Trajectory Session15

8/2/2019 Trajectory Session15

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Sharif University of Technology - CEDRA


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Robot Motion Trajectory

Generation

Hey! Where

Should I go?


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Robot Motion Trajectory Generation

We shall study Methods to:

Compute a Trajectory that describes the desired

motion of a manipulator in space

Trajectory is the time history of position, velocity,

and acceleration for each joint (i.e. degrees-of-freedom).

Trajectory Planning means the determination of the

actual trajectory, or path, along which the robot endeffector will move in Cartesian Space.


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Robot Motion Trajectory GenerationIn executing a trajectory, a manipulator moves

from its initial position to a desiredgoal (final)

position. However, the motion may be:

1. Smooth (nice and secure)

2. Rough (vibrations and damaging)

Trajectory in Cartesian Space


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Robot Motion Trajectory GenerationIn general, thetask of a robot manipulator is defined by a

sequence of points which are denoted as end points and

are stored in the robots control computer.


Motions of a robot manipulator is described as the motions

of the Tool Frame {T} relative to the Station Frame {S}.


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Robots Motion Classifications Point-to-Point (PTP) Robotic Systems: The robot moves to a

numerically defined location and then the motion is stopped.Then, the end-effector performs the required task with therobot being stationary. Upon completion of the task, the robotmoves to the next point and the cycle is repeated. (Ex: SpotWelding Operations)

Therefore; in PTP robots, the path

of the robot and its velocity while

traveling from one point to the

next is not important!

X

Y

1-Starting point

2-End point

Trajectory in PTP Cartesian Robotic

System


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Example: Consider a simple case in

which each joint of a PTP robot must

move to its end point fas fast as

possible without exceeding a maximum

admissible acceleration am and a

maximum velocity Vm.

Desired Joint Trajectory consists of

3-segments:

1. [0, t1]; based on max. acceleration

2. [t1, t2]; based on max. velocity3. [t2, tf]; based on max. deceleration

Robots Motion Classificationsa

Typical acceleration, velocity and position

of an axial motion in PTP Robot.

t

am

-am

t

t

V

Vm

0

f

t1 t2

tf1

2

0


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Robots Motion Classifications Lets assume that the time durations of acceleration

and deceleration segments are equal:{t1 = tf- t2}. The

control program must calculate the two switching

times based on the initial and final joint values (

0 =0and f) and Vm and am.

Joint Trajectory during:

1st Segment:

2nd Segment:

3rd Segment:

From above equations we can write:

1

2

11

2

,2

,2

)( taVtata

t mmmm

===

a

Typical acceleration, velocity and position

of an axial motion in PTP Robot.

t

am

-am

t

t

V

Vm

0

f

t1 t2

tf1

2

0

,)(,)()( 121211 ttVttVt mm +=+=

,2

,2

)()()(

2

112

2

222

tatV

ttattVt mmf

mm +=

+=

m

f

m

mf

m

m

m

f

mfVa

Vt

a

Vtand

VttV

+==== ,122


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Robots Motion Classifications

Continues Path (CP) Robotic Systems: In this system therobot tool performs the task while the axes of motion are

moving (both robot and the tool are moving simultaneously,and the speed of each joint can be controlled independently).

(Ex: Arc Welding Operations)

Therefore; in CP robot operations, the position of the robotstool at the end of each segment , together with the ratio ofaxes velocities, determine the generated trajectory. Ex:variations in the velocity of the arc welding result in a non-uniform weld seam thickness (i.e. an unnecessary metal built-up or even holes)


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To move the manipulatorsTool from {Tool}i to {Tool}f ,there exists an infinite numberoftrajectories.


Motions of a robot manipulator is described as the motions

of the Tool Frame {T} relative to the Station Frame {S}.

f

t

i

i: initial position of the tool

f: final position of the tool

{Tool}


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Robot Motion Trajectory GenerationTo provide more details about thedesired trajectory, we sometimesdefine some via points (intermediatepoints) throughout the path.

f

t

i

i: initial position of the tool

f: final position of the tool

{Tool}

via points


Path Points: a set of Via points plus Initial and Final points.


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In executing a trajectory, it is generallydesirable to smoothly move the

manipulator from its initial position to adesiredfinal position.

For Smooth Motion: We need to haveposition and velocity to becontinuesfunctions of time.

A Smooth Function: A function which iscontinues and has acontinues firstderivative.

Trajectory in

Cartesian Space


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Robot Motion Trajectory GenerationJoint Space Schemes: Methods of path generationin which path shapes are described in terms offunctions of joint angles.

Eachpath point (via points + initial and final points) isspecified in terms of a desired position & orientation ofthe Tool frame {T}, relative to theBase frame {B}.

In Cartesian Space a set ofvia points may be used to

take the Tool frame from initial state to a final state.

Each via and path point is converted into a set ofdesired joint angles by applying inverse kinematics.

A Smooth Function is defined for each of n-joints that

pass through the via and path points.

Thetime duration required for each motion segment isthe same for each joint, so that all joints will reach thevia point at the same time.

Joint Space Schemes

{T}i

{T}f

{i

}{f}

{B}


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Robot Motion Trajectory GenerationJoint Space Schemes:

Each via and path point is converted into a set of desired

joint angles by applying inverse kinematics.

{Tool} frame = {vector }{by Inverse Kinematics} =a set of {is and fs}

Generate a Smooth Function through all points.

