kinematics analysis of robots (part 3). this lecture continues the discussion on the analysis of the...
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KINEMATICS ANALYSIS
OF ROBOTS(Part 3)
This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots.
After this lecture, the student should be able to:•Solve problems of robot kinematics analysis using transformation matrices
Kinematics Analysis of Robots III
Example: A 3 DOF RRR Robot
Link and Joint Assignment
Link (2) Link (3)Link (1)
Revolute joint <1>
Link (0)
Revolute joint <2>
Revolute joint <3>
Example: A 3 DOF RRR Robot
Frame Assignment
Z1
Z1
Y1
Y1
X1
Example: A 3 DOF RRR Robot
Frame Assignment
Y0, Y1
X0, X1
Z0, Z1
Z2
Z2
X2
Y2
Y2
Example: A 3 DOF RRR Robot
Frame Assignment
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
Z3
X3
Y3
Y3
Example: A 3 DOF RRR Robot
Frame Assignment
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
1
2 3
Example: A 3 DOF RRR Robot
Tabulation of D-H parameters
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A B
0 = (angle from Z0 to Z1 measured along X0) = 0°a0 = (distance from Z0 to Z1 measured along X0) = 0d1 = (distance from X0 to X1 measured along Z1)= 01 = variable (angle from X0 to X1 measured along Z1)1 = 0° (at home position) but 1 can change as the arm moves
Example: A 3 DOF RRR Robot
Tabulation of D-H parameters
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A B
1 = (angle from Z1 to Z2 measured along X1) = 90°a1 = (distance from Z1 to Z2 measured along X1) = Ad2 = (distance from X1 to X2 measured along Z2) = 02 = variable (angle from X1 to X2 measured along Z2)2 = 0° (at home position) but 2 can change as the arm moves
Example: A 3 DOF RRR Robot
Tabulation of D-H parameters
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A B
2 = (angle from Z2 to Z3 measured along X2) = 0°a2 = (distance from Z2 to Z3measured along X2) = Bd3 = (distance from X2 to X3 measured along Z3) = 03 = variable (angle from X2 to X3 measured along Z3)3= 0° (at home position) but 3 can change as the arm moves
Link i Twist i Link length ai
Link offset di
Joint angle i
i=0 0 0 … …
i=1 90° A 0 1
(1=0° at home position)
i=2 0 B 0 2
(2=-0° at home position)
i=3 … … 0 3
(3=-0° at home position)
Summary of D-H parameters
Example: A 3 DOF RRR Robot
Tabulation of Transformation Matrices from the D-H table
1000
)cos()cos()cos()sin()sin()sin(
)sin()sin()cos()cos()sin()cos(
0)sin()cos(
1111
1111
1
1
iiiiiii
iiiiiii
iii
ii d
d
a
T
1000
0100
00)cos()sin(
00)sin()cos(
11
11
01
T
0,0,0 100 da
Example: A 3 DOF RRR Robot
Tabulation of Transformation Matrices from the D-H table
0,,90 211 dAa
1000
00)cos()sin(
0100
0)sin()cos(
22
22
12
A
T
1000
)cos()cos()cos()sin()sin()sin(
)sin()sin()cos()cos()sin()cos(
0)sin()cos(
1111
1111
1
1
iiiiiii
iiiiiii
iii
ii d
d
a
T
Example: A 3 DOF RRR Robot
Tabulation of Transformation Matrices from the D-H table
0,,0 322 dBa
1000
0100
00)cos()sin(
0)sin()cos(
33
33
23
B
T
1000
)cos()cos()cos()sin()sin()sin(
)sin()sin()cos()cos()sin()cos(
0)sin()cos(
1111
1111
1
1
iiiiiii
iiiiiii
iii
ii d
d
a
T
Example: A 3 DOF RRR Robot
Forward Kinematics
1000
00)cos()sin(
)sin()cos()sin()sin()cos()sin(
)cos()sin()sin()cos()cos()cos(
22
112121
112121
12
01
02
A
A
TTT
1000
)sin(0)cos()sin(
)sin())cos(()cos()sin()sin()cos()sin(
)cos())cos(()sin()sin()cos()cos()cos(
23232
121321321
121321321
2
3
0
2
0
3
B
BA
BA
TTT
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A=3 B=2 C=1
P
Example: A 3 DOF RRR Robot
What is the position of point “P” at the home position?
