Control Design

International Conference on Control, Automation and Systems 2008

Oct. 14-17, 2008 in COEX, Seoul, Korea

Fuzzy Logic PID Based Control Design and Performance for a Pectoral Fin Propelled

Unmanned Underwater Vehicle

Jason D. Geder1, John Palmisano2, Ravi Ramamurti1, William C. Sandberg1*, and

Banahalli Ratna2

1

Laboratory for Computational Physics and Fluid Dynamics, U.S. Naval Research Laboratory, Washington, DC

(Tel : +1-202-***-****; E-mail: abqix7@r.postjobfree.com)

2

Center for Bio-molecular Science and Engineering, U.S. Naval Research Laboratory, Washington, DC

(Tel : +1-202-***-****; E-mail: abqix7@r.postjobfree.com)

(Current Address) Modeling and Analysis Division, Science Applications International Corporation, McLean, VA

(Tel : +1-703-***-****; E-mail: abqix7@r.postjobfree.com)

Abstract: This paper describes the modeling, simulation, and control of a UUV in six degree-of-freedom (6-DOF)

motion using two NRL actively controlled-curvature fins. Computational fluid dynamic (CFD) analysis and

experimental results are used in modeling the fin as part of the 6-DOF vehicle model. A fuzzy logic

proportional-integral-derivative (PID) based control system has been developed to smoothly transition between

preprogrammed sets of fin kinematics in order to create a stable and highly maneuverable UUV. Two different

approaches to a fuzzy logic PID controller are analyzed: weighted gait combination (WGC), and modification of mean

bulk angle bias (MBAB). Advantages and disadvantages of both methods at the vehicle level are discussed.

Simulation results show desirable system performance over a wide range of maneuvers.

Keywords: Biomimetic pectoral fin, UUV, unsteady CFD, fuzzy logic, PID, adaptive curvature, weighted gait

combination

1. INTRODUCTION 2. VEHICLE DESIGN AND MODELING

Low-speed and high-maneuverability performance, 2.1 Vehicle design

required in near-shore and littoral zone missions, is a Our team has completed the design and construction

major weakness of current unmanned underwater of a biomimetic controlled-curvature robotic pectoral fin

vehicle (UUV) technology. To address this issue, [3]. A test vehicle (Fig. 1) has been designed to

flapping fin mechanisms have been studied to demonstrate the speed and maneuverability enabled by a

understand how certain aquatic organisms achieve their pair of these pectoral fins [4]. The hull measures 1.3

high levels of controllability and how these mechanisms high, 13 long, and 7 wide, and has a dry weight of 2.6

can be adapted to UUVs [1]. pounds making it slightly negatively buoyant.

In our previous work, we concluded that flapping

pectoral fins were the solution for low-speed,

high-maneuverability operation [2]. We designed a

biomimetic fin propulsor with actively

controlled-curvature [3], and have designed a test

vehicle that utilizes two of these fins for propulsion and

control [4].

Because vehicle simulation results show a need for

more than bang-bang (or case-based) control, a fuzzy

logic proportional-integral-derivative (PID) based

control system is developed. This novel controller

commands changes in fin kinematics for vectoring

control forces necessary to enable a stable and highly

maneuverable UUV.

This paper studies two such non-finite pectoral fin

UUV control methods. The first method is weighted

gait combination (WGC) [5]. WGC is a control Fig. 1 Test vehicle carrying two NRL fins.

method that takes several preprogrammed fin gait

motions and recombines them in real-time to form The vehicle has been integrated with onboard sensors

intermediate hybrid motions. and a custom designed microcontroller [5]. Angular

The second UUV control method studied is mean rate gyroscopes and linear accelerometers are attached

bulk angle biasing (MBAB). When a pectoral fin flaps, to three independent axes, and a pressure sensor for

it produces both lift and thrust forces. By modifying depth measurement is mounted onboard as well.

the mean angle of the flapping fin, the lift force can be Combined with signal amplifiers for high resolution

controlled without affecting forward thrust. data collection, this system provides all necessary

onboard hardware for testing fin force production and provide a more accurate representation of the forces

vehicle maneuverability. generated by the fin because the parameters used in

CFD simulation do not take into account fluid-structure

interaction.

