CALIFORNIA PATH PROGRAM INSTITUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY

Combined and for Automatic Control

R. Prohaska, P. Devlin

California PATH Working Paper UCB-ITS-PWP-98-15

This work was performed as part of the California PATH Program of the University of California, in cooperation with the State of California Business, Transportation, and Housing Agency, Department of Trans- portation; and the United States Department Transportation, Federal Administration.

The contents of this report reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California. This report does not constitute a standard, specification, or regulation.

Report for MOU 331

August 1998

ISSN 1055-1417

CALIFORNIA PARTNERS FOR ADVANCED TRANSIT AND HIGHWAYS

Combined Brake and Steering Actuator for



Automatic Vehicle Research

y

R. Prohaska and P. Devlin

University of California PATH,

1357 S. 46th St. Richmond CA 94804

Decemb er 8, 1997

Abstract

A simple, relatively inexp ensive combined steering and brake actuator

system has b een develop ed for use in automatic vehicle control research. It

allows a standard passenger to switch from normal manual op eration

to automatic op eration and back seamlessly using largely standard parts.

The only critical comp onents required are twovalve sp o ols which can

b e made on a small precision lathe. All other parts are either o the

shelf or can b e fabricated using standard shop to ols. The system provides

approximately 3 Hz small signal steering bandwidth combined with 90 ms

large signal brake pressure risetime.

1 INTRODUCTION

Exp erimental research on automated highway systems is presently somewhat

constrained by the limited availabilityofvehicles equipp ed to b e controlled by

automatic means. We present here a simple metho d for actuating and

steering on a standard passenger car. The apparatus describ ed should not b e

employed on public . It is for exp erimental purp oses only.

Throttle in the form of  have b een available for

some years now, and high p erformance actuators have b een develop ed.

Mec hanisms to allo w automatic application of brakes and steering are com-

paratively scarce, since appreciable p ower has to b e delivered and relatively

short resp onse time .1 sec or less is required for precise control of vehicle

sp eed and intervehicle spacing.

The most direct approach to delivering and controlling the substantial forces

required for steering and brake actuation is the use of hydraulic p ower assist.

Mo dern p ower steering systems already contain most of the required elements



This work was supp orted by the California DepartmentofTransp ortation

y

Permanent Address: 1384 Je erson Avenue, Akron Ohio 44313 1

and it happ ens that assist is commercially available and easily

mo di ed for automatic control.

The use of existing ABS is app ealing, but most ABS systems are carefully

optimized for cost under the assumption that activity will b e infrequent and

brief. The demands of routine braking, and the consequences of controller mis-

b ehavior, are almost certain to exceed the design capacity of an ABS system.

Obviously high p erformance develop ed for aircraft and in-

dustrial applications are available but their cost readily exceeds that of a mo d-

ern automobile. Usual high-p erformance acutators are not easily integrated

with the steering and brake systems of a pro duction vehicle since they usually

rely on constant pressure hydraulic control. In constant pressure systems the

control valves idle in the closed p osition and an accumulator stores hydraulic

energy which can b e applied to move masses quickly. In the inactive state there

is appreciable p ower dissipation from valve leakage and some sort of pressure

regulation system is required.

In contrast, standard hydraulic p ower assist schemes for automotive steer-

ing and brakes use constant ow designs; they o er acceptable p erformance at

minimum cost. In a constant ow system the control valves idle in the op en

p osition with uid circulating freely through the pump and valves. The parts

count is minimal and by careful design the idle state p ower dissipation can b e

made thermally insigni cant. However there is no accumulator, limiting the

p eak p ower available and therefore making the system resp onse slow.

Wechose to implement a constant ow actuator system. This minimizes

the mechanical disruption to the vehicle and also allows a smo oth transition

between automatic and human-controlled mo des of op eration. Figure 1 gives

an overall view of the engine compartment and makes clear that no substantial

rearrangement is required.

For the PATH steering and brake actuator system wehave used a standard

hydraulic p ower assist unit Bendix Hydro-Bo ost for the brake, the sto ckpower

steering b ox for the steering and the existing p ower steering pump. The normal

hydraulic passages b etween the integral control valves and actuator cylinders are

blo cked, and the lines brought out to external manifold blo cks containing direc-

tional valves to switchbetween normal and automatic mo de and prop ortional

valves which do the controlling in automatic mo de.

