Jacobian: Velocities and Static Forces
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Jacobian: Velocities and Static Forces
Transcript of Jacobian: Velocities and Static Forces
Jacobian: Velocities and Static Forces
What is Jacobian?
• Jacobian estabilishes – the relation between the joint velocities and
the velocity of the end-effector– the relation between the joint torque and the
force acting at the end-effector
Linear Velocity• Linear velocity of an object is the rate of
change of its position.
r C
O
x
y
z
objectxo
yo
zo
rdtdv
r
given isobject theofocity linear vel The
frame reference fixed a w.r.t object theofector position v the:
The linear velocity is a Vector!!
Angular Velocity• Angular velocity: Angular velocity of an object
describes the rate of change of its orientation.
axis-z about thelocity angular ve :axis-y about thelocity angular ve :axis- xabout thelocity angular ve :
z
yx
z
yx
More on Angular Velocity
x
y
z
00
u
0
0
00
x
y
z
x
y
z
x
y
z
runit vecto :u
More on Angular Velocity
bu
runit vecto :b
x
y
z
runit vecto :u
When an object rotates about two axes simultaneously
Angular velocity is the combination of the angular velocities due to the two rotations
Joint Velocity and Velocity of End-Effector
• The positions of the joints of a manipulator are represented by a n-dimensional vector , where n is the number of the joints.
• The joint velocity of the manipulator is given by • Denote the position and orientation of the end-
effector by – is 3 dimensional vector in planar cases (two
positions + one orientation) – is 6 dimensional vector in 3D space (3 positions + 3
orientation) • The velocity of the end-effector is given by
x
x
x
x
Jacobian• Forward kinematics:
• Differentiating the forward kinematics
function vector 16or 13 is where,)( ffx
)(
1
21
2
2
2
1
2
1
2
1
1
1
)(...
)()(............
)(...)()(
)(...)()(
)(
:matrix following theis )( where,)(
J
n
mmm
n
n
fff
fff
fff
f
nmffx
Jacobian matrix ofthe manipulator
Jacobian (cont’)
• The Jacobian matrix relates the joint velocity of a manipulator to the velocity of its end-effector
• The Jacobian matrix establishes the relation of differential motion between the joints and the end-effector.
joints theofmotion aldifferenti :effector-end theofmotion aldifferenti :
)( )(
xJxJx
)(Jx
x
12
Example
Example: Find the Jacobian matrix of a 3 DOF planar arm.
x0
y0
l1
(x,y)
l2
2
3
1
l3
yx
x
:endpoint theofn orientatio andPosition
:anglesJoint 321
321123312211123312211
slslslyclclclx
111
)( 1233123312212331221112331233122123312211
321
321
321
clclclclclclslslslslslslyyy
xxx
J
General Method for Jacobian Calculation
1. Solve the forward kinematics of the robot manipulator
2. Define the linear and angular velocity of the end-effector
3. The Jacobian matrix has the following form
10
)()(31
0
pRT effectorend
vx
)(
)(
B
px
• Consider the rotation (angular velocity) of the end-effector due to motion of joint i.
• Case A: Joint is a prismatic joint– When all the other joints do not
move, the motion of joint i does not cause rotation of the end-effector.
– Therefore, the angular velocity due to joint i is zero
?)( Calculate toHow
B
i.joint todueeffector -end theoflocity angular ve :
0i
effectorend
ieffectorend
Joint i
• Case B: Joint i is revolute– When all the other joints do not
move, the motion of joint i will cause rotation of the end-effector.
– Therefore, the angular velocity due to joint i is as follows
?)( Calculate toHow
B
i.joint todueeffector -end theoflocity angular ve :
0
ieffectorend
iii
effectorend z
Joint i
zi
i
Jacobian• The angular velocity of the end-effector is
given by
• Therefore, the Jacobian matrix is as follows:
n
iiii z
1 ii0
revolute is ijoint If 1prismatic is ijoint If 0
nn
nzzz
pppJ
02
021
01
21...
)(...)()()(
Example• A 3-DOF arm
0 l 0 0 4 0 0 0 3
0 d 0 90 2 0 0 0 1
33
2
1i11
ii-i dai
x1z1
y1 z2
x2
y2
z0
x0y0
z3
x3y3
ze
xeye
End-effector frame
100001000000
1111
10 cs
scT
100001000000
3333
32 cs
scT
10000010
1000001
22
1 dT
1000100
00100001
34
3lT
l3: distance between x3 and x4
Example• Forward kinematics
100001000000
1111
10 cs
scT
10000010
00
12111211
21
10
20 cdcs
sdscTTT
10000033
12131311213131
32
20
30
cscdcsscs
sdsscccTTT
100000
)()(
3313213131
13213131
43
300
cscldcsscssldssccc
TTT effectorend
10 z 2
0 z
30 z
p
position
Example• Then, the Jacobian is as follows:
0010000
0000)(0)(
)()()(
)()()(
)()()(
)(
1
1
1123
1123
30
320
210
1
321
321
321
cs
csdlscdl
zzz
pd
pp
pd
pp
pd
pp
J
zzz
yyy
xxx
Singularities of Jacobian• Velocity relationship
• Given a joint velocity, it is always possible to calculate the velocity of the end-effector.
• Question: Given a velocity of the end-effector, can we always calculate the joint velocity?– When n=m
)(Jx
n: the number of jointsm: the number of degrees of freedom
of the end-effector
calculated becannot exist,not does )( If)( exists, )( If
1
11
J
xJJ
Singular configurations of a manipulator are those configurations where the inverse of its Jacobain does not exist
Example of Singularities• For a 2 DOF planar arm
1212112121
2
)()()(
0When
clcllslsllJ
x0
l1
l2
2
1
x0
l1
l2
02
1
1221221112212211)(
Jacobian
clclclslslslJ
singular
The solution of the joint velocity is not unique
Redundant Manipulators• When n>m: There are more joints than the DOF of the
end-effector. In this case, the manipulator is a redundant manipulator
uniquely thesolvecannot we,)( From Jx
)( of space null in the vector :))()(( vectorarbitrary an is
))()()(()( where
))()(()(
1
JkJJIk
JJJJ
kJJIxJ
TT
0)(: :)( of space Null kJkJ
Equivalence of Forces• Consider equivalence of forces acting on the
end-effector and torque at the joints.– Equivalence: If two set of forces (torques) cause the
same motion to the manipulator, they are equivalent.• Question: Suppose that a force acting on the
end-effector. What is the equivalent torque acting at the joints of the force?
x0
l1
l2
2
1
f
x0
l1
l2
2
1
Equivalence of Forces• Equivalence of forces cause the same motion
the works done by the equivalent forces must be equal• Consider works done by the force and torque under
differential motion.
)(x :effector-end at the xmotion different the
joints, at the motion aldifferenti aFor
J
x
12
)( :fby done work TheJfxf TT
Tn
:,..., qejoint toruby done work The T
21
Equivalence of Forces• Since the works done must be equal
fJ
fJJf
T
TTT
TT
)(
))(( )(
This represents the relation between the force acting the end-effector and the torque acting at the joints
