PASSIVITY property [1] has been exploited in many nonlinear

Transcription

1 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE Natural Storage Function for Passivity-Based Trajectory Control of Hydraulic Actuators Perry Y. Li, Member, IEEE, and Meng Rachel Wang, Member, IEEE Abstract A passivity framework for hydraulic actuators is developed by considering the compressibility energy function for a fluid with a pressure-dependent bulk modulus. It is shown that the typical actuator s mechanical and pressure dynamics model can be obtained from the Euler Lagrange equations for this energy function and that the actuator is passive with respect to a hydraulic supply rate which contains, in addition to the flow work (PQ), the compressibility energy also, which has often been ignored. A storage function for the pressure error is then proposed and the pressure error dynamics are shown to be a passive twoport subsystem. Trajectory tracking control laws are then derived using the storage function. A case study is presented to compare the new passivity-based approach and the traditional backstepping approach using a quadratic pressure error term. In this example, the proposed approach has one fewer parameter to tune, is less sensitive to velocity measurement error, and requires lower feedback gains than the traditional approach. Index Terms Bulk modulus, compressibility, hydraulics, passivity, storage function. I. INTRODUCTION PASSIVITY property 1 has been exploited in many nonlinear physical domains to derive useful and robust control laws. One of the earliest and most motivating was for electromechanical manipulators. Using the mechanical systems physical energy functions and their modification as Lyapunov functions, a passivity property (with mechanical power input being the supply rate) can be derived, and from which a whole class of fixed and adaptive control laws with rigorous analysis and arbitrary gains have been obtained (see 2 and 3 as examples). This energetic passivity property of mechanical systems is a consequence of its Euler Lagrange (or Hamiltonian) structure. Thus, with the success in the mechanical domain, controls that exploit the Euler Lagrange or Hamiltonian structures have been developed for other domains as well (see e.g. 4, 5). Yet, in the area of hydraulic systems, other than approaches based on linearization and linear systems assumption, the typ- Manuscript received April 5, 2012; revised September 26, 2012 and April 1, 2013; accepted April 23, Date of publication July 4, 2013; date of current version April 11, Recommended by Technical Editor J. M. Berg. This work was supported by the National Science Foundation under Grant EEC P. Y. Li is with the Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN USA ( perry-li@umn.edu). M. R. Wang was with the Center for Compact and Efficient Fluid Power, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN USA. She is now with Eaton Corporation, Eden Prairie, MN USA ( RachelWang@eaton.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMECH ical nonlinear control approach is based on backstepping (see e.g. 6 15). In this approach, the desired force from the actuator is first designed (perhaps using passivity-based approach), and then the actuator force is controlled successively by backstepping through a cascade structure. This control structure can be modified to add robustness and performance enhancements. Usually, a simple quadratic term in the actuator force error is used in the Lyapunov function, and nonlinearities are canceled out to preserve stability. In other words, the natural and physical energetic structure of the actuator pressure dynamics have not been exploited. In and related works, passivity of the hydraulic valve is considered but passivity of an actuator with compressibility is simply assumed. There are a handful of other approaches to control hydraulic actuators. Kugi 4 and Mazenc and Richards 19 proposed a nonlinear transformation of the pressures for two-chamber actuators. For constant bulk moduli, a certain passivity property can be exhibited in the transformed system and control laws can be derived that do not require velocity feedback. However, the resulting Lyapunov function does not appear to be related to the physical energy of the system. Kemmetmuller and Kugi in 20 proposed an impedance control law based upon the immersion and invariance control technique 21. An energy Casimir method is used to define a control law in 22. Restrictive assumptions on constant bulk modulus and constant and same actuator chamber volumes are made which do not capture some very important nonlinearities. The approaches in are closest to this paper in that enthalpy from thermodynamics is used to model the power flow of an isentropic fluid. Their focus is on port-controlled Hamiltonian (PCH) systems, the use of Casimir functions, and position regulation. Also, only the case of constant bulk modulus is considered. This paper focuses on developing a passivity-based trajectory control framework, uses an Euler Lagrange formulation, and deals with the pressuredependent bulk modulus. In the presence of even a few percent of air entrainment, the effective modulus is dominated by that of air so that it is heavily pressure dependent (see e.g. Fig. 2). Our goal in this paper is to develop a passivity control framework for the mechanical-pressure dynamics in a hydraulic actuator, based upon natural physical energy. Our main motivation is to incorporate this passivity framework into the popular backstepping technique so that the natural physical property of the system can be exploited. The hope is that the robustness and ease of control can be promoted. While hydraulic fluid is not very compressible, its compressibility determines its pressure dynamics. Thus, in Section II, the density and compressible energy in the hydraulic actuator are derived. The only assumption made is that the fluid IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 1058 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014 A. Compressibility and Density Function Assumption 1 (Bulk Modulus): The constitutive relationship of the fluid is defined by its (absolute) pressure-dependent bulk modulus, β :0, ) R +, as follows 26: dρ ρ = dv V = dp β(p ) (4) Fig. 1. Two-chamber, single rod hydraulic actuator. property is determined by a pressure-dependent bulk modulus β(p ). In Section III, in accordance with this compressibility energy, a Lagrangian function is defined such that the typical actuator dynamics are derived from its Euler Lagrange equations. A passivity property is then derived from this. The energy function is modified in Section