robotics unit5 by mahendra babu of pbrvits

8/3/2019 robotics unit5 by mahendra babu of pbrvits

1/45

JacobianMethod of controlling the joints of the

manipulator in a co-ordinated fashion.

Relates the rates of the variables in one co-

ordinate system to those in another co-ordinate

system.

It allows the computation of differential

change in the tool co-ordinate frame due to the

differential change in the position of joint

variables.


2/45

Manipulator Jacobian

Matrix of differentials Describe the motion of the tool in terms

of changes in the joints

Jacobian calculated by differentiatingthe Forward Kinematic transform

Cartesian

Velocities

Joint

Velocities

ddx J

dxd1

J


3/45

Example

If a manipulator consists of n rotary joints(DOF) and

position and orientationof its tool point (Xj) at anylocation is represented in j number of co-ordinates.

Xj=f (1, 2, 3. j)

Taking the total differential

in

i

jij

i

n

i i

n

j

n

n

j

JX

fX

fffX

*

1

*

*

1

11*

*1

2

*

2

11

*

1

1*

..............

..............

Where

[Jji]-is a Jacobian


4/45

If j=n=6

Xj=f ( 1, 2, 3, 4, 5, 6)

X1=f ( 1, 2, 3, 4, 5, 6) X2=f ( 1, 2, 3, 4, 5, 6)

X3=f ( 1, 2, 3, 4, 5, 6) X4=f ( 1, 2, 3, 4, 5, 6)

X5=f ( 1, 2, 3, 4, 5, 6) X6=f ( 1, 2, 3, 4, 5, 6)

JACOBIAN

6

*5

*4

*3

*2

*1

*

1

3

1

2

3

1

2

1

1

1

*

6

*

5

*

4

*

3

*

2

1

*

.

......

......

......

.....

.....

...

f

f

fff

z

y

x

dz

dy

dx

X

X

X

X

XX


5/45

The jacobian (6X6 matrix) relates the hand

velocities to the joint velocities.

6

*

5

*4

*

3

*2

*

1

*

*

6

*

5

*

4

*

3

*

2

1

*

z

y

x

dz

dy

dx

X

X

X

X

X

X

Jacobian is a time varying quantity.

Jacobian matrix may not be a square matrix.


6/45

Example (cylindrical robot

)

r

z

(x, y, z)

x= r cos

y= r sin

z=z

dx= cos.dr - r sin.d

dy= sin.dr - r cos.d

dz=dz


7/45

Jacobian

dx = cos - rsin 0 drdy = sin - rcos 0 ddz = 0 0 1 dz

Jacobian


8/45

dr = cos sin 0 dx

d = -sin/r cos/r 0 dydz = 0 0 1 dz

Inverse of Jacobian

Inverse of Jacobian


9/45

Inverse jacobian

as function of x, y, z

dr = x/r y/r 0 dxd = -y/r2 x/r2 0 dydz = 0 0 1 dz

Inverse of Jacobian


10/45

DYNAMICS

Mathematical formulation of motion

equations Useful in motion simulation.

Useful in the design of control equations.

Useful to evaluate the kinematicstructure of robot arm.


11/45

Kinematics

Control Kinematics is the first step towards

robotic control.

Cartesian Space Joint Space Actuator Space

zy

x


12/45

The Robot System

Control System

Sensors

Kinematics

Dynamics

Task Planning

Software

Hardware

Mechanical Design

Actuators


13/45

DYNAMICS The dynamic model of the arm obtained

by 2 physical laws.

Laws of NEWTONIAN mechanics

Laws of LEGRANGIAN mechanics


14/45

DYNAMICS Motion equations of arm are obtained from

LagrangeEuler Equations.

Newton Euler Equations.

Generalised DAlemberts Equations.


15/45

DYNAMICS Forward Dynamics

Inverse Dynamics

Forces(or)Torques

Accelerationsof joints

Velocitiesof joints

Velocitiesof joints

Accelerationsof joints

Forces(or)

Torques


16/45

Lagrange Euler Equations

ni

T

q

L

q

L

dt

di

ii

.......,.........3,2,1

**

L Lagrangian function = KE PE

qi Generalised co-ordinate

Tigeneralised force/Torque applied to the system at joint i to drive thelink.

