Page 65 - Toucpad robotics C11

Think of it as a simplified version of a human arm, with a shoulder and an elbow.
y

final position

motor 2

motor 1 initial condition

0 Link 1 Link 2 x
Description:
u
It has a base, two links of lengths L and L , and two revolute joints.
l 1 2
The first joint connects the base to the first link (L ), and the second joint connects the first link to the second
l 1
link (L ).
2

The joint variables are the angles of rotation, and , relative to their respective links.
l 1 2
The goal of forward kinematics is to find the (x,y) coordinates of the end-effector.
l
u Mathematical Model: We can use basic trigonometry to determine the end-effector’s position.
The first link’s end is at (x ,y ). Using the angle , its coordinates are: x =L cos( ) y =L sin( )
l 1 1 1 1 1 1 1 1 1
The second link’s end (the end-effector) is at (x ,y ). Its position is relative to the first link’s end. The angle of the
l 2 2
second link relative to the base is ( + ). Therefore, its coordinates are: x =x +L cos( + )=L cos( )+L cos( + ) y
2
2
1
1
2
1
2
2
1
2
1
1
2
=y +L sin( + )=L sin( )+L sin( + )
1
2
2
1
2
1
2
1
1
Example:
u
Imagine a 2R manipulator with link lengths L =10 cm and L =5 cm.
l 1 2
If the joint angles are =30° and =60°.
l 1 2
We can calculate the end-effector’s position: x
l 2
=10cos(30°)+5cos(30°+60°)=10(0.866)+5(0)=8.66 cm y 2
=10sin(30°)+5sin(30°+60°)=10(0.5)+5(1)=5+5=10 cm
The end-effector’s position is (8.66,10) cm.
l
Configuration 2: The RP Planar Manipulator
The “RP” configuration refers to a planar manipulator with one revolute joint followed by one prismatic joint.
z 2
r i (t)
(t)
i
q (t)
i
z 1

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Mechanical System