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Comparative Experiments in the Dynamics and Model-Based Control of Marine Thrusters

Louis L. Whitcomb* Department of Mechanical Engineering G.W.C. Whiting School of Engineering

The Johns Hopkins University Baltimore, Maryland, 21218, USA

email: 1lwQjhu.edu

Abstract This paper examines dynamical models for bladed- propeller type marine thrusters. Previously reported thruster dynamics models are reviewed, and a new simplified model is proposed. Experimental testing of both the transient and steady-state performance of a marine thruster corroborates previously reported data, validates the simplified thruster model, and raises new questions.

1. Introduction Precise automatic position control of marine vehicles en- ables important scientific and industrial tasks such as automatic docking and station keeping; precise surveying; inspection; sample gathering; and manipulation. A grow- ing body of theoretical and experimental literature indi- cates that an essential element of improved vehicle positioning systems is improved dynamical models of the bladed thrusters commonly used to actuate dynamically- positioned marine vehicles [13, 1, 5 , 31. The control systems of present-day dynamically positioned marine vehicles typically ignore thruster dynamics completely - treating it as an unmodeled disturbance. Incorporating precise models of thruster dynamics into the feedback control systems of marine vehicles promises improved vehicle positioning. The objective of this paper is to review, experimentally corroborate, and extend the principal dynamical models that have been proposed for marine thrusters.

This paper focuses exclusively on the dynamics of elec- trically actuated thrusters typically employed on dynamically positioned vehicles, and on the region of operation typical of hovering and low-speed tracking. A variety of other engineering obstacles to high-precision vehicle positioning - most notably vehicle position and velocity sensing accuracy, update rate, and latency - fall beyond the scope of this paper. They merit continued careful atten- tion.

2. Thruster Dynamics Theory This section reviews the transient hydrodynamics of marine propellers. It provides a historical summary of recent work in lumped-parameter thruster dynamics modeling, and suggests a simplified alternative to a previously pro-

*We gratefully acknowledge the support under a postdoe toral echolamhip awarded to the Arst author by the Woods Hole Oceanographic Institution with funds provided by Devonshire Associates.

Dana R. Yoerger Deep Submergence Laboratory

Applied Ocean Physics and Engineering Department Woods Hole Oceanographic Institution

Woods Hole, Massachusetts 02543, USA email: danaQisis.whoi.edu

posed thruster model.

2.1. Basic Hydrodynamic Relations The derivation of a steady-state hydrodynamic model for propellers operating in incompressible fluid can be found in most introductory fluid dynamics texts, e.g. [lo, 7,8,2], and is only briefly summarized here.

The axial thrust force on a ducted marine propeller in steady-state operation can be equated to (i) the fluid pressure differential across the disk of the propeller and (ii) the acceleration of the fluid mass in the duct. Assuming (i) gravity effects are negligible, (ii) incompressible flow, (iii) inviscid flow, and (iv) irrotational flow, then we can apply Bernoulli's equation to streamlines upstream and downstream of the propeller to obtain

thus (pd - p,,) = $(U: - vf). Applying the linear momentum theorem to an enclosing control volume of volume 017, where 7 is an empirically determined added m888 CO- efficient, yields

and from (1) we have

(3) T = p a l v i t p + i p ( v u , 2 - U:). Applying the linear momentum theorem to a large control volume reveals that up = $(U,,, + va), thus

T = pCrl@p + pvp(vw - va). (4)

In the special case where va = 0, we can rewrite (4) in terms of the independent variable up as

T = ($714 + (2P)v; . (5 )

The above equation relates propeller thrust, TI to duct fluid axial velocity, up, and fluid axial acceleration, up. In the sequel we use the convention in [3] to write (5) in the form

where AP is an empirically determined flux coefficent. The experimentally measured value of AD will be seen to be close to the theoretically predicted value of 2.

