introduction to dynamics analysis of robots (part 5)
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INTRODUCTION TO
DYNAMICS ANALYSIS
OF ROBOTS(Part 5)
This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another.
After this lecture, the student should be able to:•Solve problems of robot instantaneous motion using joint variable interpolation•Calculate the Jacobian of a given robot•Investigate robot singularity and its relation to Jacobian
Introduction to Dynamics Analysis of Robots (5)
Summary of previous lecture
n
T
n
n
n
n
z
y
x
P J
vvv
vvv
vvv
v
v
v
v
2
1
)(2
1
34
2
34
1
34
24
2
24
1
24
14
2
14
1
14
0/
Jacobian for translational velocities
n
T
n
T
z
y
x
P
n
T
z
y
x
P JJ
a
a
a
aJ
v
v
v
v
2
1
)(2
1
)(0/
2
1
)(0/
Instantaneous motion of robots
So far, we have gone through the following exercises:
Given the robot parameters, the joint angles and their rates of rotation, we can find the following:
1. The linear (translation) velocities w.r.t. base frame of a point located at the end of the robot arm
2. The angular velocities w.r.t. base frame of a point located at the end of the robot arm
3. The linear (translation) acceleration w.r.t. base frame of a point located at the end of the robot arm
4. The angular acceleration w.r.t. base frame of a point located at the end of the robot arm
We will now use another approach to solve the angular velocities problem.
Jacobian for Angular Velocities
100034333231
24232221
14131211
112
01
0
vvvv
vvvv
vvvv
TTTTT nP
nnP
In general, the position and orientation of a point at the end of the arm can be specified using
333231
232221
131211
332313
322212
312111
333231
232221
131211
)()(,)(
vvv
vvv
vvv
tR
vvv
vvv
vvv
tR
vvv
vvv
vvv
tR T
3
133
3
123
3
113
3
132
3
122
3
112
3
131
3
121
3
111
333231
232221
131211
332313
322212
312111
333231
232221
131211
333231
232221
131211
)()(
iii
iii
iii
iii
iii
iii
iii
iii
iii
T
vvvvvv
vvvvvv
vvvvvv
vvv
vvv
vvv
vvv
vvv
vvv
tRRt
132312221121
331332123111
233322322131
3
112
3
131
3
123
21
13
32
3
2
1
)(
vvvvvv
vvvvvv
vvvvvv
vv
vv
vv
t
iii
iii
iii
3
133
3
123
3
113
3
132
3
122
3
112
3
131
3
121
3
111
333231
232221
131211
iii
iii
iii
iii
iii
iii
iii
iii
iii
vvvvvv
vvvvvv
vvvvvv
Jacobian for Angular Velocities
nn
ijijijijij
n
n
ijijijijijnij
vvv
dt
dvv
dt
dv
dt
dv
dt
dv
dt
dvvfv
22
11
2
2
1
121 ),,,(
Jacobian for Angular Velocities
nnnn
nn
nn
nn
vv
vv
vv
vv
vv
vv
vv
vv
vv
vvvv
vvvv
vvvv
vvvvvv
2333
2232
2131
2232
3322
2
3221
2
31123
1
3322
1
3221
1
311
2333
22
331
1
33
2232
22
321
1
3221
312
2
311
1
311
2333223221311
nnnn
vv
vv
vv
vv
vv
vv
vv
vv
vv
vvvvvv
3313
3212
3111
2332
1332
2
1231
2
11133
1
1332
1
1231
1
112
3313321231112
Similarly:
Jacobian for Angular Velocities
nnnn
vv
vv
vv
vv
vv
vv
vv
vv
vv
vvvvvv
1323
1222
1121
2132
2312
2
2211
2
21113
1
2312
1
2211
1
213
1323122211213Similarly:
n
nnn
nnn
nnn
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
t
2
1
1323
1222
1121
13
2
2312
2
2211
2
2113
1
2312
1
2211
1
21
3313
3212
3111
33
2
1332
2
1231
2
1133
1
1332
1
1231
1
11
2333
2232
2131
23
2
3322
2
3221
2
3123
1
3322
1
3221
1
31
3
2
1
)(
Jacobian for angular velocities
)(0/
AP J
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A=3 B=2 C=1
P
What is the Jacobian for angular velocities of point “P”?
