Passification of an electrohydraulic two-stage pressure control servo ...
Proceedings of the American Control Conference Anchorage, AK May 8-10, 2002
Passification of an Electrohydraulic Two-stage Pressure Control Servo-valve Kailash Krishnaswamy' and Perry Y. Li2 Department of Mechanical Engineering University of Minnesota 111, Church St. SE,Minneapolis, MN 55455 (kk,pli}@me .umn .edu Abstract
has been applied to enhance the safety and human friendliness of many mechanical and electromechanical systems, for example [l, 2,3].
Passive systems have the beneficial property that their interconnections with other passive systems result in necessarily stable systems. This vmue of passive systems can be exploited with hydraulic systems to build owerful machines that are safe and human friendly. A digculty in a plying the passivity concept to electrohydraulic systems is i a t they are not intnnsically passive. In a previous paper, passification techniques were developed to render single-stage directional control valves as passive two-port devices with a supply rate related to the wer in ut at the hydraulic port. In this paper, we extend passiffcation technique to two stage pressure control servo-valves. The assified two-stage valve avoids the flow rate and bandwidthyimitations that exist for single-stage valves.
While the inherent passivity property of the above mentioned applications is well known, the passivity property of electrohydraulic systems was not stuQed until recently. It is shown in [4] that the single-stage four way directional control valve is not intrinsically passive. Structural modification of valve design (such as adding leaka e and pressure feedback), as well as active passification tecfnique that utilized feedback control to achieve suitable spool dynamics, are also proposed in [SI to render the four way single-stage directional control valve a passive two-port device. A critical element in the passification process is the use of load pressure feedback on the spool, either implemented structurally or via feedback control. The pro osed active passification technique has been implementefon a direct acting proportional control valve and applied in the control s stem for passive bilateral teleo eration of a single degree off?eedom hydraulic actuator p61 via a motorized joystick. The teleoperation experiment achieved 1) a specified ower scaling between human operator input and the mackne output and 2) a kinematic scaling between the motions of the j o stick and the hydraulic actuator. Although successful, &t single-stage proportional control valve is inherently limiting in terms of its bandwidth and flow rating due to the limited physical capabilities of the solenoids used to move the spool. To overcome this limitation, two-stage valves are needed since stronger hydraulic forces can be used to move the spool quickly.
1 Introduction Hydraulics are useful in applications where high ower density, good flexibility and portability are needel. Our research interests focus on making hydraulic systems safer and more human friendly in directly human controlled applications, and in applications that involve the machines interacting with uncertain or delicate environment. Roughly speaking, an energetically passive system is a system that apparently does not generate any energy of its own. It has the desirable property of ensuring stability whenever cou led with other strictly passive systems. This property enatles passive systems to operate robustly in uncertain physical environments which consist mainly of passive or stnctly passive objects. A passive system is also mherently safer than a non-passive system because the amount of energy that it can impart on its work environment is limited. Hence the possibility of a assive hydraulic machine unintentionally damaging an o\ject, for example rupturing an underground pipeline while earth moving, is limited.
In this pa r, we develop an active passification technique using feexack control for a generic two-stage ressure control servo-valve that uses a pressure control pifot and a single spool pressure control main stage. As a two-stage valve it has the potential to overcome the performance limitations of single-stage valves. A descriphon of the valve can be found in [7]. We choose to assify pressure control servovalves because pressure fee&ack, which is critical for passification,is often already present in their structuraldesigns. This is generally not available in flow control servo-valves, which are designed to be insensitive to load ressures. The bond graph perspective for developin passifcation control law and dynamics for single-stage vafves, developed in [SI will be extended in this paper.
We say that a dynamical system with input u ( t ) and output y ( t ) is passive with respect to a supply rate s(u,y) E & ! if s ( u ( t ) , y ( t ) ) is a L1, integrable function with respect to t and given a set of initial conditions, there exists a constant, c so that for all time, t and for all inputs, U(.),
The rest of the paper is organized as follows. Section 2 presents the d y n m c s of the various subsystems of the pressure control valve and formulates the passive control problem. Section 3 presents the passification algorithm. Section 4 develops a teleoperation control for the low-frequency approximated passified ressure control valve and simulation results are also provi&d. Section 5 contains discussion and concluding remarks.
For physical systems and for our purpose, useful supply rates are those associated with actual power into the system. For example, if effort and flow variables are used as input and output variables, the inner product of the input and the output is a sup ly rate related to actual power. Eq. (1) means that the tot$ energy that can be extracted from a passive system, is at most the initial energy, 3.Passivity
0-7803-7298-0/02/$17.00 0 2002 AACC
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... .._
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Figure 2: A magnified schematic of the left pilot pressure chamber with the flapper *........ '& ........................ %' QLI
9
" :1
where K~ = C d W E and for pS> sgn(x,)PL,
Figure 1: The pressure control servo-valve
2 Dynamic model and problem formulation A similar expression can be obtained for the unusual case when Ps < sgn(x,)PL. Note that K,(x,,Pt) > 0 for all values of x,. A useful interpretation of Fq. (3) is that of an electrical circuit consisting of an ideal flow source Kqxvand a shunt leakage conductance given by Kt(x,, PL).
A schematic of the pressure control servo-valve being considered is shown in Fig. 1. It is a two-stage servo-valve with a pressure control pilot stage. As the electrical current input from the command source to the valve pilot (coil) is varied, the flapper angle changes which in turn differentially varies ssures PI and 9 to the boost stage. For example, as PI increases and 9decreases. :=PI - P2 causes the spool to move to the right hence opening the source at pressure Ps to the port 'B' through the right chamber of the boost stage. The port 'A' o ens to tank through the left chamber of the boost stage. &us, fluid tends to flow out of Port B and return through Port A. The flow rate out and returnin to the valve depends on the spool displacement, x, and #e load pressure PL := Pa - f i . The work pressures Pa and f i are structurally fed-back to the opposite sides of the metering lands. Hence, in steady state, the spool is stabilized at a point such that the load pressure, PLequals the pilot differential pressure AP. The fact that the load pressure is structurally fed-back to the s 001 is an essential element in our now develop mathematical modpassification process. els of the vanous subsystems in the pressure control valve, Fig. 1.
