Proceedings of the ASME/BATH 2015 Symposium on Fluid Power & Motion ControlFPMC2015

October 12-14, 2015, Chicago, Illinois, United States

FPMC2015-9618

PASSIVITY BASED BACKSTEPPING CONTROL FOR TRAJECTORY TRACKINGUSING A HYDRAULIC TRANSFORMER

Sangyoon LeeERC for Compact and Efficient Fluid Power

Department of Mechanical EngineeringUniversity of Minnesota

Minneapolis, Minnesota 55455Email: leex6261@umn.edu

Perry Y. LiERC for Compact and Efficient Fluid Power

Department of Mechanical EngineeringUniversity of Minnesota

Minneapolis, Minnesota 55455Email: lixxx099@umn.edu

ABSTRACTThrottling loss is a major contributor to the low system ef-

ficiency in hydraulic systems. Hydraulic transformers can po-tentially be an energy efficient, throttle-less control approachfor multi-actuators systems powered by a common pressure rail(CPR). The transformer transforms the input CPR pressure to thedesired pressure of the actuator instead of throttling it. Regener-ative energy can also be captured. For transformers to be use-ful, they must also have good control performance. This paperpresents a a passivity based trajectory tracking controller for ahydraulic actuator driven by a transformer consisting of two me-chanically coupled variable displacement pump/motors. In ad-dition to controlling the motion of the actuator, the transformerspeed can also be regulated at the most efficient operating speed.The nonlinear controller is designed using a Lyapunov functionthat is based upon a recently discovered natural energy storagefunction for hydraulic actuators. Experimental results validatethe efficacy of this controller.

1 INTRODUCTIONHydraulic power transmission offers multiple benefits over

competing technologies including an order of magnitude higherpower density than electric systems, relatively low cost, fast re-sponse, and flexible packaging. Thus, hydraulic actuators are of-ten used in applications that demand power, precision and com-pactness. However, typical hydraulic systems suffer from lowsystem efficiency with the use of throttling valves for control.

Various methods to improve the efficiency of hydraulic systemshave been researched in recent years. These include load sensing(LS) control and displacement control. Yet another approach isto use a common pressure rail (CPR) and a hydraulic transformerfor each individual actuator to transform the CPR pressure to therequired pressure of the actuator. Since hydraulic transformersdo not rely on throttling for control, it can improve over LS sys-tems especially when the various actuators have very differentpressure requirements.

A hydraulic transformer consists of a hydraulic pump and ahydraulic motor connected mechanically. By varying the relativedisplacements of the pump and of the motor, hydraulic power atone pressure/flow at the input port is converted to another pres-sure/flow at the output port and vice versa. A hydraulic trans-former used in place of a servo valve can eliminate throttling lossand can allow for energy regeneration through four-quadrant op-eration, increasing the overall efficiency of the system.

Although a hydraulic transformer can be configured by con-necting two pump/motors mechanically, there has been a focusedresearch effort, in the past decade, to develop a different con-figuration, known as INNAS hydraulic transformer (IHT) (seee.g. [1–4]). By using a rotatable 3-ported port plate, IHT com-bines the pumping and motoring pistons into a single rotatinggroup and the transformation ratio is determined by the rotationof the port plate. For a transformer to be useful, it must also havegood control performance in addition to efficiency. However,most previous works are concerned with the design of hydraulictransformers. Only a few papers discuss the control aspect. For

1 Copyright c© 2015 by ASME

PA

PB

D1 D2

x

P1, A1

P2, A2

PT

QA QB

FL

FIGURE 1. Schematic of hydraulic transformer: PM-1

PA

PB

D1 D2

x

P1, A1

P2, A2

QB

QA

PT

FL

FIGURE 2. Schematic of hydraulic transformer: PM-2

example, Werndin and Palmberg [5,6] presented design conceptsnecessary to control the IHT at low speed. They used a modelbased estimator and a feed-forward control in parallel with a PIcontroller to simulate an IHT driving a hydraulic cylinder. Vaelet al [7] qualitatively laid out various possible hydraulic systemsto be used in their experiment on an excavator. Ahn and Ho [8]presented a robust controller based on disturbance observer forregulating the position of a hydraulic cylinder driven by a tradi-tionally configured transformer where two pump/motors are cou-pled together.