Example: Jacobian for Angular Velocities
Given:
1000
)sin()sin(1)cos()sin(
)sin())cos()cos(()cos()sin()sin()cos()sin(
)cos())cos()cos(()sin()sin()cos()cos()cos(
3223232
13221321321
13221321321
CB
CBA
CBA
TnP
133
2312
3
2211
3
2113
2
2312
2
2211
2
2113
1
2312
1
2211
1
21
333
1332
3
1231
3
1133
2
1332
2
1231
2
1133
1
1332
1
1231
1
11
233
3322
3
3221
3
3123
2
3322
2
3221
2
3123
1
3322
1
3221
1
31
)(
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
vv
J A
Example: Jacobian for Angular Velocities
3
3132
2
31
1
31
3231
)cos(
0
)sin(
vv
v
v
3
3232
2
32
1
32
3232
)sin(
0
)cos(
vv
v
v
0
1
3
33
2
33
1
33
33
vvv
v
Example: Jacobian for Angular Velocities
3
11321
2
11
3211
11
32111
)sin()cos(
)cos()sin(
)cos()cos(
vv
v
v
3
12321
2
12
3211
12
32112
)cos()cos(
)sin()sin(
)sin()cos(
vv
v
v
3
13
2
13
11
13
113
0
)cos(
)sin(
vv
v
v
3
21321
2
21
3211
21
32121
)sin()sin(
)cos()cos(
)cos()sin(
vv
v
v
3
22321
2
22
3211
22
32122
)cos()sin(
)sin()cos(
)sin()sin(
vv
v
v
3
23
2
23
11
23
123
0
)sin(
)cos(
vv
v
v
Example: Jacobian for Angular Velocities
0)1,1( 231
3322
1
3221
1
31)(
vv
vv
vv
J A
)sin()2,1(
0)sin()sin()sin()cos()sin()cos(
)2,1(
1)(
3213232132
232
3322
2
3221
2
31)(
A
A
J
vv
vv
vv
J
)sin()3,1(
0)sin()sin()sin()cos()sin()cos(
)3,1(
1)(
3213232132
233
3322
3
3221
3
31)(
A
A
J
vv
vv
vv
J
00)cos()sin()sin()sin()cos()sin(
)1,2(
3232132321
331
1332
1
1231
1
11)(
v
vv
vv
vJ A
)cos(0)cos()cos()cos()sin()sin()cos(
)2,2(
13232132321
332
1332
2
1231
2
11)(
vv
vv
vv
J A
)cos(0)cos()cos()cos()sin()sin()cos(
)3,2(
13232132321
333
1332
3
1231
3
11)(
vv
vv
vv
J A
Example: Jacobian for Angular Velocities
1)1,3(
)sin()sin()sin()cos()sin()cos()cos()cos()cos()cos(
)1,3(
)(
11321321321321
131
2312
1
2211
1
21)(
A
A
J
vv
vv
vv
J
00)sin()cos()cos()sin()cos()cos()sin()sin(
)2,3(
321321321321
132
2312
2
2211
2
21)(
v
vv
vv
vJ A
00)sin()cos()cos()sin()cos()cos()sin()sin(
)3,3(
321321321321
133
2312
3
2211
3
21)(
v
vv
vv
vJ A
Example: Jacobian for Angular Velocities
Example: Jacobian for Angular Velocities
001
)cos()cos(0
)sin()sin(0
11
11)(
AJ
What is after 1 second if all the joints are rotating at
3,2,1,6
it
i
0/3
5236.0
9069.0
5236.0
5236.0
5236.0
5236.0
001
866.0866.00
5.05.00
001
866.0866.00
5.05.00
001
)cos()cos(0
)sin()sin(0
)(0/0/3
11
11)(
A
P
A
J
J
The answer is similar to that obtained previously using another approach! (refer to the example on relative angular velocity)
Clarification
Why 0/0/3 P
Note: every point on the link will rotate at the same angular velocity! However, the linear velocities at different points on the link are not the same!