The dynamics of the spool are given by: €3, = A ~ M - A ~ P L
(4)
where E is the spool mass, AP is the pilot difference and PL is the load pressure. A1 is the cross s e c t i o n r z z the spool land A2 is the cross sectional area of the end of the rod on which the lands are mounted. Note that the natural dynamics of the spool are forced by the load pressure feedback unlike in 141. In reality. in addition to the differential pilot and load prespt) other s ring and damping forces manifested sures (0, by the steady-state anftransient flow forces act on the spool. Interested readers are referred to [9, 81 and the references therein. Taking these forces into consideration, a more accurate model of the spool is given by:
d
€3V+b&+kr, =AlAP-A&
2.1 Spool dynamics We use the notation that x, > 0 when the spool is displaced to the right, Fig. 1. QL is the fluid flow throu the load attached to the valve and it is positive when it ows out of Port 'B' and into Port 'A'. For simplicity, we assume that the valve is connected to a double-ended actuator so that outgoing and incoming flows are the same and are 'ven by QL. we also assume that the t a d pressure w 0. %e flow rate through the valve for a given spool displacement x, is given by [8]:
(5)
which can be re-written as:
tP
+ kxyT + A ~ P L U
&fv= A1 AP - bxv
(6)
2.2 Pilot pressure chamber dynamics The pressure (PI in Fig. 2) in the pilot chamber is determined by the flapper position, xf. For a detailed treatment of the dynamic model, readers are referred to [ 10, 113. We linearize the dynamics of the pressure chambers about the equilibrium point (xf,P1,2) = (O,P,). The linearized dynamics of the pilot pressure chambers are given by:
where Cd. w and p are the dischar e coefficient, area gradient and fluid density. Following [&, Eiq. (2) can be written as: (3) QL = K 4 ~ y - Kt (xv PL)PL 4832
,
where
=
8 (a +i *) cd
~
X
O t
Y = &d,
;;
p c
----e-
and Po is the steady state pressure in the pilot chambers when the flapper is in the null position. It is given by
Po =
rg5pgl-L - -
NoN--p-vE "*I."* ..................................................................
.D@
o&!&
+ spoa
sowow
it ................................................................
1 ...................................................
PS
i
.Pm4E.wYAkYrr.
+ (e>,.
1
Using the definition of AP, we can combine equations (7), (8) as:
2.3 Pilot stage model A quasi-static model for the pilot flapper is used in this par. This approximation is acceptable if the time constant or the pilot flapper dynamics is much smaller than that of other subsystems in the valve. The approximation is motivated by [lo], where the dynamics of a two-stage two spool valve using a similar pilot stage were analyzed. It was found that the quasi-static flap r model reproduces well the dominant valve dynamics. flapper is assumed to be in quasistatic equilibrium with the pressure, the flow forces at the nozzle, spring and solenoid torque. The quasi-static model is given by:
r
k
C:
.....---
............. ------....--
Spool
HydraullCY
3 Valve feedback passification methodology The passivity property of a ressure control two-stage servo-valve, Fig. 1 is a n a l y J u s i n g a bond graph. This analysis is an extension of the technique proposed for single-stage directional control valves [5]. A bond graph of the pressure control servo-valve after a duality transformation for the pilot flapper stage is shown in Fig. 4. The valve is re resented as the interconnection of the combined solenoid-fPapper-pilot chamber stage (1 1). the spool dynamics stage (6) and the h draulics stage (3). Note that in Fig. 4 there are two signarbonds. This demonstrates that the pressure control servo-valve is not passive. The passification conce t in [5] is to replace the two signal bonds with Power ScaEng Transformers (FTF) or Power Scaling Gyrators (PGY)and choosing the coordinate transformations / control that make this replacement possible. We will use a similar approach here. Causal properties of the newly roposed (pseudo) bond graph elements: PTF / PGY, folPow the regular two-port transformers / gyrators 1121 except that that there is a non-unity scaling factor which relates the power between the two ports. For example, for a PTF(r,q) with transformer modulus r and power scaling factor q, the causal relationship is given as
(10)
9,Pi = 9and yi = F.
2.4 Problem formulation The pressure control servo-valve is a two-port device as shown in Fig. 3 (top). The load pressure PL and the load flow rate QL are the effort and flow variables of the 'load port' and solenoid current i and pilot differential pressure AP are the equivalent effort and flow variables of the 'command port'. The control goal is to find an alternate pair of effort and flow variables (ic,y) for the command port so that the pressure control servo-valvebehaves like a two-port passive device with respect to the following supply rate:
+ (llv)icY,
S, :7
Figure 4: Active bond graph of the 2-stage pressure control servo-valve
ptc+yti
s((PL,QL), ( i c , r ) )= -&QL
:
----------I-
By combining the quasi-static pilot stage (10) with the dynamics of the pressure chambers (9). we have:
where ht = h+
8
L....
PlM+ S o h d d
where i is the current into the solenoid, G is a linear current gain, K is the apparent spring stiffness rate and A,, is the apparent nozzle area.
Q = -A@-
] !i
.............................................................................
Figure 3: The feedback control concept to valve passification
(9)
Gi = Kxf +A,,AP
iu, :.
el := re2; f2 := q
f 1
where ej is the effort and fi is the corresponding flow.
(12)
On replacing the signal bonds in Fig. 4 with PTF's we get a passive pressure control servo-valve whose bond graph is shown in Fig. 5 . According to the bond graph, the dynamics
where v E $+ is some scaling. The first term in the supply rate is the physical power input to the valve from the load port and the second term represents the power input to the valve from the command port.
C:€
1
The concept of the passificationcontrol scheme is illustrated in Fig. 3. We determine a feedback current iact and a coordinate transformation y so that with a control law i = iac ic, the valve behaves like a passive two-port device with reto the supply rate (12). To determine ic,y we anyze the pressure control servo-valve using n graphs. Bond graphs are useful because of the inherent concepts of power and energy embedded in them.
