Screw Pump for Electro-Hydrostatic Actuatorthat Enhances Backdrivability

Hiroshi Kaminaga, Hirokazu Tanaka, Kazuki Yasuda, and Yoshihiko Nakamura

Abstract— Force sensitivity and backdrivability are vitalfunctionality of actuators to be used in robots that physicallyinteract with humans. Electro-Hydrostatic Actuator (EHA) is atype of a hydraulic actuator with backdrivability. To improvebackdrivability of an EHA, reduction of friction, especiallystatic friction is important. This, however, is difficult becausemost of the hydraulic pumps require sliding contacts betweenmechanical components. A viscous screw pump is a variationof a viscous pump that transfers mechanical kinetic energy tofluidic kinetic energy with viscous friction of the fluid. Since thisclass of pumps does not require mechanical contacts betweenthe rotor and the stator, they are pumps with least staticfriction. In this paper, design and development of a screwpump targeted for use in an electro-hydrostatic actuator toimprove the backdrivability of the actuator system is presented.Pressure-Flow discharge performance of the developed pumpand backdrivability performance when combined with a vanemotor were evaluated.

I. INTRODUCTION

Backdrivability is one of the vital features of actuatorsystem in realizing force sensitivity and enhancing controlla-bility. This, however, is not easy due to transmission frictionand reflected inertia. It is widely known that from the linkside, motor side friction becomes N -times lager (here N isthe reduction ratio), and reflected inertia of the motor rotorbecomes N2-times lager when seen from the link side. Thisrelation is fixed in gear drives, which are most commonlyused in robot actuators; making it worse, they often havelarge N .

Series elastic actuation[1] is one of the classical methodthat decouples motor side and link side dynamics with aspring connected in series to the gear drive output. Bydecoupling motor side and link side dynamics, SEA (SeriesElastic Actuator) overcame the curse of motor side frictionand reflected inertia. SEA and its variations are widely usedin robot systems that physically interact with human[2], [3],[4]. However, SEAs suffer from complicated vibration modesinduced by the spring that becomes more relevant whensoft springs were used to enhance backdrivability and forcesensitivity.

Kaminaga et al.[5] developed low friction hydrostatictransmission and constructed highly backdrivable EHA

This work was supported by Research Grant from Fluid Power Tech-nology Promotion Foundation, Grant-in-Aid for Scientific Research (S)(No.20220001) of the Japan Society for the Promotion of Science, andGrant-in-Aid for Young Scientists (B) (No.90571571) of the Japan Societyfor the Promotion of Science.

H. Kaminaga, H. Tanaka, K. Yasuda, and Y. Nakamura arewith Graduate School of Information Science and Technology,The University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku, Tokyo, Japankaminaga@ynl.t.u-tokyo.ac.jp

Fig. 1. Basic Structure of Single Axis Screw Pumps

(Electro-Hydrostatic Actuator). EHA decouples motor sideand link side dynamics with internal leakage of the hydrauliccomponents. From its nature of being series dissipativeactuator[6], EHA is less likely to oscillate.

In the EHA developed in [5], authors tried to reducemechanical contacts as much as possible. The EHA used inthe knee power assist [7] has minimum mechanical contactsin the hydraulic systems; contacts are at axis packing andtrochoid pump, both of them being inevitable from itsprinciple.

In this paper, we present the design methodology ofconstructing pump that minimizes the mechanical frictionby eliminating the gear mesh in the pump. Viscous typescrew pump was examined from its principle of not havingmechanical contact within the pump. Basic characteristicsand the design methodology of screw type viscous pump wasexplained. Prototype was developed and its characteristics onbasic pump functionality and backdrivability when used inEHA were evaluated.

