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Tracking Environmental Isoclines using PolygonalFormations of Submersible Autonomous Vehicles

Shahab Kalantar, Uwe Zimmer

To cite this version:Shahab Kalantar, Uwe Zimmer. Tracking Environmental Isoclines using Polygonal Formations ofSubmersible Autonomous Vehicles. 6th International Conference on Field and Service Robotics - FSR2007, Jul 2007, Chamonix, France. �inria-00258741�

Tracking Environmental Isoclines using Polygonal Formations of Submersible Autonomous VehiclesShahab Kalantar and Uwe Zimmer

Australian National University,Department of Information Engineering, Canberra, ACT 0200, Australiashahab.kalantar@rsise.anu.edu.au

Summary: Knowledge of iso-contours of the underwater terrain can be usedto reconstruct it using interpolation. Identifying a set of isoclines can be moreefficient and less time-intensive than sweeping a large area. In this paper, wepropose a system where a small number of agile underwater vehicles coopera-tively maintain a polygonal formation on a plane above the terrain and use fieldvalues measured by the individual robots to locally reconstruct the field usinginterpolation schemes. The formation then tracks a desired iso-contour of thefield by tracking the corresponding curve on the reconstructed field.

1 IntroductionTraditionally, to create abathymetric map of theunderwater terrain em-ploying AUV’s, a singlebulky vehicle (moving ata certain depth) is towedto follow a path com-posed of a set of paralleltransects (covering a largearea) and record altitudevalues. The recorded in-formation is then used toapproximate the field inthat region (see, e.g., 3.). An alternative approach is that the vehicle autono-mously guide itself along lines of constant altitude. A set of these recorded iso-contours can then be used to reconstruct the field using Hermite interpolation14.. An obvious benefit of such an approach is that only regions with high var-iability need to be covered. Given the local gradient and Hessian, this is an easytask to do. One only needs to climb the gradient path and follow the direction

Fig. 1. Serafina underwater vehicle.

orthogonal to field gradient once at the desired isocline. In reality, though,these information are not readily available (have to be estimated) and are some-times even meaningless. In 5., a single vehicle uses a history of past field val-ues. In 6., UUV-gas theory is used where we only need to determine whetherwe are inside or outside the region surrounded by the iso-contour. A similar ap-proach is described in 7.. In 4., the trajectory of a vehicle is modeled as adamped sinosoid whose parameters are adjusted to stabilize the vehicle on thecontour. Rather than using a single vehicle, we can employ a formation of com-municating vehicles. This way, local characteristics of the field can be comput-ed more easily 9.. In 8., a four-vehicle system is described which estimates thegradient and the curvature to track a contour and adjust its size to minimize es-timation error.In this paper, we propose a polygonal for-mation and interpolation for local recon-struction based on only static instantane-ous measurements of the field at thepositions of the robots. The interesting as-pect of this method is that the representa-tion of the desired isocline in the recon-structed field can be used to convert theproblem of tracking tangent to the true iso-cline to that of tracking a well-definedpath. This is because, at each particulartime instant, a reconstructed picture of thefield is made available which contains a smooth curve representing the isoclne.Accordingly, robust methods for tracking such curves can be employed.As models for our physical platforms, we use Serafina 2., a small fast fivethruster vehicle equipped with a suite of sensors (figure 1). Ongoing research isaiming at low-bandwidth short-range long-wave radio communication 12., op-tical communication 10., scheduling strategies 11., and acoustic range andbearing measurement 13.. In this paper, we will not be concerned with these is-sues. Furthermore, we will consider a very simple motion model for the vehi-cles, i.e., that of a non-holonomic unicycle moving on the plane:

, , , (1)where and are, respectively, desired linear and angular velocity in-puts of the yaw-surge controller. Additionally, a heave controller keeps the ve-hicle at the desired depth and the roll and pitch controllers maintain these an-gles at zero. We also assume that the ambient flow is negligible or iscounteracted by the controller.

Fig. 2. Polygonal formation.

x· t( ) v t( ) θ t( )cos= y· t( ) v t( ) θ t( )sin= θ· t( ) ω t( )=v t( ) ω t( )

2 Initializing and maintaining polygonal formationsIn this section, we explain how a polygonal formation (figure 2) of co-planarvehicles can be created and maintained. Given a desired radius , a deforma-tion of the corresponding regular polygonal formation , composed of ve-hicles , , with planar positions, denoted by

, and collective state, can be defined by the formation function 15.