I:1
1
R:&
1
+
5""'
L6
figure 5: Passive bond graph of the servo-valve
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2-stage pressure
control
Integrating the above inequality completes the proof.
of the pressure control valve are given by:
The problem with implementing the feedback control iaa given in Eq.(17) is that it involves derivatives of load pressure which are not exactly known. Also note that the transformed statey is uncertain in PL (14). We partially solve the implementation problem by redefining y as follows:
+
where Elk = q. Using the storage function Wideai = $$ &?&y2 it can be shown that this valve is passive Q L ) ) = %icy with respect to the supply rate S( (ic,y),(PL, PLQL. Note that the states [ X , , ~ ~ , AinP the ] active bond graph Fig. 4 are transformed to the new states [xu,z,y)of the passive bond graph Fig. 5 according to the following coordinate transformation:
+
fl
where denotes the best estimate of the argument. We also design a reduced order observer [131 which ensures that the error bztween the actual and estimated pressure rate PL := [PL- &] is bounded. The dynamics of the reduced order observer are given by:
+
z := iv K,PL k A ' y := i+ -x, E
E
= AI -U+ E
2
+ --z&b
Xob
= -LPL
+& -
A
(%K, - %)PA+Kq&
(19)
Yob = P L = L f k + X o b r
(14)
where L is the scalar observer gain. Consider a new storage function,
Usin these coordinate transformations, the actual dynamics ofthe pressure control servo-valve can be re-written as:
Differentiate the storage function with respect to time and substitute the pressure control servo-valve ,dynamics (13, the reduced order observer dynamics (19) and the implementable feedback current (20) given below,
and&=((EhlK,+ where61 = (h1[bKq-Az]+A1P~K,)/A~ bK, -AZ)/A1. The objective is to regulate the current input i = iacr i, into the actual pressure control valve (15) so that it behaves like the assive pressure control valve (13). The procedure is form$& in the following theorem.
Further, on substituting for x, using (3) and noting that Kt(xv,PL) > 0, we get:
+
Theorem 1 The pressure control servo-valve. whose dynamics are concisely expressed in (15) assuming a quasistatic model of the solenoid stage (10) and linearized dynamics of the pilot pressure stage (1 1); is passive with respect to the supply rate AiYi kP1
s((icly),(pL,Qd) = -icy
-PLQL
(16)
ifthe feedback current into the solenoid, iact = i - i, is chosen so that
i,
can be where Q = (bK, -Az+EK,L)/(kPI) and L, chosen so that 52 is negative semi-definite. The last term in (21) is the uncertain power input to the observer. Since the observer dynamics are given by
h o t : The result is straight forward on choosing the storage function Wad =
by choosing L to be arbitrarily large, the term fi$~ can be made arbitrarily small. So that for the supply rate
1 E Ai -4 + -2 + -y2. 2 2k 2kP1
AlYl kP1
Taking the time denvate of W,, and substituting the control iactgiven in (17). and (3) for xu results in the following inequality:
s((ic19),(P~,Q~)) = -ic9-ptQ~, the pressure control servo-valve ensures:
@ 5 s((ic,9)1(PL,QL)) +&& 4834
(22)
and 6 ~ + 8 0~as L -b 00. Using the notation with which we defined passivity in section 1, a dynamic system is said to be nearly passive with respect to a supply rate L I , 3 s(u(t),y(t))E 9(,,if for all time t < Tf,for all inputs u(t) and given a set of initial conditions, there exists c such that, L's(u(r),y(r))dT2 -c2 - 6(L)t. where 6(L) can be made arbitrarily small by choosing L appropriately. The above analysis of the pressure control servo-valve is now summarized in the following theorem.
Figure 6: A teleoperation setup using the passified pressure control servo-valve
Theorem 2 The pressure control servo-valve, whose dynamics are concisely expressed in (15) assuming a quasistatic model of the solenoid stage (10) and linearized dynamics of the pilot pressure stage (1 1); is nearly passive with respect to the supply rate given in (22) if the feedback current into tke solenoid, iact is chosen as given by (20)and the estimate & is the output of the reduced order observer whose dynamics are given by (19).
4 Teleoperationand simulation results Readers are referred to [a] for a detailed design of teleoperation control involving a single-stage directional control valve. Here we use the teleoperation control to portray the passivity propert of the ressure control servo-valve. We also show that passized two-stage valve achieves a higher passive teleoperation bandwidth. The control ensures that the teleo ration system behaves like a passive rigid common m e c k i c a l tool between a human and the environment. 4.1 Passive bilateral teleoperation A schematic of the teleoperation control is illustratedin Fig. 6. The teleoperation setup includes the following subsystems: 1. The Motorized joystick is a pivoted inertia and its dynamics are given by:
de
Remark 1 When the nearly passive valve is used to drive a load like a hydraulic actuator (as is ofen the case), the pressure dynamics of the load features as a state equation containing measurable states and so Pr. can be accurately determined Hence, it will be possible to achieve exactpassijication of the pressure control servo-valve. When PL in (15) is exactly estimated and canceled by the solenoid feedback current id, theorem 2 ensures the passivity property of the pressure control valve. Such a passive valve can be interconnected with other passive elements, an actuator and a mechanical robot for example, thus leading to overall passive hydraulic machines. These passive hydraulic machines can be used in high performance tasks while ensuring enhanced operation safety and stability. The tridiagonal structure of the state evolution matrix (15) with positive entries on the upper diagonal and negative entries in the lower diagonal characterizes the system's ability to transfer energy bilaterally.
