Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSOutline:
5.1 INTRODUCTION5.2 NOTATION FOR TIME-VARYING POSITION AND ORIENTATION5.3 LINEAR AND ROTATIONAL VELOCITY OF RIGID BODIES5.4 MOTION OF THE LINKS OF A ROBOT5. 5 VELOCITY "PROPAGATION" FROM LINK TO LINK5.6 JACOBIANS5.7 SINGULARITIES5.8 STATIC FORCES IN MANIPULATORS5.9 JACOBIANS IN THE FORCE DOMAIN
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSIntroduction
• In this Chapter the following concepts will be studied: – The linear and angular velocity of a rigid body to analyze the
motion of the manipulator– Forces that act on a rigid body (application on static forces of a
manipulator) Jacobian matrix relations
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Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSIntroduction
• Velocity analysis between Cartesian Space and Joint Space
1
2
n
vJ
Jacobian MatrixTransformation/Mapping
1
12
n
vJ
If J is singular it is not invertible
Singular point/configuration(Important issue in robot design
that should be avoided)When θ2 = 0/180
singular configuration
The Jacobian is important to determine the torque in the joints needed to be applied to give a specific contact force at the End-Effector (Static problem)
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSNotations
• Differentiation of a position vector (Linear Velocity)
• ≡ the velocity of a position vector• ≡ Linear velocity of a point in space represented by the position vector. Derivative is made relative to frame {B} (Frame of Differentiation) Also the velocity vector is expressed in the same frame {B}
To express the velocity in any other frame {A}
≡ The velocity of point (position vector Q) relative the frame {B} expressed in frame {A}.
(draw frames for explanations) 4
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSNotations
• Differentiation of a position vector (Linear Velocity) – Often used the velocity of the origin of a frame {C} (for example)
relative to the universe frame {U}.
Remember that in the differentiation is made relative to {U} and expressed in {U}
– as v is small-letter the differentiation is made relative/ respect to {U}, but the velocity is expressed in {A}.
• Example:
OC origin of frame {C}UC OCv V
UOCV
ACv
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Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSNotations
• Example 5.1:
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Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSNotations
• The Angular Velocity Vector Angular velocity vector of a body ( is for a point)
As a frame represents the orientation of the body ≡ rotational velocity of the
frame. Rotational velocity of frame {B} relative to frame {A}:
- it’s direction represents the instantaneous axis of rotation of {B} relative to {A} - it’s magnitude represents the speed of rotation. the angular velocity of {B} relative to {A} expressed in
{C} In the universe frame {U}
ACv
AB
C AB
UC C 7
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSLinear and rotational velocities of rigid bodies
1. Linear velocity: (Translation only)
– Translation only the orientation of {B} relative to {A} is not changing
8
BQ
,A A A B
A B BQ P R Q
A A A BQ OB B QV V R V
,A
A BP
AQ
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSLinear and rotational velocities of rigid bodies
2. Rotational velocity: (Rotation only)Two frames {A} and {B} have the same origin for all the time only the relative orientation is changing in time
is fixed in {B}
What is the velocity of point Q in {A}
If is changing in {B}
AB
BQ
0BQV
9
A A AQ B
A A BB B
V Q
R Q
BQ
?AQV
AA A A B BQ B B Q
A A B A BB B B Q
V R Q V
R Q R V
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSLinear and rotational velocities of rigid bodies
3. Linear and rotational velocity at the same timeThe same as point 2, however frame {B} is moving relative to frame {A}, i.e. A
