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Longitudinal Motion Planning and Control of a Swimming Machine Saroj Saimek and Perry Y. Li Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis MN 55455 {saimek,pli}@me.umn.edu Abstract— We propose a practical maneuvering control

ago, Sir James Gray put forward the so called Gray’s para-

strategy for an aquatic vehicle (AV) that uses an oscillat-

dox which stimulated much research and controversy [6],

ing foil as a propulsor. The challenge of this problem lies in the need to consider the hydrodynamic interaction as well

[8], [23]. The paradox suggests that if the resistance of an

as the underactuated and non-minimum phase natures of

actively swimming dolphin is equal to that of a rigid model

the AV system. The control task is decomposed into the

towed at the same speed, the dolphin’s muscles must be ca-

off-line step of motion planning and the online step of feedback tracking. Optimal control techniques are used to compute a repertoire of time scalable and concatenable motion primitives. The complete motion plan is obtained by concatenating time-scaled copies of the primitives. The computed

pable of generating at least seven times more power than is typical of mammalian muscle [6]. In recent years, Triantafyllou et. al. [23], [3], [2] pro-

optimal motion plans are regulated by a controller that con-

posed an explanation of the Gray paradox. They propose

sists of a cascade of linear quadratic regulator, input-output

that fish are able to utilize the energy that exists in the ed-

feedback linearization and sliding mode control. Time vary-

dies of an oncoming flow by repositioning the vorticities. In

ing LQ controllers can also be time scaled and concatenated. Therefore, they can be computed before hand. The pro-

addition, the fish’es oscillating swimming motion induces

posed strategy has been experimentally validated for longi-

flow relaminization which serves to reduce the body drag

tudinal maneuvers.

[22]. Using an articulated robotic fish, they were able to

Keywords: Aquatic vehicles, oscillating foil, swimming ma-

demonstrate, experimentally, indeed, the power required

chine, optimal control, hydrodynamics, motion planning,

to propel an actively swimming fish-like body is lower than

motion primitives, Linear Quadratic Control.

the power required to tow a rigid fish-shaped object [3]. Inspired by the swimming motion of an aquatic animal,

I. Introduction

oscillating foil has therefore been proposed as an alternative

Study of fish locomotion can be traced back as early as

propulsor to the conventional screw propeller for aquatic

the fourth century B.C. by Aristotle [8]. Nearly 70 years

vehicles [23], [14], [4]. These mimic Carangiform swimming in which only the last third of the body participates in the

Submitted to the International Journal of Robotics Research, 2002. Portion of the paper was presented at the 2001 American Control Conference. This research is partially supported by the University of Minnesota Graduate School Grant-in-aid program. Please send all correspondence to Professor Perry Li.

undulatory motion. In addition to being potentially more efficient [2], an oscillating foil may also be more stealthy. Also, as thrust can be generated in all directions, vehicles propelled this way may also be more maneuverable. Harper

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(Fig. 1). The challenges lie in the need to consider the hydrodynamic interaction with the structure, and the the under-actuated and non-minimum nature of the AV system. In our approach, a 2-D potential flow model is assumed [21]. Free vortices and central vortices are initially ignored and treated as disturbances. Because the resulting system model is under-actuated and non-minimum phase (with motion of the AV as output), conventional nonlinear control approaches cannot be applied directly. Our strategy is to decompose the control task into two stages: motion planning and tracking control. Fig. 1. The the University of Minnesota Aquatic Vehicle Setup.

The motion planning problem is solved off-line via optimal control method. While it is theoretically possible to

et al. [7] proposed the design of an optimal spring constant

use optimal control to determine the motion plan for every

to actuate the oscillating foil. Kelly et al. [12], [11] derived

single maneuver, this is impractical because solving optimal

the equations of motion of a submerged foil in standard

control problem is computationally expensive and cannot

control-affine form via the method of reduced Lagrangian in

be done online. To reduce the complexity of the prob-

a geometric framework. The experimental setup was later

lem, maneuvers will be constructed by concatenating the

modified by Mason and Burdick [15], [16]. Lateral and ro-

time scaled copies of a finite set of basic maneuvers, called

tational degrees of freedom and a rectangular plate to sim-

motion primitives. By allowing the motion primitives to

ulated a submerged body are added. They also experimen-

be time scaled, time shifted, and concatenated, it will be

tally verified a mathematical model based on quasi-steady

possible to accomplish maneuvers at different speeds and

uniform flow. Morgansen et al. [19] proposed geometric

at different ranges. Thus, the need to calculate the mo-

methods to compute the gait for forward and turning mo-

tion primitives for each basic maneuver at every speed is

tions. Yamaguchi and Bose compared the propulsion per-

avoided.

formances of a tanker propelled by an oscillating foil and

To track the planned motion, a cascade control which

by a conventional screw propeller [24]. They found that

consists of finite-time linear quadratic regulator (LQR),

the flexible oscillating foil has a higher quasi-propulsive ef-

input-output linearization and sliding mode control is pro-

ficiency than a conventional screw propeller, but a con-

posed. LQR determines the required heaving and pitching

ventional screw propeller has a higher quasi-propulsive ef-

acceleration signals to achieve tracking, while feedback lin-

ficiency than a rigid oscillating foil. Kato considered the

earization determines the actual motor inputs to achieve

control of pectoral fin like mechanism as a propulsor [10] to

those accelerations. Sliding mode control enhances the ro-

achieve rendezvous and docking with an underwater post

bustness to model uncertainty and disturbances. In this

in water currents. Beside Carangiform swimming robots,

paper, the above motion planning and control strategy is

there are also research that focus on Anguilliform swim-

implemented and tested for the longitudinal motion of an

ming (in which the whole body participates in undulatory

experimental AV system (Fig. 1).

motion) such as eel-like robots [17].

