Toward Safe and Human Friendly Hydraulics: The Passive Valve

Transcription

1 Perry Y. Li Department of Mechanical Engineering, University of Minnesota, Church St. SE, Minneapolis MN Toward Safe and Human Friendly Hydraulics: The Passive Valve Hydraulic systems, as power sources and transmissions, offer many advantages over electromechanical or purely mechanical counterparts in terms of power density, flexibility, and portability. Many hydraulic systems require touching and contacting the physical environments; and many of these systems are directly controlled by humans. If hydraulic systems are passive, they would be both safer to interact with, and easier for humans to control. In this paper, it is shown that a critical hydraulic component, the directional control valve, is not passive. However, the directional valve, as a one-port or a two-port device can become passive if appropriate spool valve dynamics are imposed. Methods to passify the valve for both first-order and second-order spool dynamics are considered. In the case of second-order spool dynamics, a passive method that relies on hardware modification, and an active feedback method, are proposed. S I Introduction Hydraulic systems, as power sources and transmissions, offer many advantages over electromechanical or purely mechanical counterparts in terms of power density, flexibility, and portability. Many hydraulic systems require touching and contacting the physical environments. It is critical that these systems can safely interact with the environments without unintentionally damaging the environments or themselves. The environment and the hydraulic system form a closed-loop system when they interact with each other. The closed loop system can be unstable even if the environment and the hydraulic system are individually stable. Many of these systems are also directly controlled by humans in the work environment. A typical example is a construction worker operating a hydraulic boom-and-bucket to perform an earth digging task. It is, therefore, important that these systems are natural and easy for the human operator to control. Both the safety and the human friendliness aspects of these applications can be enhanced if the system can be shown to be passive. Roughly speaking, a passive system behaves as if it does not generate energy, but only stores, dissipates, and releases it. A passive system is inherently safer than a nonpassive system because the amount of energy that it can impart on the environment is limited. The well-known passivity theorem ensures that a passive system can interact stably with any strictly passive system. The latter includes a wide variety of physical objects and environments. The inherent safety that passive systems afford has been exploited in machines that interact with humans such as smart exercise machines 2,3, bilateral teleoperated manipulators 4,5, and Cobots 6. It is also exploited in the passive velocity field control PVFC methodology 7 0, which by ensuring that the closed loop system is passive, enables mechanical systems to become coordinated such as for contour following in machining operation or multiple robots cooperating and at the same time interact safely with the often ill characterized environment. Passive systems are also potentially easier to control. One reason for this is that some form of stable haptic feedback of the environment to the human user will be quite natural to achieve for a passive system 4. Moreover, users will be able to use the familiar concept of power while executing a manipulation task. Based on the observation that passive systems form the basis of almost all artificial learning and adaptive schemes see and references therein, it can also be argued that passive tools are potentially easier than non-passive ones for users to learn to use. Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division July 5, 999 Associate Technical Editor: S. Nair. A key premise of many passivity based control techniques is that the underlying system possesses some passivity property. While it is true that mechanical systems are passive, the passivity properties of hydraulic devices have not previously been investigated. The objective of this paper is to investigate the passivity property of a key component in a hydraulic system-the directional control valve, and to propose modification and control so that the valve becomes passive. Specifically, it will be shown that the directional control valve, as a one-port, is not passive with respect to the hydraulic power input at the hydraulic port. Using an analogous electrical circuit, the source of the nonpassivity is identified. Subsequently, we will show that the valve can become passive if appropriate first or second order spool dynamics are imposed. This passification method can be extended to the two-port valve the extra port is for the command signal. In the case of the second order spool dynamics, the valve can be passified by making hardware modification to the valve, or by using active feedback compensation. The rest of this paper is organized as follows. Passivity is defined in Section II. In Section III, we show that the one-port directional control valve is not passive. In Sections IV and V, we propose first-order spool dynamics so that the one-port valve and the two-port valve are passive. Methods to preserve the passivity of the valve in the presence of second order spool dynamics are presented in Section VI. Simulation results are presented in Section VII. Sections VIII and IX contain discussion and concluding remarks. II Passivity and Supply Rates Consider a system with input u and output y. A supply rate for a system is some function s:u,ysu,yr. For physical systems, useful supply rates are those associated with power input into the system. For example, if the pair of input and output variables for a system are the effort and flow variables, then a supply rate which is the power input is the inner product between the effort and flow variables. Following 2, a system is said to be passive with respect to the supply rate s(u,y) if, for a given initial condition, there exists a constant c so that for all time t and for all input u( ) 0 t su,ydc 2. () If s(u,y) is the power input into the system, then Eq. says that no matter how one manipulates the input u(t), the maximum 402 Õ Vol. 22, SEPTEMBER 2000 Copyright 2000 by ASME Transactions of the ASME

2 amount of energy that one can extract from the system is limited by the constant c 2 which can be interpreted to be the initial energy stored in the system. III Four-Way Directional Control Valves are not Passive Four-way directional valves are central to electro-hydraulic technology. A typical four-way valve is shown in Fig.. A hydraulic device, such as a hydraulic actuator piston or a hydraulic motor, is connected to device ports A and B. The constant supply pressure P s 0 is typically supplied by a pump and an accumulator, and the return line is connected to the reservoir at pressure P 0 0. By stroking the spool in the four-way valve, flow is metered into and out of the device ports A and B. For example, when the spool is stroked to the left as shown in Fig., flow is metered into port A from the pump and out of port B back to the reservoir. When the spool is stroked to the right, flow in the reverse direction occurs. For simplicity, assume that the valve will be connected to devices that cannot store fluid volume. This assumption excludes single ended actuators that have different cap side and rod side areas. However, it allows us to analyze the interaction between the valve and the device as a one-port with the pressure difference, P L P A P B and the flow rate QQ Q 2 being the corresponding effort and flow variables. We further assume that the four-way valve is critically lapped, matched, and symmetric. Under these assumptions, it easy to show that the pressure at the two terminals P A and P B are symmetrically located from P s /2 3, in the sense that P s P s P 0 P A P B ; P A P B P L. This means that P s P L 2P A and P s P L 2P B so that as P L increases from 0, P A and P B increases and decreases symmetrically from P s /2. Under these assumptions, the relationship between the flow rate and the pressure difference can be readily derived using the standard orifice relationship 3: QQ L x v,p L ª Cd wx vp s x v x v P L; C d wx v x v x v P LP s ; sgnx v P L P s sgnx v P L P s where C d 0 is the orifice coefficient, 0 is the fluid density, w0 is the gradient of the orifice area with respect to the spool position, and x v is the spool displacement from the center position. In normal operation, P L P s so that only the first case statement in Eq. 2 is normally used. We include the abnormal case P s sgn(x v )P L for mathematical completeness. The graph of Q L (x v,p L ) is depicted in Fig. 2. A natural supply rate to consider for the four-way valve is (2) Fig. 2 FlowÕpressure relationship of a typical matched, critically lapped, four-way directional valve. Top: normal case sign x v P L ÏP s ; bottom: exceptional case: sign x v P L ÐP s. Ps Ä3000 psi, wä0.3 in, C d Ä0.66, and Ä000 kgõm 3. sp L,QªP L Q (3) which is the hydraulic power input exerted by the hydraulic device on the valve. Proposition. For each nonzero fixed spool position x v 0, the four-way directional valve is not passive with respect to the supply rate given in Eq. (3). Proof: Let x v 0 be a nonzero spool position. Choose a constant port pressure trajectory P L (t)p c such that 0sgn(x v )P c P s. From Eq. 2, define p 0 ªP L (t) Q(t)P c Q L (x v,p c ) 0, 0 t P L Qdp 0 t. Fig. Four-way, three land directional valve Thus, any amount of energy M0 can be extracted by waiting for a long enough time tm/p 0 because then, 0 t P L QdM. Therefore, there is no constant c0 that will satisfy Eq. for all t0. This proves that the valve is not passive. Notice that p 0 in the proof above is the power withdrawn from the valve. This proposition shows that for each constant spool Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2000, Vol. 22 Õ 403

3 voltage x v K q /K t and output resistance (/K t) is not as useful for interpreting Eq. 4 because of the way that P L would affect the voltage source and the output resistance. Fig. 3 A Analogous electrical circuit for the four way valve. B Circuit is passified by replacing the current source by an inductor. position x v, it is possible for a device connected to the valve to extract an infinite amount of energy from the valve. A component which allows an infinite amount of energy to be extracted from it is clearly not passive with respect to the supply rate given by power input into the component. We will now define an analogous electrical circuit for the fourway directional valve which will be useful in understanding the rest of this paper. Let us rewrite Eq. 2 as where Q L x v,p L Q 0 x v K tx v,p L P L (4) is the flow rate through the valve when P L 0, and Q L K tx v,p L x ª0 P v,l P L dl L C d wx v dl (6) 0 2P s signx v P L is the mean gradient of the flow with respect to the port pressure. Notice that K t(x v,p L ) is the negative slope of the line joining the two points (0,Q 0 (x v )) and (P L,Q L (x v,p L )) in Fig. 2. Notice also that Eq. 4 is exact and it is not a linearization of Eq. 2. Lemma. For each x v, the mapping P L Q L (x v,p L ) is monotone non-increasing. Thus, for any (x v,p L ), K tx v,p L 0. Moreover, K t(x v,p L )0 if and only if x v 0. Proof: The validity of this key lemma can be seen graphically from Fig. 2; or analytically by noticing that Q L /P L 0 and Q L /P L 0 if and only if x v 0. Because of Lemma, we can interpret Eq. 4 using an analogous electrical circuit shown in Fig. 3A which consists of an ideal current source Q 0 (x v )K q x v and a nonlinear output shunt conductance K t(x v,p L ). The output current and voltage of the circuit are Q and P L, respectively. If we define the supply rate to this system to be the negative of the power output of the circuit, then it is equivalent to the supply rate Eq. 3 for the four-way valve. From Fig. 3A, it is clear that it is the current source, not the nonlinear shunt resistance, which is responsible for the circuit not being passive. We remark that although possible, the Thevenin s equivalent circuit 4 involving a voltage source with (5) IV Passifying the One-Port Valve We now propose a method to make the four-way valve passive. The key idea is to make the spool position x v or the current source dynamically related to the port pressure P L. In essence, the current source will be replaced by an inductor. Consider the ideal spool dynamics Bẋ v AP L. (7) We can think of this as the dynamics of a light and heavily damped spool acted on by a force AP L. One way to generate the force AP L is to utilize the actual pressure forces to act on the spool via direct structural feedback. This can be achieved by making the spool lands have different areas see Fig. 4. For example, if we make the center land area A in smaller than the outlying land area A out, so that AA out A in, then the pressure force acting on the spool would be AP L. Alternatively, this force can be generated by an actuator such as a second stage valve or a linear motor that acts on the spool, using pressure measurements in the two chambers. In this section, we will show that the spool dynamics Eq. 7 passifies