In between via points, the shape of the path, althoughrather simple in joint space, is complex if described inCartesian space.

Joint Space schemes are easiest to compute, since there isusually no problem withsingularities of the mechanism.

Joint Space Schemes

{T}i

{T}f

{i

}{f}

{B}

via


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Cubic Polynomials:

By applying inverse kinematics.

{Tool}i = {is }{Tool}f = {fs }

We need to define asmooth function for each joint

whose value at ti is i, and at tf is f

There exist many smooth functions for (t)= ?...

Joint Space Schemes

{T}i

{T}f

{i} {f}

{B}


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Robot Motion Trajectory GenerationCubic Polynomials:

For Smooth joint motion, we need toimpose at least 4-constraints on (t) as:

We need to define asmooth function foreach joint whose value at ti is i,

and at tf is f

There exist many smooth functions for(t)= ?...

f

tf

Many path shapes

ti=0

i= 0

(t)

t0)(

0)0(

)(

)0( 0

=

=

=

==

f

ff

i

t

t

&

& Continues in Velocity: Startand Stop at Zero Velocities


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5th Order Polynomials:

If we add theacceleration constraints at the beginningand end of the path; then we have 6-Constraints.

If acceleration is continues, then vibration is diminished

(i.e. if ai = af= 0) at thestart and thestopping points.

Hence, a 5th Order Polynomial would be sufficient todefine the joint motion.

5

5

4

4

3

3

2

210)( tatatatataat +++++= (6-Coefficients)


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Position, Velocity, and Acceleration Curves

&

&&

0

f

(t)

t0=0 tf

t

tf

tf

t

t

t0=0

t0=0

ai = af=0

ai = af=0

ai = af=0

Position

Velocity

Acceleration

vi = vf=0

Zero Slope


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Robot Motion Trajectory GenerationFor intermediate (via) points at which

you dont want the robot to stop,

velocity constraints at the end of

cubic polynomial is no longer zero.

t

f

tft0=0

0

(t)

&

tft0=0

f&

0&

t

ff

ff

t

t

&&

&&

=

=

=

=

)(

)0()(

)0(

0

0

4-Constraints

3

3

2

210)( tatataat +++=


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4-Coefficients

3

3

2

210)( tatataat +++=

)(1

)(2

12)(3

02033

0022

01

00

&&

&&

&

++=

=

=

=

f

f

f

f

f

ff

f

f

tta

ttta

a

a

Several Ways to specify the desired velocities at the via points:

By the user in terms of Cartesian linear and angular velocities (mapped

into joint velocities by Jacobian Inverse) The system automatically choose

The system automatically choose such that the acceleration is continues


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Robot Motion Trajectory GenerationLinear Function with Parabolic Blends:

Another choice of path shape is to moveLinearlyfrom initialjoint position 0 to thefinaljoint

position f.

Linear Function Results in:1. Velocity to bediscontinuous at

the beginning and end of motion,

2. Acceleration to bediscontinuous

Undesirable (Rough Motion)

f

tft0=

0

0

(t)

t

(Linear Function)


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To avoid rough motion, we canadd a Parabolic Blendto the linear function at each end.

AddingParabolic blends

withconstant acceleration

results in smooth change in

velocity. Linear function and two parabolic

functions are splined together so

that the entire path is continuous

inPosition and Velocity.

)( b&&

2

210)( tataat ++=

(Linear Function with Parabolic Blends)

f

tft0=0

0

(t)

t

tf--tbthalftb

half

b


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Assume both blends have the same time duration, and note

that velocity at the end of the blend must be equal to the

velocity of the linear segment.

(Linear Function with Parabolic Blends)

f

tft0=0

0

(t)

t

tf--tbthalftb

half

b

)()( speedslope

tt

tt

bh

bhbbb =

==

&&&

For theLinear segment:

For theBlendregion:

21

2

1

2

1

tan

CtCt

Ct

tCons

bb

bb

b

++=

+=

=

&&

&&&

&&

Integrating using I.C.:

0)0(

0)0(

=

=

b

b&

0

0 0


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Combining the above relations, we get:

0

2

2

1 +=

bbb

t&& at tb , and let: t=2th

b

bb

f

bh

bhbb

tt

t

ttt

+

=

=

2

12

1

2 020

&&

&& 0)( 02 =+ fbbbb ttt &&&&

Where:

t : Desired duration of motion

: Acceleration acting during the blend regionb&&


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b&&

Given any f, 0, and t, we can follow any of the paths

given by choices of and tb which satisfy equation *. .b&&

0)( 02

=+ fbbbb ttt&&&&

You may pick and solve for tb ?

b

fbb

b

ttt

&&

&&&&

2

)(4

2

0

22

=0

2

0

0

22)(4

)(4t

tf

bfbb

&&&&&&


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= Equality means linear segment has zero length (Path is composed

of two blends with equivalent slope).> As the acceleration increases, the blend region becomes

shorter and shorter. In the limit when , we are back to

the simple linear interpolation case.

b&&

b&&

You may pick and solve for tb ?

b

fbb

b

ttt

&&

&&&&

2

)(4

2

0

22

=0

2

0

0

22)(4

)(4t

tf

bfbb

&&&&&&

b&&


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Adding via points:

(t)

tf(Linear Function with Parabolic Blends)

f

t0=0

0

t

Documents

Trajectory Session15