LQR technique is used as the baseline controller because

In this paper we consider the maneuvering control of

of its ease of computation and its ability to handle time

an aquatic vehicle (AV) propelled by an oscillating foil

variation that results from Jacobian linearization about a

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85cm Floating mechanism Platform

α

R 7cm









 









 





 15cm 





 







 



Rotating hub 





 Peduncle 




 
  
 
 (Heaving  
 (unactuated) link) 









 
 
 
 
 
  
 6cm 
 
 
 
  Joukowski foil 

 
 
 
 

10cm

Pitching motor Heaving motor (center)

a MicroMo 3863 series D.C. motor (max torque 177oz-in)

Pitching pulley/belts Heave angle

φ

φ

pitch angle

with a 43:1 gearhead. The pitching (θ) motion is actuated by a MicroMo 3557 series D.C. motor (max torque 74.2ozin) with a 25:1 gearhead via two 1:1 belt/pulley sets. The mechanism and pulley sets are designed around a floating mechanism such that the pitching and heaving motions can be independently driven by their respective D.C. motors so that the rotation of the each motor corresponds exactly to the heaving and pitching angles. The heaving and pitching angles are monitored by optical encoders with resolutions

Fig. 2. Top view of the swimming machine (aquatic vehicle) at the University of Minnesota. Only the 30cm spanned Joukowski foil

of 512 × 43/(2π) lines / rad, and 512 × 25/(2π) lines / rad

is submerged in water. The system has 4 DOFs with only the

respectively.

heaving and pitching motions of the Joukowski foil being directly actuated. The floating mechanism enables the pitching pulley to move independently of the heaving motor.

The platform is above water and can slide along a pair of rails on a set of linear bearings. However, for the purpose of this paper, it is locked in place. The rails are connected

trajectory. Other linear control techniques cannot usually

to a rotating hub at the center of a 1.75m diameter water

handle possibly large time variation easily. Generic nonlin-

tank. The rotation of the platform simulates the longitu-

ear control schemes for state tracking like input-state lin-

dinal motion (α) of the vehicle, and the sliding radial mo-

earization would require finding solutions for the Frobenius

tion on the rail (if unlocked) simulates the lateral motion

partial differential equations, which is a relatively difficult

(R). The longitudinal and radial motions are monitored

task.

by rotary and linear optical encoders with resolutions of

The rest of the paper is organized as follow. Section II describes the experimental AV system. Section III presents the dynamic model of the aquatic vehicle and the hy-

6144/(2π) lines/rad and 1024 lines/in respectively. Notice that lateral and longitudinal motions of the platform are not directly actuated.

drodynamic interaction. In section IV, the motion plan-

The heaving and pitching motors are under torque (cur-

ning strategy for longitudinal maneuvers and examples of

rent) control mode so that as far as the control design is

planned motions are given. Section V presents the track-

concerned, torque input is the control input. Controller im-

ing control strategy to ensure that the planned motion is

plementation is carried out using Matlab Realtime Toobox

tracked. Section VI presents some experimental results.

(Humusoft, CZ). The allows us to conveniently code our

Section VII contains some concluding remarks.

controller as Matalb M-files at the expense of a relatively long sampling time (8ms) which translates ultimately to

II. Experimental Setup

limitation in control performance.

Figures 1 and 2 are a picture and a schematic of our

Our experimental setup is inspired by the Caltech de-

experimental AV setup. It consists of a two link actuated

sign [12], [11] except that it is much smaller. The smaller

mechanism mounted on a platform. The second link is the

scale accentuates un-modeled effects such as uneven gravi-

propulsor which is a submerged symmetric Joukowski foil

tational effects, friction in the longitudinal and lateral mo-

[18] which has a cord length of 0.06m, a maximum thick-

tions, waves and finiteness of the water tank. Our control

ness of 0.008m, and a vertical span of 0.3m. The heaving

system must therefore be robust enough to mitigate these

(φ) motion of the Joukowski foil is directly actuated by

effects.

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III. System Model

which, using (2) can be written as

We assume that the fluid flow is two-dimensional, inviscid, incompressible, and irrotational and also that the fluid is at rest at infinity. Under these assumptions, the flow field and the hydrodynamic forces and torque that act on the foil are determined using potential flow theory [18], [21]. The presence and effects of free and central vortices are ignored for the purpose of motion planning and controller design, and are treated as disturbances. This simplifies the motion planning and control significantly. The estimation and the purposeful incorporation of the vortices in the control design is an ongoing research [13].

U

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    V      Ω dω  = Q1 (ζ, ζ1 (t), . . . , ζnk (t))  . dz  γ1     ..   .    γ nk

for some Q1 (ζ, ζ1 (t), . . . , ζnk (t)) ∈