IV so that it can be used for control purposes by making it a function of the pressure error instead of absolute pressure. Using this error storage function and a feedforward action, an error passivity property is elicited. In Sections V and VI, a control design case study is presented in which the proposed passivity approach and the traditional backstepping approach are compared experimentally in a trajectory tracking task. It is shown that the passivity-based approach is less sensitive to velocity measurement errors, requires one fewer control gain to tune and needs less aggressive control gains to achieve the same level of performance. Section VII presents an example to compare the effect of using a constant bulk modulus when the actual bulk modulus is pressure dependent. Notation: Both absolute pressures and gauge pressures are used. Absolute pressures are denoted without lettered subscripts (P ). Gauge pressures are denoted with a subscript g (P g ). Pressure errors are denoted with. II. FLUID COMPRESSIBILITY, DENSITY, AND ENERGY A two-chamber, single rod actuator (see Fig. 1) is typically modeled as M p ẍ := A 1 P 1g A 2 P 2g + F load (1) β(p 1 ) 1g := (V 10 + A 1 x) Q 1 A 1 ẋ (2) β(p 2 ) 2g = (V 20 A 2 x) Q 2 + A 2 ẋ (3) where x is the actuator displacement, M p is the combined actuator/load inertia, A 1 and A 2 are actuator capside and piston side areas, P 1g = P 1 and P 2g = P 2 are the gauge pressures in each of the chambers, is the ambient pressure, β(p ) is the pressure-dependent bulk modulus, V 10 and V 20 are the volumes of the actuator chambers and hoses when x =0, and Q 1 and Q 2 are the flows into the chambers. To show how (1) (3) are related to the Euler Lagrange equations for a suitable storage function, we first focus on the energy contained in the volume of the compressed fluid. where V and P are the volume and pressure of a fixed fluid mass m, and ρ(p ):=m/v (P ) is the fluid density. We assume that for all P 0, ), β(p ) β > 0. Given β(p ), define the function g(p 2,P 1 ) as the integral of (4) over the pressure limits P 1 and P 2 ρ(p2 ) P2 dp g(p 2,P 1 ):=ln = ρ(p 1 ) P 1 β(p (5) ) where ρ(p ) is the fluid density at pressure P. The function g(p 2,P 1 ) in (5) satisfies the group and inverse properties under addition: P 1,P 2,P 3 0, g(p 3,P 2 )+g(p 2,P 1 )=g(p 3,P 1 ) (6) g(p 1,P 2 )= g(p 2,P 1 ). (7) Using these properties and definition (5), we have the following results. Theorem 1: Let the pressure-dependent liquid bulk modulus be β(p ) > 0, where P 0, ). Then, the pressure-dependent densities ρ( ) at any pressures P, P 1 0 satisfy ρ(p )=ρ(p 1 ) e g(p,p 1 ). (8) Since g(p, P) =0 and g(p 2,P 1 ) > 0 for all P 2 >P 1,ρ: P ρ(p ) is a monotonic function. Hence, the inverse function ρ 1 ( ) exists on the achievable density range. Proof: Equation (8) is readily obtained by taking exponential of both sides of (5). Monotonicity is a consequence of β( ) > 0 in the integral expression in (5). B. Pressure Dynamics The actuator pressure dynamics in (2) and (3) can be obtained from mass flow and displacement dynamics via the coordination transformation provided by the density, chamber volume, and fluid mass relationship: m 1 =(V 10 + A 1 x)ρ(p 1 ) (9) m 2 =(V 20 A 2 x)ρ(p 2 ). (10) Differentiating (9) and (10) w.r.t. time, we obtain (V 10 + A 1 x) dρ(p 1) dp ρ(p 1 )A 1 ẋ = ṁ 1 (V 20 A 2 x) dρ(p 2) 2 ρ(p 2 )A 2 ẋ = ṁ 2. dp 2

3 LI AND WANG: NATURAL STORAGE FUNCTION FOR PASSIVITY-BASED TRAJECTORY CONTROL OF HYDRAULIC ACTUATORS 1059 Using the bulk modulus definition (4) and rearranging, β(p 1 ) 1 = ρ(p 1 )(V 10 + A 1 x) ṁ 1 ρ(p 1 )A 1 ẋ β(p 2 ) 2 = ρ(p 2 )(V 20 A 2 x) ṁ 2 + ρ(p 2 )A 2 ẋ. Equations (2) and (3) are obtained by defining the volumetric flow rates as Q 1 := ṁ1 ρ(p 1 ), Q 2 := ṁ2 ρ(p 2 ). (11) C. Compressibility Energy To derive the compressibility energy in a fluid, consider a fluid of mass m in an ambient pressure. Its volume V and pressure P are related by V ρ(p )=V 0 ρ 0 = m where ρ 0 and V 0 are the density and volume of the fluid at the datum pressure. Excluding the work done by the ambient pressure, the extra work needed to compress the fluid from (,V 0 ) to pressure (P, V ) is W (m, P )= = V V 0 m/v (P )dv (P )V ρ=ρ 0 ρ m/v (P ) dρ = m ρ=ρ 0 ρ 2 (12) (P ) = m β(p )ρ(p ) dp (13) where V = m/ρ(p ) with ρ( ) defined in (8), ρ(p )dv = Vdρ (since m is fixed), and the bulk modulus in (4) were used in the last two equalities. P is the integration dummy variable. Note that with P = ρ 1 (m/v ), the work input in (13) can be expressed in terms of V as well. Define W V (P g, ) and W m (P g, ) as the volumetric energy density and the gravimetric energy density, respectively, at gauge pressure P g := P, relative to ambient reference pressure by W V (P g, )= W (m, P ) V W m (P g, )= W (m, P ) m. Theorem 2: Relative to ambient pressure, the volumetric energy density W V (P g, ) and the gravimetric energy density W m (P g, ) are given by (P )ρ(p ) W V (P g, )= β(p )ρ(p dp ) = e g(p,p ) 1 dp (14) dρ W m (P g, )= (P ) β(p )ρ(p ) dp = 1 ρ(p ) e g(p,p ) 1 dp. (15) Proof: Since W V (P g, )=W(m, P )/V and W m (P g, )=W(m, P )/m, the first equalities in (14) and (15) are immediate from (13). For the second equality in (14), it is obtained by the substitution ) ρ(p ) eg(p,p β(p )ρ(p = ) β(p ) and by integrating by parts = deg(p,σ) dσ W V (P g, )= (P ) d = (P )e g(p,p ) P + σ =P dσ eg(p,σ) σ =P dp e g(p,p ) dp = P g + e g(p,p ) dp. The second equality in (15) is obtained by noting that W m (P g, )=ρ(p)w V (P g, ). Note that both W V (P g, ) and W m (P g, ) are fluid properties that are only dependent on pressures. Remark 1: 1) W V (P g, ) and W m (P g, ) are proper energy functions in that W V (P g, ) 0 and W m (P g, ) 0 for all P = P g + 0, are zero when at the ambient pressure: W V (0, )=0, and W m (0, )=0, and are positive definite with respect to the gauge pressure P g.thisis why we choose to use P g in the argument instead of P. 2) When the bulk modulus β is a constant, the density, gravimetric energy density, and volumetric energy density (relative to )are ρ(p )=ρ 0 e (P )/β (16) ( W V (P g, )=β e P g /β 1+ P ) g (17) β W m (P g, )= β ( e P g /β 1+ P ) g (18) ρ(p ) β where ρ 0 is the fluid density at the ambient pressure. These expressions are compatible with the internal energy function in 23. 3) From the Taylor expansion of (17) for P g β, wesee that W V (P g, ) P g 2 2β is essentially a quadratic function of P g when β is constant. When β(p ) is pressure dependent, we can define the pressure-dependent mean bulk modulus, β(p, ) to express the energy density quadratically with P g : β(p, ):= (P ) 2 (19) 2W V (P g, ) so that W V (P g, )=Pg 2 /(2 β(p, )).