2.2 Equations of motion

A model of the physical system is formulated to aid

in control system development and vehicle performance

studies. Our unique UUV propulsion system makes

use of actively controlled elastic fin deformations that

produce forces at any desired vector [5, 6]. The rest of

the vehicle is rigid and can thus be defined by standard

rigid body equations of motion [7],

v f 0 = m 0 + v0 + rG + rG (1),

t

r

v ~ ~

m 0 = I 0 + I 0 + mrG 0 + v, (2)

t 0

r

where m is mass, 0 is the translational state vector, is

the rotational state vector, rG is the center of gravity Fig. 2 Mean fin generated thrust as a function of the

location, I0 is the inertia tensor, f0 is the external forces controllable parameters, bulk fin stroke amplitude and

vector, and m0 is external moments vector. fin curvature, both non-dimensional. Max. bulk angle:

From Eqs. (1)~(2) we have derived the complete 85 (blue) biased by +0.1N, 65 (green), 45 (red)

equations of motion and computed the prototype vehicle biased by -0.1N.

coefficients from CFD analysis, vehicle geometry, and

mass placement [4].

2.3 Fin Forces

The external forces and moments on the vehicle, f0

and m0, from Eqs. (1)~(2) include those generated by

the NRL fins. A combination of 3-D unsteady CFD

computations [2] and experimental measurements [6]

are used to determine force time-histories generated by

the fins for a range of fin kinematics.

Initial computational studies to optimize the

performance of the NRL fin [3, 8] provided a basis for

determining which fin parameters to analyze as

potential control variables. These controllable fin

parameters include bulk fin stroke amplitude, fin

mean stroke position ( mean), deflection of each of the

four actuated fin ribs ( 1-4), and flapping frequency.

Fig. 3 Mean fin generated lift as a function of the

Unsteady CFD simulations were completed across a

controllable parameters, bulk fin stroke amplitude and

wide span of fin motions across all controllable

fin curvature, both non-dimensional. Max. bulk angle:

parameters [2]. Non-dimensional scale factors, k and

85 (blue) biased by +0.1N, 65 (green), 45 (red)

k, were used to define the values of the controllable

biased by -0.1N.

parameters, and, as,

2.4 Preprogrammed Gaits

(radians) = 1.2514 k, (3)

A specific set of fin kinematics is called a gait, and

(radians) = 1 = 0.1152 k, (4)

ideally, an infinite number of these gaits would be

programmed onto the onboard vehicle microcontroller.

where is a measure of the fin curvature from leading

This would ensure an exact gait to match any force

to trailing edge, and is defined by the maximum tip

vector commanded by the vehicle controller. However,

deflection of the leading edge rib.

this being infeasible, instead we must carefully choose

To evaluate UUV controller performance in

only a few preprogrammed gaits that the vehicle

simulation, a full range of fin forces need to be known.

navigation control system can draw from [5].

Therefore, curve-fitting equations were derived to map

Therefore, the control system uses combinations of this

fin kinematics to stroke averaged forces (Figs. 2~3).

limited set of gaits to generate the infinite number of fin

After validating the experimental force measurement

force vectors needed to perform six degree-of-freedom

accuracy [3, 5], a library relating force data to specific

(6-DOF) maneuvers. The results from Figs. 2~3

fin gaits has been built [6]. These experimental results

identify trends in force output which help determine the certain onboard sensors, such as video cameras for

preprogrammed gaits. object imaging, blurring low-light imaging and adding

Four gaits have been chosen, including a maximum significant noise to the inertial measurement unit.

forward thrust gait (Kf), a maximum reverse thrust gait

(Kr), and maximum upward and downward thrust gaits

(Ku and Kd, respectively) (Fig. 4). Any desired force

vector (Kc) for each fin can be achieved through

weighted combinations of preprogrammed gaits [5], and

then maps of these combinations to forces through the

equation, Kc = f K f, Kr, Ku, Kd . (5)

Note that all gaits are chosen to be entirely uncoupled

allowing each gait to be independently optimized and

studied to improve the overall controller. Extensive

experimental testing has been carried out to determine

optimum gaits for generating maximum forward and

reverse thrust, and maximum positive and negative lift

[6].

