- Email: [email protected]
Vibration Damping in Flexible Fingers of an Underwater Robot Hand Via Sliding Modes
Cupyright © 1996 IFAC 13th Triennial Wurld Congress. San Franciscu. USA
Ib-03 2
VIBRATION DAMPING IN FLEXmLE FINGERS OF AN UNDERWATER ROBOT HAND VIA SLIDING MODES
G. Bartolini*, W. Capnto*, M. Cecchi*, A. Ferrara*, L. Fridman+
*Department of Communication. Computer and System Sciences~University of Genova Via Opera Pia 13 16145 Genova -lrALY rel.- +39/0 3532207 - Fax.' +39 103532948 E-mail: [email protected]
+ School of Mathematical Sciences - TeI Aviv University Ramat-Aviv 69978 rei Aviv - ISRAEL rel.- 972-3-6408031 - Fax .' 972-3-6409357 E-mail.' [email protected]
Abstract. The aim of the present paper is to formulate a simple model of a flexible robotic structure and to propose a class of control strategies based on the sliding mode concept. In the first part, a simplified model of a finger of an unconventional gripper is presented. In the second part the problem of suppressing vibrations in flexible structures by means of the sliding mode technique is considered. Then an adaptive control strategy allowing to damp oscillations also in presence of delay is proposed, Simulation examples are provided to complement the theoretical discussion. Keywords. Discontinuous control, Vibration dampers, Sliding mode, Delay analysis, Robot control.
I. INTRODUCTION!
The work presented in this paper is relevant to the project AMADEUS (Advanced MAnipulator for DEep Underwater Sampling, EEC MAST program) whose goal is that of achieving good underwater sampling capabilities by using advanced grippers with flexible fingers. For the mechanical realization of the fingers of the AMADEUS gripper the so-called "elephant's trunk" technology (Davies, 1994), which appears to be particularly suitable for operating underwater, has been adopted. This technology allows the motion of the fingers (or tentacles, [Work supported by Project MAST AMADEUS under contract "MAS2funded by the European Community.
cr91~OOI6"
for very wide structures) without requiring the presence of rotative or prismatic joints. The aim of this paper is to define a model for the flexible finger of the AMADEUS gripper and design a control strategy for damping its vibrations around the equilibrium configurations. More precisely, a lumped parameters formulation of the model will be presented. This model relies on the assumption that the whole mass of the system is concentrated at the free cnd of the finger. Moreover, linear relationships for elasticity have been adopted. In the second part of the paper, the problem of suppressing vibrations in flexible structures, and in particular in the robotic element considered, by means of sliding mode control will be dealt with. Sliding mode control gives rise
73
to stable and robust closed loop systems in case of finitedimensional systems without delay (Utkin, 1992; Drakunov and Utkin, 1990). Yet, in many real cases, with robotic structures of the type considered, it is impossible (0 obtain an instantaneous control action , because of the intrinsic presence of delays which cannot be neglecled (Bartolini, et al., 1995). The delay does not allow one to
make an ideal sliding mode control and leads to oscillations. According to recent theoretical results (Shustin. et al.. 1994; Fridman, el al., 1993), a control strategy that
enables to damp oscillations also in presence of delay will be proposed in this work. It is characterisod by the fact that gain of the classical sliding mode technique is adaptively adjusted explicitly taking into account the delay of propagation of the actuation forces . The application of the
Fig. I. Robotic finger.
algorithm to the flexible robotic structure, whose model is the object of the first part of this work. turns out to be particularly effective as shown by simulation.
2. FORMULAnON OF TIlE LUMPED PARAMETERS MODEl.
The peculiar robotic finger consists of three tubes, the ends of which are connected to two plates (Fig. I ). One of these is assumed to be the basis of the whole system and is therefore supposed to be fixed. while the other one is the
Fig.2. Local coordinate systems placed on the free plate. v
free end . The goal is to control the position of the free end
of the finger by acting on the pressures of the oil (or of any other actuation means) which is present within each tube. To formulate a simple lumped parameters model, it is necessary to introduce some simplificative assumpti ons. More precisely. the linear theory of elasticity is applied (Novacki, 1963). Each tube in the system can be considered as a rod with circular scction undergoing axial, transversal and torsional defonnalions. Choosing the local coordinate systems as indicatcd in Fig.2. the relationship between displacements and rotations of one end of a rod and the forces and torques due to the defonnations can be cxprc.'ised in Ihe following way
i = 1, 2, 3
v
0·· ....................