In this paper, we investigate the control performance of ahydraulic actuator controlled using a traditionally configured hy-draulic transformer (two pump/motors connected mechanically)in which both displacements can be used as control input. Whilea transformer can be constructed with only one pump/motor dis-placement being variable, with both displacements being ad-justable, both the transformation ratio and the transformer speedcan be controlled simultaneously. We propose a passivity basedback-stepping control strategy to enable a hydraulic actuator toprecisely track a desired trajectory, and to control the rotationalspeed of the transformer. This extra degree of freedom can beused to optimize efficiency and to avoid transformer stalling. Theproposed controller can be adapted to all three traditionally con-

PB

D1D2

x

P1, A1

P2, A2

PA

QB

QA

PT

FL

FIGURE 3. Schematic of hydraulic transformer: PM-3

figured transformers that differ by port connections. Comparedwith our previous work [9], this paper uses the natural energystorage function in [10] instead of a quadratic function in thedefinition of the Lyapunov function to achieve better robustness.The controller performance is also validated experimentally inthis paper.

In section 2, the dynamics of the transformer controlled hy-draulic actuator system is presented. In section 3, the controllerdesign is presented. Experimental results are presented in Sec-tion 4. Section 5 contains concluding remarks and future works.

2 System DescriptionIn a typical hydraulic system, a directional servo valve is

placed between the pressure source (often constant) and the ser-vice to throttle down any excess pressure. This is a major sourceof hydraulic system inefficiency. Such losses can be eliminatedby using a hydraulic transformer. In the circuit shown in Fig. 1,a hydraulic transformer with two pump/motor units (PM trans-former) is used in place of a servo valve to control the flow rateinto the cylinder chamber carrying a vertical mass load. By con-trolling the displacement ratio of right hand side pump/motorin the PM transformer, velocity and the position of the cylin-der can be tracked. To recover the gravitational potential to thecommon pressure rail (CPR) while the load is being lowered,the pump/motor can go over-center or run in opposite direc-tion. By controlling the displacement ratio of the left hand sidepump/motor, the shaft speed of the transformer can be controlledby varying the power/torque balance within the transformer. Ifmore power is injected to the transformer than what is requiredby the load trajectory, the transformer speed will increase. Whenmore power is present in the transformer than what is needed, theextra potential can be recovered back to the CPR.

In Fig. 1, pump/motor unit 1 serves as a motor and unit 2serves as a pump when the energy is delivered into the hydrauliccylinder. Their roles reverse when recovering regenerative loads.The two pump/motors need to be controlled simultaneously in or-

2 Copyright c© 2015 by ASME

der to achieve trajectory tracking performance while re-capturingthe energy from the gravitational potential to attain high effi-ciency. Figs. 2 and 3 illustrate two other PM transformers thatdiffer in the ways that the ports are connected. The PM-2 con-figuration in Fig. 2 is more adept at pressure bucking whereasthe PM-3 configuration in Fig. 3 is more adept at pressure boost-ing. In all three configurations, by adjusting the displacements ofthe 2 pump/motors, the flow requirements and the energy/torquebalance of the transformer can also be simultaneously satisfied.This feature could be used to prevent a transformer from stallingin low speed region and to operate transformer at the most opti-mal speed.

2.1 System DynamicsThe inertia dynamics of the hydraulic cylinder in Figs. 1-3

are:

mx =−bx+P1(t)A1−P2A2 +FL (1)

where m is the mass of the cylinder rod and load, x is the verti-cal position of the cylinder load mass, A1 and A2 are respectivelythe cap side and rod side areas of the hydraulic actuator, b is theviscous friction coefficient, and FL(t) is a load force that encap-sulates any external load including gravity, environment forcesand un-modeled dynamics.