T = ($7)ir, + (APPa)VPlVPl* (6)

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UNIT

Model1 Model2 Model3

meter ma m radian NevltC?n N . m radlsec

N/ma N/ma N/m2 m/sa m/sa m/sa

kdm3

radians kg . ma amp vdts - N.m

O Y "

rad/.===

9 ohm

1 2

dynamicd sa Q dynamical R,vp Vm dynamical vp n 1

P radius of propellor thruster duct m a thruster duct length prop blade average pitch prop shaft thrust prop shaft torque prop angular velocity density of water ambient fluid pressure pressure, prop upstream side pressure, prop downstream side ambient axial fluid velocity axial fluid velocity at prop axial fluid velocity in wake prop efficiency (0 5 added mass coefficient flux coefficient fluid/blade angle of attack inertia, prop and shaft motor drive current motor drive voltage motor torque constant motor viscous friction constant

5 1.0)

motor back-emf constant motor winding resistance

Figure 1. Notation

I NAME I TYPE I STATES I 1" I 0- I Model 0 I static maD I none I R I 0 I

2.2. Thruster Model 1 It is well known, e.g. [?I, that under bollard-pull conditions a symmetrical propeller's steady-state axial thrust, TI is proportional to the square of the propeller's rotational velocity, Q. We will find it convenient to write this quadratic relationship in the form

T = pAR2q2 tan (P)~RIRI. (7) where 9 is the propeller efficiency coefficient, p is the propeller pitch, a is the propeller area, and p is the fluid density. In [13] the authors propose a first-order nonlinear dynamical model in which prop rotational velocity is the state variable, and propeller torque is the control input. This model employs the additional assumptions

0 Va = 0 zero ambient fluid velocity, 0 vu, = vp: flow velocity equal at prop and in wake, a 7 = 1.0. no added m w , and

up = q tan (p)R: propeller rotational speed is proportional to the fluid axial flow velocity. The proportion-

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ality is governed by a slip model based on a propeller efficiency coefficient, r ) , and the propeller pitch, p.

The authors employ an energy balance analysis equating axial power expended at the propeller disk, T-up , to power expended on the prop shaft, Q R,

The authors apply these assumptions directly to (3) to obtain the differential equation of motion with independent variable of propeller angular velocity, and input of shaft torque

Finally, the model proposed in [13] omits transient momentum balance terms to write instantaneous propeller thrust using the steady-state relation (7).

2.3. Thruster Model 2 In [3] the authors propose a nonlinear thruster dynamics model b& on a closer examination of the electro- mechanical dynamics and hydrodynamics of the thruster. In this model, the propeller and fluid dynamics are a p proximated by a two-dimensional second order nonlinear dynamical system in which the state variables are (i) axial fluid velocity and (ii) propeller rotational velocity. The control input is motor drive voltage. This model employs the additional assumptions:

0 v, = 0 zero ambient fluid velocity and 0 prop thrust, torque, and velocity are related by the

lift and drag forces on the wing-like blades of the propeller according to specified lift and drag relations.

The model proposed in [3] has three parts: First, assume the thruster motor/prop combination possesses electro- mechanical dynamics as follows:

h = I-'[kti, - kfs2 - Q] (10)

where I is the motor and prop moment of inertia; kt is the motor torque constant; kf is the motor viscous friction constant; and Q is the propeller rotational torque due to hydrodynamic loading.

In the experiments reported in [3], the authors employed a voltage amplifier to drive the thruster motors. They explicitly modeled the motor electro-mechanical dynamics. By assuming the motor inductance to be negligible, the authors use the relation im = (Vm - kemjS2)/R to re-write (10) with voltage input Vm. The voltage-controlled eqw tion of motion can then be written

h = - K i n + k2Vm - k3Q (11)

where the constants kl, k2, and k3 are given by

kl = I-'[R-'kemf + kf] k3 =I-lR-' (12) k3 =I-'.

This 1-D first order linear differential equation has state J z , command input V,, and exogenous input Q due to

the propeller hydrodynamic loading. Second, the linear momentum equation (6) is rewritten

This 1-D second order nonlinear differential equation has state U,, and exogenous input T due to the propeller hydrodynamic loading.

Third, the authors write the propeller torque and axial thrust as a function relating T and Q to propeller blade lift, L, and drag, D.

T = L . cos(@) - D . sin(@) Q = 0.7rL sin(@) + D . cos(8) (14)

where ut = 0.7rR 0 = ATANP(up, ut) a = p - e (15) v =Jm.