the four-way valve. We postpone, to a later section, the detailed discussion of how to deal with the reality that the spool dynamics are in fact second order. To see intuitively how Eq. 7 may successfully make the fourway valve passive, notice the compensatory effect of the spool dynamics when P L increases: an increase in P L would normally extract more power from the valve if Q were to remain the same. However, with the spool dynamics in Eq. 7, the magnitude of x v, hence the flow rate Q, are decreased. The spool dynamics in Eq. 7 therefore limits the power. As we shall see, the spool dynamics also limits the available energy. Theorem. The nonlinear four-way valve Eq. (4) with the spool dynamics given by Eq. (7) is passive with respect to the supply rate s(p L,Q)P L Q, the power input into the valve by the hydraulic device. Proof: Consider the storage function candidate Wt 2 Bx 2 vt. Using Eq. 7, its time derivative is given by Ẇtx v AP L A K q Q 0 x v P L where in the second equality, we utilized 5 to express x v in terms of Q 0 (x v ). Applying Eq. 4, weget Ẇt A QP K L K tx v,p L P 2 L A QP q K L. (8) q Integrating this expression, we obtain Fig. 4 Structural pressure feedback using different land areas 404 Õ Vol. 22, SEPTEMBER 2000 Transactions of the ASME

4 t sp L,Qd K q 0 A WtW0 K q A W0. This shows that the nonlinear four way valve with the spool dynamics given by Eq. 7 is passive with respect to the supply rate given by s(p L,Q)P L Q. The effect of the spool dynamic in Eq. 7 on the analogous electrical circuit is to replace the current source by an inductor with inductance LB/AK q Fig. 3B. The dynamics of the new circuit is L di L dt P L Qi L K tx v,p L P L where the spool position x v is identified with the inductor current via i L K q x v. It is obvious that the new circuit is passive. V Passivity of the Two-Port Valve A passive one-port hydraulic valve is not very useful since it only reacts to the environment but cannot be commanded. We now introduce a passive two-port valve in which the addition port can be used as a port for external control, such as the direct or teleoperated control by a human. We modify the spool dynamics Eq. 7 so that it can interact with an additional external force F x Bẋ v AP L F x (9) QQ 0 x v K tx v,p L P L (0) where Q 0 (x v ) and K t(x v,p L ) are given by Eqs. 5 and 6 and B0 is the damping. Notice that the spool is now affected by both the hydraulic port pressure P L and the additional control command F x. The valve is now a two-port device. The first port of this system interacts with the hydraulic device via the effort and flow variables (P L,Q). The additional port interacts with a control unit via the variables (F x,x v ). For this two-port system we can think of P L and F x as the input variables, and Q and x v as the output variables. Proposition 2. The modified two-port nonlinear valve given by Eqs. 9 0 is passive with respect to the supply rate given by sp L,F,Q,x v A K q P L QF x x v () where K q C d wp s / and A is the pressure feedback constant in 9. Proof: Using the storage function candidate W(t)/2Bx v 2,we obtain ẆAP L x v F x x v. Using a similar procedure in the proof of Theorem to rewrite the first term on the right-hand side, we have Ẇt A QP K L K tx v,p L P 2 L F x x v A QP q K L q F x x v sp L,F,Q,x v where K t(x v,p L ) is given in Eq. 6. The passivity property ensued on integration. where 0 is the spool inertia. These dynamics can be realized by using a force motor to generate the force F x, and using either structural pressure feedback or sensor/actuator feedback to realize AP L. The second order dynamics generate an approximation of the desired spool dynamics Eq. 9 because as 0, the slow dynamics is given by Bẋ v F x AP L which is the desired spool dynamics in Eq. 9. The question we must now address is how the passivity property that results from the ideal spool dynamics Eq. 9 can be preserved when the spool dynamics becomes Eq. 2. We propose two methods to ensure the passivity of the hydraulic valve in the presence of nonzero spool inertia. In the first method, we utilize an output redefinition technique, which amounts to physically introducing an additional fluid leakage and a stiff interaction spring. In the second method, active feedback control is applied to enforce the passivity. A Passive Passification Method. In this method, we make two modifications to the two-port valve. First, we add an extra leakage between the fluid chambers Fig. 5 such that q leak sgn(p L )P L with being a leakage coefficient. This can be done by simply adding an orifice across the outlets, or by punching a hole between the two middle chambers in the valve. The second modification to the valve system is to place a spring with spring constant K between the spool and the actuator that provides the force F x. The outputs of the two-port valve are correspondingly modified to: Remark. Q mod K q x v K tx v,p L P L P L P L P L (3) yªx v K F x. (4) y, called the interaction displacement for the force F x, is the displacement of the end of the spring at which F x is applied (Fig. 5). 