Robot Trajectory Planning

7/24/2019 Robot Trajectory Planning

http://slidepdf.com/reader/full/robot-trajectory-planning 1/15

Chapter 5Trajectory Planning

5.1 INTRODUCTION

In this chapters…….

• Path and trajectory planning means the ay that a ro!ot is

mo"ed

#rom one location to another in a controlled manner.

• The se$%ence o# mo"ements #or a controlled mo"ement

!eteenmotion segment& in straight'line motion or in se$%ential

motions.

• It re$%ires the %se o# !oth (inematics and dynamics o#

ro!ots.




5.) P*T+ ,-. TR*/CTOR0

• Path * se$%ence o# ro!ot con2g%rations in a partic%larorder

itho%t regard to the timing o# these con2g%rations.

• Trajectory It concerned a!o%t hen each part o# the pathm%st !e attained& th%s speci#ying timing.

3ig. 5.1 -e$%ential ro!ot mo"ements in apath.




5.4 OINT'-P*C/ ,-. C*RT/-I*N'-P*C/ D/-CRIPTION-

• oint'space description' The description o# the motion to !e made !y the ro!ot !y its joint"al%es. ' The motion !eteen the to points is %npredicta!le.

• Cartesian space description' The motion !eteen the to points is (non at all times andcontrolla!le. ' It is easy to "is%alie the trajectory& !%t is is di6c%lt to ens%rethat

sing%larity.

3ig. 5.) -e$%ential motions o# aro!ot

3ig. 5.4 Cartesian'space trajectory 7a8 The trajectoryspeci2ed in Cartesian coordinates may #orce the ro!ot to r%ninto itsel#& and 7!8 the trajectory may re$%ires a s%dden




5.9 :*-IC- O3 TR*/CTOR0 P;*NNIN<

• ;et=s consider a simple ) degree o# #reedom ro!ot.

• >e desire to mo"e the ro!ot #rom Point * to Point :.

• ;et=s ass%me that !oth joints o# the ro!ot can mo"e at the

ma?im%m rate o# 1@ degreeAsec.

• ;et=s ass%me that !oth joints o# the ro!ot can mo"e at the

ma?im%m

rate o# 1@ degreeAsec.

3ig. 5.9 oint'space nonnormaliedmo"ements

• Move the robot from A to B, to run both jointsat their maximum angular velocities.

• After 2 [sec], the lower link will have finished itsmotion, while the uer link continues for another! [sec].

• "he ath is irregular and the distances traveledb# the robot$s end are not uniform.





3ig. 5.5 oint'space& normalied

mo"ementso# a ro!ot ith to degrees o##reedom.

• Both joints move at different seeds, but move

continuousl# together.

• "he resulting trajector# will be different.

• ;et=s ass%me that the motions o# !oth joints are normalied

!y a

common #actor s%ch that the joint ith smaller motion ill

mo"eproportionally sloer and the !oth joints ill start and stop

their

motion sim%ltaneo%sly.





3ig. 5.B Cartesian'space mo"ements

o# a to'degree'o#'#reedomro!ot.

•

%ivide the line into five segments and solvefornecessar# angles and at each oint.

• "he joint angles are not uniforml# changing.

• ;et=s ass%me that the ro!ot=s hand #ollo a (non path

!eteen point

* to : ith straight line.• The simplest sol%tion o%ld !e to dra a line !eteen points

* and :&so called interpolation.





3ig. 5. Trajectory planning ith an acceleration'deceleration

regiment.

• &t is assumed that the robot$s actuators arestrong enough to rovide large forces necessar#

to accelerate and decelerate the joints as needed.

• %ivide the segments differentl#.

• "he arm move at smaller segments as we seed u at

the beginning.• <o at a constant cr%ising rate.

• Decelerate ith smaller segments asapproaching point :.

• ;et=s ass%me that the ro!ot=s hand #ollo a (non path !eteen point * to : ith

straight line.• The simplest sol%tion o%ld !e to dra a line !eteen points * and :& so called

interpolation.





3ig. 5. :lending o# diEerent motion segments ina path.

• Blend the two ortions of the motion at oint B.

• Ne?t le"el o# trajectory planning is !eteen m%ltiple points #or

contin%o%s mo"ements.

• -top'and'go motion create jer(y motions ith %nnecessary stop

3ig. 5.F *n alternati"e scheme #or ens%ring that thero!ot ill go thro%gh a speci2ed point d%ring !lending o#motion segments. To "ia points D and / are pic(ed s%chthat point : ill #all on the straight'line section o# the

segment ens%ring that the ro!ot ill pass thro%gh point:.