.!
,
,
..-.....,'i \-----<>
20···
o
~
.......... .!
Fig.3. Coordinate system placed on the barycentre of the free plate. Due to the peculiar structure of the robotic element considered, a coordinate system is chosen placed in the barycentre of the free plate (Fig.3). Now, considering elementary displacemenls of the plate along its axes, the resulting displacements of nodes I, 2. 3 along their axes can be found
(I)
where ti and mj are respectively the forces and torques
vectors acting at the tip of the ;-th rod (node); di and
'Pi
are
respectively the displacements and rotations of the coordinate system on node i. and K is the stiffness matrix whose coefficients need to be experimentally identified.
(2)
where QJ, Q" Q3 are the kinematic matrices. Equations (1) and (2) yield the relationship between forces and torques due to the deformations of each rod and the movements of the plate. i.e.,
74
(3)
As far as the actuation forces are concerned. it can be observed that their direction is a]ways perpendicular to the plate plane. In fact, they are caused by the oil pressure in
then effectively constrained to lie within a certain subspace of the state space. The system is thus formally equivalent to an unforced system of lower order, called "equivalent system", whose (;haracleristic polynomial can be arbitrarily selected by the designer. The equivalent system must be asymptotically stable to ensure that the state approaches the state space origin within the sliding mode.
the tubes, so the actuation torque will have null component
along the x-axis. In order to obtain the equilibrium equations for each elastic element of the structure it is
Let us consider the equation r-K,a =
necessary to find out the resultant of the actuation forces
M~
(6)
(and torques) acling on each free node. Lel us call r the actuation forces vedor, that is
(4)
To apply sliding mode theory, we need to represent system (6) in state variable form. Let a be the equilibrium point vector due to the input force r , then M is the difference vector satisfy ing
where F, T are, respectively. the forces and torques acting on the plate. The equilibrium equation of the whole system
MAii = -K,Aa +Bu
can be written as
Since M is invertible, one obtains Aa :::: - KAa + :Du
where uT = [u
u
I
5
u
6
l,
since only the first, the fifth
and the sixth degrees of freedom are actually controllable (5)
Extending these results to the dynamic case we can keep on considering the lumped parameters hypothesis and so suppose that the masses and inertial moments of the tubes
are very small compared to those of the plate. Then, the inertial term Mii can be added to the r.h.s. of equation (5). Due to the particular choice of the coordinate system of the free plate (Fig.3), the inertial matri x M is diagonal.
3. THE SLIDING MODE CONTROL APPROACH
To achieve the goal of vibration damping, a discontinuous control law is used, based on the sliding mode technique (Utkin, 197H). This approach enables to control the system also in case of parametric uncertainties and external disturbances. The essential feature of a variable structure control system is the non-linear feedback control which has a discontinuity on onc Of more manifolds in the state space (Zi nober, 1994). Thus the structure of the feedback system is altered as its state crosses each discontinuity surface. As a consequence, the closed loop system is said to be a variable structure control (VSC) system. The design of a
(see vector r). Then, setting x T = [M T
Aa T],
one
obtains
(7)
Since matrix K has the same structure as the stiffness matrix K (as far as the position of null entries is concerned), one can split system (7) inlo three independent SUbsystems, that is l)
First response mode (xJl
11)
Second response mode (xW\ol X2
= Xg
x6 =
VSC system e ntails the choice of the switching surfaces ,
x l2
the specification of the discontinuous control functions and
.i8 = -kn X 2 - k26 X6 _ _
the determination of the switching logic associ ated with
xI2 -= -k 62x2 - k 66x6
the discontinuity surfaces. Due to the action of the control law the system tends la reach the discontinuity surface. Once the state starts sliding, the motion of the system is
75
I
+-u.
J,
(8)
Note that u, acts on the rotation angle '1', and , indirectly, on the displacement ur of the plate at the end of the finger. u
x5 : : .t) I
0
0
eu
o
-0.13
X3 = x9
E
u
.Q
o
05
5
-1
-0.2
0
0.5
-0.4
0
'"
x9 = -[33-"3 - [3,X5
xII
'8 0·'8 g o:a -O';B og
o.s E
UIl Third response mode (x,J,l
I = -kSj xl - k55 x S +
JUs
1!
y
0
~M'
-0.5
"()I.6
.,
Also in this case the coupling between the displacement and the rotatio n angle 'Pr is apparent.
