KADOMS04

KADOMS KRIM, WIMIR, AGH Kraków 1

Katedra Robotyki i Mechatroniki

Akademia Górniczo-Hutnicza w Krakowie

Wojciech Lisowski

4

Trajectory planning

Kinematics and Dynamics of Mechatronic Systems


Problems:

Motion path and trajectory

Classification of constraints in the trajectory

planning

Technique of planning of the joint trajectories

Scaling of the joint trajectories


Motion Path – a set of points in space, which are attained by an end-

effector with some (demanded or not demanded) orientation during

carrying out of some operation (Geometry).

Parameterization of the motion path with respect to time leads to

definition of the Motion Trajectory of the end-effector (Kinematics).

The planned trajectory influences:

- cycle time

- energy consumption

- quality of manufacturing

- effectiveness of cooperation of robots with other robots, machines

or devices

- effectiveness of obstacle avoidance.


Motion at each point of a spatial curve that the manipulator end-

effector follows during operation might be defined by the following

kinematic parameters:

- position vector P

- orientation vector

- translational velocity vector V

- angular velocity vector

- translational acceleration vector A

- angular acceleration vector

The most often only the position is defined.

Less often the orientation and translational velocity of an end-effector

is planned.

The applied robot programming technique determines how complex

the trajectory planning algorithm is.

- on-line programming (teaching)

- off-line (direct) programming


Trajectory planning diagram:

The result of the trajectory planning is the description of the motion

path in terms of joint variables q(t) or less often in terms of Cartesian

coordinates: P(t), (t), v(t), w(t), a(t), e(t).

TRAJECTORY

PLANNER

pd, d, vd, wd,ad, ed

p(t), (t), v(t), w(t), a(t), e(t)

GEOMETRICAL

CONSTRAINTS

KINEMATIC

AND DYNAMIC

CONSTRAINTS

JOINT TRAJECTORY

CARTESIAN

TRAJECTORY

)(),(),( tqtqtq


Geometrical constraints concern:

- locations in space of points of the end-effector path

- locations of obstacles in the workspace.

Kinematic constraints concern:

- values of velocities, accelerations, jerks as well the higher order

derivatives of displacement with respect to time.

- parameters of driving systems in a form of relationship between

velocity/acceleration and driving torque/force.

Dynamic constraints concern:

- values of driving torque/force

- dynamic loads for a manipulator structure


Applicable geometrical and kinematic constraints are of the type of:

- continuity

- differentiability

- limits of extreme values of a trajectory parameters.

Fast changes of values of kinematic parameters of discontinuity

and/or non-differentiability type:

- cause vibrations of manipulators

- limit precision of motion

- cause increase of an operation cycle

- cause fast wear of elements and subassemblies of manipulators.

When planning of trajectories takes into account constraints

corresponding to limits of driving forces, it is necessary to introduce

the dynamic model of a manipulator during the trajectory planning.


In case of concurrent planning and tracking of trajectories at each time

sample the current corrections of the planned trajectory parameters are

generated.

It requires high computational power and for majority of types of

operations used in the engineering practice it is unjustified.

Usually the trajectory planning stage is carried out separately from the

trajectory tracking stage. The planning precedes the tracking.

During tracking the planned values of the trajectory parameters cannot

be updated any more.

The tracking errors are minimized by application of an appropriate

controller.


The main practical reason against application of Cartesian trajectories

in practice is a lack of sensors capable to determine positions and

orientation of an end-effector in the Cartesian space that can be reliably

applied in the industrial workshop conditions

Without the sensors the Tracking of the Cartesian trajectories requires

transformation of geometrical and kinematic parameters

- from the joint coordinates space to the Cartesian space in order to

determine the control error

- from the Cartesian space to the joint coordinates space in order to

determine the correction of the control.

With the sensors:

- from the Cartesian space to the joint coordinates space in order to

set the correction of the control for each single axis controller.

-This approach is:

- complex

- computationally demanding

- (usually) unjustified from the economical point of view.


In practice the most often the trajectory planning in the joint coordinates

space is used

To plan the joint trajectories usually the two following types of the spline

functions are used:

- polynomial splines

- trigonometric splines.

The main advantage of the joint trajectories:

Simplicity of trajectory planning algorithms and ease of trajectory

tracking by control systems of majority of industrial robots.

The main drawback of the joint trajectories:

Lack of coordination of motions of the consequent joints

(unpredictable shape of the resultant Cartesian trajectory that is a

composition of a set of the joint trajectories).


