Motion Programming

8/12/2019 Motion Programming

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Trajectory Planning


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Trajectory Planning

Goal: to generate the reference inputs to theGoal: to generate the reference inputs to themotion control system which ensures that themotion control system which ensures that themanipulator executes the planned trajectorymanipulator executes the planned trajectory

Motion controlsystem

RobotTrajectory planning system

torques

Position, velocity, acceleration


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Path and Trajectory

Path: the locus of points in the joint space or inPath: the locus of points in the joint space or inthe operational spacethe operational spaceTrajectory: a path on which a time law isTrajectory: a path on which a time law isspecified in terms of velocities and/orspecified in terms of velocities and/oraccelerationsaccelerations

t


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Path and Trajectory

Trajectory planning algorithm

Path description

path constraints(obstacles)

constraints imposed byrobot dynamics (smooth)

(limits, not modeled

resonant modes)

Joint (end-effector)

trajectories in terms of position, velocity andacceleration


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Path and Trajectory

Specification of geometric pathSpecification of geometric pathExtremal points, possi le intermediateExtremal points, possi le intermediatepoints,geometric primitives interpolating the pointspoints,geometric primitives interpolating the points

Specification of motion time lawSpecification of motion time lawTotal trajectory time, maximum velocity andTotal trajectory time, maximum velocity andacceleration, velocity and acceleration at points ofacceleration, velocity and acceleration at points ofinterestsinterests


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Joint Space Trajectory

Inverseinematics

algorithm

Trajectory parameters in

operation space

Joint (end-effector)trajectories in terms of position, velocity and

acceleration

Trajectory parameters in joint space Trajectory

planningalgorithmInitial and final

end-effectorlocation, traveling

time, etc!


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Joint Space Trajectory

!e"uirements for joint space trajectory!e"uirements for joint space trajectoryplanning algorithmplanning algorithm

The generated trajectory e easy to computeThe generated trajectory e easy to compute

Position and velocity #acceleration$ e continuousPosition and velocity #acceleration$ e continuousfunction of timefunction of time%ndesira le effects e minimi&ed%ndesira le effects e minimi&ed

Point'to'point motionPoint'to'point motion(oving from an initial to a final joint configuration in(oving from an initial to a final joint configuration ina given time ta given time t ff


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Point-to-point Motion

)igure *'+ from )u, Gon&ale& and ee

#tf $#t+$

#t- $

#t. $ ift'off0

Set'down0

)inal

1nitial


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Polynomial interpolationPolynomial interpolation

Example: initial and final position and velocity eExample: initial and final position and velocity egiven2given2


"##

#)( at at at at q nn nn ++++=


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Example: initial and final acceleration also eExample: initial and final acceleration also egiven2given2

Six constraints #initial and final position, velocity andSix constraints #initial and final position, velocity and

accelerationacceleration3rder at least five3rder at least five



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Trape&oidal velocity profileTrape&oidal velocity profile

4irectly verifying whether the velocity and4irectly verifying whether the velocity andacceleration violate the mechanical limitsacceleration violate the mechanical limits



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5rea enclosed y the velocity profile 5rea enclosed y the velocity profile

given accelerationgiven acceleration



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Given the following conditions:Given the following conditions:initial and final position are giveninitial and final position are giveninitial and final velocity are set to &eroinitial and final velocity are set to &ero(aximum velocity and acceleration are given(aximum velocity and acceleration are given

6hat7s the6hat7s the minimumminimum traveling time8traveling time8


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Path Motion

4isadvantages of single high order polynomial4isadvantages of single high order polynomial 5 suita le num er of low order polynomials 5 suita le num er of low order polynomials


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Operation Space Trajectory

9ot easy to predict end'effector motion due to9ot easy to predict end'effector motion due toinematics nonlinearityinematics nonlinearity

Path motion planning similar to joint spacePath motion planning similar to joint space

4ifferent method if the end'effector motion has to4ifferent method if the end'effector motion has tofollow a prescri ed trajectory of motion such asfollow a prescri ed trajectory of motion such asline, circle, etc2line, circle, etc2


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Two-link Planar Arm

Parameters


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;ams

(otion programming historically(otion programming historically

associated with mechanical camsassociated with mechanical cams;onstant speed rotation of camshaft;onstant speed rotation of camshaftconverted to varia le linearconverted to varia le linear

displacement of valve #or other devicedisplacement of valve #or other deviceattached to cam follower$attached to cam follower$ < ;amshafts in auto engines #all * stro es$;amshafts in auto engines #all * stro es$ < Sewing machine #older mechanical style$Sewing machine #older mechanical style$


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Pictures of ;ams

http://www2http://www2 howstuffwor showstuffwor s 2com/camshaft.22com/camshaft.2 htmhtm

1ndustrial1ndustrial

;ar Engines;ar Engines
http://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htmhttp://www.howstuffworks.com/camshaft1.htm


