Spatial models of an unconstrained rigid body
In this chapter the model of an unconstrained rigid body under spatial motion will be derived for three applications: Rigid body attached to the base by a translational spring-damper element. Spatial servo-pneumatic parallel robot. Model equations of a sp
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Springer-V erlag Berlin Heidelberg GmbH
ONLINE LIBRARY
http://www.springer.de/engine/
Hubert Hahn
Rigid Body Dynamics of Mechanisms 2
Applications
With 228 Figures
'Springer
Professor Dr. Hubert Hahn Universität Gh Kassel Regelungstechnik und Systemdynamik, FB Maschinenbau Mönchebergstraße 7 D-34109 Kassel Germany
e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographieinformation published by Die Deutsche Bibliothek Die Deutsche Bibliotheklists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in theInternetat ..+f(p,v) + qc(p,v),
(2.3a)
jJ
(2.3b)
=:Cf with
g(p) = 0
(2.3c)
as the constraint position equation, which is written in an implicit form. Consider P;nct E ~np-nc as the vector ofthe minimal (independent) Cartesian coordinates of p with for planar mechanisms and for spatial mechanisms,
(2.4a)
and Pctep E ~nc as the vector of the dependent coordinates. Then V;nct can be written in the form
Pind
and
(2.4b) (2.4c)
Pind
= Prind · P
Pind
=
Vind
= Prvind ·V= Prind
Vind
= Prind
Pr ind
·
P
(2.4d) (2.4e)
·V
(2.4f)
·V
with the projection matrix p .
av m_d E ap. ___ ,_ ~ ap -
rmd . -
av
JH.np-nc,np
(2.4g)
or apind 1lßP1 Prind
=
(
.
, ... ,
.
8pinct
1IPnp
. .
' .' . apind np - nc I apl ' ... ' OPind np - nc I Pnp 0
•
0
) •
2.2 Model equations in symbolic DE form
13
Here Prvind has been chosen as Prind in (2.4e) and (2.4f). Then, the vectors of the dependent coordinates and velocities are P dep = ( Pctep
1l • • • l
(2.4h)
Pctep nc ) T E JR."c
Pdep
=
P
(2.4i)
Pctep
= Prdep · P
(2.4j)
vdep = vdep
P,-dep ·
(2.4k)
Prdep. V
= P,-ctep · v
(2.41)
with the projection matrix
P rdep .._- 8pdep -_ 8vdep 8p 8v
E
mnc,np
(2.4m)
_!!.3!._
= ( 8pind ) T
(2.4n)
__!!3!._
= (
m,.
Then
8p
=
__!!E_
=
pT d =
8pind
rm
8p
8vind
and =
pT rdep
8pdep
8v dep
8pctep ) 8p
T
(2.4o)
'
and the following relations hold: P
= 88p . Pind + __!!E_ . Pdep = 8 Pind
Pctep
pT
rmd
.
Pind
+ pTrdep . Pctep
(2.4p)
Vinct
:r + prdep. Vctep·
(2.4q)
and
8v
V=->)--. uvind
V;nd
+ 8v
->)--.
Vctep
:r
= prind.
uvdcp
The implicit constraint position equation (2.3c) will now be written in the explicit form (2.5a)
w Pind
w f--+
P
=
h(pinJ
with the nc components h; of h as the solutions Pctep j =Pi
= h;(p,nJ
(2.5b)
of the nc independent, consistent and smooth constraint position equations (2.3c). The time derivative of (2.5a) is jJ = hPind (p,nJ. Pind with hPind (p,nJ :=
8 ~;.inJ E md
]R_np,np-nc.
(2.5c)
2. Model equations in symbolic DAE and DE form
14
Due ta (2.4d), (2.3a), and (2.5a), Pind = Prind · P = Prind · T(p)
(2.5d)
·V
ar
Pind = Prind · T(h(pind))
(2.5e)
·V.
Introducing the relatian (2.5f) which defines the matrix T,nct(P,nJ, yields, tagether with (2.5e): Tind(Pind) · Vind =Prim!· T(h(p,"J)
·V
Assuming that T,nct(P,nd) is a regular matrix, this yields v,nd = T;;;;(P,nJ · Prind · T(h(p,"J) · v.
(2.5g)
Tagether with (2.4c), this yields thc rclat