•

'ecif# two via oint % and ( before and after oint B




5.5 OINT'-P*C/ TR*/CTOR0 P;*NNIN<

• +o the motions o# a ro!ot can !e planned in joint'space ith

controlled characteristics.

• Polynomials o# diEerent orders• ;inear #%nctions ith para!olic !lends

5.5.1 Third'Order Polynomial Trajectory Planning

• "he initial location and orientation of the robot is known, and using the inversekinematic e)uations, we find the final joint angles for the desired osition and

orientation.3

32

210)( t ct ct cct +++=θ

iit θ θ =)(

f f t θ θ =)(

( ) 0it θ =G

0)( = f t θ

•

&nitial *ondition

2321 32)( t ct cct ++=θ

• +irst derivative of theol#nomial of e)uation

ii ct θ θ == 0

)(3

32

210)( f f f f t ct ct cct +++=θ

1( ) 0it cθ = =G

032)( 2321 =++= f f f t ct cct θ

•

'ubstituting the initialand final conditions



Example

It is desired to have the first joint of a six-axis robot go from initial angle of 30 o to

a final angle of 75o in 5 seconds. Using a third-order polynomial calc!late the

joint angle at " # 3 and $ seconds.

2 30 1 2 3( )t c c t c t c t θ = + + +

0(0) 30cθ = =

1(0) 0cθ = =G





• -peci#y the initial and ending accelerations #or a segment.

• To %se a 2#th'order polynomial #or planning a trajectory& the

totaln%m!er o# !o%ndary conditions is B.

5.5.) 3i#th'Order Polynomial Trajectory Planning

• *alculation of the coefficients of a fifthorder ol#nomial with osition, velocit# and a acceleration boundar# conditions can be ossible with

below e)uations.

5

5

4

4

3

3

2

210)( t ct ct ct ct cct +++++=θ 2

321 32)( t ct cct ++=θ

35

2432 201262)( t ct ct cct +++=θ





• ;inear segment can !e !lended ith para!olic sections at the

!eginning and the end o# the motion segment& creating

contin%o%s position and "elocity.• *cceleration is constant #or the para!olic sections& yielding a

contin%o%s

"elocity at the common points * and :.

5.5.4 ;inear -egments ith Para!olic :lends

3ig. 5.14 -cheme #or linear segments ith para!olic!lends.

2210

21)( t ct cct ++=θ

t cct 21)( +=θ

2)( ct =θ

22

21)( t ct i += θ θ

t ct 2)( =θ

2)( ct =θ





• The position o# the ro!ot at time t @ is (non and %sing the

in"erse(inematic e$%ations o# the ro!ot& the joint angles at "ia points

and at

the end o# the motion can !e #o%nd.• To !lend the motion segments together& the !o%ndary

conditions o#

each point to calc%late the coe6cients o# the para!olic

segments is

%sed.

5.5.9 ;inear -egments ith Para!olic :lends and ,ia Points

• Ha?im%m alloa!le accelerations sho%ld not !e

e?ceeded.





• Incorporating the initial and 2nal !o%ndary conditions

together ith

this in#ormation ena!les %s to %se higher order polynomials inthe

!elo #orm& so that the trajectory ill pass thro%gh all

speci2ed points.• It re$%ires e?tensi"e calc%lation #or each joint and higher

order

polynomials.

5.5.5 +igher Order Trajectories

• Com!inations o# loer order polynomials #or diEerent

segments o# the

trajectory and !lending together to satis#y all re$%ired

!o%ndaryconditions is re$%ired.

nn

nn t ct ct ct ct cct ++++++=

−−

11

33

2210)( θ




5.B C*RT/-I*N'-P*C/ TR*/CTORI/-

• Cartesian'space trajectories relate to the motions o# a ro!ot

relati"e to

the Cartesian re#erence #rame.• In Cartesian'space& the joint "al%es m%st !e repeatedly

calc%lated

thro%gh the in"erse (inematic e$%ations o# the ro!ot.• Comp%ter ;oop

*lgorithm-/ &ncrement the time b# t0t1 t.

∆

-2/ *alculate the osition and orientation of the hand based on the selectedfunction for the trajector#.

-!/ *alculate the joint values for the osition and orientation through theinverse kinematic e)uations of the robot.

-/ 'end the joint information to the controller.

-3/ 4o to the beginning of the loo

Documents

Robot Trajectory Planning