4 1060 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014 It will be shown that the Euler Lagrange equation of this Lagrangian for generalized coordinate x recovers the actuator dynamic equation (1). The Euler Lagrange equations for coordinates m 1 and m 2, on the other hand, define the effort variables complementary to the flow input ṁ 1, ṁ 2. The effort variables in turn define the supply rates for the hydraulic ports. Recall that the pressure dynamics in (2) and (3) have been obtained in Section II from ṁ 1, ṁ 2, ẋ via a coordinate transformation given by the density function. Define the Lagrangian as L(q, q) :=T (x, ẋ) W act (m 1,m 2,x) (20) where the kinetic coenergy for the actuator/load inertia M p is T (x, ẋ) = 1 2 M pẋ 2 (21) the potential energy (relative to the ambient pressure )is Fig. 2. Bulk modulus β(p ) (solid lines) and mean bulk modulus, β(p, ) (dotted lines, defined in (19) with =0.1 MPa) at 1%, 10%, and 30% airentrainment. Here, the bulk modulus model based on 27 is used to illustrate the general trend. Other bulk modulus models can also be used. If β( ) is positive and bounded over 0,P, then so is β(, ). The point and mean bulk moduli β(p ) and β(p, ) are plotted in Fig. 2, which shows that β(p, ) is much smaller than β(p ) especially at high level of air entrainment. 4) Fig. 2 also illustrates that typically the point and mean bulk moduli are nondecreasing functions of pressure. This property will be used to derive some useful bounds for the mean bulk density in Proposition 1. III. LAGRANGIAN AND PASSIVITY OF A SINGLE ROD ACTUATOR In this section, we define a Lagrangian function for the twochamber actuator with an inertia load using the compressibility energy function derived in the previous section. The generalized coordinates are q =(x, m 1,m 2 ), where m 1 and m 2 are the fluid masses in the respective chambers, and x is the displacement of the actuator. Fluid inertance effects are considered negligible. The mechanical effort input is F load and the hydraulic inputs are the mass flow rates u 1 = ṁ 1,u 2 = ṁ 2 (or equivalently, the volume flow rates, Q 1,Q 2 ). Note that valves, pumps, and other components would interact with the actuator via these input flow variables and their complementary effort variables. Since inertance effects are neglected, the Lagrangian is not a function of ṁ 1, ṁ 2. The dynamics of m 1 and m 2 are trivially the integrals of ṁ 1 and ṁ The compressible actuator system is analogous to a mass spring system in which the velocity of the free end of the spring is an input to the system (analogous to ṁ i or Q i ), the positions of the mass and of the spring s free end are the configuration variables (analogous to x and m i respectively). Note that the dynamics of the free-end position is first order. W act (m 1,m 2,x)=m 1 W m (P 1g, )+m 2 W m (P 2g, ) =(V 10 + A 1 x) W V (P 1g, ) +(V 20 A 2 x) W V (P 2g, ) (22) where P 1g = P 1 and P 2g = P 2 are the chamber gauge pressures obtainable from (x, m 1,m 2 ) via (9), (10), and ρ(p ), the invertible density function in (8). The Euler Lagrange formula in the x coordinate is d T dt ẋ T x + W act x = F load which gives M p ẍ = F load x V 1(x)W V (P 1g, ) m 1 x V 2(x)W V (P 2g, ) m 2 where V 1 (x) =V 10 + A 1 x and V 2 (x) =V 20 A 2 x. Using (56) in Lemma 2 (in Appendix A), the last two terms are (P 1 ) dv 1 dx = A 1(P 1 )=A 1 P 1g (P 2 ) dv 2 dx = A 2(P 2 )= A 2 P 2g. Hence, we recover (1) M p ẍ = F load + A 1 P 1g A 2 P 2g. (23) The dynamics of m 1,m 2 are ṁ 1 = u 1 = ρ(p 1 )Q 1 (24) ṁ 2 = u 2 = ρ(p 2 )Q 2. (25) Equations (23) (25) together with (8) (10) form the dynamics of the hydraulic actuator. m 1 and m 2 dynamics are first order because the Lagrangian is not a function of ṁ 1, ṁ 2. Let Ψ m (P 1g, ) and Ψ m (P 2g, ) be the generalized effort (force) variables; the Euler Lagrange equations for the m 1 and

5 LI AND WANG: NATURAL STORAGE FUNCTION FOR PASSIVITY-BASED TRAJECTORY CONTROL OF HYDRAULIC ACTUATORS 1061 m 2 coordinates are, for i =1, 2, d T T + W act(q) dt ṁ i m i m i =Ψ m (P ig, ). m j i,x Since T is not a function of ṁ 1 and ṁ 2, evaluating the above with the help of (58) in Lemma 2, we obtain, for i =1, 2, 1 ρ(p i ) P ig + W V (P ig, ) = Ψ m (P ig, ). (26) Accordingly, if the input is the volume flow rate (Q 1,Q 2 ) in (11), the corresponding output effort variables are, for i =1, 2, Ψ(P ig, ):=P ig + W V (P ig, ). (27) Power inputs to two hydraulic ports in terms of mass or volume flows are, for i =1, 2, Ψ m (P ig, ) ṁ i =Ψ(P ig, ) Q i. (28) Theorem 3: The actuator system (1) (3) is energetically passive with respect to the supply rate F load ẋ +Ψ(P 1g, )Q 1 +Ψ(P 2g, )Q 2 mechanical power input hydraulic power input (29) where Ψ(P 1g, ) and Ψ(P 2g, ) are the output variables associated with the hydraulic ports in (27), i.e., c Rsuch that for all input functions Q 1 ( ),Q 2 ( ),F load ( ) and t 0, t F load ẋ +Ψ(P 1g, )Q 1 +Ψ(P 2g, )Q 2 dτ c 2. 0 (30) Moreover, the total energy function or the Hamiltonian H(x, ẋ, P 1g,P 2g ):= 1 2 M pẋ 2 +(V 10 + A 1 x)w V (P 1g, ) +(V 20 A 2 x)w V (P 2g, ) (31) is 1) positive definite in (ẋ, P 1g,P 2g ) and 2) is a storage function. Proof: H(x, ẋ, P 1g,P 2g ) is positive definite in (ẋ, P 1g,P 2g ) because W V (P g, ) is positive definite in P g (see Remark 1), V 1 (x) > 0 and V 2 (x) > 0, and M p > 0. Differentiating H(x, ẋ, P 1g,P 2g ) in (31) w.r.t. time Ḣ =ẋ(f load + P 1g A 1 P 2g A 2 )+ W act x ẋ + W act m ṁ. Utilizing (56) in Lemma 2 and (26) 2 Ḣ =ẋ(f load +P 1g A 1 P 2g A 2 ) (P 1g A 1 P 2g A 2 )ẋ +ṁ 1 Ψ m (P 1g, )+ṁ 2 Ψ m (P 2g, ) (32a) = F load ẋ +Ψ(P 1g, ) Q 1 +Ψ(P 2g, ) Q 2. (32) Integrating it over time 0,t), H(t) H(t =0)= t 0 F load ẋ +Ψ(P 1g, )Q 1 +Ψ(P 2g, )Q 2 dτ 2 Since the actuator dynamics are Euler Lagrange equations and H(x, ẋ, P 1,P 2 ) is the Hamiltonian for L(q, q), the following relation is in fact automatically true. The calculations may be instructive however. Fig. 3. W V (P g, )/P g or P g /(2 β(p, )) at 1%, 10% and 30% air-entrainment. and using the fact that H(t) 0, the desired passivity property (30) is obtained with c 2 = H(t =0). Remark 2: 1) The energetic supply rate in (29) consists of mechanical power input and hydraulic power inputs. The mechanical power input ẋf load is as expected. 