Mq=Fq+T,
2. The Hydraulic actuator dynamics (assuming incompressible fluid flow and no leakages) are given by:
A g P = QL A p P ~= FW
sZ+;s+;
k1
(27)
where A, is the cross section area, x, is the piston position, PL is the load pressure, Few is the environment force acting on the actuator and QL is the load flow. It is easy to show that the actuator is passive with respect to the supply rate sacruator = ApPbp - & w X p . We seek a controller &at ensures 1) coordination of the kinematic scaled joystick and the hydraulic actuator and 2) passivity of the teleoperation system with respect to a su ply rate that includes the power input by the human and l e work environment. The details of the controller design cannot be presented here due to space constraints. The control technique is similar to [6] and uses the DC gain of the passified pressure control servo-valve for design. The controller incorporates an internal state and has a skew-symmetric feedback structure thus naturally emitting bilateral energy transfer between the joystick, hyfraulic actuator and the intemal state. The controlleralso ensures that the joystick and hydraulic actuator are coordinated when operated at low frequencies. 4.2 Simulation results Simulations are performed under two operating conditions in the presence of sensor noise. 1) The piston of the hydraulic actuator is unconstrained and the simulation is intended to demonstrate the coordination ability of the controller. 2) The piston reacts against a wall through a com-
where
[
(26)
where F9 is the motor control input and Tq is the human input force. The joystick is passive with respect to the suppl rate sj.y.tick = p~,cj+ p ~ , 4where p is a scalar power scal ing.
If exact passification of the pressure control servo-valve is achieved, then the closed loop transfer matrix from inputs (i&) to outputs ( 9 , ~is~given ) by:
021(s)=y1
.
,
and s is the Laplace variable. The low frequency approximated dynamics of the valve, (23)-(24) are given by:
These dynamics are used in the simulated application of the pressure control servo valve in a teleoperauon framework.
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flapper) are now involved. The method to achieve feedback passification of the pressure control valve was motivated using bond graphs. The information needed to implement the proposed algorithm are the spool velocit the differential chamber pressure, the load pressure. In a&ition, load pressure rate & is estimated using an open loop observer. Simulation results are presented for a motorized joystick driven teleo eration control system. Results indicate improved peArmance of two-stage passified valves over single-stage passified valves. The advantage of feedback passified valves is that they can interconnect with other assive hydraulic devices, like an actuator or a motor anZensure the passivi property of the combined system. Such systems, which nd daily applications in a variety of industries, being passive will interact safely with the environment which may consist of the human. These safe interactions would encourage the possibility of the use of hydraulics in applications that demand both safety and performance.
Figure 7: Coordinated tracking of the teleoperation system in the absence of environmental force on the actuator, Fe, = 0
x
References [ l ] J. E. Colgate, W. Wannasu hoprasit, and M. A. Peshkin, “Cobots: Robots for colkboration with human o erators,” Proceedings of the ASME Dynamic Systems and &ntrol Division, vol. 58, pp. 433439, 1996. [2] P.Y. Li and R. Horowitz, “Control of smart exercise machines, part 1: problem formulation and non-adaptive control,” IEEWASME Transactions on Mechutronics, vol. 2, pp. 237-247,1997. [3] P. Y. Li and R. Horowitz, “Application of passive velocity field control to contour following problems,” Proceedings of 1996 IEEE Conference on Decision and Control, vol. 1, pp. 378-385, 1996. 141 P. Y. Li, “Towards safe and human friendly hydraulics: The passive valve,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, pp. 402409, 2000. [5] P.Y.Li and R. E Ngwompo, “Passification of electrohydraulic valves using bond graphs,” Proceedings of 15th
Figure 8: Tracking in the presence of a spring between the piston end and a wall (top). The human input (Tq) and the environment force (F&) (bottom). pliance and the passivity property of the passified valve is analyzed. The following values are chosen for the various parameters. Ps = 3ooOpsi., A, = 0.06in2, w = OSin, A, = 0.32in, A = 0.048in2, m = 0.5Zb, b = 2Olb./in.s, k = 200lb./2. G = 10, K = 1O00, ph = 0.031Zb./in3 andx, = 0.08in. With these parameters, the bandwidth of a passified single-stage directional control valve proposed in [4] would be 50 r d s , where as the bandwidth of the passified two-stage valve proposed in this paper is 71 rads. This is because, unlike the single-stage valve whose bandwidth was only determined the bandwidth of the 2by the damping to mass ratio stage valve is determined by both the spool damping and stiffness, and (24).
&,
IFAC World Congress. to appear, July 2002.
[6] P. Y.Li and K. Krishnaswamy, “Passive bilateral teleoperation of an electrohydraulic actuatorusing an electrohydraulic passive valve,” Proceedings of the American Control Conference, vol. 5 , pp. 3932-3937, Jul. 2001. [7] W. Anderson, “Controlling electrohydraulic systems,” Marcel Dekker. New York, 1988. [8] H. E. Merritt, “Hydraulic control systems,” John Wiley & Sons, 1967. [9] K. Krishnaswamy and P. Li, “On using unstable electrohydraulic valves for control,” ASME Journal of Dynamic Systems, Measurement and Control, to appear, March 2002. [lo] P. Y. Li, “Dynamic redesign of a flow control servovalve using a pressure control pilot,” Proceedin s of the 2001 ASME-IMECE - VoLDSC. NYC, NY, 2001. Afko to appear in ASME JDSMC. [113 S . T. Tsai, A. Akers, and S . J. Lin, “Modeling and dynamic evaluation of a two-stage two-spool servovalve used for pressure control,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 113, no. 4, pp. 709-713, 1991. [12] . D.Karnopp and R. Rosenber “System dynamics: A unified approach,” John Wiley anfkons, NZ, 1975. [ 131 C.F. Franklin, J. D.Powell, and A. E-Naeini, “Feedback control of dynamic systems,” Addison Wesley, 3 e d . 1995.
2,
Almost rfect coordinated tracking was achieved in the absence o K n environmental load Fig. 7. To study the second case we consider a scenario where a double ended actuator is constrained between two walls through a spring on both sides of the piston. Notice from the top plot of Fig. 8 that tracking is achieved even in the presence of an environmental force. Notice that when the human input force is zero, (when t E (4,s) seconds, for exam le) the spring pushes the This demonstrates that actuator back and hence the joystic!. the controller was able to ensure that the teleoperation setu behaves like a common rigid passive mechanical tool witf: which both the human and the environment interact.