II. VISCOUS SCREW PUMP

Screw pumps are viscous type pump that does not haveany gear meshing in the pump. As in other types of viscouspumps as [8], [9], they transfer the mechanical energy fromthe pump to the fluid by shear stress between the rotor andthe fluid. Screw pump has following benefits: the fluid pathalong the rotor is spiral and long that enable screw pumpsto produce high pressure, and the fluid path follows the pathon the rotor with maximum speed that enable screw pumpsto generate high flow rate. The basic structure of the pumpis illustrated in Fig. 1. The rotating screw transfers kineticenergy to the fluid at the surface. They are often used ininjection extruders, axis seals, concrete pumps, and greasepumps that operate in one direction and with high viscosityfluid.

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The principle of the pump is simple that carries the fluidalong its groove of the thread by the shear stress betweenthe screw and the fluid, but not many theoretical studies havebeen done[10], [11], [12], [13]. We follow the method ofAsanuma[11], [14] due to its simple linear result.

The application of the screw pump was limited to veryhigh viscosity fluid as mentioned above. One of the reasonswas that the amount of the gap was large due to the machin-ing precision that increased the leakage loss of the pumpsignificantly; the leakage resistance is inverse proportionalto cubic of the gap amount.

Today, with the contribution of the improvement in themachining precision, gap amount can be reduced to tens ofmicrons. This fact enables us to consider high pressure usageof such screw pumps with normal hydraulic oil.

Screw pumps have following advantages over gear pumps:1) Small friction. Since there is no gear meshing in

this pump, mechanical contact can be minimized, thatcontributes in reduction of static friction.

2) No pulsation. Since the pumping is fully continuous,there is no pulsation in pressure and flow rate. Thisfeature contributes to more stable and accurate pressurecontrol.

3) High speed operation. Since majority of modern highpower motors are high speed, higher pump operationrequires less reduction ratio before the pump.

Feature 1 and 3 are expected to enhance backdrivability whenthe pump is used in EHA via reduction of friction. Feature2 is expected to realize smooth torque output.

From the discussion above, screw pumps might serve asa suitable device in EHA to realize smooth force control.

III. GOVERNING EQUATIONS OF SCREW PUMPS

A. Basic Fluidic Property of Screw Pumps

In screw pump shown in Fig. 2, screw rotates in thecylinder (hereafter denoted as a sleeve). Fluid receives shearstress from the rotating screw that accelerates the fluid alongthe screw groove. The grooves are separated to each otherby ridges.

We assume x axis with the direction of the groove andtangential to the sleeve surface, having its origin on the sleevesurface. z axis is taken in the direction of the groove depthand having its origin coinciding x axis. y axis is chosen sothe x − y − z axes form a right hand system. We assumex− y − z coordinate system is fixed to the screw.

When the diameter of the screw is sufficiently larger thanthe depth of the groove, thus dt >> hc, flow in the groovecan be approximated with the laminar flow between twoparallel planes. Nomenclature of the parameters are listed inTable I. The groove length lc then becomes lc = lt/ cos θc,with the width of wc and depth of hc. We assume the fluidto be Newtonian with viscosity μ and non-compressive 1.Analysis on the fluid is done assuming the screw is fixedand cylinder is moving in opposite direction. Letting p(x)

1This assumption is reasonable since hydraulic oil is sufficiently incom-pressive in the pressure range we use in EHA, that is below 6MPa.

TABLE INOMENCLATURE

Description ParameterScrew length lt

Bore diameter of the sleeve dtGroove width wc

Groove depth hc

Ridge width wb

Number of helices in the thread nt

Lead angle of the screw θc = asin(wc+wbπdtε

nt)

Rotation speed of the screw ω1

Rotatioal torque of the screw τ1Discharge pressure p1

Viscosity μDensity of the fluid ρDynamic viscosity ν = μ/ρ

Fig. 2. Parameters in Screw Pumps

denote the pressure at point x, and the pressure differencealong the groove is constant, Navier-Stokes equations of themodel become as follows.