2) The hydraulic power inputs Ψ(P ig, )Q i consists of the flow work P ig Q i, which is commonly assumed for hydraulic systems 26, but also the compressibility energy associated with the flow, W V (P gi, )Q i, which is often not considered (except in the enthalpy in 23). 3) The presence of the compressibility term also implies an asymmetry in changing the stored energy by either varying the actuator volume (A i ẋ) or by the withdrawal of fluid with the same volume Q i. For the former, the change per unit volume is P ig, and for the latter, it is Ψ(P ig, ).This difference can be seen by comparing the second and third sets of term in (32a). 4) The hydraulic output effort variables Ψ(P 1g, ) P 1g and Ψ(P 2g, ) P 2g if W V (P ig, )/P ig = Pig /(2 β(p i, )) 1 for i =1, 2, where β(pi, ) is the mean bulk modulus in (19). They reduce to P 1g and P 2g as the fluid becomes incompressible (β( ) ). 5) Fig. 3 shows that the discrepancy term W V (P g, )/P g (or P g /(2 β(p, ))) is less than 5.5% over the entire working pressure range even with 30% (extreme case) air entrainment. IV. STORAGE AND PASSIVITY FOR THE PRESSURE ERROR We now propose a storage function for the pressure error. This will be used for controlling pressure and piston velocity according to some desired reference trajectories. Let P d (t) and r(t) be the reference pressure and velocity. Correspondingly, let the pressure error and velocity error be (t) =P (t) P d (t) (33) e v (t) =ẋ(t) r(t). (34) We define pressure error storage density similarly as W V (,P d ) in (14) except that the reference pressure P d (t)

6 1062 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014 is used instead of the ambient pressure W V (,P d ):= = ρ(p )(P P d ) P d ρ(p )β(p )) dp e g(p,p ) 1 dp. (35) P d Similar to W V (P g, ),W V (,P d ) is positive definite w.r.t. and can be expressed as a quadratic function in. Hence, following (19), we can define the mean bulk modulus over the range P d,p as β(p, P d ):= (P P d) 2 2W V (,P d ). (36) The following proposition establishes bounds on β(p, P d ). Proposition 1: The mean bulk modulus over the range P d,p satisfies β(p, P d ) max σ P d,p β(σ)e g(p,σ) β(p, P d ) min σ P d,p β(σ)e g(p,σ). If β( ) is monotonically increasing, then the above reduce to β(p, P d ) max(β(p ),β(p d )e g(p,p d ) ) which is equivalent to β(p, P d ) min(β(p ),β(p d )e g(p,p d ) ) β(p ) β(p, P d ) β(p d )e g(p,p d ), if P P d β(p d )e g(p d,p ) β(p, P d ) β(p ), if P d P. The proof of this result is given in the Appendix B. It shows particularly that β(p, P d ) is bounded from zero if β( ) is, and is close to β(p d ) as P P d. Define now the pressure error storage function for the pressure chamber as W p (x,,p d ):=V (x)w V (,P d ). (37) For an actuator chamber with volume V (x), piston position x, pressure P, and in-flow rate Q(t), the pressure dynamics are P = β(p ) Q A(x)ẋ (38) V (x) where A(x) = dv dx is the effective piston area. Using (38) and formulae similar to Lemma 1 and Lemma 2, it can be shown that d dt W p(x,,p d )= + WV (,P d ) Q A(x)ẋ Ψ(,P d ) V (x) Note that Ψ(,P d ) 0 for all if W V (,P d ) = e g(p,p d ) 1 P d. (39) 2 β(p, P d ) > 1. Fig. 4. W v (,P d )/ versus P = P d + for various P d and entrained air content is 30%. This is indeed the case as established by the following proposition. Proposition 2: For all P, P d 0, ), 2 β(p, P d ) min e g(p d,p ) 1, 0 > 1. It approaches 1 when P d and P =0.Also, 2 β(p, P d ) max e g(p,p d ) 1, 0 e g(p,0) 1. The proof is given in Appendix C. Fig. 4 illustrates (with the bulk modulus model in 27) the function P W V (,P d )/, where the entrained air content is assumedtobe30% (extreme case). Note that W V (,P d )/ 0as 0. Moreover, W V (,P d )/ <0for <0. As indicated in Proposition 2, W V (,P d )/ is smallest when P d is large and P = 0. Even so, this is always greater than 1. Fig. 4 shows that over the entire range P, P d 1, 350 bar, W V (,P d )/ 0.5, 0.1. The range is smaller for smaller or for lower air entrainment. For a trajectory-tracking task, the reference actuator velocity r and reference pressure P d can be determined using a backstepping procedure. The following theorem shows that the function W V (,P d ) is indeed an appropriate storage function for showing that the pressure error dynamics is a passive 2 port system. It also provides the required control to accommodate the reference velocity and desired pressures. Theorem 4: Using the input flow control law, Q := A(x)r + V (x) P β(p ) d γ p + Q (40) Q d where A(x) =dv (x)/dx is the piston area, with a sufficiently large γ p > 0, the error storage function given by (37) W p (x, P, P d ):=V (x)w V (,P d )

7 LI AND WANG: NATURAL STORAGE FUNCTION FOR PASSIVITY-BASED TRAJECTORY CONTROL OF HYDRAULIC ACTUATORS 1063 satisfies the error passivity property d dt W p(x, P, P d ) + WV (,P d ) Q A(x)e v (41) Ψ(,P d ) where e v is the velocity error in (34), so that t Ψ(,Pd ) Q A(x)e v dτ Wp (x(t 0 ),P(t 0 ),P d (t 0 )). t 0 Furthermore, the output function Ψ(,P d ) is a positive function of in that Ψ(0,P d )=0and Ψ(,P d ) 0. Proof: Q d in (40) is designed from (39) by assuming that the pressure error output Ψ(,P d ), and the P d feedforward gain term e g(p,p d ) 1 1/β(P d ). Any discrepancies are then handled by the pressure feedback γ p. To wit, consider the latter approximation. Observe that the d feedforward gain term in (39) can be written, using the mean value theorem, as e g(p,p d ) 1 e g(σ,p d ) = β(σ) β(p d ) where σ P d,p. The approximation is obtained by evaluating σ at P d. The approximation error increases continuously from 0asσ goes from P d to P. Thus, there exists μ(p, P d ) 0 such that e g(σ,p d ) 1 β(σ) β(p d ) μ(p, P d) e g(p,p d ) 1 