5 Discussion and conclusion
In this paper, we have developed a feedback control algorithm that ensures a two-stage pressure control servo-valve behaves like a near1 passive two- ort device. Results per- taining to simuladapplication oPthe passified valve m a teleoperation setting are presented. The advantage of twosta e valves over single stage valves is that higher bandwifth and flow rates can be achieved. The complexity that results from the two-stage valve is that multiple subsystems (main spool, pressure chamber, pilot 4836
Passification of an Electrohydraulic Two-stage Pressure Control Servo-valve Kailash Krishnaswamy' and Perry Y. Li2 Department of Mechanical Engineering University of Minnesota 111, Church St. SE,Minneapolis, MN 55455 (kk,pli}@me .umn .edu Abstract
has been applied to enhance the safety and human friendliness of many mechanical and electromechanical systems, for example [l, 2,3].
Passive systems have the beneficial property that their interconnections with other passive systems result in necessarily stable systems. This vmue of passive systems can be exploited with hydraulic systems to build owerful machines that are safe and human friendly. A digculty in a plying the passivity concept to electrohydraulic systems is i a t they are not intnnsically passive. In a previous paper, passification techniques were developed to render single-stage directional control valves as passive two-port devices with a supply rate related to the wer in ut at the hydraulic port. In this paper, we extend passiffcation technique to two stage pressure control servo-valves. The assified two-stage valve avoids the flow rate and bandwidthyimitations that exist for single-stage valves.
While the inherent passivity property of the above mentioned applications is well known, the passivity property of electrohydraulic systems was not stuQed until recently. It is shown in [4] that the single-stage four way directional control valve is not intrinsically passive. Structural modification of valve design (such as adding leaka e and pressure feedback), as well as active passification tecfnique that utilized feedback control to achieve suitable spool dynamics, are also proposed in [SI to render the four way single-stage directional control valve a passive two-port device. A critical element in the passification process is the use of load pressure feedback on the spool, either implemented structurally or via feedback control. The pro osed active passification technique has been implementefon a direct acting proportional control valve and applied in the control s stem for passive bilateral teleo eration of a single degree off?eedom hydraulic actuator p61 via a motorized joystick. The teleoperation experiment achieved 1) a specified ower scaling between human operator input and the mackne output and 2) a kinematic scaling between the motions of the j o stick and the hydraulic actuator. Although successful, &t single-stage proportional control valve is inherently limiting in terms of its bandwidth and flow rating due to the limited physical capabilities of the solenoids used to move the spool. To overcome this limitation, two-stage valves are needed since stronger hydraulic forces can be used to move the spool quickly.
1 Introduction Hydraulics are useful in applications where high ower density, good flexibility and portability are needel. Our research interests focus on making hydraulic systems safer and more human friendly in directly human controlled applications, and in applications that involve the machines interacting with uncertain or delicate environment. Roughly speaking, an energetically passive system is a system that apparently does not generate any energy of its own. It has the desirable property of ensuring stability whenever cou led with other strictly passive systems. This property enatles passive systems to operate robustly in uncertain physical environments which consist mainly of passive or stnctly passive objects. A passive system is also mherently safer than a non-passive system because the amount of energy that it can impart on its work environment is limited. Hence the possibility of a assive hydraulic machine unintentionally damaging an o\ject, for example rupturing an underground pipeline while earth moving, is limited.
In this pa r, we develop an active passification technique using feexack control for a generic two-stage ressure control servo-valve that uses a pressure control pifot and a single spool pressure control main stage. As a two-stage valve it has the potential to overcome the performance limitations of single-stage valves. A descriphon of the valve can be found in [7]. We choose to assify pressure control servovalves because pressure fee&ack, which is critical for passification,is often already present in their structuraldesigns. This is generally not available in flow control servo-valves, which are designed to be insensitive to load ressures. The bond graph perspective for developin passifcation control law and dynamics for single-stage vafves, developed in [SI will be extended in this paper.
We say that a dynamical system with input u ( t ) and output y ( t ) is passive with respect to a supply rate s(u,y) E & ! if s ( u ( t ) , y ( t ) ) is a L1, integrable function with respect to t and given a set of initial conditions, there exists a constant, c so that for all time, t and for all inputs, U(.),
The rest of the paper is organized as follows. Section 2 presents the d y n m c s of the various subsystems of the pressure control valve and formulates the passive control problem. Section 3 presents the passification algorithm. Section 4 develops a teleoperation control for the low-frequency approximated passified ressure control valve and simulation results are also provi&d. Section 5 contains discussion and concluding remarks.
For physical systems and for our purpose, useful supply rates are those associated with actual power into the system. For example, if effort and flow variables are used as input and output variables, the inner product of the input and the output is a sup ly rate related to actual power. Eq. (1) means that the tot$ energy that can be extracted from a passive system, is at most the initial energy, 3.Passivity
0-7803-7298-0/02/$17.00 0 2002 AACC
4831
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... .._
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Figure 2: A magnified schematic of the left pilot pressure chamber with the flapper *........ '& ........................ %' QLI
9
" :1
where K~ = C d W E and for pS> sgn(x,)PL,
Figure 1: The pressure control servo-valve
2 Dynamic model and problem formulation A similar expression can be obtained for the unusual case when Ps < sgn(x,)PL. Note that K,(x,,Pt) > 0 for all values of x,. A useful interpretation of Fq. (3) is that of an electrical circuit consisting of an ideal flow source Kqxvand a shunt leakage conductance given by Kt(x,, PL).
A schematic of the pressure control servo-valve being considered is shown in Fig. 1. It is a two-stage servo-valve with a pressure control pilot stage. As the electrical current input from the command source to the valve pilot (coil) is varied, the flapper angle changes which in turn differentially varies ssures PI and 9 to the boost stage. For example, as PI increases and 9decreases. :=PI - P2 causes the spool to move to the right hence opening the source at pressure Ps to the port 'B' through the right chamber of the boost stage. The port 'A' o ens to tank through the left chamber of the boost stage. &us, fluid tends to flow out of Port B and return through Port A. The flow rate out and returnin to the valve depends on the spool displacement, x, and #e load pressure PL := Pa - f i . The work pressures Pa and f i are structurally fed-back to the opposite sides of the metering lands. Hence, in steady state, the spool is stabilized at a point such that the load pressure, PLequals the pilot differential pressure AP. The fact that the load pressure is structurally fed-back to the s 001 is an essential element in our now develop mathematical modpassification process. els of the vanous subsystems in the pressure control valve, Fig. 1.