In case of Rotary Joint qi = i

Prismatic Joint qi = di


17/45

Lagrange Euler Equations

y0

x0

z0

iri

Ti

rzyx

z

y

x

i 1

1

iri be the point fixed and atrest in linkiand expressed inhomogeneous co-ordinatesw.r.to the ith frame


18/45

0ri be the same point irifixed and at rest in linki and expressed in homogeneous co-ordinatesw.r.to the base frame

0ri=0Ai

iri

Where 0Ai= 0A11A22A3..(i-1)Ai

1000

cossin0

sinsincoscoscossin

cossinsincossincos

),(),0,0()0,0,(),(

)1(

1)1(

iii

iiiiiii

iiiiiii

ii

n

iiiiiii

d

a

a

A

ZRotdTransaTransXRotA


19/45

0Vi=

i

I

i

j

i

I

I

AQq

A

Q

11

0000

0000

0001

0010

The partial derivativeof0Ai w.r.to. qjCan becalculated by matrix Qi

For revolute joint

For prismatic joint

Velocity of joint

0000

0000

0001

0010

0

1

*

1

*0

00

I

j

iij

i

j i

i

jIJ

i

j i

i

j

j

i

i

i

ii

Q

q

AWhereU

rqU

rqq

A

rAdt

dr

dt

d


20/45

Kinetic Energy

rp

T

ir

n

i

i

p

i

r iipi

rp

T

ir

i

p

T

i

i

i

i

i

r

ipii

T

iii

iiii

qqUJUK

ICENERGYtotalKINET

qqUrdmrUdKK

dmVVdK

dmzyxdK

**

1 1 1

**

1 1

2

*

2

*

2

*

..

2

1

....2

1

*

2

1

21

Ji


21/45

Potential Energy

n

i i

i

ii

iii

rAgmP

ENERGYTIALtotalPOTEN

ni

rgmP

1

0

0

..

..........1,0

..


22/45

n

i ii

iirpT

ir

n

i

i

p

i

r iipi rAgmqqUJUL

PEKEFunctionLLegrangian

10

**

1 1 1 ....2

1

n

i i

i

ii

T

ji

n

ij

j

k

j

m jjkmk

T

ji

n

ij

j

k jjkirAgmUJUqUJUT

1

0

1 1

**

1...

Lagrangian Formulation

ni

T

q

L

q

L

dt

di

ii

.......,.........3,2,1

**

Torque


23/45

Matrix form ofLagrangian Formulation

T(t)= D[q(t)] q(t) + h[q(t), q(t)] + C[q(t)]

ni

CqqhqDT in

k

n

k

n

m mkikmkiki

.............................................................2,1

1 1 1

****


24/45


Newtons Equation:Force = mass X acceleration

Euler Equation

ocityAngularVel

sorInertiaTenIIIN

c

cc

2*

.moment


25/45

Newton-Euler equations

IIrp

Irp

I

lnmf

nf

lm

m

EquationsEuler'EquationNewtons

TorqueNetForceNet

MomentumAngularMomentumLinear

InertiaMass


26/45

Moment of Inertia- measure of massdistribution of a rotated body with 1 DOF

Inertia Tensor - measure of massdistribution of a rotated body with 3 DOF

Inertia Tensor - generalization of scalarmoment of inertia.

Inertia Tensor


27/45

Inertia Tensor

dmyxyzdmxzdm

yzdmdmzxxydm

xzdmxydmdmzy

I

)(

)(

)(

22

22

22

dmzyIxx )(22I: Inertia Tensor:

Diagonal terms :moments of inertia

Off-diagonal terms :

products of inertia

dmxyIxy )(


28/45

To calculate the joint torques requiredto cause the motion the following

parameters are required.

Position

Velocity

Acceleration

Mass distribution of robot arm



29/45

Outward Iterations to computevelocities and accelerations

Rotational velocity

Linear & rotational acceleration of centre ofmass of each link.