Uz
Assuming that X2, X(i. X1 2 and X3. Xs. XII and Xj, X7 are available it is possible to define the sliding surfaces S" S"
S,
o
05
.a.re
0.5
'"
]o~ 0
0
0.5
0
0.5
'" FigA. Closed loop system with zero initial conditions and f, = ID Kg, f, = f, = 20 Kg .
S6S. < -K' must be satisfied (Utkin, 1992). Choosing SI:::: x7 + clXt
Ss::::
xII +c5 x S
+c3x 3
= x l 2 +c6 x 6 +cZ x 2
S6
and the control1aws -y ,sign(S,)
", =
u, = -y ,sign(S,)
u6
= -y 6 sigll (S6)
(9)
In particular, considering the second response mode. if a control law U6 exists such that the sliding surface S6 is reached in finite time, then system (8) is equivalent to the reduced order system
(10)
The characteristic polynomial of system (11) is
which is Hurwitz if and only if coefficients
C2. C6
are
k"
chosen such that c 6 > 0 and 0< c 2 < -:;:-c6 -
k"
In order to obtain S6
=
0 the so-caned "reaching condition"
with Y>k62Ix21+k66lx61+c2Ix81+c6IxI21+K2 the above condition holds. In the same manner the control law u, for the third response mode is designed. In FigA the behaviour of the closed loop system obtained by applying the sliding mode approach is shown. In this case it is assumed to apply the action forces f, = ID Kg, f, = f, = 20 Kg. Sirnulations presented in this paper have been realized using Mat lab/Simulink software on a PC 486, In particular a Runge-Kutta algorithm with a very small integration step has been chosen.
4. DISCONTINUOUS CONTROL IN DELAYED SYS1EMS Up to now. the assumption that the observation of the slate variables was without delay and that the corresponding control forces would reach immediately the plate placed at the end of the tubes was made. In practice, it will be impossible to obtain an instantaneous control action due to the propagation effect. Indeed, let us consider the closed loop system of the previous section with zero initial conditions, f, = ID Kg, f, = f, = 20 Kg, and assume that the observed variables are affected by a time delay 't, In Fig.5 and Fig.6 displacemenl' and rotations trends are reported. It is apparent that an increment in time delay increases oscillations amplitudes because of the inefficiency of the control action. Referring '0 recent studies (Shustin, el aI" 1994; Fridman,
76
'8 EJ D
to
0S
us ~
[]
o
·0.5
[]
~
~
-0.5 .
o
0.5
-,
a
·OA
0.5
'"
'"
~·~EJ
e 0:8 0'·
~-O .OA
-0.5
-DJE
-1
-ocs
o
'()2
0.5
~
0
OS
o .,g
0
0
0.5
0
0.5
'"
'8 8 g
-0.5
0:8 o
0.5
[] 0.5
0
5 -05
§ -0 .2
-1
[]
~~EJ
~ ,()_04
-O.S
-{HE
-1
[]
05
-0-4
0.5
'"
!o
-OCE
Definition: a solution of a DDeS is called "steady mode" if it has constant frequcncy function. Theorem : any solution of a DDCS coincides with some steady mode after some time interval.
Fig.5. Closed loop system with zero initial conditions and f, = 10 Kg, f, = f, = 20 Kg, t = 0.0005 sec.
~ []
This is a function valued in NvIO, +00 J, which counts the zeros of v(t) on intervals of length t.