The dynamics of the cap-side pressure P1 are given by thecompressibility of the fluid in the cylinder and hose:

P1 =β (P1)

V10 +A1x(QB−A1x) (2)

where QB is the flow rate into the cap side chamber to be pro-vided by the transformer, V10 is the volume in the capside cham-ber and hose when the actuator is at the position x = 0, and β (P1)is the pressure dependent bulk modulus [11]. The rod side is con-nected to the lower common pressure rail so, P2 = PT , which isassumed to be constant.

The capside flow is supplied (or absorbed) by the hydraulictransformer which consists of a pair of variable displacementhydraulic pump/motors. The pump/motors are connectedmechanically, and two of the ports, one from each pump/motor,are connected together. The transformer dynamics are governedby the common rotational inertia J and the torque appliedby the pump/motors. The input, output and tank ports arelabeled as A, B and T . By permuting the port connections,the three configurations in Figs. 1-3 can be obtained. Eachconfiguration will have different flow capability and efficiencycharacteristics. The transformer rotational speed (ω) dy-namics, input flow (QA), output flow (QB), and (ideal) pressuretransformation ratio (λ ) for the three configurations are given by:

PM-1 (Fig. 1):

Jω = (PA−PT )D1

2πu1− (PB−PT )

D2

2πu2−Btω−Tloss

QA = ω · D1

2πu1 +Qleak′

QB = ω · D2

2πu2−Qleak

λ (u1,u2) =D1u1

D2u2

(:≈ PB−PT

PA−PT

)(3)

PM-2 (Fig. 2):

Jω =−(PA−PB)D1

2πu1− (PB−PT )

D2

2πu2−Btω−Tloss

QA =−ω · D1

2πu1 +Qleak′

QB = ω ·(−D1

2πu1 +

D2

2πu2

)−Qleak

λ (u1,u2) =D1u1

D1u1−D2u2

(4)

PM-3 (Fig. 3):

Jω =−(PA−PT )D1

2πu1 +(PA−PB)

D2

2πu2−Btω−Tloss

QA = ω ·(−D1

2πu1 +

D2

2πu2

)+Qleak′

QB = ω · D2

2πu2−Qleak

λ (u1,u2) =−D1u1 +D2u2

D2u2

(5)

where D1 and D2 are the maximum volumetric displacements ofthe pump/motor units in m3/rev, u1 and u2 ∈ [−1,1] are controlinputs which are the normalized displacements, Bt is the damp-ing coefficient, Qleak′ , Qleak and Tloss are the lumped volumetricloss at the A and B ports and the mechanical loss inside the trans-former due to friction. These losses are generally configuration,pressure and speed dependent. λ is the input to output flow trans-formation ratio, which, in steady state, is also the output to inputpressure transformation ratio when losses are absent.

3 Control StrategyThe control objective is for the actuator position x(t) to track

a reference trajectory xd(t) subjected a load FL, while regulatingthe hydraulic transformer speed at ωd(t).

In all 3 configurations in Figs. 1-3, PA and PT are the highand low pressures of the common pressure rails which are as-sumed constant. Overrunning load and cavitation are assumed

3 Copyright c© 2015 by ASME

not to occur as would be the case when the gravity load is suf-ficiently large and speed is sufficiently slow. Otherwise, a di-rectional control valve can be added between the transformerand the actuator. In the proposed approach, the desired veloc-ity, force, pressure of the actuator, and finally the required flowto the actuator are successively defined and controlled via passiv-ity backstepping. Unlike feedback linearization or backsteppingthat uses a generically defined quadratic lyapunov function [9]where active cancellation of specific terms are needed, passivity-based approach uses a natural energy inspired Lyapunov functionsuch that the cancellation is done automatically due to the struc-tural property of the system. This results in improved perfor-mance, robustness against model uncertainties, and fewer gainsto tune [10].

