KADOMS06

KADOMS KRiM, WIMiR, AGH Kraków 1

Katedra Robotyki i Mechatroniki

Akademia Górniczo-Hutnicza w Krakowie

Wojciech Lisowski

6

Spatial distribution of mass.

Evaluation of kinetic and potential energy

Kinematics and Dynamics of Mechatronic Systems


Problems:

inertia tensor

simplified spatial mass distribution patterns

kinetic energy

potential energy

Description of the links’ spatial mass distribution

Inertia Tensor Ii, elements are determined about central axes (the ones

that cross the mass centre)

MRD KRIM, WIMIR, AGH Kraków 3

dmyxI

dmzxI

dmzyI

m

ii

izz

m

ii

iyy

m

ii

ixx

22

22

22

dmzyI

dmzxI

dmyxI

i

m

i

iyz

i

m

i

ixz

i

m

i

ixy

Cartesian (3D) coordinates are used

𝐼𝑥𝑥 −𝐼𝑥𝑦 −𝐼𝑥𝑧−𝐼𝑦𝑥 𝐼𝑦𝑦 −𝐼𝑦𝑧−𝐼𝑧𝑥 −𝐼𝑧𝑦 𝐼𝑧𝑧


model radius

[mm]

wall

thickness

[mm]

Ixx

[kgm2]

*10-3

Iyy, Izz

[kgm2]

*10-3

A - - 0.0 41,67

B 22 22 0.5 41,90

C 49 5 4.3 43,84

D 118 2 27.4 55,36

z

y x

Analysis of influence of simplifications of spatial mass distribution

on the mass inertia moments - an aluminum rod example

A. Infinitely thin rod

B. Solid rod

C. Hollow rod

D. Thin walled hollow rod mass 2 kg lenght 0.5 m


z

y x

Ixx/Izz from solid rod to thin walled hollow rod of 1 mm wall thickness


Link No. 1

Link No. 2

Link No. 3

Link No. 4

Example of asymmetrically integrated elements


m

Tl

i

l

i

i dmrrJ

Pseudoinertia Matrix Ji

mm

l

m

l

m

l

m

l

m

l

m

ll

m

ll

m

l

m

ll

m

l

m

ll

m

l

m

ll

m

ll

m

l

i

dmdmzdmydmx

dmzdmzdmzydmzx

dmydmzydmydmyx

dmxdmzxdmyxdmx

J2

2

2

Homogeneous coordinates used for description

of an i-th link expressed with use of the mass inertia moments evaluated

with respect to the the link local coordinate frame axes x, y and z. It

does not vary during manipulators’ motion


Mass moments of inertia with respect to ij planes

relationships

𝐼𝑥𝑥 = 𝐽𝑥𝑦 + 𝐽𝑥𝑧

𝐼𝑦𝑦 = 𝐽𝑥𝑦 + 𝐽𝑦𝑧

𝐼𝑧𝑧 = 𝐽𝑥𝑧 + 𝐽𝑦𝑧


iiiiiii

ii

izziyyixx

iyzixz

iiiyz

izziyyixx

ixy

iiixzixy

izziyyixx

i

mzmymxm

zmIII

II

ymIIII

I

xmIIIII

J

2

2

2

dmyxI

dmzxI

dmzyI

m

l

i

l

i

izz

m

l

i

l

i

iyy

m

l

i

l

i

ixx

22

22

22

where:

dmzyI

dmzxI

dmyxI

l

i

m

l

i

iyz

l

i

m

l

i

ixz

l

i

m

l

i

ixy

m

i dmm

dmzzm

dmyym

dmxxm

m

i

i

ii

m

i

i

ii

m

i

i

ii


Simplifed spatial distribution of mass

Effective analysis of dynamics of mechanisms requires simplification

of the description (a model) of their spatial mass distribution.

The following simplifications are considered:

a particle – placed in a centre of mass of a link (inertia moments of

the link neglected)

a prismatic rod – in case of slim links rotating about an axis located

at one end of a link and perpendicular to its length (inertia moment

with respect to the longitudinal link axis is neglected)

a disc – for a rotating (the first) link of a mechanism (e.g. a column

of a manipulator) – there is considered only the inertia moment with

respect to the principal central inertia axis (the one perpendicular to

the motion plane).