These equations employs standard %mped parameter" simplifying approximation, e.g. [lo] p. 137, to compute lift and drag on the propeller blades based on the characteristic radius and pitch of the blades at a point 0.7r from the propeller axis. Neglecting rotational flow, the characteristic tangential fluid velocity at the blade, ut = 0.7rR, is determined by the prop rotation speed. The lift force L acts on the blades parallel to the steamlines of oncoming fluid. The drag force D acts on the blades perpendicular to the steamlines of oncoming fluid. In the classical flu- ids literature, e.g. [lo], lift and drag forces for a foil are normally computed as

where CL(^) and CD(Q) are the empirically tabulated functions of the angle of attack a. In [3] the authors in- troduce a truncated fourier approximation for Ci(a) and Ca(a) as follows:

where CL^^^ and Coma, are experimentally determined scalar coefficients.

2.4. Thruster Model 3 The electmmechanical dynamics of in Model 2, from [3], were included to more closely approximate the experimental thruster apparatus reported therein - which incorpo- rated a voltagedriven DC thruster motor. It has been our experience, however, that full-ocean depth thrusters are often actuated by Sphase DC brushless electric motors equipped with commutating c u m t amplifiers. Our o p erational underwater robot systems (e.g. JASON, ABE, HYLAS [ll, 12, 61 ) employ high-bandwidth closed loop current amplifiers to drive the thruster motors, and are thus able to precisely command motor electrical current.

Using current emplifiers allows us to employ the motor model (10) directly, obviating the need to model motor electrical parameters such as winding resistance, winding inductance, and back-emf. Moreover, them commonly

available current amplifiers are usually equipped with high- resolution resolvers measuring propeller shaft position. Thus instrumented, a thruster motor can easily be configured as a closed-loop shaft velocity servo which delivers motor currents as required to track a commanded propeller rotational velocity.

Enabled by this, we propose a simplified version of Model 2 in which the propeller and fluid dynamics are approximated by a one-dimensional first order nonlinear dynamical system in which the state variable is axial fluid velocity and the control input is propeller rotational velocity. This model employs the following assumptions:

0 U , = 0 zero ambient fluid velocity, 0 prop thrust and torque are related by the lift and

drag forces on the wing-like blades of the propeller according to specified lift and drag relations, and

0 the thruster model input is the propeller rotational velocity. In the actual system, a velocity servo guar- antees exact tracking of the desired propeller velocity regardless of hydrodynamic torque and thrust loads.

First, as in Model 2, the linear momentum equation (6) is rewritten as per (13), resulting in a 1-D second order nonlinear differential equation with state up and exogenous input T due to the propeller hydrodynamic loading.

Second, as in Model 2, we write the propeller torque and axial force as function of a lumped parameter propeller blade lift, L, and drag, D, as per (14) and (17). Model 3 includes axial flow dynamics, but omits rotational prop and motor dynamics - a simplification of the second-order model originally proposed in [3].

3. Thruster Dynamics Experiments This section reports a series of experiments conducted under a variety of conditions to compare actual thruster performance to that predicted by the models of the previous section.

3.1. Experimental Setup These experiments employed a new brushless electric thrusters designed for the 6000 Meter JASON ROV incorporating MOOG model 304-14OA frameless windings and magnets and resolver feedback in a custom oil- compensated housing. The motors were driven by an ELMO model EBAF15/160 %phase commutating 20KHz PWM current amplifier. Under the inductive load of the motor, the current amplifier was configured to track current commands with a 2ms time constant. The current limit wm set at 9 Amps at a supply voltage of 120V for a maximum power of about 1KW. The tests employed VE- TUS propeller of diameter 24cm and pitch 22.5 degrees, surrounded by a duct of length 12.7 cm and diameter 26cm. The propeller is symmetrical in forward and re- verse motion. The propeller was mounted directly on the motor shaft; no gearbox was employed. Resolver feedback was converted to quadrature by an AD2S82 on-board the EBAF15/160 to a resolution of 4096 counts per shaft mechanical revolution.

For these experiments a thruster was mounted on test stand designed for maximum rigidity under thrust loading. The stand transferred thrust to a JR3 force sensor

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with full-scale range of +/- 200 pound-force. The JR.3 amplifier converted the thust load to a proportional +/- 10 volt analog output, with internal analog low-pass filters set to a 163 Hz cutoff frequency. The stand held the motor in a horizontal-shaft configuration, submerged l meter in a cylindrical test-tank measuring 5 meters in diameter and 3.5 meters in depth. The fluid flow velocity was measured at 20 H z with a 3-axis BASS acoustic flow-meter, [9], mounted 30cm downstream of the propeller along its central axis.