Kr Ku

Kc

Fig. 5 Vehicle response to steady level flight command

with bang-bang control

Kf

Kd

Fig. 4 Preprogrammed gait forces (red vectors) used to

achieve desired force vectors (blue vectors).

3. VEHICLE CONTROL

3.1 Bang-bang (or case-based) control

Before designing the fuzzy logic PID based control

algorithm for the vehicle, we first analyzed the

performance of a simple bang-bang control algorithm.

This algorithm limits the fins to switching discretely

between only the four preprogrammed gaits (Figs. 6, 8).

Vehicle simulations employing the bang-bang control

approach exhibit undesirable highly oscillatory motions

dictating the need for development of a more Fig. 6 Control response to steady level flight command

sophisticated control technique. with bang-bang control

The four experimentally selected gaits, previously

mentioned for producing each of the major thrust and The limitations of bang-bang control are displayed on

lift vectors, have been modeled as baseline sets for both a broader scale in lateral turning maneuvers (Fig. 7)

of the NRL fins. where we see vehicle roll, pitch, and yaw (as

We can see in the z-position response (Fig. 5) that the defined in Fig. 1) oscillate with peak-to-peak excursions

vehicle exhibits undamped oscillatory behavior with in excess of 60 degrees. These results show that

bang-bang control. This is also apparent in the pitch bang-bang control is unable to provide the precise

angle (as defined in Fig. 1) response which is maneuvering required of the UUV.

marked by 60 degree peak-to-peak oscillations. This

high magnitude oscillation would preclude the use of

For example, we linearize about steady level flight at a

constant forward speed and determine control gains for

depth response using the algorithm,

u z = K zp e z + K w w + K zi e z, (6)

where uz is the control output, Kzp is the proportional

gain, Kw is the derivative gain, Kzi is the integral gain, ez

is the vertical position error, and w is the vertical

velocity. Similar control outputs are found for the

other states in the forms,

u x = K xp e x + K u u + K xi e x

u y = K yp e y + K v v + K yi e y

u = K p e + K q q + K i e

Fig. 7 Vehicle response to yaw command with

u = K p e + K r r + K i e

bang-bang control

(7)

3.2.1 Weighted Gait Combination (WGC) Method

The individual PID control outputs for various

vehicle states, Eqs. (6)~(7), are then combined to

compute the total control value for each of the fins. In

the WGC method, these control values are computed as

shown in Eq. (8).

u LEFT _ FIN = u x + u y + u z + u + u

(8)

u RIGHT _ FIN = u x u y + u z + u u

In this weighted gait method, the output control

values for the left and right fins are then mapped to

percentages of the four preprogrammed gaits using a

membership function. Using vertical position control

in forward flight as an example (Fig. 10), the

membership function determines weighting of three of

Fig. 8 Vehicle response to yaw command with

the sets of preprogrammed gaits that contribute to the

bang-bang control

control output kinematics for each fin.

3.2 Fuzzy Logic PID Based Control

To improve stability and damp the oscillatory

behavior of the vehicle, a continuous transition between

the four preprogrammed gaits is necessary. One way

to program this type of control is using a fuzzy logic

system in which two or more preprogrammed fin gaits,

each optimized for producing a desired force vector, are

combined to create an intermediate gait. Fig. 9

outlines the design steps necessary to the creation of a

controller for the NRL test vehicle.

Fig. 10 Membership function for vertical position

controller in forward flight.

Fig. 9 Representative control block diagram.