The prop ortional valves are op erated by analog electronic circuits delivering

a few watts of electrical p ower. Closed lo op control is achieved by adding a

pressure sensor on the brake system and a p osition sensor on the steering system.

The use of analog electronics on the inner lo op may seem primitive at rst, but

it greatly simpli es initial setup and testing. The analog circuit can

b e easily replaced by a digital one if more sophisticated control is desired.

The constant ow control valves use standard constant-pressure sp o ol valve

b o dies but custom sp o ols. Valve b o dies are essentially imp ossible to craft by

non-sp ecialists but an acceptable sp o ol can b e made by a skilled machinist with

little more than a go o d lathe and some patience. 2

Figure 1: View of the left side of the engine compartment showing minimal

disruption of the sto ck con guration. The only changes are removal of the cold

air intake duct and removal of the original op erated cruise control.

2 Technical Description.

The complete hydraulic diagram is shown in Figure 2. All valves are in p ositions

corresp onding to human control, with no command applied. There is a free path

from the pump, through the brake b o oster into the steering gear, whence the

uid is returned to the tank. The lter and co oler are omitted for clarity.

The descriptions b egin with normal human control of the vehicle and then

elab orate on changes required to p ermit automatic control.

2.1 Brake Servo and Actuator

Brake actuation is achieved through use of a Bendix Corp oration Hydro-Bo ost

brake servo, mo di ed for external control. The Bendix Hydro-Bo ost is a

constant owhydraulic brake servo designed for placement upstream of standard

. Since it was meant to b e substituted in place of vacuum b o osters

in existing vehicle designs it is relatively easy to adapt using b olt plates, spacers

and p edal ro ds to a variety of . Some cars which are equipp ed with Hydro-

Bo ost from the factory include late '70s Eldorados, mid '80s Lincoln

Mark 7 LSCs and mid to late '80s GM diesels. The units cost b etween $300 and 3 Bendix Hydro-Boost unit Apply Chamber

Booster

pressure sense transducer piston to brakes Pedal Rod power brake control valve

port added to power internal apply steering controls not chamber pump shown steering selector valves

brake selector valves

Pressure relief power steering valve rack piston

power Brake steering actuator control control valve valve steering actuator control valve

steering selector valve

Figure 2: Complete hydraulic diagram for the combined actuators. See the

sp o ol construction discussion for a more intuitive representation of the control

valves.

$500 as replacement parts.

In manual op eration, the Hydro-Bo ost receives ow from the p ower steering

pump. With no p edal e ort, uid travels freely through the Hydro-Bo ost control

valve and enters the steering system. The apply chamber is vented to the tank,

and an isolation valve allows the steering system to develop normal op erating

pressure without inadvertently applying brakes. When the driver applies the

brakes the p ower brake control valve senses displacement of the brake p edal

closing the vent, op ening the isolation passage and restricting uid ow from

the pump to develop pressure in the apply chamb er. The b o oster piston applies

force to the brake master cylinder, while at the same time a sense piston on the

p edal input ro d opp oses the driver's fo ot. When the hydraulic force develop ed on

the p edal overcomes the driver's e ort, the p ower brake control valve returns to

neutral p osition, halting the pressure rise and giving the driver tactile feedback.

When the driver releases the p edal, a relatively soft spring returns the valve

to the vent p osition, closing the isolation valve and releasing pressure in the

apply chamb er. The hydraulic force gain is simply the ratio of apply piston 4

Figure 3: Figure 3. The brake manifold blo ck, lo oking toward the rear of the

car.

area to sense piston area, typically ab out 10. The p edal ro d and apply piston

are linked by a pushro d, providing manual brake application in the eventof

hydraulic failure. The sense piston corrects for back pressure generated by the

steering system, whichwould otherwise app ear if the driver applied the brakes

lightly and then turned the steering .

To apply the brakes via an external control valve one must blo ck the vent,

bypass the isolation valve and restrict pump ow to develop pressure. Releasing

the brakes is accomplished by restoring isolation and venting. All three functions

are accomplished within the manifold blo ck illustrated in Figure 3.

The function of the sense piston on the p edal ro d is taken over by an elec-

tronic pressure transducer mounted on the brake master cylinder output. It's

visible in the photo b etween the manifold blo ck and the brake uid reservoir.