The host computer was a 486 class PC equipped with a quadrature interface, 12 bit Analog I/O, and a 100 KHz hardware clock. The amplifier, sensors, and host computer were extensively shielded and opto-isolated to minimize electromagnetic interference. The controllers described herein were executed at 500Hz. The data was sampled and logged at 100Hz, with the exception of fluid flow which was sampled at 20Hz. Thrust, torque, and fluid velocity data are reported and plotted unfiltered - no numerical smoothing or post-processing has been employed for these signals. Propeller rotation velocity was obtained by numerically differentiating the raw propeller position data, and smoothing the result with a first order recursive low- pass digital filter with lOms time constant.

3.2. Model 1 Evaluation This section investigates the validity of Model 1 by com- paring experimental thruster response to that predicted by numerical simulations of Model 1. The numerical simulation of Model 1 employed actual measured values for all physical plant parameters (e.g. r, U , I , p , p).

3.2.1. Model 1: Steady-State Response Figure 3 shows experimental steady-state data from the thrusters described in Section 3.1. As predicted by the steady-state relation (7), Figure 3 (top) shows the actual steady-state thruster force to be proportional to the square of the propeller angular velocity:

. Wl N rad2/sec2 T = 0.0375

This data confirms the well known quadratic relation between propeller rotational speed and thrust under steady- state bollard-pull conditions [7]. As predicted by (9), Fig- ure 3 (middle) shows actual steady-state thruster torque to be proportional to the square of the propeller angular velocity:

. QlRl N - m Rad2/Sec2 Q = 0.00232

Finally, as predicted by the combination of (9) and (7) Figure 3 (bottom) shows actual steady-state thruster force to be proportional to propeller torque.

N N - m T = 16.1- . Q

Each of the three experimentally determined steady-state relations (18), (19), and (20), can be equated to its theoretical counterpart to yield a numerical value for q.

First, equating (7) with the experimentally determined (18) and solving for q yields 81 = 0.49. Second, the stable fixed point of (9) occurs when h = 0. Equating this k e d

1-n 8h.1 v* S q l " d . c u h I u a ~ ~ M U t W ~ ~ S ............................................. ................................. ................, 'T-- !

Figure 3. Steady-State Experimental Data: Propeller Thrust vs Angular Velocity Squared (top). Propeller Torque vs Angular Velocity Squared (middle). Pro- peller Thrust vs Torque (bottom).

point to the experimetally determined (18) and solving for q yields = 0.81, Third, combining this with (7) yields the theoretical relation Q = 4qR tan @)T which can be equated with the experimentally determined relation (20) to yield = 0.53.

These three values for q differ substantially, indicating that while Model 1 is qualitatively descriptive of steady-

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state thruster performance, it is not a quanititatively accurate model of steady-state thruster performance.

3.2.2. Model 1: Transient Response HOW well does Model 1 represent the transient dynamics of an actual thruster? The comparison of experimental transient response and theoretical transient response (via numerical simulation) for Model 1 is complicated by the (theoretically disallowed) conflicting values of q obtained experimentally in the previous section. As an ad-hoc so- lution, in the numerical simulation we have employed ql in the simulation of the thrust map (7) and in the simulation of the differential equation (9). The V parameter determines the trensient response of (9) but does not enter into its steady-state behavior of Model 1. In the simulations we set V to be the actual duct volume.

Figure 4 shows the actual thruster response (left column) and simulated Model 1 thruster response (right column) to identical step torque inputs. The rise time for both thrust and propeller rotation speed is similar for the actual and simulated model responses. The actual thruster response exhibits significant overshoot in thrust, as reported in [l, 5,3], as well as overshoot in propeller rotation speed. This overshoot is not observed in the Model 1 simulation - indeed the Model 0 equations are structurally incapable of generating thrust or prop speed overshoot for any set of plant parameter values. Is this discrepency between experiment and Model 1

simulation unique to this particular thrust value? Figure 5 (left) shows the experimental thrust and prop rotation velocity responses to 18 different step torque commands. Figure 5 (right) shows the corresponding numerical simulation of Model 1 responses. The data coriform that Model 1 numerical simulations fail to accurately correspond to actual experimental thruster transient responses over a wide range of operating conditions.

We conclude that Model 1 is not an accurate model for thruster transient response. This corraborates independent thruster dynamics experiments first reported in [4, 5, 31 which demonstrated the need for more elaborate thruster models to accurately represent thruster transient response.