2 It is possible to bound the nonlinear leakage by a linear leakage if P L is known to be bounded. Let M (,P L ) ª/ PL. Then whenever P L P L, q leak P L M P L. 3As 0 and /K 0, the modified outputs Q mod andyin Eqs. (3) (4) converge to Q and x v, respectively, which are the port flow variables for the two-port valve in Section V with the ideal first-order dynamics (9). The following theorem states that with suitable leakage and spring constant K, the modified valve is passive with respect to a modified supply rate. Moreover, the behavior of the modified valve approaches the behavior of the valve with the first order spool dynamics when the inertia of the spool is small. VI Passivity of Valves With Second Order Spool Dynamics Spools have inertia, so their dynamics are necessarily second order. In this section, we consider spool dynamics of the form ẋ v ẍ v 0 0 B x v ẋ v 0 F x AP L, (2) Fig. 5 Valve with extra leakage and added spring Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2000, Vol. 22 Õ 405

5 Theorem 2. Consider the four-way directional valve which is modified by a leakage, and an interaction spring K, and whose output equations are given by Eqs. (3) (4). Suppose that the spool dynamics are second order and are given by Eq. (2). The two-port valve has the following properties: It is passive with respect to the supply rate ż 2 B z 2 B F xap L (8) (9) s new P L,F x,q mod,y A K q P L Q mod F x y, (5) if the matrix NB2 A 0 A A M 0 K q K, where M is the linear leakage as defined in Remark, is positive definite. Thus, the compliance /K and leakage M (and hence ) can be chosen to be arbitrarily small when spool inertia 0. 2 Let Q i (t)q(t) and x v,i (t)x v (t) be the flow rate and the spool displacement trajectories of the two-port valve with firstorder ideal spool dynamics, described by Eqs. (9) (0), given a pair of bounded pressure and interaction force trajectories input (P L (t),f x (t)). Let Q mod (t) and y(t) be the modified flow and interaction displacement of the two-port valve given by Eqs. (2) and (3) (4), which has been modified by leakage and interaction spring K, under the same inputs (P L (t),f x (t)). Assume that x v (0)x v,i (0) and ẋ v (0)0. Under these conditions, as, /K, 0, y x v,i 0, Q mod Q i 0. Proof:. Let us define a new flow variable, and a new supply rate, Q lin ªK q x v M P L (6) s mod P L,F x,q lin,yª A K q P L Q in F x y. Notice that s mod ((P L,F x ),(Q lin,y))s new ((P L,F x ),(Q mod,y)) because of Remark. Thus, to show passivity with respect to s new ((P L,F x )(Q mod,y)), it suffices to show that 0 t s mod P L,F x,q lin,ydtc 2. Using a set of new coordinates (z,z 2 ): z z 2 0 B B x v ẋ v, x v ẋ v 0 B z z 2, the spool dynamics and the output equations after scaling become ż B F xap L (7) Now consider the storage function W 2 z 2 2 z 2 2. Differentiating with respect to time, (20) Ẇ B z 2 2 z 2 B F xap L z B F xap L. (2) Inspired by the formulation of the KYP lemma with a direct feedthrough term, we rewrite the multipliers to P L z 2 and F x z 2 in Eq. 2 in terms of the coefficients that multiply z 2 in Eqs. 9 and 20, respectively A B C 22, B C 222. (22) Hence, A/B and /B. Substituting Eq. 22 into Eq. 2, and subsequently replacing C 2 z 2 and C 22 z 2 using 9, 20, we have K A A M BẆz 2 P L F x B2 z 2 A 0 P L K q F x 0 s mod P L,F x,q lin,y (23) So that BẆs new P L,F x,q mod,y. The passivity property is attained after integration. 2. Because z (0)x v,i (0)0 and z (t) and x v,i (t) satisfy the same differential equation Eq. 9 and Eq. 7, x v (t)x v,i (t) z 2 (t). However, from Eq. 8, z 2 B 2 F x AP L. Thus, as /B 2 0, x v ( ) x v,i ( ). Moreover, as 0, the leaked flow Q mod in 3 approaches Q in 4. Similarly, as /K 0, y( ) x v ( ) in 4 which in turn approaches the ideal spool displacement x v,i ( ). The advantage of the passive passification method is that the structural pressure feedback, as well as the extra leakage and interaction spring, can all be implemented by using hardware components. Sensing/actuation is not needed. However, the extra leakage can lead to inefficiency, and the extra interaction spring may introduce sluggishness when using the interaction displacement y as feedback signal. To get an idea of how much leakage is necessary, consider a m 2 in. long steel spool with diameter d 406 Õ Vol. 22, SEPTEMBER 2000 Transactions of the ASME