0.05
'"
0.5
=max{t''; rlv(t') = O}
[]
o.,g
Relying on the above theorem and to other considerations (Shustin, et al., 1994), it can be stated that steady modes are an extension of the notion of sliding modes to DDCS. Since constant frequency solutions have also constant amplitudes. it is necessary to design an algorithm th at, observing the distribution of zeros of the sol ution (Le. its frequenc y), enables to reduce the solution amplitude. If the following lemma holds, then the solution x(t) of (11) is bounded and the maximum time delay compatible with stability can he evaluated. Lemma : let, in system (J I)
0.5
'"
~
F(O, V) " 0
of '5.k , ox
-
0 Il3
[]
0.5
[]
[]
..,
I
"f Icp U) < -,
then all solutions of (I I) with
et al.. 1993), where the problem of stability and stabilization in discontinuous delayed control systems is addressed, let us summari ze some important properties of systems with delay which are analogous to the one referred to in this paper. Consider the system
(11 )
where YE R" . In particular. according to Shustin. et al. (1994), onc has: Definition : let v(t) be a t-delayed output in a DDeS (Discontinuous Delayed Control System). Define the "frequency function" of v(t) as
t
e [-t,O]
k
tl.5
Fig.6. Closed loop system with zero initial conditions and f, = 10 Kg, f, = f, = 20 Kg, t = 0.00 I sec .
X(t ) = -y . sign[x(t -t)]+ F[x(t) , V(t )] { \' (1) = AV+Bx
In 2 t
~ (O)I < y(2e
-I«
- I)
k satisfy the inequality
Ix(t)1< y(e
I«
-I) . Under the above k assumptions, the following adaptive algorithm can be designed with the aim of damping permanent oscillations of the considered robotic finger .
OF klx (O)I _ I.Choose: k ;o,-,ao > - I« ,to OX 2e - I
=T
2. Wait the first zero crossing of x(t-t) after
1,..
3. Taking into account the distribution of zeros of x(t-t) in the interval [0, lit. estimate the next zero crossing ~+I of x(1-t). At
a
77
I =
t;+l update the gain ex; according to
=a .p.
1+11/+1
, p,=
-1
(2)
eH, - I« ,0,=° 0 1+2e - 1 i
where
80 is
In Fig. 7 and in Fig. 8 the behaviour of increasing values of the time delay
"t
X3,
O~b
o~~
the observation error. Then, iterate steps 2 to 3.
.a.5
x" S, for
are shown in the cases
of constant (I st column) and adaptive (2nd column) gain. The satisfactory effect of the adaptive component to damp
02
0
.a~~
the vibrations is apparent.
02
0
·0.5
04
0.4
·5
An unconventional flexible robotic structure is considered
0.2
0
0.4
.00:E 0 0.2
0.4
.,~E
~t-
5. CONCLUSIONS
0.2
0
0.4
0
0.2
0.4
s:
0.4
s x 10
in this paper. Its lumped parameters model is first
1~ f
formulated and a sliding mode control strategy with adaptive gain is proposed. Simulation evidence shows that sliding mode control is particularly efficient in vibration
0
0.2
damping also if a time delay affects the controllable degrees of freedom of the system. Fig. 7. Closed loop system with zero initial conditions and f, \0 Kg, f, f, 20 Kg, " = 0.0005 sec.
=
REFERENCES
= =
Bartolini, G., A. Ferrara and V.1. Utkin (1995). Adaptive sliding mode control in discrete time systems.
A"tomatica, 31,769-773. Davies, B. (1994). The elephant trunk gripper. Internal Report, Herriott Watt University. Drakunov, S.V. and V.1. Utkin (1990). Sliding mode in dynamic systems. International Journal of Control, 55, 1029 - 1037. Fridman, L.M., E.M. Fridman and E.1. Shustin (1993). Steady modes in an autonomous system with break and delay. Differential EqUfJtions. 29, No. 8. Nowacki. W. (1963). Dynamics of elastic systems. Chapman & Hall Ltd., London. Shustin, E.I., E.M. Fridman and L.M. Fridman (1994).
o~~
o~~
·0.5
.as
0
0.2
.a~~ 0
.,~F 0
0.4
02
04
02
0.4
and
Lyapunov
Techniques,
0.2
0.4
.oo:~ 0
0.2
0.4
0 x 10$
0.2
0.4
0
02
.,:F 1~ f
Sliding modes, stability and stabilization in discontinuous delay control systems. Proceedings of the Workshop on Robust Control via Variable
Structure
0
161-165,
~
04
Benevento, Italy.
Utkin, V.l. (1978). Sliding modes and their application in variable structure systems. Mir publisher, Moscow. Utkin, V.I.(l992). Sliding modes in control and
Fig. 8. Closed loop system with zero initial conditions and f, = 10 Kg, f2 = f, 20 Kg," 0.001 sec.
=
optimization. Springer - Verlag, Berlin.
