.m
2012
18th International Conference
ENGINEERING MECHANICS 2012
pp. 28–29
Svratka, Czech Republic, May 14 – 17, 2012
Paper #323
INPUT
T SHAPIN
NG CONT
TROL OF
F ELECT
TRONIC CAMS
P. Beneš*, M . Valášek***, O. Marek
k***
Abstrract: The papeer deals with the input shapping control of electronic ca
ams that elimiinates their reesidual
vibrattions. The moodels of seveeral different kinds of eleectronic camss are describbed, i.e. the simple
tradittional one, thee serial one, the
t parallel onne, the multi-input one. Then the input sshaping contrrol and
its geeneralization is
i described. The generalizzation means the shaper off arbitrary tim
me length and
d/or of
arbitrrary rate com
mbined with the set of shhaping functio
ons of reentrry kind. It iss demonstrateed and
explaiined that som
me more comp
plex shaping functions in comparison with the simpple Heavisidee pulse
shapeers are more robust
r
against model misallignment. Thiss generalized input shapingg control is applied
a
to diff
fferent kinds off electronic ca
ams
Keyw
words: Input shaping, elecctronic cams, residual vibra
ation, reentry commands
1. Intrroduction
Conventtional cam drives
d
in modern machinnes can be replaced
r
by properly conntrolled serv
vomotors.
This conncept is gennerally called
d an electronnic cam and
d can be furrther dividedd into severaal groups
accordinng to system structure – e.g.
e serial, paarallel or multi-input elecctronic cams.. The deman
nd for fast
and preccise positioniing is cruciall in all mentiioned cases but
b it could be
b easily corrrupted by thee residual
vibrationn. To suppreess the unwaanted dynam
mics of flexib
ble system the standard control inpu
ut can be
reshapedd in such a way
w that it do
oesn’t excite flexible mod
des or, moree generally, th
that all the en
nergy put
into flexxible modes is
i completely
y relieved at the end of th
he travel (Miiu, 1989). Thhe differencee between
original unshaped annd shaped sig
gnal as well aas the respon
nse of the two
o-mass modeel is shown in Fig. 1.
Fig. 1:Com
mparison of sshaped and unshaped
u
control input.
ntrol input sh
haping
2. Con
The conntrol input thhat ensures no-vibration
n
positioning has to fulfilll some neceessary condittions. For
the systeem describedd using state space formuulation as
(1)
,
these connditions can be derived in
n the form
*
**
***
Ing. Peetr Beneš: Fakuulta strojní, ČVU
UT v Praze; Teechnická 4; 166 07, Praha 6; CZ, e-mail: petr.b
.benes@fs.cvut.cz
Prof. Ing. Michael Valášek, DrrSc.: Fakulta strojní, ČVUT v Praze; Technická 4; 166 07, Prah
ha 6; CZ,
e-maill: michael.valassek@fs.cvut.cz
Ing. O
Ondřej Marek: VÚTS,
V
a.s.; U Jeezu 525/4, 461 19, Liberec; CZ
Z, e-mail: ondreej.marek@vutss.cz
Beneš P., Valášek M., Marek O.
29
|
(2)
,
⋅
where Ul(s) is the finite time Laplace transform (Miu, 1989) of the l-th input, bl is the corresponding
column of B matrix, t1 and t2 is the start and the finish time, n is the number of inputs. The solution
ul(t) in the time domain is the inverse Laplace transform of Ul(s).
This approach leads to the control input in the form of pre-computed curve. However if it is
rewritten to the form of a dynamical block it acts like a filter that transform any arbitrary signal to
no-vibration one (Beneš & Valášek, 2008). And in contrast with patented input shaping technique by
Singhose & Seering (1990) the length of this shaper is not dependent on the system natural frequency
and can be set arbitrary.
3. Robustness
15
15
10
10
5
5
U(s)
U(s)
Being a feed-forward method all control shaping techniques need precise system models. The
vibration suppression is in fact caused by placing zeros of the control input into the poles of the
system. Therefore incorrect system model causes that the control input is not design properly and
vibrations are not canceled. To increase robustness to modeling errors it is possible to formulate
additional constrains that either introduced more zeros to the control input or increase the order of
existing ones. The price for that is the increase of necessary acting force or longer settling time.
Comparison of spectral analysis of a standard shaper and a robust one is in Fig. 2.
0
−5
−10
−10
0
−5
0
10
Real(s)
−10
0
Imag(s)
10
−10
−10
0
Real(s)
10
0
−10
10
Imag(s)
Fig. 2: Spectral analysis of a) standard shaper b) robust shaper.
4. Conclusions
The presented approach to control of electronic cams and other flexible systems combines advantages
of two different control shaping techniques. It produces command shapers of arbitrary length with
reentry property as well. It is opened to formulation of additional constrains that ensure robustness to
modeling errors.
Acknowledgement
The research has been supported by the Czech Science Foundation, project No. GAP101/11/2110.
References
Beneš, P. & Valášek, M. (1990) Input Shaping Control with Reentry Commands on Prescribed Duration.
Applied Computational Mechanics, Vol. 2, no.2, pp.227-234.
Miu, D.K. (1993) Mechatronics, Electromechanics and Contromechanics. Springer-Verlag, New York.
Singer, N. & Seering, W. (1990) Preshaping Command Inputs to Reduce System Vibration. Journal of Dynamics
Systems, Measurements and Control, Vol. 112, pp.76-82.