6 m (0.3 in.). Suppose that C d /0.02 m 2 /N /2 /s (70 in. 2 /lbf/s. and wd. The mass of the spool is kg. Suppose also that P s KPa (3000 Psi) and the pressure feedback gain A0.(d/2) 2. For passivity, we must have M ()AK q /(/B) m 4 s (/B) 2. We get (/B) 2 P L in S.I. units. As a comparison, if P L689 kpa 00 psi and B/000 rad/s, the required leakage is This corresponds to an orifice area of m 2, which is highly significant. Notice however that the leakage can be significantly reduced if the maximum load pressure or the time constant /B decrease. B Active Passification. The passive passification technique has several drawbacks. For example, a possibly large additional leakage, which introduces inefficiency, is needed. Moreover, the damping B must be sufficiently large. This also implies that the area differential A for the pressure feedback need to be correspondingly increased to maintain sensitivity. If the pressure feedback is to be achieved using structural feedback, then the spool size must be increased. The spool inertia may then have to increase even more. To overcome these difficulties, an active feedback method is proposed. We define the spool dynamics to be ẍ v Bẋ v AP L F x F act (24) where F act is the active passification control to be defined. Let we get, K q x v Q 0 x v QK t x v,p L P L, Ẇ A K q P L QF x x v A K q K t x v,p L P L 2 from which the passivity result is obtained. In the case of the active passification method, there is no need for B to be large. It is however required that the derivative of F x AP L can be well estimated or bounded. The spool dynamics resulting from Eqs. 24 and 26 are no longer the ideal first-order spool dynamics in Eqs. 9. Indeed, if the estimates of the derivatives of P L and F x are good, the transfer function from F x AP L to x v will be given by: sb/ Hs BssB/B/. Notice that introduces a spring force which has the largest effect at low frequencies. At high frequencies, the H(s) approximates /Bs which is the first order dynamics in Eq. 9. VII Simulation To illustrate the property of the passified four-way valve, we simulate the situation when the valve is used to operate a piston. We investigate only the nonpassive valve and the valve passified zªẋ v B F xap L. Then the spool dynamics become: Bẋ v ż 0 B 0 B/ x v z F act d dt F x AP L Now define the control F act to be of the form: F x AP L B. (25) F act Bx v B Ḟ AṖL x g2 tsgnz (26) where is a positive constant, g 2 (t) is some dominating gain to be chosen, and denotes the best estimate of the argument. Theorem 3. Consider the spool dynamics given by (24) with the control law F act given by Eq. (26). If g 2 (t) in Eq. (26) is defined so that g 2 tsgnztḟ AṖL x t Fˆ xtaṗ L t, then the four-way two-port valve is passive with respect to the supply rate sp L,F x,q,x v ª A K q P L QF x x v. (27) Notice that this supply rate is exactly the one in Eq. for the valve in Section V with the first order spool dynamics. Proof: Define the storage function W/2Bx 2 v /2z 2 and taking its time derivative Ẇx v F x AP L x v Bz B z2 z F act Ḟ xaṗ L. B Thus, if F act is chosen as given in the theorem, Fig. 6 wall Four way valve controlling a piston that interacts with a Using the fact that Ẇx v F x AP L x v B z2. Fig. 7 Pressure trajectory that occurs when a piston controlled by a nonpassive four-way valve interacts with a wall Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2000, Vol. 22 Õ 407