OBV
10
A A A A B A BQ OB B B B QV V R Q R V
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSMotion of the links of a robot (notations)
≡ linear velocity of the origin of frame {i} relative to the reference frame {O} (fixed), expressed in {i}
≡ angular velocity of frame {i} relative to the reference frame {O} (fixed), expressed in {i}
ii
iiv
iiv
iO
ii
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSVelocity propagation from link to link
ii
11
iiv
iiv
11
ii
Starting from the base the velocity of any link (i+1) equal to the previous link (i) + the relativeVelocity between (i+1) and (i)• Angular velocity propagation
Note that: is the third column of
1 1
11 1 1 1 1 1
i i ii i i
i i i ii i i i i iz R z
11
0
0
1
iiz
11 1 1 1
1 1 1 11 1 1 10 0 1
i i i ii i i i i
Ti i i i i ii i i i i i i i
R z
R z R
11 1i i
i iR z 1i
i R
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSVelocity propagation from link to link
ii
11
iiv
iiv
11
ii
• Linear velocity propagation
A ≡ 0; Q ≡ i+1; B ≡ i;
A A A A B A BQ OB B B B QV V R Q R V
0 0 0 01 , 1
01 , 1
1 , 1
1 11 , 1
ii i i i i i
ii i i i i i
i i i ii i i i i
i i i i ii i i i i i
V V R P
v v R P
v v P
v R v P
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSVelocity propagation from link to link
• Prismatic joint
A ≡ 0; Q ≡ i+1; B ≡ i;
0 0 11 , 1 1 1 1
11 , 1 1 1 1
1 11 , 1 10 0
i ii i i i i i i i i
i i i i i ii i i i i i i i
Ti i i i ii i i i i i i
v v R P R z d
v v P R z d
v R v P d
A A A A B A BQ OB B B B QV V R Q R V
≠ 0
0 0 0 11 1 1 1 1 1
A B i i iB Q i i i i i i i iR V R V R z d R z d
1
1 11
i ii i
i i ii i iR
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSVelocity propagation from link to link
2Z
3Z
2X
3X
1 0ˆ ˆ,X X
0 1 2, ,O O O
3O
1 0ˆ ˆ,Z Z
1 0ˆ ˆ,Y Y
2Y3Y
three • Example
Chapter 5: Jacobians: Velocities and Static Forces
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ROBOTICS
• Example (Solution)From the following can be extracted:1
ii T
i ai-1 αi-1 θi di
1 0 0 θ1 0
2 0 90 θ2 0
3 a2= Const. 90 0 d3
Velocity propagation from link to link
1 1
1 101
0 0
0 0
0 0 1 0
0 0 0 1
c s
s cT
2 21 12 2
2 2
0 0
0 0 1 , 0
0 0
c s
R P
s c
22 23 3 3
1 0 0
0 0 1 ,
0 1 0 0
a
R P d
2 2 2 11 1 0 2
1 0 0 2 2 2 2 1
1 1 1 2 2
0 0 0 0 0
0 0 ; 0 0 0
0 1 0
c s s
R s c c
3 3 23 2 2R
1 13
3 2
1 1
s
c
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
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• Example (Solution)Velocity propagation from link to link
2 1 2 11 2 3
1 2 2 1 3 2
1 2 2 1
0
0 ;
s s
c
c
1 1 0 0 01 0 0 0 0,1
22
2 1 23 3 2 2 2
3 2 2 2 2,3 2 1 3
3 2 3
0
0 ;
0
0
0 ;
0
0 1 0 0 0
0 0 0 1 0
0 1 0 0
v R v P
v
s a
v R v P c d
d d
2 3 2 33
3 2 3 2 1 3 2 1 2
2 1 3 2 1 2 3 2 3 3
1 0 0 0
0 0 1 0
0 1 0
d d
v a s d c a
s d c a d a d
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
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• Example (Solution)
– Expressed in {0} frame
Velocity propagation from link to link
2 13
3 2
2 1
s
c
2 33
3 2 1 3 2 1 2
2 3 3
d
v s d c a
a d
0 0 33 3 3R
0 0 33 3 3v R v End of the solution
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
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• Example (Solution)– One can put in the following form
Velocity propagation from link to link2 1
33 2
2 1
s
c
3 3
3 3 and v
2 33
3 2 1 3 2 1 2
2 3 3
d
v s d c a
a d
3
2 3 2 213
3323
233
2
0 0
0 0
0 1
0 0
0 1 0
0 0
d
s d c a
av
sd
c
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSJacobians
• If we have the function:
• Or, in vector form • The differentiation of yi can be considered as follows:
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSJacobians
• Or, simply, in matrix form • Remark: for nonlinear functions f1, f2, …, fn of X the partial
derivative is also a function of X
Jacobian maps/transform velocities from X to YAs X=X(t) (time dependent) J=J(X) is also time dependent.