The dynamics of the spool are given by: €3, = A ~ M - A ~ P L
(4)
where E is the spool mass, AP is the pilot difference and PL is the load pressure. A1 is the cross s e c t i o n r z z the spool land A2 is the cross sectional area of the end of the rod on which the lands are mounted. Note that the natural dynamics of the spool are forced by the load pressure feedback unlike in 141. In reality. in addition to the differential pilot and load prespt) other s ring and damping forces manifested sures (0, by the steady-state anftransient flow forces act on the spool. Interested readers are referred to [9, 81 and the references therein. Taking these forces into consideration, a more accurate model of the spool is given by:
d
€3V+b&+kr, =AlAP-A&
2.1 Spool dynamics We use the notation that x, > 0 when the spool is displaced to the right, Fig. 1. QL is the fluid flow throu the load attached to the valve and it is positive when it ows out of Port 'B' and into Port 'A'. For simplicity, we assume that the valve is connected to a double-ended actuator so that outgoing and incoming flows are the same and are 'ven by QL. we also assume that the t a d pressure w 0. %e flow rate through the valve for a given spool displacement x, is given by [8]:
(5)
which can be re-written as:
tP
+ kxyT + A ~ P L U
&fv= A1 AP - bxv
(6)
2.2 Pilot pressure chamber dynamics The pressure (PI in Fig. 2) in the pilot chamber is determined by the flapper position, xf. For a detailed treatment of the dynamic model, readers are referred to [ 10, 113. We linearize the dynamics of the pressure chambers about the equilibrium point (xf,P1,2) = (O,P,). The linearized dynamics of the pilot pressure chambers are given by:
where Cd. w and p are the dischar e coefficient, area gradient and fluid density. Following [&, Eiq. (2) can be written as: (3) QL = K 4 ~ y - Kt (xv PL)PL 4832
,
where
=
8 (a +i *) cd
~
X
O t
Y = &d,
;;
p c
----e-
and Po is the steady state pressure in the pilot chambers when the flapper is in the null position. It is given by
Po =
rg5pgl-L - -
NoN--p-vE "*I."* ..................................................................
.D@
o&!&
+ spoa
sowow
it ................................................................
1 ...................................................
PS
i
.Pm4E.wYAkYrr.
+ (e>,.
1
Using the definition of AP, we can combine equations (7), (8) as:
2.3 Pilot stage model A quasi-static model for the pilot flapper is used in this par. This approximation is acceptable if the time constant or the pilot flapper dynamics is much smaller than that of other subsystems in the valve. The approximation is motivated by [lo], where the dynamics of a two-stage two spool valve using a similar pilot stage were analyzed. It was found that the quasi-static flap r model reproduces well the dominant valve dynamics. flapper is assumed to be in quasistatic equilibrium with the pressure, the flow forces at the nozzle, spring and solenoid torque. The quasi-static model is given by:
r
k
C:
.....---
............. ------....--
Spool
HydraullCY
3 Valve feedback passification methodology The passivity property of a ressure control two-stage servo-valve, Fig. 1 is a n a l y J u s i n g a bond graph. This analysis is an extension of the technique proposed for single-stage directional control valves [5]. A bond graph of the pressure control servo-valve after a duality transformation for the pilot flapper stage is shown in Fig. 4. The valve is re resented as the interconnection of the combined solenoid-fPapper-pilot chamber stage (1 1). the spool dynamics stage (6) and the h draulics stage (3). Note that in Fig. 4 there are two signarbonds. This demonstrates that the pressure control servo-valve is not passive. The passification conce t in [5] is to replace the two signal bonds with Power ScaEng Transformers (FTF) or Power Scaling Gyrators (PGY)and choosing the coordinate transformations / control that make this replacement possible. We will use a similar approach here. Causal properties of the newly roposed (pseudo) bond graph elements: PTF / PGY, folPow the regular two-port transformers / gyrators 1121 except that that there is a non-unity scaling factor which relates the power between the two ports. For example, for a PTF(r,q) with transformer modulus r and power scaling factor q, the causal relationship is given as
(10)
9,Pi = 9and yi = F.
2.4 Problem formulation The pressure control servo-valve is a two-port device as shown in Fig. 3 (top). The load pressure PL and the load flow rate QL are the effort and flow variables of the 'load port' and solenoid current i and pilot differential pressure AP are the equivalent effort and flow variables of the 'command port'. The control goal is to find an alternate pair of effort and flow variables (ic,y) for the command port so that the pressure control servo-valvebehaves like a two-port passive device with respect to the following supply rate:
+ (llv)icY,
S, :7
Figure 4: Active bond graph of the 2-stage pressure control servo-valve
ptc+yti
s((PL,QL), ( i c , r ) )= -&QL
:
----------I-
By combining the quasi-static pilot stage (10) with the dynamics of the pressure chambers (9). we have:
where ht = h+
8
L....
PlM+ S o h d d
where i is the current into the solenoid, G is a linear current gain, K is the apparent spring stiffness rate and A,, is the apparent nozzle area.
Q = -A@-
] !i
.............................................................................
Figure 3: The feedback control concept to valve passification
(9)
Gi = Kxf +A,,AP
iu, :.
el := re2; f2 := q
f 1
where ej is the effort and fi is the corresponding flow.
(12)
On replacing the signal bonds in Fig. 4 with PTF's we get a passive pressure control servo-valve whose bond graph is shown in Fig. 5 . According to the bond graph, the dynamics
where v E $+ is some scaling. The first term in the supply rate is the physical power input to the valve from the load port and the second term represents the power input to the valve from the command port.
C:€
1
The concept of the passificationcontrol scheme is illustrated in Fig. 3. We determine a feedback current iact and a coordinate transformation y so that with a control law i = iac ic, the valve behaves like a passive two-port device with reto the supply rate (12). To determine ic,y we anyze the pressure control servo-valve using n graphs. Bond graphs are useful because of the inherent concepts of power and energy embedded in them.