11 rv
r121 rv
r2
Getting the Angular Acceleration
n
A
n
A
z
y
x
P
n
AP JJJ
2
1
)(2
1
)(0/
2
1
)(
3
2
1
0/
If the joint angular acceleration for 1, 2, …, n are 0s then
n
A
z
y
x
P J
2
1
)(0/
Example: Getting the Angular Acceleration
Example: The 3 DOF RRR Robot:
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A=3 B=2 C=1
P
What is after 1 second if all the joints are rotating at
3,2,1,6
it
i
0/3
Getting the Angular Acceleration
001
)cos()cos(0
)sin()sin(0
11
11)(
AJ
000
)sin()sin(0
)cos()cos(0
1111
1111)(
AJ
All the joints angular acceleration for 1, 2, …, n are 0s:
0
2742.0
4749.0
5236.0
5236.0
5236.0
000
2618.02618.00
4534.04534.00
0/
z
y
x
P
The answer is similar to that obtained previously using another approach! (refer to the example on relative angular acceleration)
Transformation between Joint variables and the general motion of the last link
We can combine the Jacobians for the linear and angular velocities to get:
n
A
T
n
z
y
x
P
P
A
T
J
JJ
v
v
v
v
J
JJ
2
1
)(
)(2
1
3
2
10/
0/
)(
)(
Example: Transformation between Joint variables and the general motion of the last link
Example: The 3 DOF RRR Robot:
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A=3 B=2 C=1
P
What is the Jacobian for the 3 DOF RRR robot?
Example: Transformation between Joint variables and the general motion of the last link
001
)cos()cos(0
)sin()sin(0
11
11)(
AJ
)cos()cos()cos(0
)sin()sin()sin())sin()sin(()cos())cos()cos((
)sin()cos()cos())sin()sin(()sin())cos()cos((
32322
32113221322
32113221322)(
CCB
CCBCBA
CCBCBA
J T
001
)cos()cos(0
)sin()sin(0
)cos()cos()cos(0
)sin()sin()sin())sin()sin(()cos())cos()cos((
)sin()cos()cos())sin()sin(()sin())cos()cos((
11
11
32322
32113221322
32113221322
)(
)(
CCB
CCBCBA
CCBCBA
J
JJ
A
T
Jacobian and Singularities
n
T
n
n
n
n
z
y
x
P J
vvv
vvv
vvv
v
v
v
v
2
1
)(2
1
34
2
34
1
34
24
2
24
1
24
14
2
14
1
14
0/
We know that
z
y
x
n
n
n
PT
n
v
v
v
vvv
vvv
vvv
vJ
1
34
2
34
1
34
24
2
24
1
24
14
2
14
1
14
0/
1)(2
1
The above is true only if the Jacobian is invertible. From algebra, we now that a matrix cannot be inverted if its determinant is zero (i.e. the matrix is singular)
Example: Jacobian and Singularities
Example: The 3 DOF RRR Robot:
Y0, Y1
X0, X1
Z0, Z1
Z2
X2
Y2
Z3
X3
Y3
A=3 B=2 C=1
P
Investigate the singularities of the 3 DOF RRR robot
Example: Jacobian and Singularities
)cos()cos()cos(0
)sin()sin()sin())sin()sin(()cos())cos()cos((
)sin()cos()cos())sin()sin(()sin())cos()cos((
32322
32113221322
32113221322)(
CCB
CCBCBA
CCBCBA
J T
23232
231231211232
23232
231231211232
)(
)()(
)()()det(
CcCcBc
sCcsCcsBccCcBcA
CcCcBc
sCssCssBssCcBcAJ T
}{)(
}{)()det(
232312
2321232312
23211232
232312
2321232312
23211232)(
cscCscBCccscCcsBCccCcBcA
cssCscBCscssCCcsBssCcBcAJ T
}{)(
}{)()det(
232123211232
232123211232)(
scBCccsBCccCcBcA
scBCsCcsBssCcBcAJ T
Example: Jacobian and Singularities
}{)(
}{)()det(
232123211232
232123211232)(
scBCccsBCccCcBcA
scBCsCcsBssCcBcAJ T
}){()det( 23221232
21232
21232
21232
)( scBCccsBCcscBCsCcsBsCcBcAJ T
)()()det(
}){()det(
3232)(
232232232)(
sBCCcBcAJ
cBCssBCcCcBcAJT
T
0)det(0
0)det(0)()(
3
)(232
T
T
Js
JCcBcA
Under these two conditions, we cannot determine the joint angular velocities using the Jacobian
This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another.
The following were covered:•Robot instantaneous motion using joint variable interpolation•The Jacobian of a given robot•Robot singularity and its relation to Jacobian
Summary
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