β(p d ) μ(p, P d) 2. Let W V denote W V (,P d ). Using this, ẋ = r + e v, and the control law (40), (39) becomes Ẇ p Q d dv dx r V (x) P β(p d ) dv d dx e v +(μ(p, P d ) V P d ) 2 + W V Q d +( + W V )( Q γ p ) A(x)e v +( + W V ) Q ( γ p 1+ W ) V (μ(p, P d )V P WV d ) Q d 2 2. Since W V = 1/(2 β(p, P 2 d )) is finite and from Proposition 2, W V > 1, we can choose γ p sufficiently large such that the last term is negative. Remark 3: 1) Theorem 4 shows that the pressure error dynamics is a passive two-port subsystem with the mechanical port power being A e v, where A is the force error and e v is the reference velocity error; and the hydraulics port power being Ψ(,P d ) Q. 2) If the mechanical system is passive with respect to a compatible supply rate A e v, then a cascade of the pressure Fig. 5. Interconnections of compatible passive blocks within the actuator control system. error with the mechanical system results in a passive system and the Ae v term in (41) is canceled out by a similar term in the mechanical system supply rate. This interconnection of passive blocks is illustrated in the pressure error dynamics block in Fig. 5 and can be stabilized simply by the introduction of damping. 3) The control term in (40) consists of: a) a feedforward term; b) a damping term to passify the feedforward term; and c) an additional input. The feedforward term in turn consists of components for r and P d. Note that velocity (ẋ) feedback is not needed. 4) Since Ψ(,P d ) is a positive function, Q defined to be a negative function of, will result in further stabilization. 5) If designed by backstepping, P d will contain some error terms. These may be compensated by large enough feedback gains, similar to the approach in the desired compensation control law (DCCL) 3. Two-Chamber, Single Rod Actuator Theorem 4 can be applied to the two-chamber, single rod actuator in (1) (3) by duplicating W p (x, P, P d ) in (37) for the two chambers. Let V 1 (x) =V 10 + A 1 x and V 2 (x) =V 20 A 2 x. Let the desired pressures for the chambers be P d1 and P d2, and the reference velocity be r(t). Assuming that each chamber flow can be controlled independently. Using the flow as suggested in Theorem 4, d dt V 1 (x)w V ( 1,P d1 )+V 2 (x)w V ( 2,P d2 ) 1 1+ W V ( 1,P d1 ) 1 ( 1 A 1 2 A 2 )e v Q W V ( 2,P d2 ) 2 Ψ( 1,P 1d ) Q 1 +Ψ( 2,P 2d ) Q 2 (F F d )e v where the actuator force and desired actuator force are F := A 1 P 1 A 2 P 2 F d := A 1 P d1 A 2 P d2. Hence, the two-chamber actuator is shown to be passive with the mechanical supply rate being the product between the force error and reference velocity error; and the hydraulic supply rates at both ports being Ψ( ig,p di ) Q i. V. CONTROL DESIGN EXAMPLE In this section, we illustrate the use of the pressure dynamics storage function for the design of trajectory tracking control laws. For simplicity, we consider a one-chamber, spring-loaded Q 2

8 1064 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014 By successively applying (42) and (44), we also have d = 1 A M( ẍ d λ p ë) F load + Kẋ d K p ė K v ė v = 1 A M ẍ d F load + Kẋ d +f(e, e v, ) (45) d 1 Fig. 6. Single-chamber spring-loaded linear (duty ratio control) actuator for the rotary PWM valve in 30. The rotary PWM valve spool is connected to the actuator on the left-hand side. actuator in Fig. 6. It is used for the axial positioning or duty ratio control of a self-spinning rotary three-way PWM on/off valve The PWM valve alternately ports flow from the inlet to one of the two outlet ports as it rotates and its axial position determines the duty ratio. Pairing the PWM valve with a fixed displacement pump/motor, a virtually variable displacement pump/motor can be created. The dynamics of the actuator and the moving mass are Mẍ = AP K(x + x 0 )+F load P = β(p ) (Q Aẋ) V 0 + Ax where M 0.5 kg,a=1.74 cm 2 is the equivalent piston area (which is the area difference between the right and left hand side pistons), K 2594 N/m is the spring rate, Kx N is the preload, and V 0 60 cc is the chamber dead volume. Suppose that Q(t) is the input (it is implemented using a threeway flow control valve) and the objective is for x(t) x d (t), where x d (t), ẋ d (t), ẍ d (t) and ẍ d (t) are smooth and available. Following a typical passivity-based robot motion control design procedure 3, let e = x x d be the tracking error and define the reference velocity, and the reference velocity error as r := ẋ d λ p e; e v := ẋ r =ė + λ p e (42) where λ p > 0. The desired pressure is obtained from P d = 1 A M(ẍ d λ p ė) F load + K(x d + x 0 ) K p e K v e v (43) where K p > 0 and K v > 0. Then, we have the following error dynamics: Mė v = (K + K p )e K v e v + A (44) where = P P d. Then, for W mech := 1 2 Me2 v (K + K p)e 2 dw mech dt = K v e 2 v λ p (K + K p )e 2 + Ae v i.e., the mechanical system is passive w.r.t. the supply rate Ae v. where f(e, e v, )=α e e + α ev e v + α P for some αe,α ev, α P. Next, we take into account pressure dynamics using the proposed passivity approach, and for comparison, the traditional backstepping approach. Passivity approach Define an augmented Lyapunov function using the proposed pressure error energy density function W total = 1 2 Me2 v (K + K p)e 2 +(V 0 + Ax)W V (,P d ). (46) Following Theorem 4, write Q = Q d + Q. Then, dw total = K v e 2 v λ p (K + K p )e 2 + dt Ae v + Q Aẋ (47) + W V (,P d )Q V (x) e g(p,p d ) 1 P d = K v e 2 v λ p (K + K p )e 2 + Q d Ar V (x) P B(P, P d ) d + W V (,P d )Q d + 1+ W V (,P d ) Q (48) where B(P, P d ) is defined from e g(p,p d ) 1 1 = B(P, P d ) (49) which is shown to exist by using the mean value theorem. We are now ready to design Q d to compensate only for the terms related to the trajectory Q d = Ar + V (x) P β(p d ) d1 (50) where r(t) is the reference velocity defined in (42) and P d1 is given in (45) Ẇ total K v e 2 v λ p (K + K p )e 2 V (x) β(p d ) f(e, e v, )+ 1+ W V (,P d ) +(μ(p, P d )V (x) P d + ɛ(p, P d ) Q d ) 2 κ where μ(p, P d ) > 0,ɛ(P, P d ) > 0 are defined from μ(p, P d ) 1/B(P, P d ) 1/β(P d ) ɛ(p, P d ) W V (,P d )/ 2 =1/(2 β(p, P d )). Q