(2)

where , ,

, , (3)

, . (4) is the centre of mass and is a vector of size with all the entries equal

to . The operations on the indexes are modulus , so that . Any rootof gives a polygonal formation which is unique up to transla-tion and rotation. More technically, is a singleton.The position of the centre of mass breaks the translational ( ) symme-try and the angle (which can be defined as the orientation of the forma-tion) breaks the rotational ( ) symmetry. Letting denote an -ballaround , an -neighborhood of a formation is defined by .A number of control strategies have been proposed in the literature for creatinga polygonal formation from an arbitrary aggregate of robots. In 17., cyclic pur-suit strategy is proposed for obtaining a regular polygon at equillibrium. In thisscheme, each robot pursues with appropriate control inputs amd

. Once the vehicles have converged to a sufficiently small neighborhood ofa polygonal formation, they enter a formation keeping phase. The control rulesfor this phase have to make sure that the formation does not deviate from anideal polygon too much, or, in other words, it stays within . Wewill use the concept of virtual leaders proposed in 15.. We design the desiredposition for the non-holonomic vehicle as where de-notes the trajectory of , parametrized by , and

, (5)

. (6)

rP

P NRi i 0 … N, , 1–=

qi t( ) xi t( ) yi t( ),( )T= ℜ2∈qP t( ) q0 t( ) … qN 1– t( ), ,[ ]T=

GP qP t( )( ) x̃ t( ) rP1N–( )T x̃ t( ) rP1N–( )

ϑ t( ) 2πN------1N–⎝ ⎠

⎛ ⎞T

ϑ t( ) 2πN------1N–⎝ ⎠

⎛ ⎞

+=

xi t( ) qi t( ) q t( )–= x̃i t( ) xi t( )=

ϑi t( ) cos 1– n xi t( )( ) n xi 1+ t( )( )⋅( )= n v( ) v v⁄=

x̃ t( ) x̃0 t( ) … x̃N 1– t( ), ,[ ]T= ϑ t( ) ϑ0 t( ) … ϑN 1– t( ), ,[ ]T=q t( ) 1N N

1 N qN q0=GP q t( )( ) 0=

GP1– 0( ) SO 2( ) ℜ2⊗( )⁄

q t( ) ℜ2

θ0 t( )SO 2( ) Bε u( ) ε

u ℜN∈ ε Bε GP1– 0( )( )

Ri Ri 1+ vi

ωi

Bε GP1– 0( )( )

Ri qidt( ) pi si( )= pi si( )

qidt( ) si

q· id t( )si∂∂ pi si( )s· i=

si∂∂ pi si( ) GP qP t( )( )qi t( )∇– β qd t( ) q t( )–( )+=

is the tangent to the trajectory. is the speed of traversal of the tra-jectory. can be regarded as the virtual leader of . is the desiredposition of the centre of mass and should be designed according to application.It plays the role of the virtual leader for the whole formation. Note that this con-trol rule effectively decouples formation keeping and maneuvering which isvery desirable. Control rules for non-holonomic vehicles (15.) can be used fortracking. Initially, . The trajectory for can be representedas the parametrized curve such that

. (7)

will be designed in later sections. The integrity of the formation ismaintained by regulating the speed of traversal of the paths , i.e. . In15., expressions for and are given.

3 Field interpolation by polygonal formationsInterpolation on polygons can be done by barycentric coordinates (18.,19.). Letthe domain denote the inside of the (assumed convex) polygonalformation, with boundary . Let be a given point. Any vector of realnumbers is called the generalized barycentriccoordinates of if (1) the coordinates have linear precision, i.e., they can re-produce a linear function exactly, , (2) the coordinates arenon-negative and bounded to guarantee no under- or over-shooting in the coor-dinates, , (3) the coordinates form a partition of unity to assure con-stant precision and to make the formulation invariant to both translation and ro-tation, , (4) the coordinates are infinitely differentiable with respectto their arguments to ensure smoothness when a node is moved, ,

pi si( )∂ si∂⁄ s· i

qidt( ) Ri qd t( )

qd t0( ) q t0( )= qd t( )p0 s0( )

q·

d t( )s0∂∂ p0 s0( )s·0=

p0 s0( )∂ s0∂⁄pi si( ) s· i

s· i s·0

Fig. 3. (a) Interpolation on a convex polygon. (b) Construction of the interpolated isocline.