In addition to specifying the required flow (QB) to controlthe cylinder motion, the net torque (Utotal) on the transformer willalso be specified. The required flow to the cylinder and the nettorque are then simultaneously satisfied by decomposing theserequirements into appropriate settings for the two displacementsof the hydraulic transformer.

3.1 Cylinder Flow Requirement - QBIn this subsection, we design required QB in Eq. (2) such that

x(t)→ xd(t), where xd , xd , xd , and...x d are assumed to be smooth

and available. The passivity approach in [10], summarized be-low, is taken for this purpose. The readers are referred to [10] fordetails.

Let e := x−xd be the tracking error and define the referencevelocity, and the reference velocity error as

r := xd−λpe (6)ev := x− r = e+λpe (7)

where λp > 0. Then, by designing the desired pressure to be:

Pd :=1

A1(mr+br−FL +A2P2−Kpe−Kv1ev) (8)

where Kp > 0, and Kv1 > 0, the reference velocity error dynamicsbecome:

mev =−Kpe−Kvev +A1P (9)

where P := P1−Pd and Kv = Kv1 + b. With the Lyapunov (orstorage) function

Wmech :=12

me2v +

12

Kpe2

Wmech =−Kve2v−λpKpe2 + PA1ev

(10)

the mechanical system can be seen to be passive with respect tothe supply rate PA1ev.

Next, pressure dynamics is taken into account by augment-ing the Lypaunov (or storage) function Wmech with the pressureerror energy: V1(x)WV (P,Pd) where V1(x) := (V10 +A1x),

WV (P,Pd) :=∫ Pd+P

Pd

[eg(Pd+P,P′)−1

]dP′ (11)

is the volumetric pressure error energy density associated withcompressing the fluid from pressure Pd to Pd + P with

g(Pd + P,Pd) :=∫ Pd+P

Pd

dP′

β (P′)(12)

and β (P′) is the bulk modulus at pressure P′ (see [10] for details).Hence, the augmented Lyapunov function is:

Wtotal =12

me2v +

12

Kpe2 +V1(x)WV (P,Pd) (13)

Using the property [10]:

ddt

[(V1(x)WV (P,Pd)

]=[P+WV (P,Pd)

]QB− PA1x−V1(x)

[eg(P1,Pd)−1

]Pd

and writing QB = Qd + Q, we have:

Wtotal =−Kve2v−λpKpe2 + PAev + PQB− PAx

+WV (P,Pd)QB−V1(x)[eg(P1,Pd)−1]Pd

=−Kve2v−λpe2 + P

[Qd−A1r− V1(x)

B(P1,Pd)Pd

]+WV (P,Pd)Qd + P

[1+

WV (P,Pd)

P

]︸ ︷︷ ︸

>0

Q

(14)

where B(P,Pd) is defined from

[eg(P1,Pd)−1] =1

B(P1,Pd)P. (15)

By successively applying Eq. (6) and Eq. (8),

Pd =1

A1

[m(

...x d−λpe)− FL−Kpe−Kvev]

(16)

=1

A1

[m

...x d− FL]

︸ ︷︷ ︸Pd1

+ f (e,ev, P) (17)

where f (e,ev, P) = αee+αevev +αPP for some αe, αev, αP.