An example of the simplified spatial mass distribution of a link

A particle mi

si is a distance of the particle from the i-th local frame origin (in –x

direction)

xi

yi

zi

si

mi

2

iiiyy smI

2

iiizz smI

iiii smxm

iii

iiii

i

msm

smsm

J

00

0000

0000

002

iiiiiii

ii

izziyyixx

iyzixz

iiiyz

izziyyixx

ixy

iiixzixy

izziyyixx

i

mzmymxm

zmIII

II

ymIIII

I

xmIIIII

J

2

2

2


An example of the simplified spatial mass distribution of a link

xi

yi

zi

½ai

ai

mi

2

3

1iiiyy amI

2

3

1iiizz amI

iii

iiii

i

mam

amam

J

002

10000

00002

100

3

1 2

A slim rod

iiiiiii

ii

izziyyixx

iyzixz

iiiyz

izziyyixx

ixy

iiixzixy

izziyyixx

i

mzmymxm

zmIII

II

ymIIII

I

xmIIIII

J

2

2

2


Evaluation of the Kinetic Energy with use of the geometrical model

based on the homogeneous transformation matrix

A position vector of a l-th particle of an i-th link with respect the its local

coordinate frame (homogeneous coordinates)

and with respect of the reference frame:

The velocity vector of the l-th particle of the i-th link with respect to

the reference frame:

Tl

i

l

i

l

il

i

zyxr 1,,,

l

i

ll rTr00

liil

i

i

li

ili

ili

illl

rdt

dAAArAAA

rAAArAAArTdt

dr

dt

dvv

......

.........

2121

2121

00


The differential relationships indicate that:

Derivatives of Tj matrices with respect to the joint variables qk:

jkkkjk

k

kk

k

j

jk AAQAAAAAq

AAAA

q

TU

............ 121121

kkk

k

k

kk qAQdt

dq

q

A

dt

Ad

0000

0000

0001

0010

Q

where:

0000

0000

0000

1000

aQ

0000

1000

0000

0000

dQ

0000

0010

0100

0000

Q

jjj

jAQ

q

A


0

1

1 j

k

kkjk

TQTU

jk

jk

l

ii

k

kikl rqUv

1

0

The linear (translational) velocity vector of the l-th particle

of the i-th link


Kinetic Energy Ki of the i-th link is composed of a sum of dKil

components

m

Tl

i

l

i

i dmrrJ

The total kinetic energy:

kj

n

i

i

j

i

k

T

ikiij

n

i

kj

T

ik

i

j

i

k

iij

n

i

i qqUJUTrqqUJUTrKK

1 1 11 1 11 2

1

2

1

n

i

iiaTr1

A

dmrqUrqUTrdmvvTrdmzyxdK

Ti

k

l

i

kikl

ii

j

jij

Tlllllil

11

222

2

1

2

1

2

1

i

j

i

k

kj

T

ik

T

l

i

l

i

ijil qqUrdmrUTrdK1 12

1

kj

T

il

i

j

i

k m

Tl

i

l

i

ijili qqUdmrrUTrdKK 1 12

1


An example – a RRP manipulator geometrical model

x 0

y 0

z 0

x 1

y 1

z 1

x 2 y 2

z 2

x 3 y 3

z 3

1000

0100

0

0

1111

1111

1

SaCS

CaSC

A

1000

0100

0

0

2222

2222

2

SaCS

CaSC

A

1000

100

0010

0001

3

3d

A

1000

0100

0

0

111221212

111221212

2

SaSaCS

CaCaSC

T

1000

100

0

0

3

111221212

111221212

3d

SaSaCS

CaCaSC

T


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5translational velocity - absolute value

time [s]

[m/s

]

the RRP manipulator example – the kinetic energy

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

[s]

[J]

-1

0

1-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.3

-0.28

-0.26

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

yx

z

m1=4 kg m2=2 kg m3=1 kg


The total potential energy:

iiiisii sTgmrgmP 0

The Potential Energy of the i-th link

Tzyx gggg 0,,, 000

n

i

n

i

ii

T

ii sTgmPP1 1

Position vector si of the centre of mass of the i-th link with respect to

the i-th local coordinate system

Documents

KADOMS06