I:1
1
R:&
1
+
5""'
L6
figure 5: Passive bond graph of the servo-valve
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2-stage pressure
control
Integrating the above inequality completes the proof.
of the pressure control valve are given by:
The problem with implementing the feedback control iaa given in Eq.(17) is that it involves derivatives of load pressure which are not exactly known. Also note that the transformed statey is uncertain in PL (14). We partially solve the implementation problem by redefining y as follows:
+
where Elk = q. Using the storage function Wideai = $$ &?&y2 it can be shown that this valve is passive Q L ) ) = %icy with respect to the supply rate S( (ic,y),(PL, PLQL. Note that the states [ X , , ~ ~ , AinP the ] active bond graph Fig. 4 are transformed to the new states [xu,z,y)of the passive bond graph Fig. 5 according to the following coordinate transformation:
+
fl
where denotes the best estimate of the argument. We also design a reduced order observer [131 which ensures that the error bztween the actual and estimated pressure rate PL := [PL- &] is bounded. The dynamics of the reduced order observer are given by:
+
z := iv K,PL k A ' y := i+ -x, E
E
= AI -U+ E
2
+ --z&b
Xob
= -LPL
+& -
A
(%K, - %)PA+Kq&
(19)
Yob = P L = L f k + X o b r
(14)
where L is the scalar observer gain. Consider a new storage function,
Usin these coordinate transformations, the actual dynamics ofthe pressure control servo-valve can be re-written as:
Differentiate the storage function with respect to time and substitute the pressure control servo-valve ,dynamics (13, the reduced order observer dynamics (19) and the implementable feedback current (20) given below,
and&=((EhlK,+ where61 = (h1[bKq-Az]+A1P~K,)/A~ bK, -AZ)/A1. The objective is to regulate the current input i = iacr i, into the actual pressure control valve (15) so that it behaves like the assive pressure control valve (13). The procedure is form$& in the following theorem.
Further, on substituting for x, using (3) and noting that Kt(xv,PL) > 0, we get:
+
Theorem 1 The pressure control servo-valve. whose dynamics are concisely expressed in (15) assuming a quasistatic model of the solenoid stage (10) and linearized dynamics of the pilot pressure stage (1 1); is passive with respect to the supply rate AiYi kP1
s((icly),(pL,Qd) = -icy
-PLQL
(16)
ifthe feedback current into the solenoid, iact = i - i, is chosen so that
i,
can be where Q = (bK, -Az+EK,L)/(kPI) and L, chosen so that 52 is negative semi-definite. The last term in (21) is the uncertain power input to the observer. Since the observer dynamics are given by
h o t : The result is straight forward on choosing the storage function Wad =
by choosing L to be arbitrarily large, the term fi$~ can be made arbitrarily small. So that for the supply rate
1 E Ai -4 + -2 + -y2. 2 2k 2kP1
AlYl kP1
Taking the time denvate of W,, and substituting the control iactgiven in (17). and (3) for xu results in the following inequality:
s((ic19),(P~,Q~)) = -ic9-ptQ~, the pressure control servo-valve ensures:
@ 5 s((ic,9)1(PL,QL)) +&& 4834
(22)
and 6 ~ + 8 0~as L -b 00. Using the notation with which we defined passivity in section 1, a dynamic system is said to be nearly passive with respect to a supply rate L I , 3 s(u(t),y(t))E 9(,,if for all time t < Tf,for all inputs u(t) and given a set of initial conditions, there exists c such that, L's(u(r),y(r))dT2 -c2 - 6(L)t. where 6(L) can be made arbitrarily small by choosing L appropriately. The above analysis of the pressure control servo-valve is now summarized in the following theorem.
Figure 6: A teleoperation setup using the passified pressure control servo-valve
Theorem 2 The pressure control servo-valve, whose dynamics are concisely expressed in (15) assuming a quasistatic model of the solenoid stage (10) and linearized dynamics of the pilot pressure stage (1 1); is nearly passive with respect to the supply rate given in (22) if the feedback current into tke solenoid, iact is chosen as given by (20)and the estimate & is the output of the reduced order observer whose dynamics are given by (19).
4 Teleoperationand simulation results Readers are referred to [a] for a detailed design of teleoperation control involving a single-stage directional control valve. Here we use the teleoperation control to portray the passivity propert of the ressure control servo-valve. We also show that passized two-stage valve achieves a higher passive teleoperation bandwidth. The control ensures that the teleo ration system behaves like a passive rigid common m e c k i c a l tool between a human and the environment. 4.1 Passive bilateral teleoperation A schematic of the teleoperation control is illustratedin Fig. 6. The teleoperation setup includes the following subsystems: 1. The Motorized joystick is a pivoted inertia and its dynamics are given by:
de
Remark 1 When the nearly passive valve is used to drive a load like a hydraulic actuator (as is ofen the case), the pressure dynamics of the load features as a state equation containing measurable states and so Pr. can be accurately determined Hence, it will be possible to achieve exactpassijication of the pressure control servo-valve. When PL in (15) is exactly estimated and canceled by the solenoid feedback current id, theorem 2 ensures the passivity property of the pressure control valve. Such a passive valve can be interconnected with other passive elements, an actuator and a mechanical robot for example, thus leading to overall passive hydraulic machines. These passive hydraulic machines can be used in high performance tasks while ensuring enhanced operation safety and stability. The tridiagonal structure of the state evolution matrix (15) with positive entries on the upper diagonal and negative entries in the lower diagonal characterizes the system's ability to transfer energy bilaterally.
Mq=Fq+T,
2. The Hydraulic actuator dynamics (assuming incompressible fluid flow and no leakages) are given by:
A g P = QL A p P ~= FW
sZ+;s+;
k1
(27)
where A, is the cross section area, x, is the piston position, PL is the load pressure, Few is the environment force acting on the actuator and QL is the load flow. It is easy to show that the actuator is passive with respect to the supply rate sacruator = ApPbp - & w X p . We seek a controller &at ensures 1) coordination of the kinematic scaled joystick and the hydraulic actuator and 2) passivity of the teleoperation system with respect to a su ply rate that includes the power input by the human and l e work environment. The details of the controller design cannot be presented here due to space constraints. The control technique is similar to [6] and uses the DC gain of the passified pressure control servo-valve for design. The controller incorporates an internal state and has a skew-symmetric feedback structure thus naturally emitting bilateral energy transfer between the joystick, hyfraulic actuator and the intemal state. The controlleralso ensures that the joystick and hydraulic actuator are coordinated when operated at low frequencies. 4.2 Simulation results Simulations are performed under two operating conditions in the presence of sensor noise. 1) The piston of the hydraulic actuator is unconstrained and the simulation is intended to demonstrate the coordination ability of the controller. 2) The piston reacts against a wall through a com-
where
[
(26)
where F9 is the motor control input and Tq is the human input force. The joystick is passive with respect to the suppl rate sj.y.tick = p~,cj+ p ~ , 4where p is a scalar power scal ing.