4 Copyright c© 2015 by ASME

Now compensating only for the terms related to the trajec-tory, Qd is designed to be:

Qd = A1r+V1(x)β (Pd)

Pd1 (18)

where r(t) is the reference velocity defined in Eq. (6) and Pd1 isgiven in Eq. (17). Using this term:

Wtotal ≤−Kve2v−λpe2 + P

[1+

WV (P,Pd)

P

]Q

− PV1(x)β (Pd)

f (e,ev, P)

+(µ(P1,Pd)V (x)Pd + ε(P1,Pd) |Qd |)︸ ︷︷ ︸κ

P2

(19)

where µ(P1,Pd)> 0, ε(P,Pd)> 0 satisfy:

µ(P1,Pd)|P| ≥ |[1/B(P1,Pd)−1/β (Pd)]|ε(P1,Pd)≥WV (P,Pd)/P2

Note that the term PA1ev from the mechanical system Eq. (14)has been canceled out automatically by the term from the pres-sure error dynamics.

Finally, since it can be shown that[1+WV (P,Pd)/P

]> 0,

we design Q=−λ3P such that the overall control law for desiredflow into the piston chamber is:

QB = A1r+V1(x)β (Pd)

Pd1−λ3P (20)

Using the notation Vβ to denote V1(x)2β (Pd)

gives rise to

Wtotal ≤−(e ev P

)Mpass

eevP

(21)

where

Mpass :=

λpKp 0 αeVβ

0 Kv αevVβ

αeVβ αevVβ λ3−κ +2αPVβ

with λ3 = λ3(1+

WV (P,Pd)P ). Thus, for λ3 > 0 sufficiently large,

Mpass is positive definite and (e,ev, P) converge to (0,0,0) expo-nentially. This implies that the bounded un-modeled disturbancewould only cause bounded effects.

Despite the analysis being a little involved, the control lawin (20), (6), (7), (16) and (17) is quite straightforward. Moreover,inaccuracies or ignorance in the estimation of Pd1 and β (Pd) inEq. (20), could be treated as disturbances with negligible effectsafter proper controller tuning.

3.2 Transformer Speed ControlSince the displacements of both pump/motors in the hy-

draulic transformer can be manipulated, an additional control ob-jective other than controlling the cylinder motion can be speci-fied. Here, we impose that the transformer speed should track anarbitrary profile ωd(t). ωd(t) can be designed to prevent stallingor to optimize the operating efficiency of the transformer.

From (3)-(5), the speed dynamics of the three transformerconfigurations can be written as:

Jω =Utotal−Btω−Tloss (22)

where

Utotal =

(PA−PT )

D12π

u1− (PB−PT )D22π

u2 PM-1−(PA−PB)

D12π

u1− (PB−PT )D22π

u2 PM-2−(PA−PT )

D12π

u1 +(PA−PB)D22π

u2 PM-3(23)

is the total torque acting on the transformer by the pump/motorunits. Given the reference shaft speed for transformer ωd(t), anappropriate Utotal is needed to drive the transformer speed ω tothe desired speed. Here we use a simple PI control with feedfor-ward:

˙ωI = ω := ω−ωd

Utotal = Jωd−Kpω−KIωI(24)

Additional terms can also be added to compensate for dampingand mechanical loss.

3.3 Displacement inputsHere, we determine u1 and u2 to work simultaneously to

provide the desired torque in Eq. (24) and the desired flow QBin Eq. (20).

For each transformer configuration, u1 and u2 could besolved simultaneously using the flow equations in Eqs. (3)-(5)and Eq. (23) as follow:PM-1:[

u1u2

]=

[0 ω · D2

2π

(PA−PT )D12π−(PB−PT )

D22π

]−1 [QB

Utotal

](25)

PM-2:[u1u2

]=

[−ω · D1

2πω · D2

2π

−(PA−PB)D12π−(PB−PT )

D22π

]−1 [QB

Utotal

](26)

PM-3:[u1u2

]=

[0 ω · D2

2π

−(PA−PT )D12π

(PA−PB)D22π

]−1 [QB

Utotal

](27)

Notice that mechanical and volumetric losses were ignored inthese expressions. Additional terms to compensate for thesecould also be added to improve performance.

5 Copyright c© 2015 by ASME

FIGURE 4. Transformer based control is tested on the pitch axis ofthis experimental setup.

