Добавил:
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
Jack H.Integration and automation of manufacturing systems.2001.pdf
Скачиваний:
86
Добавлен:
23.08.2013
Размер:
3.84 Mб
Скачать

page 372

12.4 DYNAMICS FOR KINEMATICS CHAINS

There are a variety of common methods,

-Euler-Lagrange - energy based

-Newton-Euler - D’Alembert’s equations

12.4.1 Euler-Lagrange

• This method uses a Lagrangian energy operator to calculate torques

L

( θ

i

, ω ) = K

i

P

i

 

L( θ ,ω ) = Ki Pi

i

 

 

 

 

i

 

 

 

 

 

 

 

 

d

 

 

 

 

 

 

 

 

 

 

 

 

----

 

 

 

 

 

L

 

 

L

 

=

Q

 

dt

∂ ω

i

 

i

 

∂ θ i

 

 

i

 

 

 

i

where,

L = lagrangian

K = kinetic energy of link ‘i’ P = potential energy of link ‘i’ Q = forces and torques

• For a typical link,

page 373

Ki =

miVCiT VCi

T

Iiω i

---------------------- + ω

i

 

2

 

 

where,

m = mass of link i

V = velocity of center of mass of link i omega = angular velocity of link i

I = mass moment of inertia of link i

Pi = migTRCi

where,

g = gravity vector

R = displacement from base of robot to center of mass of link

• If we have used matrices to formulate the problem, we use the Jacobian to find velocities.

VCi = J( θ )ω

ωi

Consider the example below,

page 374

 

0.2m

 

1m

θ 2

TCP

(xT, yT)

1m

 

Mlinks = 5kg

Ilinks

= 10

θ 1

Mtool = 0.5kg

Itool

= 1

(xb, yb)