## Transcribed Text

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22
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3,
SOLVING THE INVERSE KINEMATICS USING THE
1
NEWTON-RAPHSON METHOD
d,
Q2
di
Consider the manipulators of figure 1 with do = 30cm, d1 =
30em. d2 25cm.
We want to use the Newton-Raphson method to solve the
a
inverse kinematics problem. We consider two cases:
The task space is the end-effector's location (FerBer 2c).
A,
The task space is (x. Penzel. where O2 is the end-
effector's orientation
The desired homogenous transformation for the end-effector
is given by
0.50
-0.87
39.69
0.87
0.50
34.33
1.00 20.00
(1)
Flg.
1.00
The Newton-Raphson method is a numerical method used to
c) Write code to implement the Newton-Raphson
solve nonlinear equations of the general form f(r) 0. The
method and solve the inverse kinematics problem
method is iterative where the solution can be calculated using
d) What are the joint variables that allow to put
the end-effector at the desired location re, and
(2)
orientation de?
c) Verify your results using the forward kinematics
where J is the Jacobian matrix calculated at
equations
1) Find the direction and angle of rotation from the homo-
geneous transformation HO.
In both cases, feel free to compare your result with built-in
function fsolve or equivalent functions.
2) Case I
PROBLEM 2: KINEMATICS AND INVERSE KINEMATICS OF
a) Write the equations relating Te- to the con-
RRP MANIPULATOR
figuration variables 01,02 ds and the link lengths
do,d1,d2-
Figure 2 shows a simple RRP manipulator with B1 = 50cm
b) Find the Jacobian matrix of the system
and B2 = 30cm. The configuration shown in the figure
c) Write code to implement the Newton-Raphson
(so/lzs) corresponds to 01 = 90°. A suggested representation
of the axes is shown in the figure. Feel free to use other
method and solve the inverse kinematics problem
d) What are the joint variables that allow to put the
representations.
end-effector at the desired location?
1) What are the configuration variables?
e) Verify your results using the forward kinematics
2) Write the equations for the position of the end-effector
equations
in the base reference frame.
3) Write equations for the configuration variables in terms
of and the lengths of the links. You should
Case 2
be able to simplify the equations and solve analytically.
3)
a) Write the equations relating is Parze to the con-
4) We want to put the end-effector at position
figuration variables 01,02, d3 and the link lengths
(Ocm. 34. 33cm, (50cm). find the configuration variables.
do,d1,d2-
For this question, use the results of the previous
b) Find the Jacobian matrix of the system
question to solve analytically by hand.
2%
2,
a
0,
23
shoulder
4
R²
H.,
B,
yo
waist
Flg.
5) Use fsolve function or your Newton-Raphson code to
solve for the configuration of the robot that puts the
end-effector at the following locations
a) Zr) = (0cm, 34.33cm, 50cm),
b) Ze) = (30cm, 50cm, 50cm),
c) = (30cm, 10cm, 10cm).
6) Use Matlab or equivalent tools to represent the robot at
each one of these configurations. Represent the prismatic
joints by squares and the revolute joints by circles.
7) Now we want to study the velocity kinematics of the
manipulator, find Je such that
r.
He
(3)
ze
8) Calculate J, at configuration (@1,02.d)
=
33cm)
9) Determine the speed of the configuration variables so
that Yes 2.) Sem/s 3cm/s, 2cm/6) at the con-
figuration of the previous question.
2

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