TABLE 1. Experimentation parameters

Parameter Notation Value

Cylinder mass m 100 kg

Load FL -981 N

Viscous damping b 5000 N/m · s

Piston Cap Area A1 11.87 cm2

Piston Rod Area A2 5.1 cm2

Supply (gauge) pressure PA 5.5 MPa

Return (gauge) pressure PT 0 MPa

4 Experimental ResultsThe controller in Section 3 has been experimentally imple-

mented on the pitch axis of the robotic device shown in Fig. 4.The prototype hydraulic transformer used was constructed bymechanically connecting two 3.15cc micro-piston pump/motors(Fig. 5). Lead screws, stepper motors and encoders are addedto actuate the swashplates and to adjust the displacements of thepump/motors. The transformer in all three configurations in Fig.1-3 have been tested. Various system parameters are summarizedin Table 1. Although the orientation of the hydraulic cylinder inFig. 4 does vary, the variation is small. This allows us to esti-mate the effective mass and damping coefficient (referred to theactuator) as constants.

4.1 Cylinder trajectory TrackingFirst, the controller is tested with two different sinusoidal

trajectories on the PM-1 setup (Fig. 1). One trajectory has ahigher amplitude (0.04m) and lower frequency (0.4π rad/s), andthe other has a lower amplitude (0.015 m) and higher frequency(0.7π rad/s). The transformer speed is to be regulated at 196

FIGURE 5. Prototype transformer based upon two 3.15 cc micro-piston pump/motors

TABLE 2. Fast and slow sinusoidal trajectory tracking with fixedtransformer speed on PM-1. RMS errors in position, pressure and trans-former speed.

e [mm] P [MPa] ω [rad/s]

Slow 0.88 0.041 2.31

Fast 0.9 0.0480 2.45

rad/s and 167 rad/s. Results for these two cases are shown inFig. 6 and Fig. 7 respectively and RMS errors in position, pres-sure and transformer speed are shown in Tab. 2. The perfor-mance with both trajectories are similar. RMS motion errors ofless than 1mm and transformer speed errors of less than 1.3% areachieved for both trajectories.

4.2 Transformer Speed TrackingNext, the controller is tested with a smoothed trapezoidal

motion trajectory on the PM-1, 2, 3 setups while the desiredtransformer operating speed is varied arbitrarily. Tracking re-sults are plotted on Figs 8-10. RMS errors are summarized intable 3. The RMS motion errors are within 1mm for all threeconfigurations whereas the transformer speed errors are slightlylarger than the case when the desired speed is a constant. Whilethe performances of all 3 configurations are very similar, PM-1and PM-3 have slightly better motion control performance thanPM-2, whereas PM-1 and PM-2 have slightly better transformerspeed control performance than PM-3.

These experiments demonstrate that the proposed control al-gorithm can be used to simultaneously control the trajectory andthe transformer operating speed. The latter could be used in thefuture to optimize the transformer operation.

4.3 Effect of Parameter UncertaintyEffects of uncertainty in the effective mass and damping co-

efficient are tested. Instead of using the best known parametersin Tab. 1, other parameters in Tab. 4 are used in the controllerinstead. Control gains are, however, kept the same. Results with

6 Copyright c© 2015 by ASME

65 70 75 800

0.05

0.1

[m]

Actuator

65 70 75 800.4

0.6

0.8

1

[MP

a]

Pressure

65 70 75 80185

190

195

200

205

[ra

d/s

ec]

Transformer Speed

65 70 75 80−1

−0.5

0

0.5

1

Control inputs

Desired

Actual

Pd

Pb

Desired

Actual

u1

u2

FIGURE 6. Low frequency large amplitude trajectory tracking forPM-1

the PM-1 configuration are summarized in Tab. 4 and select casesare plotted in Fig. 11. As expected, uncertainties in mass anddamping values result in larger error and the position error tendsto be an bias. However, in all cases but one, the RMS positionerror does not increase by more than 1.7mm, and in all cases, theRMS transformer speed error does not increase by more than 0.8rad/s. To improve on these errors, an adaptive control schemethat estimates the mass and the damping coefficient (or similarly,by adding an integral action) can be a fruitful avenue for furtherinvestigation.