It detects the actual pressure develop ed in the brake system and generates a

feedback signal used to control the actuator sp o ol valve p osition. Excess pres-

sure whether from control valveoversho ot or from steering system op eration is

detected and relieved.

The only p oint of dissatisfaction is that driver p ower assist is lost once

automatic mo de is invoked: The apply chamber is vented unless the actuator

is trying to apply the brake. As a practical matter, a driver of normal strength

cannot rapidly stop the test car: gradual stops are p ossible but generous space 5

Figure 4: The steering actuator control valve, viewed from the front of the car.

The manifold blo ck extends ab out 4 inches b elow the control valve.

should b e allowed for initial testing.

2.2 Steering Servo and Actuator

In manual op eration hydraulic uid enters the p ower steering control valve and

is immediately vented to tank, with b oth sides of the p ower steering rack piston

at tank atmospheric pressure. When the driver applies steering e ort, defor-

mation of a torsion bar in the p ower steering control valve results in one side

of the rack piston b eing preferentially connected to pump output, the other re-

mains connected to tank, whereup on further steering e ort restricts pump ow,

elevating pressure and generating net force on the steering system. There is no

feedback to the driver, so steering feel is highly dep endentuponvalve pro le

and torsion bar sti ness.

Toinvoke automatic control of steering the normal owofhydraulic uid

through the p ower steering control valve mounted on the steering input shaft

is diverted to the steering actuator control valve. Pressure is delivered to the

rack piston through selector valves via external hydraulic lines. The steering

selector valves isolate the actuator system from the p ower steering system when

the car is in manual op eration. Figure 4 gives some sense of the layout; the

control valve is b eing p ointed to, the three black solids with hex nuts on top are 6

Figure 5: Steering gearb ox showing hoses added to rack piston cylinder.

the tops of the directional valves. Figure 5 illustrates the sort of mo di cation

required on the steering gearb ox; tting are simply tapp ed into the casting.

A steering p osition sensor in this case, a linear p otentiometer mounted on

the idler arm provided p osition feedback which is compared to the command

signal and the di erence is used to generate signals to drive the solenoids which

op erated in push-pull mo de.

If p ower fails to the selector valves the car reverts to normal p ower steering.

Power failure to the actuator control valve is equivalent to manual steering with

no p ower assist. The only ill-b ehaved p ower failure mo de is to the p osition

sensor: the car turns to one side, in this case to the right.

In automatic mo de the driver retains mechanical steering connection but

no p ower assist. As a practical matter, that means the driver has no steering:

the of the actuator exceeds that of any normal driver. In the eventof

actuator or software malfunction, the only recourse is to shut o the actuator

and recover control of the car using normal p ower assist. The transition b etween

actuator and normal p ower assist is smo oth; the only diculty is in rememb ering

to shut o the actuator. 7

3 Construction Details

The vehicle used to test our design was a 1989 Ford Crown Victoria. It was

equipp ed with factory p ower steering and a b o oster. The vacuum

brake b o oster was replaced with a Hydro-Bo ost unit mo di ed with a larger p edal

ro d eye to t the existing p edal assembly, mounted by means of a b olt plate

which allowed the somewhat strange Hydro-Bo ost b olt pattern to t the rewall.

Internal passages b etween control valve and apply chamber were blo cked and

the hydraulic signals were brought out to a separate manifold blo ck containing

directional valves which select b etween manual and automatic op eration and

the actuator control valve.

A relief valvewas included to ensure that hydraulic ow to the steering

cannot b e cut o by a saturated brake actuator. The relief pressure is lower

than that of the hydraulic pump's internal relief valve and ensures that an errant

brake command cannot disable the steering.

The steering gearb oxwas mo di ed by blo cking the normal internal passages

between the control valve and rack piston. External p orts were added to divert

the output from the p ower steering control valve to the manifold blo ck holding

the control and selector valves. Directional valves then connected the rack piston

to either the actuator control valveortothepower steering control valve.

The tank connections of b oth the steering actuator control valve and the

power steering control valvewere brought out to a common connection for rout-

ing through a lter and co oler, whence the uid returned to the tank. The lter

was simply a precaution but the co oler was necessary: continuous application

of brake whose ow then transits the steering system could easily raise uid

temp eratures to destructive levels.