7 Fig. 8 Passive four-way valve controlling a piston that interacts with a wall using the passive method. The valve spool is acted on by a force F x 4.45N lbf. The piston is assumed to be light so that it presents no resistance unless it reaches an obstacle. An obstacle in the form of a parallel combination of a spring and a damper to simulate a wall is placed in front of the piston Fig. 6. The port pressure trajectory for the non-passive valve is shown in Fig. 7. After hitting the wall, the piston relentlessly pushes forward and the pressure continues to rise until the port pressure reaches the supply pressure KPa or 3000 psi. On the other hand, for the passified valve, the pressure decreases continuously from 586 KPa 230 psi to 96.5 KPa 4 psi Fig. 8. Asthe piston hits the wall, the spool displacement decreases accordingly. To generate a higher pressure, the operator must increase the force F x. This effect is exactly what is required for human friendly operation. It should be pointed out that because of the damping model of the wall, the peak pressure is determined by the initial impact speed alone. VIII Discussion The directional control valve is often the only potentially nonpassive component in the control path of a hydraulic system. Other components such as the dynamics of the fluid due to compressibility and inertia, cylinder dynamics and leakage etc. are all passive phenomena. Therefore, once the directional control valve has been shown to be passive, passive control techniques such as the passive velocity field control PVFC in 7,8 can be applied. Safety critical applications that require interaction with humans and other delicate environments are most likely to benefit from this development. It is important to bear in mind that when a two-port system is passive, it does not imply that the scaling of power from one port to another is not possible. Power scaling can be incorporated by defining an appropriately scaled supply rate and then by making sure that the system is passive with respect to it. This concept was proposed for passive teleoperation in 4,5 and it is currently being pursued in the context of bilateral hydraulic teleoperation. In this paper, we have shown that the four-way directional valve is by itself not passive, but can be made so if appropriate spool dynamics are imposed. A passive method and an active method have been proposed to this end. The passive method can be implemented by making hardware modifications such as changing the spool land areas for structural pressure feedback, and adding an additional leakage and an interaction spring. This redefines the output from Q to Q mod so that the relative degree from P L Q mod and from F x y are both 0. It is necessary that the relative degrees be less than or equal to for passivity to hold. Moreover, the behavior of the valve is similar to the valve with the ideal first order spool dynamics when the spool inertia is small. This is convenient because one is likely to be able to assume the simpler first order dynamics while designing controllers that utilize the passified valve. However, the additional leakage required can be significant unless the damping B is high and the port pressure is low. Because a smaller leakage is needed when the pressure is small, some improvement in efficiency can be gained if the leakage can be made to be pressure sensitive, such as by using a pilot operated valve. The active passification method does not require the extra leakage or the interaction spring, but requires active feedback of F x, P L and ẋ v to compensate for the derivative of the spool force F x and the port pressure P L. It is interesting to note that several terms in the spool dynamics can also be implemented using hardware solutions. For example, Bẋ v and AP L in Eq. 24 can be implemented using a physical damper and structural pressure feedback, and Bx v in Eq. 26 can be implemented using a spring. Indeed, there is the possibility that neither the damping nor the spring terms need be explicitly implemented at all. It is because the transient and the steady flow forces 3, which naturally occur, but are neglected in Eq. 2 already contribute to the damping and spring forces on the spool. Thus, the flow forces are helpful and need not be dominated or canceled in this case. It is worth remarking that the concept advocated here of using intriguing structural design of the hydraulic valves to provide feedback so to achieve desired property is common in many hydraulic valves. For example, relief valves, pressure compensated flow control valves, and flow dividers/combiners are all valves that utilize pressure feedback on spool. However, the author is not aware of any passivity analysis that has been performed on these valves. Notice that the active feedback causes an effective lowering of the relative degree (P L Q) of the system from 2 to. This is possible because of the use of the derivative