Zinober, A.S.1. (1994). Variable Structure and Lyapunov Control. Springer-Verlag, London.
78
=
Ib-03 2
VIBRATION DAMPING IN FLEXmLE FINGERS OF AN UNDERWATER ROBOT HAND VIA SLIDING MODES
G. Bartolini*, W. Capnto*, M. Cecchi*, A. Ferrara*, L. Fridman+
*Department of Communication. Computer and System Sciences~University of Genova Via Opera Pia 13 16145 Genova -lrALY rel.- +39/0 3532207 - Fax.' +39 103532948 E-mail: [email protected]
+ School of Mathematical Sciences - TeI Aviv University Ramat-Aviv 69978 rei Aviv - ISRAEL rel.- 972-3-6408031 - Fax .' 972-3-6409357 E-mail.' [email protected]
Abstract. The aim of the present paper is to formulate a simple model of a flexible robotic structure and to propose a class of control strategies based on the sliding mode concept. In the first part, a simplified model of a finger of an unconventional gripper is presented. In the second part the problem of suppressing vibrations in flexible structures by means of the sliding mode technique is considered. Then an adaptive control strategy allowing to damp oscillations also in presence of delay is proposed, Simulation examples are provided to complement the theoretical discussion. Keywords. Discontinuous control, Vibration dampers, Sliding mode, Delay analysis, Robot control.
I. INTRODUCTION!
The work presented in this paper is relevant to the project AMADEUS (Advanced MAnipulator for DEep Underwater Sampling, EEC MAST program) whose goal is that of achieving good underwater sampling capabilities by using advanced grippers with flexible fingers. For the mechanical realization of the fingers of the AMADEUS gripper the so-called "elephant's trunk" technology (Davies, 1994), which appears to be particularly suitable for operating underwater, has been adopted. This technology allows the motion of the fingers (or tentacles, [Work supported by Project MAST AMADEUS under contract "MAS2funded by the European Community.
cr91~OOI6"
for very wide structures) without requiring the presence of rotative or prismatic joints. The aim of this paper is to define a model for the flexible finger of the AMADEUS gripper and design a control strategy for damping its vibrations around the equilibrium configurations. More precisely, a lumped parameters formulation of the model will be presented. This model relies on the assumption that the whole mass of the system is concentrated at the free cnd of the finger. Moreover, linear relationships for elasticity have been adopted. In the second part of the paper, the problem of suppressing vibrations in flexible structures, and in particular in the robotic element considered, by means of sliding mode control will be dealt with. Sliding mode control gives rise
73
to stable and robust closed loop systems in case of finitedimensional systems without delay (Utkin, 1992; Drakunov and Utkin, 1990). Yet, in many real cases, with robotic structures of the type considered, it is impossible (0 obtain an instantaneous control action , because of the intrinsic presence of delays which cannot be neglecled (Bartolini, et al., 1995). The delay does not allow one to
make an ideal sliding mode control and leads to oscillations. According to recent theoretical results (Shustin. et al.. 1994; Fridman, el al., 1993), a control strategy that
enables to damp oscillations also in presence of delay will be proposed in this work. It is characterisod by the fact that gain of the classical sliding mode technique is adaptively adjusted explicitly taking into account the delay of propagation of the actuation forces . The application of the
Fig. I. Robotic finger.
algorithm to the flexible robotic structure, whose model is the object of the first part of this work. turns out to be particularly effective as shown by simulation.
2. FORMULAnON OF TIlE LUMPED PARAMETERS MODEl.
The peculiar robotic finger consists of three tubes, the ends of which are connected to two plates (Fig. I ). One of these is assumed to be the basis of the whole system and is therefore supposed to be fixed. while the other one is the
Fig.2. Local coordinate systems placed on the free plate. v
free end . The goal is to control the position of the free end
of the finger by acting on the pressures of the oil (or of any other actuation means) which is present within each tube. To formulate a simple lumped parameters model, it is necessary to introduce some simplificative assumpti ons. More precisely. the linear theory of elasticity is applied (Novacki, 1963). Each tube in the system can be considered as a rod with circular scction undergoing axial, transversal and torsional defonnalions. Choosing the local coordinate systems as indicatcd in Fig.2. the relationship between displacements and rotations of one end of a rod and the forces and torques due to the defonnations can be cxprc.'ised in Ihe following way
i = 1, 2, 3
v
0·· ....................