30/45

Outward Iterations

Rotational velocity

)1int(

)1(

)(..)1(

1(

*

1

)1(

1

1)1(

*

)1(1

1

iocityOfJoangularvel

ieaxisOfFramZZ

iToRWiameVectorOfFrRotationalR

ocityangularvelZR

i

i

ii

i

ii

i

i

iii

i


31/45

i

i

iiii

i

iii

iii

i

iii

i

iiii

R

CisPRISMATIiJoif

ZZRR

ALisROTATIONiJof

*

)1(1

*1

1

1)1(

**

1

1)1(

*

)1(

*

)1(1

*1

)1int(

)1int(

Outward Iterations

Angular acceleration of joint (i+1)


32/45

Robot Motion Planning

Path planning

Geometric path

Issues: obstacle avoidance, shortest

path

Trajectory planning,

interpolate or approximate thedesired path by a class of polynomial

functions and generates a sequenceof time-based control set points forthe control of manipulator from theinitial configuration to its destination.

Task Plan

Action Plan

Path Plan

TrajectoryPlan

Controller

Sensor

Robot

Tasks


33/45

Path of an object

= curve in the configuration space

represented either by:

Mathematical expression, or

Sequence of points

Trajectory

= Path + assignment oftime to points along the path

Motion Planning (MP), a general term, either:

Path planning, or

Trajectory planning


34/45

Path planning

design of only geometric (kinematic) specifications of

the positions and orientations of robots

Trajectory planning

path planning + design oflinear and angularvelocities

Path planning < Trajectory planning

at path planning, dynamics of robots unimportant orneglected

path planning also used as first step in design of trajectories


35/45

Trajectory/Path

The space curve along which themanipulator moves from initial location

to final destination.

To plan the trajectory one must satisfy

the following two constraints Obstacle constraint

Path constraint


36/45

Controlling the manipulator (so that itfollows the pre planned path) is divided

into two sub problems.

Trajectory Planning

Motion control

Trajectory Planning


37/45

Trajectory Planning

Trajectory Planning methods generally

interpolate (or) approximate the desired

path by a class of polynomial and

generates a sequence of time based

control set points for the control of the

manipulator from initial location to final

destination.


38/45

Trajectory Planning

TrajectoryPlanning

ManipulatorDynamic

constraints

Pathconstraints

Pathspecifications

*

)(tq )(**

tq)(tq

Trajectory planning


39/45

Trajectory planning

Path Profile

Velocity Profile

Acceleration Profile

t0 t1 t2 tf Time

q(t0)

q(t1)

q(t2)q(tf)

Initial

Lift-off

Set down

Final

Joint i

t0 t1 t2 tf Time

Speed

t0 t1 t2 tf Time

Acceleration

Th b d di i


40/45

The boundary conditions1) Initial position

2) Initial velocity3) Initial acceleration4) Lift-off position5) Continuity in position at t16) Continuity in velocity at t17) Continuity in acceleration at t18) Set-down position9) Continuity in position at t210) Continuity in velocity at t2

11) Continuity in acceleration at t212) Final position13) Final velocity14) Final acceleration

R i


41/45

Requirements

Initial Position

Position (given)

Velocity (given, normally zero)

Acceleration (given, normally zero)

Final Position

Position (given)

Velocity (given, normally zero)

Acceleration (given, normally zero)


42/45

Requirements

Intermediate positions

set-down position (given)

set-down position (continuous withprevious trajectory segment)

Velocity (continuous with previous

trajectory segment)Acceleration (continuous with previous

trajectory segment)


43/45

Requirements

Intermediate positions

Lift-off position (given)

Lift-off position (continuous with previoustrajectory segment)

Velocity (continuous with previous

trajectory segment)Acceleration (continuous with previous

trajectory segment)


44/45

Trajectory Planning

n-th order polynomial, must satisfy 14conditions,

13-th order polynomial

4-3-4 trajectory

3-5-3 trajectory

001

2

2

13

13 atatata

02

2

2

3

3

4

4

2021

2

22

3

232

1012

2

12

3

13

4

141

)(

)(

)(

nnnnnn atatatatath

atatatath

atatatatath

t0t1, 5 unknow

t1t2, 4 unknow

t2tf, 5 unknow


45/45

ROBOT PROGRAMMING

Robots need to be taught what theyare expected to do and how they

should do it. The teachjing of the workcycle to a robot is known as robotprogramming.

Interface the control system to externalsensors

robotics unit5 by mahendra babu of pbrvits

Documents