KADOMS AGH, WIMIR, KRIM Kraków 1

Katedra Robotyki i Mechatroniki

Akademia Górniczo-Hutnicza w Krakowie

Wojciech Lisowski

9

Dynamics of manipulators.

Formulation of the model – Newton-Euler method

Kinematics and Dynamics of Mechatronic Systems

KADOMS AGH, WIMIR, KRIM Kraków 2

Problems:

Newtona-Euler’s recursive algorithm

KADOMS AGH, WIMIR, KRIM Kraków 3

Newton-Euler’s method of modification of set of DEM:

The method is composed of 2 steps:

- recursive determination of kinematic parameters starting from

the base and ending at the end-effector

- recursive determination of generalized forces starting from

the end-effector and ending at the base

The method employes Rj rotation matrices and vectors in 3D.

Number of arithmetic operations during numerical determination

of DEM is proportional to the number of the generalized coordinates

The method leads to determination of the internal forces what

is necessary to carry out deflection and stress analysis

KADOMS AGH, WIMIR, KRIM Kraków 4

The form of DEM

where:

iiim Fa

iiiiii NωIωεI

gffF iiii m 1

1

***

1 iiiiiiii fsfspnnN

KADOMS AGH, WIMIR, KRIM Kraków 5

N-E recursive algorithm

Kinematic parameters of the base:

00 ω 00 v 00 ε 00 v 00 a

Gravity acceleration vector with respect to the reference frame:

Tzyx gggg

Forward recursion– from the base towards the end-effector: i=1, 2, …, n

1

0

1ˆ

i

r

iii q bωω

1

0

11

0

1ˆˆ

i

r

iii

r

iii qq bωbεε

**

1

0

11

0

1ˆ2ˆ

iiiiii

t

iii

t

iii qq pωωpεbωbvv

**

iiiiiii sωωsεva

KADOMS AGH, WIMIR, KRIM Kraków 6

Backward recursion– from the efector towards the base: i= n, n-1, …, 1

Force and moment acting on the i-th link

Projection of fi or ni on motion axes leads to determination of:

1

0

iii bf

- driving force

- driving torque 1

0

iii bn

gfFf iiii m 1

1

***

1 iiiiiiii fsfspnNn

iii m aF

iiiiii ωIωεIN

KADOMS AGH, WIMIR, KRIM Kraków 7

Vectors pi*, si

*, and I matrix are expressed with repect to the reference

frame, and their coordinates/elements vary during the motion

Solution: express vectors pi*, si

*, and Ii matrix in the i-th local

coordinate frame

1

0

T

iii

0

PRT

T

ii

iRRR

010

0

*

0 i

isR OiCi vector

*

0 i

ipR Oi-1Oi vector

ii

iRIR

0

0

The i-th link inertia tensor expressed with

respect to the central axes parallel to the axes

of the i-th local frame

KADOMS AGH, WIMIR, KRIM Kraków 8

Forward recursion i=1, 2, … , n

Initial values:

𝐑0𝐚𝑖𝑖 = 𝐑0𝐯 𝑖

𝑖 + 𝐑0𝛆𝑖𝑖 × 𝐑0𝐬𝑖

∗𝑖 +

+ 𝐑0𝛚𝑖𝑖 × 𝐑0𝛚𝑖

𝑖 × 𝐑0𝐬𝑖∗𝑖

KADOMS AGH, WIMIR, KRIM Kraków 9

Backward recursion i = n, n-1, …, 2, 1

KADOMS AGH, WIMIR, KRIM Kraków 10

Generalized driving forces

Translational motion:

Rotational motion:

MRD KRiM, WIMiR, AGH Kraków 11

RRP manipulator example

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

𝐠 = 0,0,−𝑔 𝑇

MRD AGH, WIMIR, KRIM Kraków 12

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

Assumed spatial mass distribution:

Link No. 1 – a prismatic rod of mass m1,

the centre of mass at the centre of the rod

𝐑0𝐬1∗1 = −

𝑎12, 0,0

𝑇

𝐑0𝐩1∗1 = 𝑎1, 0,0

𝑇 𝐑01 𝐈1 𝐑1

0 =

0 0 0

0𝑚1𝑎1

2

120

0 0𝑚1𝑎1

2

12

𝐑01 𝐠 = 𝐑1

−1𝐠 =𝐶1 𝑆1 0−𝑆1 𝐶1 00 0 1

000−𝑔

=00−𝑔

1000

0100

0

0

1111

1111

1

SaCS

CaSC

A

KADOMS AGH, WIMIR, KRIM Kraków 13

Assumed spatial mass distribution:

Link No. 2 – a prismatic rod of mass m2,

the centre of mass at the centre of the rod

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

𝐑0𝐬2∗2 = −

𝑎22, 0,0

𝑇

𝐑0𝐩2∗2 = 𝑎2, 0,0

𝑇 𝐑0

2 𝐈2 𝐑20 =

0 0 0

0𝑚2𝑎2

2

120

0 0𝑚2𝑎2

2

12

𝐑02 𝐠 = 𝐑2

−1𝐠 =𝐶12 𝑆12 0𝑆12 −𝐶12 00 0 −1

000−𝑔

=00𝑔

1000

0100

0

0

111221212

111221212

2

SaSaCS

CaCaSC

T

MRD AGH, WIMIR, KRIM Kraków 14

Assumed spatial mass distribution:

Link No. 3 – a particle of mass m3 located at

the origin of the 3rd local coordinate frame

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

𝐑0𝐬3∗3 = 0,0,0 𝑇

𝐑0𝐩3∗3 = 0,0, 𝑑3

𝑇

𝐑0

3 𝐈3 𝐑30 =

0 0 00 0 00 0 0

𝐑03 𝐠 = 𝐑3

−1𝐠 =𝐶12 𝑆12 0𝑆12 −𝐶12 00 0 −1

000−𝑔

=00𝑔

1000

100

0

0

3

111221212

111221212

3d

SaSaCS

CaCaSC

T

MRD AGH, WIMIR, KRIM Kraków 15

Initial conditions:

𝐑0𝛚00 = 𝟎 𝐑0𝛆0

0 = 𝟎 𝐑0𝐯 00 = 𝟎 𝐑0𝐚0

0 = 𝟎

Etap 1+

𝐑0𝛚11 = 𝐑1 0 𝐑0𝛚0

0 + 𝜃 1

001

=001𝜃 1

𝐑01 =

𝐶1 𝑆1 0−𝑆1 𝐶1 00 0 1

𝐑0𝛆11 = 𝐑0

1 𝐑0𝛆00 + 𝜃 1

001

+ 𝐑0𝛚00 × 𝜃 1

001

=001𝜃 1

1000

0100

0

0

1111

1111

1

SaCS

CaSC

A

MRD AGH, WIMIR, KRIM Kraków 16

𝐑0𝐯 11 = 𝐑0

1 𝐑0𝐯 00 + 𝐑0𝛆1

1 × 𝐑0𝐩1∗1 +

+ 𝐑0𝛚11 × 𝐑0𝛚1

1 × 𝐑0𝐩1∗1 =

=001𝜃 1 ×

𝑎100

+001𝜃 1 ×

001𝜃 1 ×

𝑎100

=0𝑎10

𝜃 1 +−𝑎100

𝜃 2

𝐑01 𝐚1 = 𝐑0

1 𝐯 1 + 𝐑0𝛆11 × 𝐑0𝐬1

∗1 +

+ 𝐑0𝛚11 × 𝐑0𝛚1

1 × 𝐑0𝐬1∗1 =

=0𝑎10

𝜃 1 +−𝑎100

𝜃 2 +001𝜃 1 ×

−𝑎1200

+

+001𝜃 1 ×

001𝜃 1 ×

−𝑎1

2

00

=`

0𝑎120

𝜃 1 +−𝑎1200

𝜃 2