.!
,
,
..-.....,'i \-----<>
20···
o
~
.......... .!
Fig.3. Coordinate system placed on the barycentre of the free plate. Due to the peculiar structure of the robotic element considered, a coordinate system is chosen placed in the barycentre of the free plate (Fig.3). Now, considering elementary displacemenls of the plate along its axes, the resulting displacements of nodes I, 2. 3 along their axes can be found
(I)
where ti and mj are respectively the forces and torques
vectors acting at the tip of the ;-th rod (node); di and
'Pi
are
respectively the displacements and rotations of the coordinate system on node i. and K is the stiffness matrix whose coefficients need to be experimentally identified.
(2)
where QJ, Q" Q3 are the kinematic matrices. Equations (1) and (2) yield the relationship between forces and torques due to the deformations of each rod and the movements of the plate. i.e.,
74
(3)
As far as the actuation forces are concerned. it can be observed that their direction is a]ways perpendicular to the plate plane. In fact, they are caused by the oil pressure in
then effectively constrained to lie within a certain subspace of the state space. The system is thus formally equivalent to an unforced system of lower order, called "equivalent system", whose (;haracleristic polynomial can be arbitrarily selected by the designer. The equivalent system must be asymptotically stable to ensure that the state approaches the state space origin within the sliding mode.
the tubes, so the actuation torque will have null component
along the x-axis. In order to obtain the equilibrium equations for each elastic element of the structure it is
Let us consider the equation r-K,a =
necessary to find out the resultant of the actuation forces
M~
(6)
(and torques) acling on each free node. Lel us call r the actuation forces vedor, that is
(4)
To apply sliding mode theory, we need to represent system (6) in state variable form. Let a be the equilibrium point vector due to the input force r , then M is the difference vector satisfy ing
where F, T are, respectively. the forces and torques acting on the plate. The equilibrium equation of the whole system
MAii = -K,Aa +Bu
can be written as
Since M is invertible, one obtains Aa :::: - KAa + :Du
where uT = [u
u
I
5
u
6
l,
since only the first, the fifth
and the sixth degrees of freedom are actually controllable (5)
Extending these results to the dynamic case we can keep on considering the lumped parameters hypothesis and so suppose that the masses and inertial moments of the tubes
are very small compared to those of the plate. Then, the inertial term Mii can be added to the r.h.s. of equation (5). Due to the particular choice of the coordinate system of the free plate (Fig.3), the inertial matri x M is diagonal.
3. THE SLIDING MODE CONTROL APPROACH
To achieve the goal of vibration damping, a discontinuous control law is used, based on the sliding mode technique (Utkin, 197H). This approach enables to control the system also in case of parametric uncertainties and external disturbances. The essential feature of a variable structure control system is the non-linear feedback control which has a discontinuity on onc Of more manifolds in the state space (Zi nober, 1994). Thus the structure of the feedback system is altered as its state crosses each discontinuity surface. As a consequence, the closed loop system is said to be a variable structure control (VSC) system. The design of a
(see vector r). Then, setting x T = [M T
Aa T],
one
obtains
(7)
Since matrix K has the same structure as the stiffness matrix K (as far as the position of null entries is concerned), one can split system (7) inlo three independent SUbsystems, that is l)
First response mode (xJl
11)
Second response mode (xW\ol X2
= Xg
x6 =
VSC system e ntails the choice of the switching surfaces ,
x l2
the specification of the discontinuous control functions and
.i8 = -kn X 2 - k26 X6 _ _
the determination of the switching logic associ ated with
xI2 -= -k 62x2 - k 66x6
the discontinuity surfaces. Due to the action of the control law the system tends la reach the discontinuity surface. Once the state starts sliding, the motion of the system is
75
I
+-u.
J,
(8)
Note that u, acts on the rotation angle '1', and , indirectly, on the displacement ur of the plate at the end of the finger. u
x5 : : .t) I
0
0
eu
o
-0.13
X3 = x9
E
u
.Q
o
05
5
-1
-0.2
0
0.5
-0.4
0
'"
x9 = -[33-"3 - [3,X5
xII
'8 0·'8 g o:a -O';B og
o.s E
UIl Third response mode (x,J,l
I = -kSj xl - k55 x S +
JUs
1!
y
0
~M'
-0.5
"()I.6
.,
Also in this case the coupling between the displacement and the rotatio n angle 'Pr is apparent.