5 ConclusionThis paper presents a controller for a hydraulic cylinder

driven by a pump/motor transformer such that the cylinder tracksa predefined trajectory and the transformer speed is regulated at

110 115 120 125 1300

0.02

0.04

0.06

[m]

Actuator

110 115 120 125 1300.4

0.6

0.8

1

[MP

a]

Pressure

110 115 120 125 130155

160

165

170

175

180

[ra

d/s

ec]

Transformer Speed

110 115 120 125 130−1

−0.5

0

0.5

1

Control inputs

Desired

Actual

Pd

Pb

Desired

Actual

u1

u2

FIGURE 7. High frequency, small amplitude trajectory tracking forPM-1

its desired valve. The controller, which was designed based uponthe passivity property of the hydraulic actuator and a recently dis-covered natural energy storage functions [10], can be applied toall three configurations of the transformer in Figs. 1-3. Exper-imental results show satisfactory cylinder trajectory and trans-former speed regulation performance. All three transformer con-figurations have similar control performance and achieve RMSposition errors of less than 1mm.

While transformer controlled system is expected to have bet-ter energy efficiency than a servo-valve controlled system, a con-cern has been whether an adequate control performance can beobtained. Experimental results in this paper shows that indeedgood trajectory tracking performance can be achieved.

The desired transformer speed has been arbitrarily defined inthis paper, the reference speed can be determined to improve effi-ciency and to avoid stalling. In the next step, the proposed control

7 Copyright c© 2015 by ASME

65 70 75 80 85 900

0.05

0.1

[m]

Actuator

65 70 75 80 85 90

0.8

1

1.2

1.4

[MPa

]

Pressure

65 70 75 80 85 9050

100

150

200

[rad

/sec

]

Transformer Speed

65 70 75 80 85 90−1

−0.5

0

0.5

1

Time [s]

Control inputs

DesiredActual

Pd

Pb

DesiredActual

u1

u2

FIGURE 8. Trapezoidal trajectory tracking with variable desiredtransformer speed for PM-1

algorithm will be expanded to enable the transformer to operateat its most efficient region. This will involve transformer speedoptimization, consideration of energy regeneration capability ofa hydraulic transformer, and active switching of the transformerbetween the three configurations.

ACKNOWLEDGMENT

This work is performed within the Center for Compact andEfficient Fluid Power (CCEFP) supported by the National Sci-ence Foundation under grant EEC-05040834. Donation of com-ponents from Takako Industries is gratefully acknowledged.

65 70 75 80 85 900

0.05

0.1

[m]

Actuator

65 70 75 80 85 90

0.8

1

1.2

1.4

[MPa

]

Pressure

65 70 75 80 85 9050

100

150

200

[rad

/sec

]

Transformer Speed

65 70 75 80 85 90−1

−0.5

0

0.5

1

Time [s]

Cont

rol i

nput

s

DesiredActual

Pd

Pb

DesiredActual

u1

u2

FIGURE 9. Trapezoidal trajectory tracking with variable desiredtransformer speed for PM-2

REFERENCES[1] Achten, P., Fu, Z., and Vael, G., 1997. “Transforming future

hydraulics: a new design of a hydraulic transformer”. InThe Fifth Scandinavian International Conference on FluidPower SICFP ’97, p. 287ev.

[2] Achten, P. A., van den Brink, T., van den Oever, J., Potma,J., Schellekens, M., Vael, G., van Walwijk, M., and Innas,B., 2002. “Dedicated design of the hydraulic transformer”.Vol. 3, pp. 233–248.