If the vehicle is very far below desired depth, the

maximum positive lift gait is 100% used for both fins,

Linearized equations of motion were computed from

thereby saturating the controller. Similarly, if the

the 6-DOF vehicle model to determine control gains for

vehicle is very far above desired depth, the maximum

various maneuvers using classical control techniques [4].

negative lift gait is 100% used for both fins. And if the a much smoother transition to steady state compared

vehicle is within some predetermined range of desired with bang-bang control response (Fig. 5). During the

depth the output kinematics is calculated as a maneuver, the full weight percentage is on the positive

combination of the optimal positive lift, forward thrust, lift gait briefly, and then smoothly levels out to a full

and negative lift gaits. forward thrust gait weighting (Fig. 13).

3.2.2 Mean Bulk Angle Bias (MBAB) Method

An alternate to the WGC method of fuzzy logic PID

control has also been investigated in which control over

fin lift force is achieved through biasing the fin mean

bulk rotation position ( mean) up or down (Fig. 11).

This propulsion control method entirely decouples lift

from thrust, simplifying the controller design.

right left

right left

(a) (b)

Fig. 11 NRL two fin vehicle showing wing mean

position at, (a) -30 which produces a positive lift force,

and (b) +30 which produces a negative lift force.

The effectiveness of this MBAB method was

suggested by the results of the CFD studies [2]

Fig. 12 Vehicle response to step command in vertical

illustrated in Fig. 3 where we see lift increase as the fin

position with fuzzy logic PID control using WGC.

stroke is biased downwards. Control over horizontal

plane motion is still achieved through combining the

preprogrammed forward and reverse gaits for the left

and right fins as in Eq. (9).

u LEFT _ HORIZ = u x + u y + u

u RIGHT _ HORIZ = u x u y u (9)

u BULK _ BIAS = u z + u

The final step in modeling the vehicle control law

progression, for both the WGC and MBAB methods, is

determining how each gait or combination of gaits maps

to a force vector generated by each fin. Based on CFD

and experimental testing, a mapping from membership

in the various gaits to these fin generated force vectors

is made as discussed in section 2.3 [2, 6].

4. SIMULATION RESULTS Fig. 13 Control response to step command in vertical

position with fuzzy logic PID control using WGC.

Simulation was highly effective for investigating

stability and maneuverability performance of the UUV One concern with the WGC method is the difficulty

using the WGC and MBAB control methods. in controlling multiple vehicle states independently. In

The fuzzy PID controller using the WGC method was the climb maneuver response, we see z-position

tested first in simulation on the 6-DOF model controlled, but this is at the expense of an accurate

previously developed [4], producing predictable results. forward speed control (Fig. 12). We can correct for

Vehicle response to a climb maneuver (Fig. 12) displays this by scaling down the overall weight on the gait

combinations, but this would reduce the envelope in

which the vehicle can operate. More specifically, we

cannot obtain maximum lift simultaneously with

maximum thrust.

The MBAB method solves this problem. By using

this method, we are almost completely decoupling lift

forces from thrust forces when we combine gaits. This

allows us to more easily create a linear map from

control output to fin force.

Vehicle response to a climb maneuver (Fig. 14) is

analyzed using the MBAB method, but in this case

vehicle speed is held constant at 0.5 m/s. During the

maneuver the mean bulk rotation position ( mean) is at

maximum position briefly, and then slowly returns to its

level cruise position for a smooth transition in the

z-response. Meanwhile, a combination of the forward

and reverse gaits is achieved to maintain desired

Fig. 15 Control response to step command in vertical

forward speed (Fig. 15).

position with fuzzy logic PID control using MBAB.

Vehicle response to a yaw maneuver (Fig. 16) using

mean stroke bias for lift control also displays a much

smoother transition to steady state, and less oscillation

in steady state than bang-bang control response. Roll

and pitch ( and ) are limited to 3-4 degrees of steady

state oscillation, while yaw displays even less at

~1-2 degrees. Rise time is also not adversely affected,

as the control limits are saturated throughout 80% of the

yaw angle change (Fig. 17), just as they are using the

WGC method.

Fig. 16 Vehicle response to yaw maneuver with fuzzy

logic PID control using MBAB.