(a) (b)

ΩP t( ) ℜ2⊂Ω∂ p ΩP∈

α p qP,( ) α0 α1 … αN 1–, , ,[ ]T=p

p α p qP,( ) qP⋅=

0 αi 1≤ ≤

Σiαi 1=qi αi C∞∈

(5) we have , where is the Kronecker delta function. Con-ditions 1 and 2 are called affine combination. Positivity of the coordinates (con-dition 2) is called convex combination. The functions : ,

, are also called shape functions associated with .Let : represent a field defined on the plane (altitude measurements).We assume that the vehicles can measure the field at their respective positions,giving the values . An interpolant (or interpolation scheme) of

, based on polygon , is a function : defined by, where . is the in-

terpolated value of . generates a vector of barycentric coordinates andtakes the inner product of this vector and the vector composed of field values atthe vertices. The coordinates generated by should of course satisfy the aboveconditions. Condition 5 above makes sure that the interpolated value at a vertexis equal to node data: . Conditions 2 and 3 ensure that the in-terpolated values are bounded between the minimum and maximum of the nod-al values: . Along the edges of the polygon, theinterpolant must be piece-wise linear (i.e., ). This can be stated as

, , (8)where and . If is a vector of real numbers such that , then partitionof unity coordinates can be found by the formula

. (9)‘s are called (non-normalized) weight functions. There are a plethora

of methods for defining shape functions. We will use a method based on Wach-spress construction which can be used for irregular but convex polygons (thusproviding for some robustness to deviations from a perfect polygon). In 20., asimple local formula is given for when is strictly inside the polygon:

. (10)

Figure 3(a) shows the various terms used in these formulas. If is very close tothe boundary, a simple linear interpolation should be used. To reconstruct a par-ticular isoline of the interpolated field, we need to compute the gradient of

:

, (11)

To track a desired isoline value of the underlying field, we construct the cor-responding isoline of the interpolated field and track it instead. This interpolat-

αi qj qP,( ) δij= δij

αi ℜ2 ℜ2N⊗ ℜ+→i 0 … N 1–, ,= C0 p

f ℜ2 ℜ→

f qi t( )( ) fi t( )=f P f̃ Ω ℜ→f̃ p( ) αp p qP,( )TfP t( )= fP t( ) f0 t( ) … fN 1– t( ), ,[ ]T= f̃ p( )

f p( ) f̃

f̃

f̃ qi t( )( ) fi t( )=

mini fi{ } f̃ p( ) maxi fi{ }≤ ≤C0

f̃ τ( ) τfi 1 τ–( )fi 1++= q τqi 1 τ–( )qi 1++=q Ω∂∈ τ 0 1,[ ]∈ w p qP,( ) w0 p qP,( ) … wN 1– p qP,( ), ,[ ]T=

w p qP,( ) qP p1N–( )⋅ 0=

αi p qP,( ) wi p qP,( ) Σkwk p qP,( )( )⁄=wi p qP,( )

p

wi p qP,( )A qi 1– qi qi 1+, ,( )

A qi 1– qi p, ,( )A qi qi 1+ p, ,( )----------------------------------------------------------------- δi t( )( )cot λi t( )( )cot+

p qi t( )– 2-------------------------------------------------------= =

p

f̃ p( )

f̃ p( )p∇ p∂∂ f̃ p( )

p∂∂ αi p qP,( )fi t( )

i 0=

N 1–

∑= =

cd

ed isoline is our observation of the real iso-contour. Tracking this curve andmaking sure that it is accurate enough (i.e., close enough to the real one) are de-coupled in our method. The desired curve is the solution to the equation

, where . We are assuming that this solution is a unique curve.To reconstruct the curve , we first find its end-points ( and

), which lie on edges. Next, we traverse the curve sampling it along theway. We can also define a safety circle sufficientlt away from the edges. Seefigure 3(b).

4 Isocline tracking controlHere, we present three simple tracking strategies.1. The simplest way to track an isoline is to move the virtual leader (the desiredgoal for the centre of mass) towards the interpolated isoline and, at the sametime, move along it. This behaviour is captured in the control law (figure 4(a))

(12)

where if exists and is zero otherwise, is a gain, denotes thevalue of the parametrization of for which is the closest point to

. This closest point is also denoted . Also is normal to the curve at , and is the

local tangent at . The gain determines how fast the formation should re-act to normal variations in the tracked position and determines thespeed of traversal.2. To reduce chattering when close to the isocline and dicourage moving alongthe tangent when far from it, we can make and functions of the normal

f̃ p( ) cd= p Ω∈γ s t,( ) γ s0 t,( )

γ sf t,( )

Fig. 4. (a) Tracking the reconstructed isocline. (b) Lateral and longitudinal control of the forma-tion centre.

(a) (b)

s0∂∂ p0 s0( ) aγ β1Nγ τC t,( ) β2Tγ τC t,( )+( )

1 aγ–( )β3 cd f̃ q t( )( )–( ) f̃ q t( )( )q t( )∇

+=

aγ 1= γ β3 τC

γ γ t τC,( )q t( ) Qγ t( ) q t( )( )Nγ τC t,( ) n Qγ t( ) q t( )( )( )= τC Tγ τC t,( )

τC β1

γ t τC,( ) β2

β1 β1

distance. Let us define , ,, where is the sigmoidal function defined in 1.

to implement a smooth switching mechanism between the two tangential andnormal behaviours . and denote the nominal normal and tangentialspeeds. For particular values chosen for and small , if we set

, then we will have ( ) and ( ) when ( ).