9 LI AND WANG: NATURAL STORAGE FUNCTION FOR PASSIVITY-BASED TRAJECTORY CONTROL OF HYDRAULIC ACTUATORS 1065 Note that the term Ae v from the mechanical system in (47) has been canceled out by the term from the pressure error dynamics, as expected. Finally, we design Q = λ 3, so that the overall control law is Q = Ar + V (x) P β(p d ) d1 λ 3. (51) Using the notation V β to denote V (x) 2β (P d ) gives rise to Ẇ total ( e ) e e v Mpass e ṽ where λ p (K + K p ) 0 α e V β M pass := 0 K v α ev V β P (52) α e V β α ev V β λ 3 κ +2α P V β with λ 3 = λ 3 (1 + W V (,P d ) ). Hence, for λ 3 sufficiently large, the matrix aforementioned is positive definite and (e, e v, ) converge to (0, 0, 0) exponentially. This implies that the system would be input-to-state stable, and hence robust in the presence of disturbance 31. Although the control law seems to require d1 and the pressure-dependent bulk modulus β(p d ) in (51) and (45), ignorance or inaccuracies in the estimation of these terms can be treated as disturbances, which will have small but bounded effects. Basic backstepping approach In the basic backstepping approach such as in 6 15 and others, a quadratic term is typically used, instead of a physically motivated Lyapunov function component, for the pressure error dynamics: W bkstp := 1 2 Me2 v (K + K p)e 2 + λ where λ 2 is a constant tunable parameter Ẇ bkstp = K v e 2 v λ p (K + K p )e 2 + Ae v β(p ) +λ 2 Q Aẋ d. V 0 + Ax The input flow is designed as Q = Aẋ V 0 + Ax Aev P β(p ) λ d1 λ 3 (t) (53) 2 where P d1 is given in (45). So, again we have exponential convergence and robustness to disturbance if λ 3 (t) is sufficiently large, since Ẇ bkstp ( e ) e e v Mbkstp e ṽ (54) P λ p (K + K p ) 0 α e λ 2 /2 M bksp = 0 K v α ev λ 2 /2 α e λ 2 /2 α ev λ 2 /2 λ 3 (t)+α P λ 2 Fig. 7. Trajectory-tracking performances using the two controllers on a filtered trapezoidal trajectory corresponding to a full-range duty variation in 50 ms. where λ 3 (t) = β(p )λ 2 (V 0 + Ax) λ 3(t). The basic backstepping control in (53) differ from the passivity-based control in (51) in these ways. 1) Actuator volume and the bulk modulus are needed only for the feedforward term for the passivity-based control (51) but are needed for both the feedback and feedforward terms for the basic backstepping control (53). The former may have some advantage for adaptation in the presence of measurement noise. 2) The treatment of the piston velocity is different. In the passivity-based approach, only the reference velocity r is used; whereas in the traditional backstepping approach, the actual piston velocity ẋ is actively canceled, and then a velocity error e v is fedback. Intuitively, the ẋ term has a positive feedback effect as ẋ has a positive effect on input flow Q which in turn has a positive effect on piston velocity. The negative feedback of e v has the purpose of stabilizing this effect. 3) Equation (53) provides an additional gain λ 2 for tuning. It will be equivalent to (51) if λ 2 (V 0 + Ax)/β(P ) and similar λ 3 (t) are chosen for both. However, this requires knowledge of the system parameters. Moreover, since λ 2 must be a constant in (53), this equivalence can only be approximated. VI. EXPERIMENTAL COMPARISON The basic backstepping controller in (53) and the passivitybased controller in (51) are experimentally implemented and compared on the linear positioning system described in Section V. Ten percent air entrainment is assumed when estimating the bulk modulus using the model in 27. Flow input is controlled using a Moog servo valve with supply pressure at 1.3 M Pa. Valve spool dynamics are neglected. Position sensing is achieved using a linear magnetic encoder. First, the two controllers are compared on a filtered trapezoidal trajectory as shown in Fig. 7. Each controller was tuned to the best of our effort. While the best tuned control performances are similar (the RMS errors are 0.76 mm for the basic

10 1066 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014 TABLE I FEEDBACK GAINS USING THE PASSIVITY-BASED APPROACH IN (51) AND THE BASIC BACKSTEPPING APPROACH IN (53) Fig. 8. Effect of λ 2 on the trajectory-tracking performance in the presence of corrupted velocity measurement. backstepping case and 0.69 mm for the passivity-based approach), the basic backstepping controller requires significantly higher gains than the passivity-based control as shown in Table I and is more sensitive to measurement noise. The slight error for both controllers during the transient portions is likely due to poorly estimated inertia/load. Next, we investigate the benefits of the extra control gain λ 2 available in the basic backstepping controller in (53) when the velocity measurement is corrupted. To wit, a 1 Hz first-order low-pass filter is applied to the velocity measurement and a 5 Hz sinusoidal-desired trajectory is used. Control gains similar to those in Table I are applied to both controllers and a range of λ 2 is applied to the basic backstepping controller. Fig. 8 shows that tracking error for the basic backstepping controller is minimized when λ and is slightly worse than that of the passivity-based control. This is expected since in (53), velocity measurement error would corrupt the first term (ẋ) when λ 2 is too large, and the second term (e v ) when λ 2 is too small. It will be insensitive to velocity measurement error as λ 2 (V 0 + Ax)/β(P ), which is when the basic backstepping controller approximates the passivity-based control. VII. EFFECT OF PRESSURE DEPENDENT OR CONSTANT BULK MODULUS The Lyapunov analysis in previous sections allows pressure dependence of bulk modulus β(p ) to be incorporated into the control design. With the passive design, β(p ) is only needed for the feedforward term in (51) so it is important only for fast time varying tasks. This is not the case for the basic backstepping approach (53). Effect of