulability in all directions.

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