∂p

∂y=

∂p

∂z= 0 (1)

∂2vx∂y2

+∂2vx∂z2

=1

μ

∂p

∂x=

1

μ

p1 sin θclt

= c0 (2)

Here, c0 is a constant.Literature [11] gives solution to (2) under boundary con-

ditions listed in (3), in Fourier series form, as in (5). All theparameters in the equation were modified from [11] to meetSI unit because the gravitational metric system was used inoriginal literature.

vx =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Vx z = 00 z = hc

0 (y = 0, 12wc) and

hg ≤ z ≤ hc

(hg − z)Vx − z(hg − z) c02 (y = 0, 12wc) and

0 ≤ z ≤ hg

(3)where

Vx =1

2dtω1 cos θc (4)

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vx = (hc − z)Vx

hc− z(hc − z)

c02

+2

hc

∞∑n=1

[sin(

nπz

hc)cosh{(nπhc

)(wc

2 − y)}cosh(nπwc

2hc)

×{(−Vx

hg+

c0hg

2)(

hc

nπ)2 sin(

nπhg

hc)

+c0(hc

nπ)3(cos(

nπhg

hc)− (−1)n)

}](5)

B. Flow Characteristics [11]

Discharge flow rate of the pump can be calculated withthe forward flow rate induced by the friction of the fluid withthe screw and the leak flow at the ridge that is induced by thepressure difference. Discharge flow rate q1 is the differencebetween the forward flow and the leakage flow as in (7).Forward flow can be calculated by integrating vx for thesection of the groove, thus y − z plane.

q1 = nt

∫ hc

hg

∫ wc

0

vx dydz − πdth3gp1

12μlt(1− ε)(6)

=1

2K1(α, β)ntwchcVx (7)

− 1

12μlt

(K2(α, β)ntwch

3c sin θc +

πdth3g

1− ε

)p1

where K1 and K2 are dimensionless function that takesdimensionless input α = wc

hcand β =

hg

hcthat are determined

from form factor of the pump.

K1(α, β) = (1− β)2 − 8

π4αβ

∞∑n=1

1

n4sin(nπβ)

× {cos(nπβ)− (−1)n} tanh(nπα2

) (8)

K2(α, β) = (1− β)2(1 + 2β)− 24β

π4α

∞∑n=1

1

n4sin(nπβ)

× {cos(nπβ)− (−1)n} tanh(nπα2

)

− 48

π5α

∞∑n=1

1

n5{cos(nπβ)− (−1)n}2 tanh(nπα

2) (9)

Parameter ε gives the ratio of the groove and the ridge width.

ε =wc

wc + wg=

wc

πdt sin θc(10)

Flow rate - speed - discharge pressure can be concludedby (11).

q1 =1

2K12

dt2wchcω1 − 1

12

wch3c

μltK22p1 (11)

K12 = K1nt cos θc

K22 = K2nt sin θc +1

sin θc

β3

ε(1− ε)

TABLE IIDESIGN SPECIFICATION OF SCREW PUMP

Description Value UnitMaximum Discharge Pressure 1.2 MPa

Maximum Flow Rate 30× 10−6 m3/sec

C. Torque Characteristics [11]

With similar discussion to previous section, torque char-acteristics can be derived from (5). Torque acting betweenthe sleeve and the screw are divided to the ridge and thegroove. The torque acting on groove can be calculated byintegrating shear stress across the groove on sleeve surface.The total torque τ1 necessary to generate pressure of p1 atthe speed of ω1 is given as follows.

τ1 = −ntμdtlt

2 tan θc

∫ wc

0

(∂vx∂z

)z=0

dy

+μd3t lt

4hg sin θcπ(1− ε)ω1

=

{d2t4μwc

hclt(cos θc)

2

sin θcT1nt +

μd3t lt4hg sin θc

π(1− ε)

}ω1

+dt4wchcT2 cos θcntp1 (12)

T1 and T2 are dimensionless function of α and β.