Most plumbing external to the existing comp onents was constructed in two

manifold blo cks, one for steering and one for brakes. The Hydro-Bo ost unit's

need for an external tank connection was satis ed by replacing the Ford's sto ck

power steering uid reservoir with a reservoir from a Hydro-Bo ost equipp ed

Lincoln Continental Mark 7 LSC, which contained the required connection.

3.1 Sp o ol Construction

The fabrication of the valve sp o ols is a critical step in the pro ject. Manufacturers

do not supply complete valves suitable for this kind of service, and most are

relatively unco op erativeatproviding things likeinternal dimensions and sp o ol

blanks. A vendor willing to provide such help is worth cultivating if found.

The valve b o dies we used are standard Vickers parts: KTG4V-3 for the

brake, and KDG4V-3 for the steering. The valves di er most notably in that

the former is a unidirectional valve, while the latter is a symmetric design for

push-pull application.

Unfortunately Vickers declined to provide sp eci cations on the internal di-

mensions, so they had to b e measured manually. This was done by using a

vernier calip er as a depth gauge and measuring to small blo cks pressed against

the lands inside the valvecavities. The resulting accuracy was adequate when 8 T A P BT

Isolation Vent seal seal Metering taper, highly exaggerated spool motion to apply brakes

Outlet to Vent to tank Pressure to Fluid from steering

apply chamber pump

Figure 6: Stylized representation of the brake actuator control valve. Radial

clearances are highly exaggerated

the measurements were rep eated by di erent p eople and averaged. The brake

valve sp o ol and b o dy are illustrated in a stylized way in Figure 6.

In the relaxed state uid enters from the pump and exits at p ort B to the

steering gear. The land o ccupying p ort A isolates pressure develop ed down-

stream by the steering system and vents the apply chamb er to tank. As the

spool is moved to the reader's left the land in p ort A rst closes the vent, then

connects the apply chamb er to the pump at p ort B. As the leftward displace-

ment increases the metering tap er enters the passage b etween P and B, raising

pressure while the uid continues out p ort B. That pressure is communicated

to p ort A where it applies the brake.

The steering valve shown in gure 7 di ers only in that the sp o ol is sym-

metric and the isolation valve seal b ecomes bidirectional to select the direction

in which steering motion o ccurs. When idle, uid enters from the supply and is

immediately vented to tank symmetrically on b oth ends of the sp o ol. When the

spool moves left, the central land rst closes the left rack piston p ort A leaving

p ort B the right rack piston p ort connected to the pump. Further leftward

movement drives the metering tap er into the right hand tank connection to

develop pressure which communicates to the steering rack via p ort B.

The central switching land deserves sp ecial attention: One would like the

valve to o er minimal ow imp edance when the valve is idle, but also to require

minimal displacement to generate threshold pressure minumum deadband.

The establishment of a suitable compromise is to some extent a matter of trial

and error. If the land is to o wide the valve will exhibit excessive pressure drop 9 T A P BT

metering taper seal for right turn seal for left turn Metering taper

pressure to rack Vent to tank pressure to rack Fluid from Vent to piston, right

piston, left pump tank

Figure 7: Stylized representation of the steering actuator control valve. Radial

clearances are highly exaggerated

in the idle state, if it's to o narrow excessive displacement will b e required to get

any resp onse at all and that resp onse is apt to b e to o big, since the metering

tap er will already b e engaged. One strategy is to make the land to o wide

initially, then trim it until the idling pressure is acceptable ie., the uid do es

not overheat. If the minimum pressure is to o high, the metering tap er is then

to o close to the seal. That defect can b e remedied by further machining of the

tap er, at the cost of requiring more sp o ol travel to achieve maximum pressure.

When the sp o ol has b een machined to the p oint that maximum pressure cannot

be achieved the sp o ol must b e discarded and a new one fabricated.

The radial clearance and metering tap er are highly exaggerated in the gures.

The metering tap er amounts to no more than 0.010" over 0.050" of length. On

the scale of the gure such clearance is invisible.