[3] Achten, P., Van den Brink, T., Paardenkooper, T., Platzer,T., Potma, H., Schellekens, M., and Vael, G., 2003. “Designand testing of an axial piston pump based on the floatingcup principle”. In The Eighth Scandinavian InternationalConference on Fluid Power SICFP ’03, pp. 805–820.

[4] Ouyang, X., 2005. “Hydraulic Transformer Research”.

8 Copyright c© 2015 by ASME

65 70 75 80 85 900

0.05

0.1

[m]

Actuator

65 70 75 80 85 90

0.8

1

1.2

1.4

[MPa

]

Pressure

65 70 75 80 85 9050

100

150

200

[rad

/sec

]

Transformer Speed

65 70 75 80 85 90−1

−0.5

0

0.5

1

Time [s]

Control inputs

DesiredActual

Pd

Pb

DesiredActual

u1

u2

FIGURE 10. Trapezoidal trajectory tracking with variable desiredtransformer speed for PM-3

PhD Thesis, Zhejiang University, Hangzhou, China.[5] Werndin, R., and Palmberg, J.-O., 2001. “Controller de-

sign for a hydarulic transformer”. In The Fifth InternationalConference on Fluid Power Transmission and Control ICFP’01, Vol. 5, pp. 56–61.

[6] Werndin, R., and Palmberg, J.-O., 2002. “Hydraulic trans-former in low-speed operation - a study of control strate-gies”. In The 5th International Symposium of Fluid Power,JFPS ‘02, Nara, Japan.

[7] Vael, G., Achten, P., and Potma, J., 2003. “Cylindercontrol with the floating cup hydraulic transformer”. InThe Eighth Scandinavian International Conference on FluidPower SICFP ’03 Tampere, Finland, pp. 175–190.

[8] Ho, T. H., and Ahn, K. K., 2008. “A study on the posi-tion control of hydraulic cylinder driven by hydraulic trans-

TABLE 3. Trapezoidal trajectory tracking with varying desired trans-former speed on PM-1, PM-2, PM-3. RMS errors in position, pressureand transformer speed.

e [mm] P [MPa] ω [rad/s]

PM-1 0.804 0.0786 4.09

PM-2 0.866 0.0816 4.03

PM-3 0.813 0.0566 5.06

TABLE 4. Tracking results for various assumed effective mass anddamping coefficient: RMS erros in position e, pressure P and trans-former speed ω .

m [kg] b N/m · s e [mm] P [MPa] ω [rad/s]

50 0 5.4719 0.4199 3.4705

70 0 2.3840 0.2050 3.7217

100 500 1.1518 0.0993 4.4237

100 5000 0.7150 0.0474 3.8671

120 0 1.2775 0.1820 3.9412

120 50 1.4120 0.1989 4.1105

120 5000 0.9451 0.1476 4.6690

4 6 8 10 12 14 160.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11Effect of Parameter Error

Po

sitio

n [

m]

Time [s]

m:50 b:0

m:80 b:5000

m:100 b:5000

m:120 b:5000

Reference

FIGURE 11. Trajectory tracking for various parameter values

former using disturbance observer”. In International Con-ference on Control, Automation and Systems 2008, IEEE,pp. 2634–2639.

[9] Lee, S., and Li, P. Y., 2014. “Trajectory tracking controlusing a hydraulic transformer”. 2014 International Sympo-

9 Copyright c© 2015 by ASME

sium on Flexible Automation, Awaji Island, Japan.[10] Li, P. Y., and Wang, M., 2014. “Natural storage function

for passivity-based trajectory control of hydraulic actua-tors”. IEEE/ASME Transactions on Mechatronics, 19(3),July, pp. 1057–1068.

[11] Cho, B.-H., Lee, H.-W., and Oh, J.-S., 2002. “Estimationtechnique of air content in automatic transmission fluid bymeasuring effective bulk modulus”. International journalof automotive technology, 3(2), pp. 57–61.

10 Copyright c© 2015 by ASME