If exact passification of the pressure control servo-valve is achieved, then the closed loop transfer matrix from inputs (i&) to outputs ( 9 , ~is~given ) by:
021(s)=y1
.
,
and s is the Laplace variable. The low frequency approximated dynamics of the valve, (23)-(24) are given by:
These dynamics are used in the simulated application of the pressure control servo valve in a teleoperauon framework.
4835
flapper) are now involved. The method to achieve feedback passification of the pressure control valve was motivated using bond graphs. The information needed to implement the proposed algorithm are the spool velocit the differential chamber pressure, the load pressure. In a&ition, load pressure rate & is estimated using an open loop observer. Simulation results are presented for a motorized joystick driven teleo eration control system. Results indicate improved peArmance of two-stage passified valves over single-stage passified valves. The advantage of feedback passified valves is that they can interconnect with other assive hydraulic devices, like an actuator or a motor anZensure the passivi property of the combined system. Such systems, which nd daily applications in a variety of industries, being passive will interact safely with the environment which may consist of the human. These safe interactions would encourage the possibility of the use of hydraulics in applications that demand both safety and performance.
Figure 7: Coordinated tracking of the teleoperation system in the absence of environmental force on the actuator, Fe, = 0
x
References [ l ] J. E. Colgate, W. Wannasu hoprasit, and M. A. Peshkin, “Cobots: Robots for colkboration with human o erators,” Proceedings of the ASME Dynamic Systems and &ntrol Division, vol. 58, pp. 433439, 1996. [2] P.Y. Li and R. Horowitz, “Control of smart exercise machines, part 1: problem formulation and non-adaptive control,” IEEWASME Transactions on Mechutronics, vol. 2, pp. 237-247,1997. [3] P. Y. Li and R. Horowitz, “Application of passive velocity field control to contour following problems,” Proceedings of 1996 IEEE Conference on Decision and Control, vol. 1, pp. 378-385, 1996. 141 P. Y. Li, “Towards safe and human friendly hydraulics: The passive valve,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, pp. 402409, 2000. [5] P.Y.Li and R. E Ngwompo, “Passification of electrohydraulic valves using bond graphs,” Proceedings of 15th
Figure 8: Tracking in the presence of a spring between the piston end and a wall (top). The human input (Tq) and the environment force (F&) (bottom). pliance and the passivity property of the passified valve is analyzed. The following values are chosen for the various parameters. Ps = 3ooOpsi., A, = 0.06in2, w = OSin, A, = 0.32in, A = 0.048in2, m = 0.5Zb, b = 2Olb./in.s, k = 200lb./2. G = 10, K = 1O00, ph = 0.031Zb./in3 andx, = 0.08in. With these parameters, the bandwidth of a passified single-stage directional control valve proposed in [4] would be 50 r d s , where as the bandwidth of the passified two-stage valve proposed in this paper is 71 rads. This is because, unlike the single-stage valve whose bandwidth was only determined the bandwidth of the 2by the damping to mass ratio stage valve is determined by both the spool damping and stiffness, and (24).
&,
IFAC World Congress. to appear, July 2002.
[6] P. Y.Li and K. Krishnaswamy, “Passive bilateral teleoperation of an electrohydraulic actuatorusing an electrohydraulic passive valve,” Proceedings of the American Control Conference, vol. 5 , pp. 3932-3937, Jul. 2001. [7] W. Anderson, “Controlling electrohydraulic systems,” Marcel Dekker. New York, 1988. [8] H. E. Merritt, “Hydraulic control systems,” John Wiley & Sons, 1967. [9] K. Krishnaswamy and P. Li, “On using unstable electrohydraulic valves for control,” ASME Journal of Dynamic Systems, Measurement and Control, to appear, March 2002. [lo] P. Y. Li, “Dynamic redesign of a flow control servovalve using a pressure control pilot,” Proceedin s of the 2001 ASME-IMECE - VoLDSC. NYC, NY, 2001. Afko to appear in ASME JDSMC. [113 S . T. Tsai, A. Akers, and S . J. Lin, “Modeling and dynamic evaluation of a two-stage two-spool servovalve used for pressure control,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 113, no. 4, pp. 709-713, 1991. [12] . D.Karnopp and R. Rosenber “System dynamics: A unified approach,” John Wiley anfkons, NZ, 1975. [ 131 C.F. Franklin, J. D.Powell, and A. E-Naeini, “Feedback control of dynamic systems,” Addison Wesley, 3 e d . 1995.
2,
Almost rfect coordinated tracking was achieved in the absence o K n environmental load Fig. 7. To study the second case we consider a scenario where a double ended actuator is constrained between two walls through a spring on both sides of the piston. Notice from the top plot of Fig. 8 that tracking is achieved even in the presence of an environmental force. Notice that when the human input force is zero, (when t E (4,s) seconds, for exam le) the spring pushes the This demonstrates that actuator back and hence the joystic!. the controller was able to ensure that the teleoperation setu behaves like a common rigid passive mechanical tool witf: which both the human and the environment interact.
5 Discussion and conclusion
In this paper, we have developed a feedback control algorithm that ensures a two-stage pressure control servo-valve behaves like a near1 passive two- ort device. Results per- taining to simuladapplication oPthe passified valve m a teleoperation setting are presented. The advantage of twosta e valves over single stage valves is that higher bandwifth and flow rates can be achieved. The complexity that results from the two-stage valve is that multiple subsystems (main spool, pressure chamber, pilot 4836