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;am (otion Profiles ' 4!4

4well < !ise < 4well4well < !ise < 4well < initial period of no motion # dwell0$initial period of no motion # dwell0$ < rise0 to a maximum displacementrise0 to a maximum displacement < final period of no motion # dwell0$final period of no motion # dwell0$

dwell0

rise0

dwell0

Time, t

4isplacement, s

s=s max , v=-, a=-s=-, v=-, a=-


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;am (otion Profiles ' 4!!4

4well < !ise < !eturn < 4well4well < !ise < !eturn < 4well < initial period of no motion # dwell0$initial period of no motion # dwell0$ < rise0 to a maximum displacementrise0 to a maximum displacement < 1mmediately return0 to origin1mmediately return0 to origin < final period of no motion # dwell0$final period of no motion # dwell0$

dwell0

rise0

dwell0Time, t

4isplacement, s s=s max , v=-, a -

s=-, v=-, a=- return0 s=-, v=-, a=-


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;am (otion Profiles ' !!

!ise < !eturn!ise < !eturn < rise0 to a maximum displacementrise0 to a maximum displacement < 1mmediately return0 to origin1mmediately return0 to origin < 9o dwell0 < do same thing over again9o dwell0 < do same thing over again

rise0

Time, t

4isplacement, s s=s max , v=-, a -

return0s=-, v=-, a -

s=-, v=-, a -


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5ccel2'>el2'4isp2 ?.

Time, t

5cceleration, 5 @ero order, 5 = constant

Time, t

>elocity, >

Time, t

4isplacement, S

T

T

T

)irst order, >= . t

Second order, S= +t+


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5ccel2'>el2'4isp2 ?.a

Time, t

5cceleration, 5

Time, t

>elocity, >

Time, t

4isplacement, S

T

T

T

AT V =this area

e"uals this value

$

$# AT S =this area

e"uals this value


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5ccel2'>el2'4isp2 ?.

Time, sec

5cceleration, 5

>elocity, >

4isplacement, S

T

- -2. -2+ -2A

T

+B m/s +

>.>+

>A

S.S+

SA

)ind numerical valuesfor >., >+, and >A

)ind numerical valuesfor S., S+, and SA

Suita le for a rise0


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General ;urve Shape: y=Cx n

D

E

n=.

n=+

n=A

n=*

n=B

#

##

"

#

+==

+

n Kx

dx Kx Arean x n

5rea under the curve y=Cx n etween x=- and x=x . is

9ote that y . =Cx . n , so

##

####

+=

+=

n

x y

n

x Kx Area

n


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5ccel2'>el2'4isp2 ?+

Time, sec

5cceleration, 5

>elocity, >

4isplacement, S

T

- -2. -2+ -2A

T

+B m/s +

>.>+ >A

S.S+

SA



Suita le for a dwell ' rise0


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5ccel2'>el2'4isp2 ?A

Time, sec

5cceleration, 5

>elocity, >

4isplacement, S

T

- -2.B -2A

T

+B m/s +

>. >+

S.

S+

)ind numerical valuesfor >. and >+

)ind numerical valuesfor S. and S+



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5ccel2'>el2'4isp2 ?*

Time, sec

5cceleration, 5

>elocity, >

4isplacement, S

T

- -2. -2+ -2A

T

+B m/s +

>.>+ >A

S.S+

SA





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Time,sec 5

>

S

T

T

+B m/s +

>.

>+>A

S.S+

SA

)ind numerical valuesfor >* and >B

)ind numerical valuesfor S*, SB, and SF

Suita le for a rise'return0

'+B m/s + -2. -2+ -2A

-2* -2B -2F

>B>*

S*SB SF


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5nalytical Solution

Solve the previous pro lem analytically:Solve the previous pro lem analytically:


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Solve 9umerically

%se Excel and trape&oidal integration%se Excel and trape&oidal integrationtime 5cc >el 4isp

- +B - --2-. +B -2+B -2--.+B-2-+ +B -2B -2--B-2-A +B -2IB -2-..+B-2-* +B . -2-+

sec)"&!""!"(*sec

)"&!"()"!"($#

sec)"&!"(

sec)"!"( $

++= maamvmv

sec)"&!""!"(*sec

)"&!"()"!"($#

)"&!"()"!"(

++= mvvm sm s


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5 9umerical Gimmic

%se this gimmic to improve accuracy when%se this gimmic to improve accuracy whenyou have a rupt changes in accelerationyou have a rupt changes in accelerationtime 5cc >el 4isp

-2+J +B I -2KJ-2+K +B I2+B .2-B.+B

-2+KKKKKKK +B I2B---- .2.+B-2A------. '+B I2B---- .2.+B-2A. '+B I2+B--- .2.KJIB-2A+ '+B I2----- .2+I

dou le up0 at anysharp transitions


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(otion Programming ?+

!o ot Loint (otions!o ot Loint (otions


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Typical !o ot (otion

)igure *'+ from )u, Gon&ale& and ee

#tf $#t+$

#t- $#t

.$

ift'off0

Set'down0

)inal

1nitial


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Position ;onstraints

1nitial position,1nitial position, ..

Motion Programming

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