Solution:
1
0
0
1
1
3 p
11
303
0 pT
p
Example: A 3 DOF RRR Robot
1
0
0
1
1000
)sin(0)cos()sin(
)sin())cos(()cos()sin()sin()cos()sin(
)cos())cos(()sin()sin()cos()cos()cos(
123232
121321321
121321321
3
0
3
B
BA
BA
pT
1= 2= 3=0, A=3, and B=2:
1
0
0
6
1
0
0
1
1000
0010
0100
5001
11
303
0 pT
p
Example: A 3 DOF RRR Robot
Inverse Kinematics
Given the orientation and position of point “P”:
TTTTpaon
paon
paon
zzzz
yyyy
xxxx
03
23
12
01
1000
1
0
0
1
1000
)sin(0)cos()sin(
)sin())cos(()cos()sin()sin()cos()sin(
)cos())cos(()sin()sin()cos()cos()cos(
123232
121321321
121321321
3
0
3
B
BA
BA
pT
Example: A 3 DOF RRR Robot
Inverse Kinematics
Equate elements (1,3) and (2,3):
y
x
y
x
a
aa
a1
11
1tan
)cos(
)sin(
Provided that 1)( 22 yx aa
Equate elements (1,4) and (2,4):
Ap
BBAp
Ap
BBAp
yy
xx
)cos(1
)sin()sin())cos((
)cos(1
)cos()cos())cos((
1212
1212
Example: A 3 DOF RRR Robot
Inverse Kinematics
If cos(1)0, then use px to find cos(2). Afterwards, find
Otherwise use py to find sin(2) and then solve using
)cos()sin(
tan
)(cos1)sin(
2
212
22
2
)cos()sin(
tan
)(sin1)cos(
2
212
22
2
Example: A 3 DOF RRR Robot
Inverse Kinematics
Equate elements (3,1) and (3,2):
z
z
z
z
on
o
n 132
32
32 tan)()cos(
)sin(
2323
Now find 1, 2, and 3 given the orientation and position of point “P”:
1000
1001
0100
5010
1000zzzz
yyyy
xxxx
paon
paon
paon
Example: A 3 DOF RRR Robot
Inverse Kinematics
0tan1
01
1
y
x
y
x
a
aa
a
Now cos(1)=10. We use px to find cos(2):
1)cos(
1)cos(
2,3,5
12
Ap
B
BAp
x
x
0)cos()sin(
tan
0)(cos1)sin(
2
212
22
2
Example: A 3 DOF RRR Robot
Inverse Kinematics90tan)(
0
1 132
z
z
z
z
on
o
n
900902323
Y0
X0
Z0
Z3
X3
Y3
1000
1001
0100
5010
1000zzzz
yyyy
xxxx
paon
paon
paon A=3 B=2
PC=1
Forward & Inverse Kinematics Issues
Given a set of joint variables, the forward kinematics will always produce an unique solution giving the robot global position and orientation. On the other hand, there may be no solution to the inverse kinematics problem. The reasons include:•The given global position of the arm may be beyond the robot work space•The given global orientation of the gripper may not be possible given that the gripper frame must be a right hand frame
For the inverse kinematics problem, there may also exist multiple solutions, i.e. the solution may not be unique.
Forward & Inverse Kinematics Issues
Example of multiple solutions given the same gripper global position and orientation:
First solution
Second solution
First solution
Second solution
Some solutions may not be feasible due to obstacles in the workspace
Summary
This lecture continues the the discussion on the analysis of the forward and inverse kinematics of robots.
The following were covered:•Problems of robot kinematics analysis using transformation matrices