3.3. Model 2 Evaluation M o m PARAM

I : I I

VALUE

0.542 1.25 1.86 2.0 0.01 0.13 0.1271 0.39 998

Kg m2 m m rad

Kg/7n3

METHOD least-square least-square least-quare

manual manual

measurement measurement measurement measurement

Figure 7. Model 2: Parameters

This section investigates the validity of Model 2 by com- paring experimental thruster response to that predicted by numerical simulations of Model 2.

3.3.1. Model 2: Parameters The numerical simulation of Model 2 employed the parameter values given in Figure 7. Actual measured values were used for all physical plant parameters (e.g. r ,u , l , p ,p ) . Values for the derived parameters CLmaz, CDmaz, 7, Ap, and kf were calculated as follows:

1. Parameters CL,,,, CD,,,, and kf enter linearly into (14). Their values were determined by a least-square fit to experimentally measured steady-state values of Q, T, R, and v, at 9 different thrust levels.

2. The parameter Ap enters linearly into the steady- state fixed-point of (6). Its value was determined by a least-square fit to experimentally measured steady- state values of T and v, at 9 different thrust levels.

3. The parameters I and 7, which determine the rise time of (6), were chosen manually (by trial-and- error) so that the simulation’s rise time roughly cor- responded to the experimental observations.

Note the following observations:

1. The experimentally determined least-square values for CL^,, and CD,,, are plausable values. These values differ from those reported for a completely different thruster in [3], wherein the authors employed a manual parameter-fitting procedure.

2. The experimentally determined least-square fit value for Ap agrees closely with that predicted by simple hydrodynamic analysis (5).

3. The least-square fitting procedure results in a small but physically unreasonable negative parameter value for the motor linear fiiction term kf - negative damp ing. Similar results were obtained when we included a motor quadratic viscous friction term. Moreover, these negative damping terms resulted in inaccurate steady-state values of the Model 2 numerical simulations. We therefore omitted Mction terms from the motor model in both the least-square Model 2 parameter fitting and in the Model 2 numerical simulations.

3.3.2. Model 2: Steady-State Response Figure 6 (left) shows experimental thrust and prop rotation velocity responses to 18 different step torque commands. Figure 6 (right) shows the corresponding numerical simulation responses of Model 2 to the same torque commands. The data confirms that Model 2, using the model parameter values given in Figure 7, accurately pre- dicts steady-state thruster behavior. The model and experimental steady-state values for thrust, torque, and axial fluid flow all agree with good accuracy over this wide range of steady-state conditions.

We note that both the acutal experimentally in- stremented thruster and the Model 2 simulation exhibit a characteristic steady-state angle-of-attack (between blade chord and incoming fluid velocity) of approximately 0.21 radian (about 12 degrees). As shown in Figure 8, this steady-state angle of attack was observed to hold over a wide range of conditions in both experiments and simulation.

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. , . . . . . . , . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . , . . . . . . , . . . . . .

. . . . . . . . . . . .

'0 03 1 id z 2.s 3 as 4 43 s 8.rmd.

Figure 4. Thruster Torque Step Response: Experimental Data (left) and Model 1 Simulation (right): Thrust vs Time (top graph) and Prop rotational speed vs Time (bottom graph). Torque step was 9.2N m at t = 1.Osec.

. . . . . (6 .... ~ ..........; ......._.._i...........,.............(__.......... ............................... 1 ; ; . . i . j . ! ; ; . . . . j . i . . . . . . . . . . . .

N e " s

Figure 8. Steady-State Angle of Attack vs Steady- State Thrust. Experimental ("o~') and Model 2 Simu- lation ("*,').

3.3.3. Model 2: Transient Response

Referring to Figure 6 we note the following observations: Variable Time Constant: The Model 2 simulation responses do exhibit "rise time" which varies with torque

level - behavior typical of nonlinear dynamical systems. This corraborates previously reported thruster experiments, e.g. [13, 4, 5, 31. Force Overshoot: The Model 2 simulation responses in Figure 6 do not exhibit the characteristic thrust overshoot which is clearly observed in the actual thruster torque step response data. These simulations employed the plant parameter values obtained experimentally as described in Section 3.3.1. Only by using a "manually tuning" set of Model 2 plant parameters were we able to elicit charwter- istic force overshoot in the model 2 simulation (not shown) - as has been done previously in independently reported experimental studies (4, 5, 31.