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( )
( )
( )
( )
FJ X
XdY J X dX
dY dXJ X
dt dt
Y J X X
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSJacobians
• In robotics, the Jacobian relates the Cartesian velocities with joint velocities
≡ vector of joint angles ≡ vector of joint velocities
• General Case:
Instantaneous, as is changing J( ) is changing
0 06 1 6 6 6 1
00
6 1 0
( )J
v
11 12 1
21 22 20
1 2
( )
n
n
m m mn
J J J
J J JJ
J J J
# of columns = # of DoF
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
3
2 3 2 213
3323
233
2
0 0
0 0
0 1
0 0
0 1 0
0 0
d
s d c a
av
sd
c
Jacobians• For the previous equation find
3 ( )J
3 3 ( )J
3
2 3 2 2
33
2
2
0 0
0 0
0 1( )
0 0
0 1 0
0 0
d
s d c a
aJ
s
c
See the example of the book
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• Given , find
is invertible if• For some values of , could be non-invertible
singularities of the robot
• Singularity:– Boundary singularity: – Interior singularity
0 1 0( )J 0
0J 0det ( ) 0J 0J
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• Boundary singularity: All robots have singularities at the boundary or their workspace.
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• Interior singularity– Inside the workspace, away from the WS boundary. – Generally are caused by a lining up of two or more joint axes.
When a manipulator is in a singular configuration, it has lost one or more degrees of freedom!
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• Example: the Puma robot has two singular configurations (not only)
– 3 -90 (Boundary)
– 5 0 (Interior)
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICS
3
2 3 2 2 3 2 3 2 2
3
0 0
det 0 0 ( )
0 1
d
s d c a d s d c a
a
Singularities• Example: determine the
singular configurations for the robot of the previous example basingon its Jacobian that relates
Solution
3 13
3 2 3 2 2 2
3 3
0 0
0 0
0 1
d
v s d c a
a d
33 , with v
22
3
tan( ) Interior!a
d
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• See example 5.4 of the book
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSSingularities
• For the previous example if 33 2 1 5 , find
Tv
2 3 2 2
3
3
3
13
3 3 1 12 3 2 2
3
0 0 0 0
( ) 0 0 ( ) 0 0
0 1 0 1
s d c a
d
ad
d
J s d c a J
a
2 3 2 2 2 3 2 2
3 3
3 3
3 3
1 11
1 22
23
0 0 2
0 0 1
0 1 5 5
s d c a s d c a
d d
a ad dd
If we try to calculate at singular configuration
this value tends to ∞
1
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Discuss the concept of force and moment propagation– Starting from the last link to the base– Force (equilibrium)
Use the notation: ≡ force exerted by link i on link i+1 = regardless to the frame of notationall vectors must be expressed in thesame frame.
Different notationis used here
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
– Moment equation (equilibrium) Use the notation: ≡ Moment exerted by link i on link i+1 =
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Different notationis used here
Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Relation between joint [torque (motor torque) for revolute joint, or joint actuating force (linear actuator force)] and joint reactions:
Joint i is revolute– The structure of the joint hold the force does not affect joint
torque– Only the component of in the direction of affect joint torque,
i.e.:
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Chapter 5: Jacobians: Velocities and Static Forces
Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Relation between joint [torque (motor torque) for revolute joint, or joint actuating force (linear actuator force)] and joint reactions:
Joint i is prismatic– The structure of the joint holds the moment, does not affect joint
force– Only the component of in the direction of affect joint force, i.e.:
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Principle of Virtual Work!
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Example: if a force of acts at the O3, find joint torques and forces
• Solve it!
❑3F1 101 1 1
0
0
0 0 1
c s
R s c
2 21 12 2
2 2
0 0
0 0 1 , 0
0 0
c s
R P
s c
22 23 3 3
1 0 0
0 0 1 ,
0 1 0 0
a
R P d
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Recall:3
2 3 2 21 13
3 332 23
233 3
2
1 33
2 3
3
0 0
0 0
0 1
0 0
0 1 0
0 0
!!!T
d
s d c a
avJ
sd d
c
FJ
N
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Recall:3
2 3 2 21 13
3 332 23
233 3
2
1 33
2 3
3
0 0
0 0
0 1
0 0
0 1 0
0 0
!!!T
d
s d c a
avJ
sd d
c
FJ
N
of O3
on O3
For the same point
Chapter 5: Jacobians: Velocities and Static Forces
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Faculty of Engineering - Mechanical Engineering Department
ROBOTICSFORCES IN MANIPULATORS
• Recall:3
2 3 2 21 13
3 332 23
233 3
2
1 33
2 3
3
0 0
0 0
0 1
0 0
0 1 0
0 0
!!!T
d
s d c a
avJ
sd d
c
FJ
N
Of body 3
on body 3
For the same point
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