The metering tap er is computed from the known ow rate of the pump, in

this case ab out 6 liters p er minute, the maximum displacement of the sp o ol

with full coil drive and the desired maximum pressure. The desired tap er is

one which givesahydraulic pressure prop ortional to solenoid current whichwe

have approximated using 3 conical sections. The pro le is constructed by rst

cutting a 1 degree tap er on the nominal sp o ol diameter 0.470" with a length

of 0.050" measured from the small end. Next a 3.5 degree tap er is cut to a

length of ab out 0.020" from the small end. Finally, a 14.5 degree tap er is cut

to a length of 0.012" from the small end. 10

Figure 8: Brake electronic circuit, showing error amp, oscillator and pulse width

mo dulator. External connections are not shown

3.2 Electronics

The electronic circuits used are the simplest p ossible which o er reasonable

p erformance. Both brake and steering circuits are housed in small b oxes hidden

inside the front of the passenger compartmenttokeep the electronics away

from engine . Both brake and steering control circuits implement a simple

P-control, in which actuator drive is prop ortional to the di erence b etween

command input and resp onse.

The brake control circuit is the most intuitive and is illustrated in Figure 8.

The circuit consists of an error ampli er, clo ck oscillator and pulse width mo du-

lator. The mo dulated pulse width is delivered to the gate of a p ower MOSFET

which controls the brake solenoid current and hence the brake pressure.

The steering circuit is a bit less obvious in op eration since it do es not use a

xed frequency clo ck and do es not employ an explicit pulse width mo dulator.

The circuit compares p osition to command at U3B. The di erence, relativeto

midrange 5 volts is delivered to U7B and U4A, which form a phase splitter,

determining which solenoid will b e driven and how hard. The comparators at

U5A and U5B function at latches, to ensure that only one solenoid is on at a

time.

The ampli ed error is delivered to U6A and U7A, where it is compared to

the current already owing in the solenoid coil. If the current in the coil is lower

than desired transistors X1 or X2 are turned on.

The circuit lo oks like, and in principle is, a linear amplifer. The pulse width 11 U2 U1 LM117 LM7805C 2 IN OUT 3 1 IN OUT 2 +12 volts from vehicle power C1 1u ADJ C2 1u GND R39 220 1 3

R9 100k R40 1.5k R33 4 LM358 6 - steering position signal hi 30k V- R34 7 V+ steering position signal lo + U7B 100k 100k 30k 5 8 R2 R6 R35 100k R1 R5 4 4 2 - LM358 6 - LM358 command in lo 100k V- 100k V- R3 1 R7 7 V+ V+ command in hi U3A U3B 3 + 5 + 100k 8 100k 8 R4 100k R8 100k

POT R11 100k + V3 10V R37 R10 4 offset - 2 - LM358 30k V- R12 1 V+ U4A 3 + 30k 8 100k R13 100k R17 100k POT R16 R21 4 6 - LM358 R20 R36 10k V- 4 deadband R14 0.5 R18 7 2 - LM358 R32 V+ 20k V- U4B 5 + R22 1 D1N5401 10k 8 V+ 5k R19 100k + U6A D1 solA 20k 3 8 L1 Spool valve R23 100k gate threshold supply X1 MTP15N05E/MC

8 U5A + 3 V+ 1 V- - 2 100k 4 R25 LM393 100k R24 R29 4 - LM358 6 R28 8 U5B 10k V- 4 5 R15 0.5 LM358 + R26 7 2 - R38 V+ 20k V- V+ U6B 7 5 + R30 1 D1N5401 10k 8 V+ 5k V- U7A - 6 R27 100k 3 + solB 20k 8 4 D2 LM393 L2 Spool valve R31 100k gate threshold supply X2

MTP15N05E/MC

Figure 9: Steering actuator control circuit showing error amp, coil current sense

amps and direction latching comparators.

mo dulation arises from the gain of the op amps and high transconductance of

the FET output stage combined with delays asso ciated with coil inductance and

mechanical inertia. The transistors are always either full on or full o , with the

result that p ower dissipation in the transistors is nil. The resulting oscillation

is ltered out by the mass of the mechanical parts. It might fairly b e said that

the valve sp o ol is hammered into p osition by the electronics.

R36 sets a voltage whichmust b e exceeded by the error deadband b efore

any corrective action is taken. It is this feature which forces the oscillations to

halt when the desired p osition is achieved.

4 Performance

The small-signal p erformance of the steering system is illustrated in Figure 10.