3. An alternative way of moving towards the reconstructed isocline is to modelthe motion of the centre of the formation as a non-holonomic device. To do this,we can associate a heading vector with the centre. In this case, instead of deter-mining normal and tangential speeds, we should design lateral and longitudi-nal speeds. Also, instead of moving towards the closest point and sliding alongthe tangent to the curve, we should now designate a lookahead point on thecurve which would serve as the desired goal for the centre of mass of the forma-tion. Using a lookahead makes tracking more robust to fluctuations of the path(whose observation changes over time). figure 4(b) illustrates this. Denoting by

this lookahead point, the virtual centre of mass is moved according to thefollowing control laws (16.) for longitudinal and lateral movements:

(13)

and , where is a nominal speed, is a desired dis-tance between the centre of mass the lookahead and is a positive gain. de-notes the formation feedback. Note that with this control scheme, longitudinaland lateral responsive-ness play the same role as tangential and normal respon-sive-ness in the previous scheme. If they are designed carefully, it is possible toproduce smooth motion even against very rugged terrain or the presence oflarge noises.

5 SimulationsFigure 5(a) shows a simulation run for a formation with vehicles and

. We have set and . Also, and. The simulations are synchronous and the vehicles carry on identical

motions. It is furthermore assumed that the designed linear and angular veloci-ties are within the tolerance of the vehicles and there is no error in measure-ment, although there is error in interpolation because of the size of the forma-tion. As is seen, the produced path for the robots is quite smooth. The reason isthat the formation is always moving forward on the line on which the tangent tothe curve lies and the spring for normal motion is not very stiff. Figure 5(b)

β1 vNσε β, Q t( )( )= β2 vT 1 σε β, Q t( )( )–( )=Q t( ) Qγ t( ) q t( )( )= σp q, u( )

vN vT

ε 0> δ 0>β δ 1 δ–( )⁄( )ln 4ε̃⁄= β1 vN 1 δ–( )≥ β1 vNδ≤β2 vTδ≤ β2 vT 1 δ–( )≥ Q t( ) ε ε̃+≥ Q t( ) ε ε̃–≤

L

vC t( ) aγvn

ρT----- q t( ) L γ q t( ),( )– Δϕ( )s·0cos

1 aγ–( )β3 cd f̃ q t( )( )–( ) f̃ q t( )( )q t( )∇

+=

ωC t( ) aγ kΔϕ ϕ· d+( )= vn ρT

k s·0

N 5=rP 100= β1 40= β2 400= kv 5.5=kω 50=

shows the same formation in a very rugged terrain. The maximum field value is6000. Underlying smooth field values have been contaminated with a uniformnoise in the range . It is seen that the behaviour has not changed andthe paths are still smooth. Figure 5(c) shows the paths for a formation on asmooth terrain with . Figure 6(a) shows a simulation run of a forma-tion employing adaptive gain strategy. The initial position is marked by a blackdisk and the arrows show parts of the path where motion is mainly translation,along the normal, towards the curve. In this simulation, , ,

, and . Figure 6(b) shows a simulation run for aformation employing the third control strategy.

6 Conclusions and future researchIn this paper, we presented a gradient-free method for tracking iso-contours us-ing a robotic network. The basic idea was that of transforming the problem offollowing the gradient (and its conjugate) into that of tracking a curve which isan isocline of a locally reconstructed distribution. The particular polygonalshape of the formation allows the use of interpolation schemes. With this robustmethod, smooth paths can be produced even when the underlying field is veryrugged or measurements and/or actuation are noisy. There are several issueswhich merit further consideration. First of all, to minimize interpolation error,

Fig. 5. (a) Small normal forcing. (b) Rug-ged terrain. (c) High normal gain.

(a) (b)

(c)

60– 60,[ ]

β1 200=

ε̃ 5= ε 10=δ 0.00001= vN vT 200= =

the size of the formation has to be adaptively changed. Relevant initial resultswill be presented elsewhere. Secondly, the interpolation method used allowsfor some deviation from the rigid shape as long as it remains convex. We will,in the future, consider methods which can handle non-convex polygons as well.Future research should also aim at explicitly dealing with noise in various sen-sor/actuator modalities. Finally, autonomous selective sampling of isoclinesshould be addressed.

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Fig. 6. (a) Smooth switching between normal and tangential motions. (b) Non-holonomically modelled formation centre, tracking a lookahead on the reconstructed curve.

(a) (b)

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