Due to the high variability of values of polynomials of high degree

and difficulties in determination of their local extremes it is not

advisable to use polynomials of degree higher than 7 (usually the

degree from 3 to 5 is used).

In case of positioning of an end-effector the safety considerations

regarding its motion indicate the use of a path defined by at least 4 points

q t a a t a t a t

t

j i i i i i n

n

, , , , ,...

0 1 2

2

0 1

p

k s1

sn


Example formulation of the type 3-5-3 polynomial trajectory

1011

2

12

3

131 atatatath

2021

2

22

3

23

4

24

5

252 atatatatatath

3031

2

32

3

333 atatatath

time

displacement

velocity

acceleration

0

X0

V0

A0

1

X1

-

-

2

X2

-

-

3

X3

V3

A3


Normalization of the time axis:

1

1

ii

it

iii td

dt 11

1

dt

tdh

td

dt

dt

tdh

d

tdhtv i

i

ii

i

1

2

2

2

111

dt

thd

ttdt

tdh

tdt

d

d

dt

dt

tdv

d

tdvta i

ii

i

i

ii

i


The first segment of the type 3-5-3 trajectory:

01 0 Xh

01 0 Vv

01 0 Aa

101 0 ah 010 Xa

1112

2

13

1

1 231

atatat

tv 1

111 0

t

av 1011 tVa

12132

1

1 261

atat

ta 2

1

121

20

t

aa

2

2

10

12

tAa

11 1 Xh 010

2

10

1312

1 XtVtA

ah 0110

2

10

132

XXtVtA

a

1

0

0

0

1

2

1

2

11

10

11

12

13

0001

000

02

00

12

1

X

A

V

X

t

t

tt

a

a

a

a

1011

2

12

3

131 atatatath


The third segment of the type 3-5-3 trajectory:

33 1 Xh

33 1 Vv

33 1 Aa

23 0 Xh

2

013

123

111

2

33

33

23

31

32

33

tA

tV

XX

a

a

a

2333

2

3331

2333

2

3332

2333

2

3333

322

1

33

2

1

XXtVtAa

XXtVtAa

XXtVtAa

303 0 ah 230 Xa

3

3

3

2

2

33

2

33

2

333

30

31

32

33

00012

233

3332

1

A

V

X

X

tt

tt

ttt

a

a

a

a

3031

2

32

3

333 atatatath


The second segment of the type 3-5-3 trajectory:

12 0 Xh

12 0 Vv

12 0 Aa

22 1 Xh

22 1 Vv

22 1 Aa

202 0 ah 120 Xa

2122

2

23

4

24

4

25

2

2 23451

atatatatat

tv 2

212 0

t

av 2121 tVa

2223

2

24

3

252

2

2 2612201

atatatat

ta 2

2

222

20

t

aa

2

2

2122

tAa

2

212324252

2

2

21

2

21232425

2

2

121

2

212324252

612201

3451

2

tAaaat

a

tVtAaaat

v

XtVtA

aaax

2

212

212

2

11

1221

2

21

23

24

25 2

61220

345

111

tAA

tVVtA

XXtVtA

a

a

a

2

22

2

2122211223

2

22

2

2122211224

12

2

22121225

2

1

2

34610

2

37815

2

136

tAtAtVtVXXa

tAtAtVtVXXa

AAtVVtXXa

2021

2

22

3

23

4

24

5

252 atatatatatath


2

2

2

1

1

1

2

2

2

2

22

2

22

2

22

2

22

2

22

2

22

20

21

22

23

24

25

000001

00000

0002

00

2410

2

3610

7152

3815

236

236

A

V

X

A

V

X

t

t

tttt

tttt

tt

tt

a

a

a

a

a

a

0

0

1

1

32

32

11

11

aA

vV

aA

vV

233

122

011

t

t

t

tatttAA

tvtttVV

thtttHH

j

i

i

ji

j

i

i

ji

j

i

i

ji

1

0

1

0

1

0

)(

1,...,2,1 Nj

Determination of the trajectory in terms of the absolute time variable :


h [rad] h [rad]

A numerical example

time

displace

ment

velocity

acceleration

example 1

example 2

0.00

0.00

0

0

0

1.00

1.00

0.3

-

-

2.25

1.60

1.7

-

-

3.25

2.60

2

0

0

displacement:


v [rad/s] v [rad/s]

a [rad/s2] a [rad/s2]

velocity:

acceleration:


Number and type of start/end

conditions

6

(displacement,

velocity, acceleration)

6

(displacement,

velocity, acceleration)

6

(displacement,

velocity,acceleration)

Number of intermediate points

N-2

N-2

N-2

Number of continuity conditions

3*(N-2)

3*(N-2)

5*(N-2)

+ 3 and 4 derivative of

displacement

Assumed: velocities and accelerations

-

2*(N-2)

-

Number of coefficients

4N-2

6N-6

6N-6

Distribution of coefficients between

trajectory segments for N=4

4*4-2=14

14-3=11

4-3-4

3-5-3

6*4-6=18

18-3=15

5-5-5

6*4-6=18

18-3=15

5-5-5



4*5-2=18

18-4=14

4-3-3-4

6*5-6=24

24-4=20

5-5-5-5

6*5-6=24

24-4=20

5-5-5-5



4*6-2=22

22-5=17

4-3-3-3-4

3-3-3-3-3

6*6-6=30

30-5=25

5-5-5-5-5

6*6-6=30

30-5=25

5-5-5-5-5


trajectory segments for N>6

(4*N-2)-(N-1)

4-3-3- ... -3-3-4

(6*N-6)-(N-1)

5-5-5- ... -5-5-5

(6*N-6)-(N-1)

5-5-5- ... -5-5-5

An algorithm of formulation of the joint polynomial trajectories:


h [rad]

v [rad/s]

a [rad/s2]

j [rad/s3]

5-5-5-5-5-5-5 trajectory example


h [rad] v [rad/s]

a [rad/s2]

j [rad/s3]

4-3-3-3-3-3-4 trajectory example


Trigonometric type trajectory example:

40

)4cos()sin()cos()(3

1

4,,,0,,

t

taktbktaatqk

ikikiiij

h [rad]

v [rad/s]

a [rad/s2]

j [rad/s3]


Scaling of a joint trajectory:

dt

tdh

ttv i

i

i

1

2

2

2

1

dt

thd

tta i

i

i

i

vi

vi

i

i

i

G

i

t

t

t

v

t

v

V

V

max,

max,

max,

2

2

2

max,

2

max,

max,

i

ai

ai

i

i

i

G

i

t

t

t

a

t

a

A

A

velocity


After the first scaling: Re-planning of the joint trajectory

G

i

iviV

Vtt

max,

G

i

iaiA

Att

max,2


Result of an iterative trajectory planning with concurrent use of

limits for the velocity and acceleration values.

),max( aivisi ttt


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-40

-20

0

20

40

Parametry kinematyczne obrotowego złącza manipulatora

Prz

em

iesczenie

[sto

pnie

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-400

-300

-200

-100

0

Prę

dkość [

sto

pnie

/s]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-4000

-2000

0

2000

4000

6000

Czas [s]

Prz

yspie

szenie

[sto

pnie

/s2]

Example:

Trajectory Type 4-(3)-4

Q=[45,30,-10,-50,-55,-60]

T=[0.1,0.1,0.1,0.1,0.1]

wmax=200

emax=300


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-40

-20

0

20

40

Parametry kinematyczne obrotowego złącza manipulatora

Prz

em

iesczenie

[sto

pnie

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-200

-100

0

100

200

Prę

dkość [

sto

pnie

/s]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-200

0

200

Czas [s]

Prz

yspie

szenie

[sto

pnie

/s2]

Q=[45,30,-10,-50,-55,-60]

T=[0.60 0.36 0.48 0.34 0.23]


Example of superposition of 4-3-3-3-4 joint trajectories for a SCARA

type manipulator

θ1 [0,0,1,2,3,3] [rad] θ2 [1,1,2,2,1,1] [rad]

d3 [0.3,0.1,0.1,0.1,0.1,0.3] [m] θ4 [0,0,-1,-2,-3,-3] [rad]

θ1 [rad] θ2 [rad]

d3 [m]

θ4 [rad]


x [m] y [m]

z [m] Φ [rad]

Time histories of the Cartesian trajectory components


y

x

Spatial result of composition of the 4 joint trajectories for the

SCARA manipulator:


x

y

z


11 points 6 points

a SCARA manipulator

Number of the path points


Motion of an end-effector in space (via positioning or trajectory

tracking) is a problem of motion control.

In a justified case trajectory planning and tracking can be carried out

concurrently.

Date post:	17-Feb-2016
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