The vertical scale is in volts, with .5 volt one division corresp onding to ab out

6 degrees of steering knuckle angle. The vehicle was stopp ed, so the turning

forces required were larger than would prevail under normal op eration. At 12

Figure 10: Figure 10. Small-signal steering resp onse. The smo oth curveis

the command, with one vertical division corresp onding to ab out 6 degrees of

steering steering knuckle angle. The timescale is 200 milliseconds p er division.

3 hertz the lag is noticeable, at ab out 4 hertz internal mo des takeover and

command following ceases.

Figure 11 illustrates steering b ehavior at larger amplitudes. Here the limita-

tion is slew rate, ab out 80 degrees p er second measured at the steering knuckle,

as evidenced by the nearly triangular p osition plot.

This can b e help ed considerably by increasing the ow rate in the hydraulic

system, but it's already some 20 higher than normal achieved by drilling out

the ow control ori ce and further increase invites pump trouble.

Large signal brake resp onse is illustrated in gure 12. Here the command

corresp onds to slightly over 800 psi with the resp onse settling within ab out 120

milliseconds

Small signal brake b ehavior is shown in gure 13. The step input corre-

sp onds to ab out 20 psi command input, and the resp onse can b e seen to settle 13

Figure 11: Figure 11. Large signal steering resp onse, showing slew rate limita-

tion.

within ab out 100 milliseconds. but the o set b ecomes comparable to the com-

mand. This o set is an artifact of the electronics and pressure sensor, not of

the actuator itself. It happ ens that at ab out 30 psi the brakes go from "o " to

"on" as tested by hand, so the o set isn't large.

5 Lessons Learned

In general terms, the brake actuator is satsifactory, the steering actuator some-

what less so.

The brakes o er reasonable time resp onse and more than sucient maximum

amplitude; indeed, it's no problem to lo ck the . In terms of nesse they

are acceptable but only barely so. Smo oth op eration at low sp eeds demands

considerable delicacy of braking, and the combination of o sets and thresholds

exhibited by these brakes are not capable of rivaling the p erformance of a go o d 14

Figure 12: Large signal resp onse of the brake actuator. The input steps are

ab out 800 psi.

human driver. The remedies p ossible are two: reduce seal or add a

direct torque measurements at the calip er. The seals are many, and mostly

rather sp ecialized; it's not likely that low friction versions of calip er seals and

b o oster piston seal can b e obtained. Even if they could, the b ehavior of the

brake pads remains outside the control lo op. If one has the choice, a direct

measurement of brake torque is probably the b est solution. With actual brake

torque in the control lo op this actuator would b e satsifactory for any practical

application.

The steering system is b oth p o orer in p erformance relative to what's needed

and harder to improve. One would like the steering to b e faster than the normal

mo des of the car's susp ension. This system is not. One would like the steering

system to have a smo oth transient resp onse, this one do es not. As it stands,

the steering is adequate to study lanekeeping at sp eeds up to 50 miles p er hour

[1], but only barely so. A more clever inner lo op would help, but can't really 15

Figure 13: Small signal brake resp onse. The input corresp onds to only 20 psi.

solve the problems.

The limitations come from a combination of slow control valves and limited

slew rate in the steering rack piston. The mechanical resonance of the steering

wheel on the p ower steering control valve torsion bar is an unhelpful in uence

but p erformance didn't improve markedly when the was removed

ie., a column clutchwon't help. The remedies p ossible are: Increase pump ow

rate, increase electrical drive to the control valves, or go to a more conventional

hydraulic system using a constant pressure design and servovalves.

Of the three the last will probably b e the most satisfactory. Using the stan-

dard p ower steering pump to charge an accumulator isn't dicult but constant

pressure op eration presents an entertaining problem of considerable imp ort:

How do es one get backtohuman constant ow op eration when the automatic

controls are in a state of irretrievable disarray? The return to human control

mo de must b e graceful and quick, a sub ject for future investigation. 16

6 Acknowledgements

The aid, counsel and graphic talents of PATH Publications Department sta

Gerald Stone, Esther Kerkman and Bill Stone are gratefully acknowledged.

Doug Wo o d of the RFS Maintenance Department constructed the manifold

blo cks from less than p erfect drawings. Dan Marvin of Norman Racing in

Berkeley, California constructed the valve sp o ols from somewhat more nearly

p erfect drawings.

References

[1] Pro ceedings of the American Control Conference, Albuquerque, New

Mexico, June 1997, pp. 1603 - 1607. 17