Unlike previous experimental studies, our test setup provided for precise high-bandwidth control of motor current (torque) and high-bandwidth fluid axial flow measurement in addition to thrust measurement. Enabled by this in- strumentation, we were able to experimentally determine thruster parameter values for Cl, c d , kf, and Ap - rather than "manually tuning" the simulation parameters. Model 2 steady-state simulations using these experimentally me& sured plant values correspond accurately to steady-state experimental data. Interestingly, Model 2 transient simulations using these experimentally measured plant values do not correspond precisely to transient experimental data. Velocity Overshoot: Finally, the experimental step re-

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h ~ : M w T a w m ~ - . h l U W # a l a A p c p i ~ - m ...... ; ........................................ ......... ......... ................... ......... . . . ! . . . . . . qm 1 ; ... .: ......... q ...... ., ......... . . . . . . * . . . . . . . . . . . . . . .................. ........................................................................ ~ . . . . . . . . . . . . . . +;y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 1 I D ' 1 . ..... ., ..... .q .....

. . . . . . 1 ...I .... ........ ... 3 .... ........

. . . . . . . . .

m m v & d y : m ~ m p e m c - ~ ~ . \ d . a i w . p t w m T" ...... .......................................................... ; ......... ; ..................... . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . ....... ! ......... % ......... : ...... .........._.....................................I ......... : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' ' a ' * * ' ' * '

0.6 1 1.6 2 23 3 3.6 4 4.6 6 8.wd.

Figure 5. Experimental (left) and Model 1 Simulated (right) Thruster Response to 18 Different Torque Step Inputs: Thrust vs time (top graph) and Prop rotational speed vs time (bottom graph). Torque steps were evenly spaced from +11N rn to -11N - m.

sponsecr in Figure 6 exhibit a characteristic propeller velocity overshoot which is not obeierved in the simulation. This experimental "velocity overshoot" has been documented in data presented in previously reported studies, [4, 51, but its presence appeam to have been overlooked. We suggest that this "velocity overshoot" be specifically identified as a characteristic behavior of marine thrusters in the present context.

The Model 2 simulations in Figure 6 do not exhibit this characteristic propeller velocity overshoot. Moreover, we were unable to find any "manually tuned" set of parameter values which elicits propeller velocity overshoot in the simulations. We surmise that Model 2 is generically incapable of eliciting "velocity overshoot" behavior, and expect that a more detailed hydrodynamic model might exhibit this experimentally observed behavior.

Model 2: Conclusion

We conclude that Model 2, [3], is a good model - it r ep resents a significant advance over previously reported dynamical thruster models. We observe the following: First, the steady-state behavior and "variable rise time" of actual thruster response is accurately modeled by Model 2 when given accurate experimentally measured plant p& rametem. Second, Model 2 demonstrates characteristic

thrust overshoot when given manually tuned parameter values. Third, Model 2 does not appear to exhibit the propeller velocity Overshoot observed in actual experiments. We surmise that a more detailed analysis of the dynamical coupling between the thruster's hydrodynamics and the propeller and motor's rotational dynamics might yield extensions to Model 2 which accurately capture these effects.

3.4. Model 3 Evaluation Experimental testing of Model 3 is predicated on the abil- ity to instantaneously "command" the propeller velocity, R to any desired reference value, 52,. To accomplish this, we implemented a propeller velocity feedback controller of the form

&n = kF1[alTr + kfb(n - nr)] (21) where al, with units N . m/N, is an experimentdy determined constant as shown in Figure 3 (bottom), kt is the motor torque constant, and the feedback gain kfb was set as high as possible while maintaining stability. The controller was evaluated at a rate of 500Hz. Velocity was estimated by numerically differentiating the position signal, and smoothing the resulting signal with a first-order low-pass filter with a lOms time constant.

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,m ........ : (00 ........, 0 t ! ........ j

. . . . . . . : . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

'i .1 ..

. . . . . . . . . . . . . . . . . . . . 4 is ; 115 ; ;5 ; is ; i.6 6

mr.. .............................................. . . . . . . . . . . !!!?!%%1?.?..!?.?..

8aand.

ala vobc*: * TC8q.m s6ph c. . . . . . . . . . . . . . . . . . . . .

....... ........

........ ........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

0 0.5 1 1.6 2 2.5 3 5 5 4 45 5 -a

........ ; \ ; j j ; : : j ; < ................... \ ........ ,. ............................ , ......... .....................