Uz
Assuming that X2, X(i. X1 2 and X3. Xs. XII and Xj, X7 are available it is possible to define the sliding surfaces S" S"
S,
o
05
.a.re
0.5
'"
]o~ 0
0
0.5
0
0.5
'" FigA. Closed loop system with zero initial conditions and f, = ID Kg, f, = f, = 20 Kg .
S6S. < -K' must be satisfied (Utkin, 1992). Choosing SI:::: x7 + clXt
Ss::::
xII +c5 x S
+c3x 3
= x l 2 +c6 x 6 +cZ x 2
S6
and the control1aws -y ,sign(S,)
", =
u, = -y ,sign(S,)
u6
= -y 6 sigll (S6)
(9)
In particular, considering the second response mode. if a control law U6 exists such that the sliding surface S6 is reached in finite time, then system (8) is equivalent to the reduced order system
(10)
The characteristic polynomial of system (11) is
which is Hurwitz if and only if coefficients
C2. C6
are
k"
chosen such that c 6 > 0 and 0< c 2 < -:;:-c6 -
k"
In order to obtain S6
=
0 the so-caned "reaching condition"
with Y>k62Ix21+k66lx61+c2Ix81+c6IxI21+K2 the above condition holds. In the same manner the control law u, for the third response mode is designed. In FigA the behaviour of the closed loop system obtained by applying the sliding mode approach is shown. In this case it is assumed to apply the action forces f, = ID Kg, f, = f, = 20 Kg. Sirnulations presented in this paper have been realized using Mat lab/Simulink software on a PC 486, In particular a Runge-Kutta algorithm with a very small integration step has been chosen.
4. DISCONTINUOUS CONTROL IN DELAYED SYS1EMS Up to now. the assumption that the observation of the slate variables was without delay and that the corresponding control forces would reach immediately the plate placed at the end of the tubes was made. In practice, it will be impossible to obtain an instantaneous control action due to the propagation effect. Indeed, let us consider the closed loop system of the previous section with zero initial conditions, f, = ID Kg, f, = f, = 20 Kg, and assume that the observed variables are affected by a time delay 't, In Fig.5 and Fig.6 displacemenl' and rotations trends are reported. It is apparent that an increment in time delay increases oscillations amplitudes because of the inefficiency of the control action. Referring '0 recent studies (Shustin, el aI" 1994; Fridman,
76
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to
0S
us ~
[]
o
·0.5
[]
~
~
-0.5 .
o
0.5
-,
a
·OA
0.5
'"
'"
~·~EJ
e 0:8 0'·
~-O .OA
-0.5
-DJE
-1
-ocs
o
'()2
0.5
~
0
OS
o .,g
0
0
0.5
0
0.5
'"
'8 8 g
-0.5
0:8 o
0.5
[] 0.5
0
5 -05
§ -0 .2
-1
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~ ,()_04
-O.S
-{HE
-1
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05
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0.5
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!o
-OCE
Definition: a solution of a DDeS is called "steady mode" if it has constant frequcncy function. Theorem : any solution of a DDCS coincides with some steady mode after some time interval.
Fig.5. Closed loop system with zero initial conditions and f, = 10 Kg, f, = f, = 20 Kg, t = 0.0005 sec.
~ []
This is a function valued in NvIO, +00 J, which counts the zeros of v(t) on intervals of length t.
0.05
'"
0.5
=max{t''; rlv(t') = O}
[]
o.,g
Relying on the above theorem and to other considerations (Shustin, et al., 1994), it can be stated that steady modes are an extension of the notion of sliding modes to DDCS. Since constant frequency solutions have also constant amplitudes. it is necessary to design an algorithm th at, observing the distribution of zeros of the sol ution (Le. its frequenc y), enables to reduce the solution amplitude. If the following lemma holds, then the solution x(t) of (11) is bounded and the maximum time delay compatible with stability can he evaluated. Lemma : let, in system (J I)
0.5
'"
~
F(O, V) " 0
of '5.k , ox
-
0 Il3
[]
0.5
[]
[]
..,
I
"f Icp U) < -,
then all solutions of (I I) with
et al.. 1993), where the problem of stability and stabilization in discontinuous delayed control systems is addressed, let us summari ze some important properties of systems with delay which are analogous to the one referred to in this paper. Consider the system
(11 )
where YE R" . In particular. according to Shustin. et al. (1994), onc has: Definition : let v(t) be a t-delayed output in a DDeS (Discontinuous Delayed Control System). Define the "frequency function" of v(t) as
t
e [-t,O]
k
tl.5
Fig.6. Closed loop system with zero initial conditions and f, = 10 Kg, f, = f, = 20 Kg, t = 0.00 I sec .