T1(α, β) = 1+4

π2

1

αβ

∞∑n=1

1

n2sin(nπβ) tanh(

nπα

2) (13)

T2(α, β) = 1− 4

π2

β

α

∞∑n=1

1

n2sin(nπβ) tanh(

nπα

2)

− 8

π3

1

α

∞∑n=1

1

n3{cos(nπβ)− (−1)n} tanh(nπα

2) (14)

IV. MECHANICAL IMPLEMENTATION

To study the feasibility of the pump in EHA, we decidedto design a screw pump that can actuate hydraulic motor.Table II shows the design specification of the pump.

The design process is not simple since the system is highlynonlinear regarding the parameters as wc, hc, and wb. Alsothere are constraints on the fabrication feasibility. (11) and(12) were recursively used to decide the parameter values.

To reduce the inertia of the pump, we chose engineeringplastic as the screw material. The largest constraint on thefabrication was on the gap precision between the screw andthe sleeve. We chose this value first as 15 μm. For thepump to have decent volumetric efficiency, it is advantageousto use multi-helix screw. Considering the balance of flowrate and the discharge pressure, triple helix screw waschosen. Other parameters of the pump is shown in TableIII. 100W brushless DC motor was chosen to drive pump.As an unmodelled friction, we took in account of 0.1Nmadditionally. Pulley reduction was applied in between thepump and the motor.

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TABLE IIIDESIGN PARAMETER OF SCREW PUMP

Description Value UnitDynamic viscosity (ν) 100 cSt

Fluid density (ρ) 882 kg/m3

Axial screw length (lt) 62 mmBore diameter (dt) 28 mmGroove width (wc) 9 mmGroove depth (hc) 0.5 mmRidge width (wb) 1.0 mm

Screw - sleeve gap (hg) 15 μmNumber of helices (nt) 3 -Reduction before pump 2 -

Fig. 3. Pressure-Flow Rate Characteristics of Designed Screw Pump

Fig. 3 shows the simulated pressure - flow rate character-istics of the pump. This simulation includes torque - speedcharacteristics to make the simulation realistic. From thisfigure, it can be seen that the specified pressure and flowrate are both covered by the operation region.

Fig. 4 shows the designed screw pump. The screw issupported with a pair of ball bearings to realize high pre-cision rotation of the screw to maintain the precision ofthe gap between the screw and the sleeve that affects theperformance significantly. The housing have 3 ports on eachside to connect tubes and pressure sensors.

Fig. 5 shows the outlook of the developed screw pump.Major components that support pressure are made of 5000series aluminium alloy. The screw is made of ABS polymer.

Fig. 4. Cross Section of Screw Pump Design

Fig. 5. Outlook of Developed Screw Pump

Fig. 6. Pressure-Flow Rate Characteristics Test Setup

V. EXPERIMENTSA. Evaluation on Pressure - Flow Rate Characteristics

To evaluate the performance of developed screw pump,pressure to flow rate characteristics was examined. Fig. 6shows the hydraulic circuitry used in the evaluation. Themotor of the pump was driven with constant current (thustorque) while load was changed using a choke valve. Flowrate was measured using flow meter. Pump discharge pres-sure was monitored using a pair of pressure sensors located atthe ports of the pump. Test was done by changing the chokecontinuously. The rate of the change was kept sufficientlylower than the system dynamics because the pressure toflow rate characteristics is a static characteristics. 1.0MPa ofpressurization was applied to the system to avoid cavitation.

Fig. 7 and Table IV shows the result of the evaluation.From the figure, linearity of the characteristics can be ob-served. It can also be observed that the relation of the flowrate and pressure changes linearly with the applied torque.This can easily be derived from (11) and (12) by cancelingout ω1 from these equations. “LS Fit” in Fig. 7 denotes theline was drawn using the least square to a linear function.

From the simulated parameter in previous section, thecharacteristics of the pump was estimated as follows.

q1 = −25.6× 10−12p1 + 17.6× 10−6τ1 (15)

In Table IV, “Slope” denote the coefficient of the first term,thus dq1

dp1. ‘Y Intercept / Pump Torque” denotes the coefficient

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TABLE IVRESULT OF PRESSURE-FLOW RATE CHARACTERISTICS.