Fig. 17 Control response to yaw maneuver with fuzzy

Fig. 14 Vehicle response to step command in vertical

logic PID control using MBAB.

position with fuzzy logic PID control using MBAB.

AIAA-2006-3658, San Francisco, CA, 2006.

4. DISCUSSION

[3] J. Palmisano, R. Ramamurti, K. Lu, J. Cohen, W.

Sandberg and B. Ratna, Design of a Biomimetic

While the benefit of decoupling lift from thrust is

Controlled-Curvature Robotic Pectoral Fin, In

seen in vehicle results using the MBAB controller, this

Proc. of the IEEE International Conference on

advantage only applies when the fin is at or near a zero

Robotics and Automation, Rome, IT, 2007.

degree angle of attack with respect to the incoming flow.

[4] J. D. Geder, J. Palmisano, R. Ramamurti, W. C.

At higher magnitude angles, the lift and thrust are no

Sandberg and B. Ratna, A New Hybrid Approach

longer decoupled when we bias the fin mean bulk angle

to Dynamic Modeling and Control Design for a

up or down. This becomes an important issue as we

Pectoral Fin Propelled Unmanned Underwater

look forward to redesigning the UUV.

Vehicle, In Proc. of the Fifteenth International

Other results indicate that turning radius and speed

Symposium on Unmanned Untethered Submersible

can be improved by one or a combination of several

Technology, Durham, NH, 2007.

methods. These methods include improving the

[5] J. S. Palmisano, J. Geder, R. Ramamurti, W.

negative thrust gait, changing the vehicle geometry (ie.

Sandberg, B. Ratna, Real-Time Robotic Pectoral

decreasing the size of the vertical tail would facilitate

Fin CPG Using Weighted Gait Combinations,

faster yaw maneuvers), and investigating the advantages

submitted to IEEE Transactions on Robotics,

of non-linear gait weighting.

2008.

Since an extensive search for reverse thrust

[6] J. S. Palmisano, J. Geder, R. Ramamurti, W.

kinematics has already been conducted [6], a mix of

Sandberg, B. Ratna, How to Optimize Efficiency

changing the vertical tail and investigating non-linear

and Propulsion of a Controlled-Curvature Robotic

control weighting need to be considered for future

Pectoral Fin, to be submitted for publication in

generations of the vehicle. Since vehicle pitch

October 2008.

oscillations are relatively small (~3-4 degrees),

[7] T. I. Fossen, Guidance and Control of Ocean

adjusting the hull geometry to increase the pitch

Vehicles, John Wiley & Sons, New York, 1994.

moment of inertia would decrease these excursions even

[8] R. Ramamurti, W. Sandberg, J. Geder,

more. However, this would be at the expense of

Computations of Flapping Fin Propulsion for

decreasing the maximum attainable vehicle pitch angle

UUV Design, submitted to 47th AIAA Aerospace

with current fin gaits, thereby slowing vertical position

Sciences Conference, January 2009.

response. A fine balance needs to be found.

5. CONCLUSIONS

A prototype vehicle to demonstrate the propulsion

and maneuvering performance of its actively controlled

surface curvature deforming pectoral fins has been

developed. Three control techniques were

quantitatively compared for their ability to enable

hovering and maneuvering of our UUV using two NRL

flapping fins. We have described the mathematical

development and maneuvering performance obtained

using this controller. Experimentally and

computationally demonstrated fin performance provided

adequate force production for vehicle propulsion and

control, as shown in simulation results.

While bang-bang control yielded poor results in

simulation, algorithms for combining preprogrammed

fin gaits using both WGC and MBAB methods were

successful in controlling vehicle speed, position and

orientation.

REFERENCES

[1] J. E. Colgate and K. M. Lynch, Mechanics and

Control of Swimming: A Review, IEEE Journal

of Oceanic Engineering, vol. 29, pp. 660-673, July

2004.

[2] R. Ramamurti and W. C. Sandberg,

Computational Fluid Dynamics Study for

Optimization of a Fin Design, In Proc. of the 24th

AIAA Applied Aerodynamics Conference

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