~ a - - v . b * . w ~ o r q u a . p . c * - i ~ + . 7 ~ % "r ........ ~ ........................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . J

0 0.6 1 15 2 25 3 3.5 4 4 5 5 &md.

Figure 6. Experimental (left) and Model 2 Simulated (right) Thruster Response to 18 Different Torque Step Inputs: Thrust vs time (top graph) and Prop rotational speed vs time (bottom graph). Torque steps were evenly spaced from +11N. m to -11N. m.

Figure 9. Experimental and Model 3 Simulated Thrust (left) and Shaft Rotational Velocity (right). Experimental data is solid line. Simulation data is dotted line.

ing both discontinuous and constant velocity commmds. Figure 10 (bottom) shows excellent tracking performance for continuously varying velocity commands. The tracking accuracy of this type of controller is principally limited by

How good is this propeller velocity controller? Figure 9 and Figure 10 (bottom) show the reference and actual propeller velocity for this controller for different reference trajectories. Figure 9 shows good performance in track-

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f

. . . . . . . . . . . . . . .

, . . . , , . . . . . . . . . . . . . . . . . . . . . . I

1 1.6 2 2.6 3 3.6 4 4.6 5 - Figure 10. Experimental and Model 3 Simulated Thrust (top) and Shaft Rotational Velocity (bottom) Experi- mental data is solid line. Simulation data is dotted line.

(a) sampling rate, (b) shaft position sensing resolution, and (c) actuator saturation limits. In this case, highly accurate velocity tracking is possible due to the high sampling rate (~OOHS), high position accuracy (4096 counts/rev), and high actuator saturation limits (current limit set to 9 Amp, bus voltage 120 Volts).

3.4.1. Model 3: Parameters

The parameters used in the model 3 simulation were identical to those given in Section 3.3.1. The parameters kf and I do not enter into Model 3.

3.4.2. Model 3: Steady-State Response

The steady-state portions (t > 1) of the graphs Figure 9 show the Model 3 simulated and actual response to a 60rad/sec propeller rotation speed commands. Experi- mental data is shown as a solid line, simulation data as a dotted line. The steady-state portions of the experimental data exhibits excellent correspondence in thrust, torque, and propeller rotation speed to the steady-state behavior of the Model 3 numerical simulation. Similar results were observed at a variety of different commanded prop veloci- ties.

3.4.3. Model 3: Transient Response

Figure 9 shows the Model 3 transient response to closely correspond to the actual experimental response for discontinuous step velocity commands. Thrust, propeller velocity, and torque ate seen to all correspond closely.

What about continuously varying propeller velocity commands? Figure 10 shows the Model 3 simulated and actual response to time varying propeller speed commands. The left column graphs show the actual and Model 3 response to a 0.25 Hz sinusoidal velocity reference with peak magnitude 6Orad/sec. The right column graphs show the response to velocity reference varying at 1.0 Hz. Here again, the Model 3 simulation response closely matches the thruster’s actual experimental response.

We conclude that Model 3 closely approximates the actual experimentally observed thruster performance. This model represents the thruster as a single degree-of-freedom dynamical system with state U,,, input f2, and output T. The disadvantage of Model 3 is that, in practice, it assumes the existance of an accurate propeller velocity servo. The advantage of Model 3 is that it obviates the need for careful modeling and identification of thruster motor dynamics. Accordingly, Model 3 is less sensitive (in comparison to Model 2) to thruster motor defects such as coulomb, linear and quadratic friction; gearbox dynamics (when present); motor torque ripple; and torque constant nonlinearities.

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4. Conclusion References Conclusions for Models 1,2, and 3 having been given in in the previous Section, we will only summarize and amplify the following points here.

[I] J. A d m , D. Burton, and M. Lee. Dynamic charac- terization and control of thrusters for underwater vehicles. Techreport, Monterey Bay Aquarium Research Institute and Stanford University Aerospace Robotics

1. The importance of thruster dynamics in underwater vehicle modeling and control, first articulated in [13], is corroborated both by the present study and previous independent studies, e.g. [l, 4, 5, 31.

2. Model 1, [13], does not accurately represent the characteristic transient responses of thrust and propeller velocity overshoot observed in actual thruster experiments.