Pump Torque Slope Y Intercept/Pump TorquemNm 10−12m3/s/Pa 10−6m3/s/Nm

46 -16.6 21.394 -19.1 26.8142 -20.6 29.7188 -19.8 29.1

TABLE VBACKDRIVABILITY COMPARISON RESULT OF TROCHOID PUMP AND

SCREW PUMP (UNITS IN Nm)

Description Trochoid Pump [7] Screw PumpOutput Backdriving Torque < 0.2 < 0.2Total Backdriving Torque 2.2 0.7

of the second term on the right hand side of (15), thusq1τ1

∣∣∣p1=0

. The result shows stable behavior despite of change

in applied torque. The parameter discrepancy is expectedto come from the error of the gap and unmodeled frictionaround the bearings and axis seal.

B. Evaluation on Backdrivability

Realizing high backdrivability is one of the importantobjective of this study. To evaluate the backdrivability, avane type hydraulic motor used in [7] was connected to thedesigned pump.

Evaluation was performed by applying torque to the hy-draulic motor through a wire as in Fig. 8 while measuringmovement of hydraulic motor with link side encoder andpump with pump side encoder. Applied torque was measuredwith force gauge attached to the wire. 1.0MPa of pressur-ization was applied to the system to avoid cavitation.

Fig. 9 and Fig. 10 shows the movement history ofhydraulic motor and pump respectively. Both data wereacquired synchronously. In Fig. 9, the point that the markerleaves the x axis shows the torque that the output backdrivingstarted. From this figure it can be said that the outputbackdriving happens with very small torque. Fig. 10 showsthe movement of the pump induced by the pressure generatedby the movement of hydraulic motor. Similar to Fig. 9, thepoint that the markers leave x axis shows the point wheretotal backdriving started. This value is always lager thanthe output backdriving torque since output backdriving isnecessary in generating the pressure to backdrive the pump.

Table V shows the result of the evaluation and theircomparison with the case of trochoid pump being connectedinstead of the screw pump that were presented in [7]. Fromthe comparison, in either case, output backdriving torquewas too small to be measured, but the total backdrivingtoque was 1/3 of the case with trochoid pump. Hence, theefficacy of the screw pump in enhancing the backdrivabilitywas experimentally shown.

VI. CONCLUSOINS

This paper proposed a design concept and methodology touse single axis screw pump in Electro-Hydrostatic Actuators

Fig. 8. Backdrivability Evaluation Test Apparatus

Fig. 9. Output Backdrivability Evaluation

to realize smoother force control and significant improve-ment in backdrivability that is expected to play an improtantroll in robotics, especially humanoid.

1) Design concept of utilizing viscous pump to realizeactuator system with minimum mechanical contacts tomaximize backdrivability was presented.

2) Introduced the fluidic property of the screw pump, firstproposed by Asanuma [11]. The form of the equationwas modified to be used for the mechanical design.

3) Presented mechanical design of the screw pump pro-totype that realize maximum discharge pressure of

Fig. 10. Total Backdrivability Evaluation

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Fig. 7. Result of Pressure-Flow Rate Characteristics Evaluation. LS Fit shows the lines fit with least-square method.

1.2MPa and maximum flow rate of 30×10−6m3/sec.The design was based on the relationship derived insection III.

4) Pressure-flow rate characteristics were evaluated onthe developed prototype. The result showed linearbehavior and the estimated parameter from the exper-imental data showed only limited error to the valueestimated in the simulation. The discrepancies betweenthe experimental results are expected to come from themechanical precision, especially the gap amount.

5) Backdrivability was evaluated on the prototype withthe hydraulic motor used in knee power assist [7].From the result, output backdriving torque was smallerthan the measurement range as in the case of [7]. Totalbackdriving torque was reduced more than 1/3 of thecase with trochoid pump that was presented in [7].

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