X(t ) = -y . sign[x(t -t)]+ F[x(t) , V(t )] { \' (1) = AV+Bx
In 2 t
~ (O)I < y(2e
-I«
- I)
k satisfy the inequality
Ix(t)1< y(e
I«
-I) . Under the above k assumptions, the following adaptive algorithm can be designed with the aim of damping permanent oscillations of the considered robotic finger .
OF klx (O)I _ I.Choose: k ;o,-,ao > - I« ,to OX 2e - I
=T
2. Wait the first zero crossing of x(t-t) after
1,..
3. Taking into account the distribution of zeros of x(t-t) in the interval [0, lit. estimate the next zero crossing ~+I of x(1-t). At
a
77
I =
t;+l update the gain ex; according to
=a .p.
1+11/+1
, p,=
-1
(2)
eH, - I« ,0,=° 0 1+2e - 1 i
where
80 is
In Fig. 7 and in Fig. 8 the behaviour of increasing values of the time delay
"t
X3,
O~b
o~~
the observation error. Then, iterate steps 2 to 3.
.a.5
x" S, for
are shown in the cases
of constant (I st column) and adaptive (2nd column) gain. The satisfactory effect of the adaptive component to damp
02
0
.a~~
the vibrations is apparent.
02
0
·0.5
04
0.4
·5
An unconventional flexible robotic structure is considered
0.2
0
0.4
.00:E 0 0.2
0.4
.,~E
~t-
5. CONCLUSIONS
0.2
0
0.4
0
0.2
0.4
s:
0.4
s x 10
in this paper. Its lumped parameters model is first
1~ f
formulated and a sliding mode control strategy with adaptive gain is proposed. Simulation evidence shows that sliding mode control is particularly efficient in vibration
0
0.2
damping also if a time delay affects the controllable degrees of freedom of the system. Fig. 7. Closed loop system with zero initial conditions and f, \0 Kg, f, f, 20 Kg, " = 0.0005 sec.
=
REFERENCES
= =
Bartolini, G., A. Ferrara and V.1. Utkin (1995). Adaptive sliding mode control in discrete time systems.
A"tomatica, 31,769-773. Davies, B. (1994). The elephant trunk gripper. Internal Report, Herriott Watt University. Drakunov, S.V. and V.1. Utkin (1990). Sliding mode in dynamic systems. International Journal of Control, 55, 1029 - 1037. Fridman, L.M., E.M. Fridman and E.1. Shustin (1993). Steady modes in an autonomous system with break and delay. Differential EqUfJtions. 29, No. 8. Nowacki. W. (1963). Dynamics of elastic systems. Chapman & Hall Ltd., London. Shustin, E.I., E.M. Fridman and L.M. Fridman (1994).
o~~
o~~
·0.5
.as
0
0.2
.a~~ 0
.,~F 0
0.4
02
04
02
0.4
and
Lyapunov
Techniques,
0.2
0.4
.oo:~ 0
0.2
0.4
0 x 10$
0.2
0.4
0
02
.,:F 1~ f
Sliding modes, stability and stabilization in discontinuous delay control systems. Proceedings of the Workshop on Robust Control via Variable
Structure
0
161-165,
~
04
Benevento, Italy.
Utkin, V.l. (1978). Sliding modes and their application in variable structure systems. Mir publisher, Moscow. Utkin, V.I.(l992). Sliding modes in control and
Fig. 8. Closed loop system with zero initial conditions and f, = 10 Kg, f2 = f, 20 Kg," 0.001 sec.
=
optimization. Springer - Verlag, Berlin.
Zinober, A.S.1. (1994). Variable Structure and Lyapunov Control. Springer-Verlag, London.
78
=