3. Model 2, [3], represents a significant improvement in thruster dynamical modeling by explicitly incorporating the hydrodynamic effects arising fkom p r e peller rotation and axial fluid flow. This corroborates thruster dynamics experiments independently reported in [4, 5, 31. We surmise that the noted dis- crepencies between Model 2 simulations and observed experimental response might be corrected by extend- ing the model to include a more detailed analysis of the dynamical coupling between the thruster’s axial hydrodynamics and the propeller and motor’s rotational dynamics. The matter deserves careful atten- tion.

4. Model 3, a simplification of Model 2, adopts propeller velocity as the control input. We observed highly accurate correspondence between experiment and Model 3 simulation. The significance of this ob- servation is its validation of the purely hydrodynamic component of model 2, independent of all motor dynamics.

A variety of questions remain unsolved. The design and testing of model-based thrust controllers incorporating the dynamical thruster models is the subject of our current efforts. Their practical utility is complicated by two principal open questions: First is the need to instrument fluid flow velocity in the duct and (in the general case) the velocity of the ambient fluid. While it is unclear if thruster designers are willing to incorporate fragile high-bandwith flow sensors in the thruster itself, there is reason to believe that a stable nonlinear observer might be developed to accurately estimate v, using only commonly available sensor and control information. Second, model-based controllers require highly accurate plant model parameters. While these parameters can be obtained via the cumbersome off- line empirical least-squares method described herein, there is reason to believe that a stable adaptive thrust controller might be developed to automatically estimate these pa, rameters on-line.

Acknowledgments Tony Healey and Jim Newman provided thoughtful com- ments and suggestions. Andrew Bowen and Alex Boc- concelli designed the new brushless thrusters. Rob Keefe designed and built the original version of the thruster test stand. Sandy Williams graciously loaned us 8 BASS system, and maintained perfect equanimity when we broke it. The authors we grateful for their generous contributions.

Laboratory, September 1991. [2] J. A. Fay. Introduction to Fluid Mechanics. MIT

Press, Cambridge, Massachusetts USA, 1994. [3] A. J. Healey, S. M. Rock, S. Cody, D. Miles, and

J. P. Brown. Toward and improved understanding of thruster dyamics for underwater vehicles. In Proceed- ings of the 1994 Synposium on Autonomous Underwa- ter Vehick Technology, pages 340-352, Boston, MA, USA, 1994.

[4] M. B. McLean. Dynamic performance of small diameter tunnel thrusters. Master’s thesis, Naval Postgrad- uate School, Monterey, CA, USA, March 1991.

[5] D. Miles, D. Burton, M. Lee, and S. Rock. Closed loop force control of underwater thrusters. Techre- port, Monterey Bay Aquarium Research Institute and Stanford University Aerospace Robotics Laboratory, October 1992.

[6] D. A. Mindell and D. R. Yoerger. Transputer-based distributed processing for underwater robotic vehicle control. In Proceedings of the American Control Con- ference, June 1991.

[7] J. Newman. Marine Hydmdynamics. MIT Press, Cambridge, Massachusetts USA, 1989.

[8] R. S. Shevell. findumentals of night. Prentice-Hall, Englewood Cliffs, New Jersey USA, 1989.

[9] D. A. Trivett, E. A. Terray, and A. J. Williams 111. Error analysis of an acoustic current meter. IEEE Journal of Oceanic Engineering, 16(4):329-37, Octo- ber 1991.

[lo] J. D. Van Manen and P. Van Ossanen. Principles of Naval Architecture, Second Revision, Volume 11: Re- sistance, Propulsion, and Vibmtion Society of Naval Architects and Marine Engineers, Jersey City, New Jersey USA, 1988. E. V. Lewis, Editor.

[ll] L. L. Whitcomb and D. R. Yoerger. A new distributed real-time control system for the jason underwater robot. In Proc. IEEE International Workshop on Intelligent Robots and Systems, Yokohama, Japan, July 1993. IEEE.

[12] D. R. Yoerger, A. M. Bradley, and B. B. Walden. Autonomous benthic explorer, deep ocean scientific auv for seafloor exploration: Untethered, on station one year without support ship. Sea Technology, pages 50-54, January 1992.

[13] D. R. Yoerger, J. G. Cooke, and J. E. Slotine. The influence of thruster dynamics on underwater vehicle behavior and their incorporation into control system design. IEEE Journal of Oceanic Engineering, 15(3):167-178, June 1990.

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