Chapter 5: Jacobians: Velocities and Static Forces Faculty of Engineering - Mechanical Engineering...

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



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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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)



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



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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ROBOTICSNotations

• Example 5.1:

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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



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




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

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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




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


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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


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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


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ii

11

iiv

iiv

11

ii

• Linear velocity propagation

A ≡ 0; Q ≡ i+1; B ≡ i;


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


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• 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


≠ 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


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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


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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



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



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



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


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ROBOTICSJacobians

• If we have the function:

• Or, in vector form • The differentiation of yi can be considered as follows:



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


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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


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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


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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


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• Boundary singularity: All robots have singularities at the boundary or their workspace.


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• 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!


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• Example: the Puma robot has two singular configurations (not only)

– 3 -90 (Boundary)

– 5 0 (Interior)


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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


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• See example 5.4 of the book


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• 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 ∞

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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




– Moment equation (equilibrium) Use the notation: ≡ Moment exerted by link i on link i+1 =

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Different notationis used here




• 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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• 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!


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• 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


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• 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


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• 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


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• 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

Chapter 5: Jacobians: Velocities and Static Forces Faculty of Engineering - Mechanical Engineering...

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Transcript of Chapter 5: Jacobians: Velocities and Static Forces Faculty of Engineering - Mechanical Engineering...