information of the input and/or the use of discontinuous control sliding mode in the active passification control law in Eq. 26. Notice also that the dynamics of the valve no longer approximates the ideal first order valve in the low frequency range. Therefore, full order dynamics of the valve needs to be considered while designing the controllers that incorporate the actively passified valve. The supply rates Eqs., 5, and 27 considered for the two-port valves with either the first or second spool dynamics are not entirely related to the mechanical power input. Indeed the port variables for the command port are (F x,x v )or(f x,y) where yx v F x /K. The inner product of these have units of work Joules, not power. Supply rates that are related to mechanical power input are useful for human interaction. This inadequacy can be remedied when the valve is used in a teleoperated setting. In this case, arbitrary power scaling can also be achieved. IX Conclusions The passivity of a four-way directional control valve is investigated. It is shown that the valve is not passive with respect to the hydraulic power input but becomes so if the spool takes on first order dynamics analogous to that of an electrical inductor. A twoport valve that can accept command inputs can be similarly passified. Because spool dynamics are naturally second order, a passive technique and an active feedback technique are proposed to preserve the passivity property of the valve in the presence of second order dynamics. We are currently pursuing research to rigorously incorporate hydraulics technology into passive teleoperation as proposed in 4,5 and to extender type human augmentation topologies 5. Nomenclature A equivalent area for pressure feedback B valve damping C d discharge coefficient 408 Õ Vol. 22, SEPTEMBER 2000 Transactions of the ASME

8 F x, F act spool stroking forces Q L, Q load flow K q ideal flow gain K t(x v,p L ) equivalent shunt conductance P 0 return tank pressure P A, P B pressures in valve chambers P L load Pressure P L bound on load pressure P L P s supply pump pressure Q 0 (x v )ªK q x v equivalent ideal flow source Q mod modified flow via passive passification s(, ) various supply rates w valve area gradient x v spool displacement active passification feedback gain, M nonlinear and linear leakages spool mass hydraulic oil density References Vidyasagar, M., 993, Analysis of Nonlinear Dynamic Systems, 2nd Ed., Prentice Hall, NJ. 2 Li, P. Y., and Horowitz, R., 997, Control of smart exercise machines: Part. problem formulation and non-adaptive control, IEEE/ASME Trans. Mechatron., 2, No. 4, pp Li, P. Y., and Horowitz, R., 997, Control of smart exercise machines: Part 2. self-optimizing control, IEEE/ASME Trans. Mechatron., 2, No. 4, pp Li, P. Y., 998, Passive control of bilateral teleoperated manipulators Proceedings of the 998 American Control Conference, pp Lee, D., and Li, P. Y., 2000, Passive feedforward approach to bilateral teleoperated manipulators, ASME IMECE Symposium on Haptic Interfaces and Teleoperators, to appear. 6 Colgate, J. E., Wannasuphoprasit, W., and Peshkin, M. A., 996, Cobots: robots for collaboration with human operators, Proceedings of the ASME Dynamic Systems and Control Division, DSC. Vol. 58, ASME, New York, NY, pp Li, P. Y., and Horowitz, R., 999, Passive velocity field control of mechanical manipulators, IEEE Trans. Rob. Autom., 5, No. 4, pp Li, P. Y., and Horowitz, R., 996, Application of passive velocity field control to contour following problems, Proceedings of the 996 IEEE Conference on Decision and Control, Dec., Vol., Kobe, Japan, pp Li, P. Y., 999, Adaptive passive velocity field control, Proceedings of the 999 American Control Conference, June, San Diego, CA, Vol. 2, pp Yamakita, M., Yazawa, T., Zheng, X. Z., and Ito, K., 998, Application of passive velocity field control to cooperative multiple 3-wheeled mobile robots, IEEE Int. Conf. Intel. Rob. Syst., Vol., pp Horowitz, R., 993, Learning control of robot manipulators, ASME J. Dyn. Syst., Meas., Control, 5, pp Willems, J. C., 972, Dissipative dynamical systems, part : General theory, Arch. Ration. Mech. Anal., 45, pp Merritt, H. E., 967, Hydraulic Control Systems, Wiley, NY. 4 Smith, R. J., 984, Circuits, Devices and Systems, 2nd Edition, Wiley, NY. 5 Kazerooni, H., and Guo, J. H., 993, Human extenders, ASME J. Dyn. Syst., Meas., Control, 5, pp Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2000, Vol. 22 Õ 409