the inaccurate estimation of β(p ) can be analyzed as disturbances into the system. To some extent, it can be compensated if a sufficiently high pressure feedback gain λ 3 is used. Whether a constant β can suffice instead of β(p ) in the control law depends on the magnitude and rate of pressure variation in the task. To shed the light on this effect, a simlulation of the spring-loaded actuation system in Section VI is conducted to track a 3Hz, 1cm amplitude sinusoidal profile with the passivity- Fig. 9. (Top) Desired pressure trajectory and variation of bulk modulus. (Bottom) Position tracking error with passivity-based control with various constant β, correct pressure dependent β(p ), and no feedforward term (β = ). Feedback gains are reduced to amplify effects: λ p =1,K p =1,K v =1,and λ 3 = based control using either a constant β, the correct β(p ) and no feedforward term (β = ). Here, the β(p ) varies between 2 kbar and 12 kbar (see Fig. 9). Fig. 9 shows the tracking errors with β =8kbar (RMS = cm), β =12kbar (RMS = cm), with correct β(p ) (RMS = 0.0 cm), and no feedforward term, i.e., β = (RMS = 0.185cm). As expected, controllers with constant β perform worse than the controller with β(p ). VIII. DISCUSSION AND CONCLUSION In this paper, a hydraulic actuator is shown to be a passive two-port system. The supply rate associated with the mechanical port is the mechanical power, whereas the supply rate associated with the hydraulic port consist of the flow work P g Q and of W V (P g, )Q. The latter is the compressibility energy carried in the flow. The storage function is obtained from the compressibility energy of the fluid in the actuator. The modified form of the storage function can be used for pressure error dynamics so that control laws can be derived without canceling the essential nonlinearities associated with the varying chamber volume and the possibly uncertain, pressure dependent, bulk modulus for the stabilizing term. Thus, the passivity control framework that has been so successful for mechanical systems can be extended to hydraulic systems. Particularly, the popular backstepping approach can be enhanced so that it is more robust, easier to tune, and less sensitive to velocity measurement error. The storage function presented for the two-chamber actuator in Section IV deals with each chamber individually and requires keeping track of the individual chamber pressures. The control law there also requires specifying the desired pressure for each chamber. It would be advantageous if the storage function can be formulated using actuator force alone while ensuring that the internal dynamics are stable. In addition, the possibly pressure-dependent bulk modulus may not be known. Indeed, the topic of understanding the behavior of the fluid when pressure varies greatly is receiving significant attention in the context of digital hydraulics (see e.g. 32). So, control schemes that robustly compensate for this uncertainty/ignorance or adaptively estimate it are fruitful avenues for further investigation.

11 LI AND WANG: NATURAL STORAGE FUNCTION FOR PASSIVITY-BASED TRAJECTORY CONTROL OF HYDRAULIC ACTUATORS 1067 Preliminary result for the adaptive control of a two-chamber actuator has recently been obtained 33. Experimental comparison of the proposed control approach with approaches other than backstepping would also be interesting. Nevertheless, the passivity-based hydraulic control concept has been applied successful to displacement control 34 of hydraulic actuators, as well as to hydraulic bilateral teleoperation 35. APPENDIX A MISCELLANEOUS FORMULAE Lemma 1: For the volumetric energy density W V (P g, ) in (14), dw V (P g, ) dp = 1 β(p ) W V (P g, )+(P ). Proof: This is obtained by directly differentiating (14) and using (14) in substitution. Lemma 2: Consider a displacement-dependent volume V (x) filled with a fluid of mass m. Let P be the pressure and V (x)w V (P g, ) be the compressibility energy. 1) If m is fixed and x is varying, P x = β(p ) dv (x) (55) m V dx x V (x)w V (P g, ) dv (x) = (P ) m dx. (56) 2) If x is fixed and m is varying, P m = β(p ) x V (x)ρ(p ) = β(p ) (57) m m V (x)w V (P g, ) = 1 x ρ(p ) (W V (P g, )+(P )). (58) Proof: 1) To obtain (55), differentiate V (x)ρ(p (x)) = m with respect to x when m is constant, making use of (8) so that dv (x) ρ(p ) P ρ(p )+V(x) dx β(p ) x =0. m 2) To obtain (56), note that m V (x)w dv (x) V (P g, ) x = dx W V (P g, ) + V (x) dw V (P g, ) P dp x. m The desired result is obtained by substituting Lemma 1 and (55) into above. 3) To get (57), differentiate V (x)ρ(p (x)) = m with respect to m when x is constant, making use of (8) so that V (x) dρ P dp m = V (x) ρ(p ) P x β(p ) m =1. x 4) To obtain (58), note that x V (x)w V (P g, ) m = V (x) dw V P dp m. x The result is obtained by substituting Lemma 1 and (57) into the above. APPENDIX B PROOF OF PROPOSITION 1 Proof of Proposition 1: Using the first expression in (35) for W V (,P d ): W V (,P g(p,p ) e d )= P d β(p ) (P P d )dp Hence, W V (,P d ) W V (,P d ) β(p, P d )= max σ P d,p min σ P d,p 2 /2 W V (,P d ) β(p, P d ) e g(p,σ) 2 β(σ) e g(p,σ) β(σ) β(σ) min σ P d,p max σ P d,p e g(p,σ) β(σ) e g(p,σ) If β( ) is a nondecreasing function of pressure, β(σ)e g(p,σ) is also nondecreasing so that the max and min are attained at the boundaries. Hence, max(β(p ),β(p d )e g(p,p d ) ) β(p, P d ) min(β(p ),β(p d )e g(p,p d ) ). APPENDIX C. PROOF OF PROPOSITION 2 Proof: For the lower bound, 2 β(p, P d ) = W V (,P d ) P = d e g(p,σ) 1 dσ (P P d ) min e g(p,σ) 1 σ P d,p = mine g(p,p),e g(p,p d ) 1 = min0, (1 e g(p d,p ) ). We have used the fact that g(p, σ) is monotonically decreasing in σ. The final expression is minimized when P =0and P d as desired. The upper bound is derived similarly 2 β(p, P d ) max e g(p,σ) 1 σ P d,p max(0, e g(p,p d ) 1). This is maximized when P d =0and P. REFERENCES 1 A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control. London, U.K.: Springer-Verlag, 2000.

12 1068 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE D. E. Koditschek, The application of total energy as a lyapunov function for mechanical control systems, in Dynamics and Control of Multibody Systems, P. S. K. J. E. Marsden and J. C. Simo, Eds. Providence, RI, USA: Contemporary Mathematics, 1989, pp N. Sadegh and R. Horowitz, Stability and robustness analysis for a class of adaptive controllers for robotic manipulators, Int. J. Robot. Res.,vol.9, no. 3, pp , A. Kugi, Non-Linear Control Based on Physical Models, (Series Lecture Notes in Control and Information Sciences). New York, NY, USA: Springer-Verlag, 2000, no R. Ortega, A. Loria, P. Nicklasson, and H. Sira-Ramirez, Passivity Based Control of Euler-Lagrange Systems. New York, NY, USA: Springer- Verlag, A. Alleyne, A systematic approach to the control of electrohydraulic servosystems, in Proc Amer. Control Conf., Philadephia, PA, USA, pp A. G. Alleyne and R. Liu, Systematic control of a class of nonlinear systems with application to electrohydraulic cylinder pressure control, IEEE Trans. Control Syst. Technol., vol. 8, no. 4, pp , Jul M. Sirouspour and S. E. Salcudean, Nonlinear control of hydraulic robots, IEEE Trans. Robot. Autom., vol. 17, no. 2, pp , Apr H. M. Kim, S. H. Park, J. H. Song, and J. S. Kim, Robust position control of electro-hydraulic actuator systems using the adaptive back-stepping control scheme, Proc. Inst. Mech. Eng., J. Syst. Control Eng., vol. 224, pp , G. Li and A. Khajepour, Robust control of a hydraulically driven flexible arm using backstepping technique, J. Sound Vib., vol. 280, pp , C. Guan and S. Pan, Nonlinear adaptive robust control of single-rod electro-hydraulic actuator with unknown nonlinear parameters, IEEE Trans. Control Syst. Technol., vol. 16, no. 3, pp , May C. Kaddissi, J.-P. Kenne, and M. Saad, Indirect adaptive control of an electrohydraulic servo system based on nonlinear backstepping, IEEE/ASME Trans. Mechatronics, vol. 16, no. 6, pp , Dec M. Choux and G. Hovland, Adaptive backstepping control of nonlinear hydraulic-mechanical system including valve dynamics, Model. Identif. Control, vol. 31, pp , A. Mohanty and B. Yao, Integrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband, IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp , Aug A. Mohanty and B. Yao, Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates, IEEE Trans. Control Syst. Technol., vol. 19, no. 3, pp , May P. Y. Li, Towards safe and human friendly hydraulics: The passive valve, Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 122, no. 3, pp , Sep P. Y. Li and R. F. Ngwompo, Power scaling bondgraph approach to the passification of mechatronic systems with application to electrohydraulic valves, Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 127, no. 4, pp , P. Y. Li and K. Krishnaswamy, Passive bilateral teleoperation of a hydraulic actuator using an electrohydraulic passive valve, Int. J. Fluid Power, vol. 5, no. 2, pp , F. Marzenc and E. Richard, Stabilization of hydraulic systems using a passivity property, Syst. Control Lett., vol. 44, pp , W. Kemmetmuller and A. Kugi, Immersion and invariance-based impedance control for electrohydraulic systems, Int. J. Robust Nonlinear Control, vol. 20, pp , A. Astolfi and R. Ortega, Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems, IEEE Trans. Autom. Control, vol. 48, no. 4, pp , Apr S. Sakai and S. Stramigioli, Passivity based control of hydraulic robot arms using natural casimir functions: Theory and experiments, in Proc. Int. Conf. Intell. Robots Syst., Nice, France, 2008, pp A. Kugi and W. Kemmetmuller, New energy-based nonlinear controller for hydraulic piston actuators, Eur. J. Control, vol. 10, pp , G. Grabmair, K. Schlacher, and A. Kugi, Geometric energy based analysis and controller design of hydraulic actuators applied in rolling mills, in Proc. Eur. Control Conf., Cambridge, U.K., 2003, pp G. Grabmair and K. Schlacher, Energy-based nonlinear control of hydraulically actuated mechanical systems, in Proc. 44th IEEE Conf. Decis. Control, Seville, Spain, 2005, pp H. E. Merritt, Hydraulic Control Systems. New York, NY, USA: Wiley, J. Yu, Z. Chen, and Y. Lu, The variation of oil effective bulk modulus with pressure in hydraulic systems, Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 116, no. 1, pp , H. Tu, M. Rannow, J. Van de Ven, M. Wang, P. Li, and T. Chase, High speed rotary pulse width modulated on/off valve, in Proc. ASME Int. Mech. Eng. Congr. Expo, Washington, DC, USA, Nov. 2007, pp H. C. Tu, M. B. Rannow, R. M. Wang, P. Y. Li, T. R. Chase, and J. D. Van de Ven, Design, modeling, and validation of a high speed rotary PWM on/off hydraulic valve, Trans. ASME, J. Dyn. Syst., Meas. Control, vol. 134, no. 6, p , Nov R. Wang, H. Tu, M. Rannow, P. Li, and T. Chase, Direct displacement control of hydraulic actuators based on a self-spinning rotary on/off valve, presented at the 2011 Int. Fluid Power Exhib. (IFPE), Las Vegas, NV, USA, Mar H. K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ, USA: Prentice-Hall, J. D. Van de Ven, On fluid compressibility in switch-mode hydraulic circuits Part I: Modeling and analysis, Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 135, no. 2, p , M. Wang and P. Y. Li, Passivity based adaptive control of a two chamber single rod hydraulic actuator, presented at the 2012 Amer. Control Conf., Montreal, QC, Canada, Jun M. Wang and P. Y. Li, Direct displacement control of hydraulic actuators based on a self-spinning rotary on/off valve, presented at the 2012 ASME Dyn. Syst. Control Conf., Fort Lauderdale, FL, USA, V. Durbha and P. Y. Li, A nonlinear spring model of hydraulic actuator for the passive controller design in bilateral teleoperation, presented at the 2012 Amer. Control Conf., Montreal, QC, Canada, Jun Perry Y. Li (S 87 M 95) received the B.A. and M.A degrees in electrical and information sciences from Cambridge University, Cambridge, U.K., in 1987, the M.S. degree in biomedical engineering from Boston University Boston, MA, USA, in 1990, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, CA, USA, in During , he was on the research staff at Xerox Corp., Webster, NY, USA. He joined the University of Minnesota in 1997 and is currently a Professor in the Department of Mechanical Engineering and the Deputy Director of the NSF Center for Compact and Efficient Fluid Power, Minneapolis, MN, USA. His research interests include design and control of mechanical and fluid power systems, interactive robotics, hydraulic hybrid vehicles, underwater vehicles, energy storage, and printing processes. Dr. Li received the 2000 Japan/USA Symposium on Flexible Automation Young Investigator Award. Meng Rachel Wang (M 12) received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2006, and the M. S. degree in electrical engineering from the University of Minnesota, Minneapolis, MN, USA, in 2010, where she is currently working toward the Ph.D. degree in mechanical engineering. From 2006 to 2012, she was a Research Assistant with the Department of Mechanical Engineering, University of Minnesota. Her research interests include modeling, control, and estimation of hydraulic systems. Since 2012, she has been a Lead Engineer with Global Research and Development, Control System Solutions, Eaton Corporation, Eden Prairie, MN, USA.