Coordinate-free Isoline Tracking in Unknown 2-D Scalar Fields

Fei Dong, Keyou You, Senior Member, IEEE, Jian Wang

Abstract— The isoline tracking of this work is concernedwith the control design for a sensing robot to track a givenisoline of an unknown 2-D scalar filed. To this end, we proposea coordinate-free controller with a simple PI-like form usingonly the concentration feedback for a Dubins robot, which isparticularly useful in GPS-denied environments. The key idealies in the novel design of a sliding surface based error term inthe standard PI controller. Interestingly, we also prove that thetracking error can be reduced by increasing the proportiongain, and be eliminated for circular fields with a non-zerointegral gain. The effectiveness of our controller is validatedvia simulations by using a fixed-wing UAV on the real datasetof the concentration distribution of PM2.5 in an area of China.

I. INTRODUCTION

The isoline tracking refers to the tactic that a mobile robotreaches and then tracks a predefined contour in a scalar field,which is widely applied in the areas of detection, exploration,monitoring, and etc. In the literature, it is also named as curvetracking [1], boundary tracking [2], [3], level set tracking [4].In fact, it covers the celebrated target circumnavigation as aspecial case [5]–[7].

Compared with the static sensor networks, it is moreflexible and economical to utilize mobile sensors to collectdata or track target. The methods for isoline tracking byrobots have been applied to many practical problems, e.g.,exploring environmental feature of bathymetric depth [3],tracking boundary of volcanic ash [8], tracking curve of seatemperature [9], and monitoring algal bloom [10].

Roughly speaking, we can categorize the methods forisoline tracking depending on whether the gradient of thescalar field can be used or not. The gradient-based methodis extensively used to the extreme seeking problem, whichsteers a robot to track the direction of gradient descending(ascending) to reach the minimizer (maximizer) of a scalarfield [9], [11].

If the explicit gradient is not available, many worksfocus on the problem of gradient estimation, which mainlyinclude two main strategies: (i) a single robot changesits position over time to collect the signal propagation atdifferent locations; and (ii) multiple robots collaborate toobtain measurements at different locations at the same time.

*This work was supported in part by the National Natural ScienceFoundation of China under Grant 61722308. (Corresponding author: KeyouYou)

F. Dong and K. You are with the Department of Automation, andBeijing National Research Center for Info. Sci. & Tech. (BNRist), TsinghuaUniversity, Beijing 100084, China. E-mail: dongf17@mails.tsinghua.edu.cn,youky@tsinghua.edu.cn.

J. Wang is with the Department of Electronic Engineering, TsinghuaUniversity, Beijing 100084, China. E-mail: jian-wang@tsinghua.edu.cn.

For the case (i), Ai et al. [12] show a sequential least-squaresfield estimation algorithm for a REMUS AUV to seek thesource of a hydrothermal plume. Moreover, the stochasticmethod for extreme seeking is also gradient-based, the ideabehind which is to approximate the gradient of the signalstrength and to use this information to drive the robot towardsthe source by adding an excitatory input to the robot steeringcontrol [13], [14]. For the case (ii), a circular formation ofrobots is adopted in [15], [16] to estimate the gradient offields. Moreover, a provably convergent cooperative Kalmanfilter and a cooperative H∞ filter are devised to estimate thegradient in [9] and [11], respectively.

In many scenarios, robots cannot obtain its position andcan only measure the signal strength at the current location ofthe sensor, i.e., the measurement in a point-wise fashion [4].Thus, it is impossible to estimate the field gradient, and re-searchers turn to exploiting gradient-free methods. A slidingmode approach is proposed for target circumnavigation by[17] and then is adopted to similar problems, e.g., level setstracking [4], boundary tracking [18], and etc. Without a rigor-ous justification, they address the “chattering” phenomenonby modeling dynamics of the actuator as a simplest form ofthe first-order linear differential equation in implementation.A PD controller is devised in [19] for a double-integratorrobot to track isolines in a harmonic potential field. Besides,a PID controller with adaptive crossing angle correction isshown in [20]. Furthermore, there are some heuristic methodsfor isoline tracking, e.g., sub-optimal sliding mode algorithmof [21].

In this paper, we propose a coordinate-free controller ina PI-like from for a Dubins robot to track a desired isolineby using only the concentration feedback. That is, we donot use any field gradient or the position of the robot, whichrenders our controller particularly useful in the cases that thefield gradient is hard to be estimated or the GPS positionof the robot is unaccessible. Our key idea lies in the noveldesign of a sliding surface based error term in the standardPI controller. Similar to the standard PI controller, we showthat the final tracking error can be reduced by increasing theproportion gain, and be eliminated for circular fields witha non-zero integral gain. For the case of smoothing scalarfields, we explicitly show the upper bound of the steady-state tracking error, which can be reduced by increasingthe proportion gain. To validate the effectiveness of ourcontroller, we adopt a fixed-wing UAV to track the isoline ofthe concentration distribution of PM2.5 in an area of China.An extended version of this work is presented in [22], whichfurther investigates the cases of a single-integrator robot anda double-integrator robot.

2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)October 25-29, 2020, Las Vegas, NV, USA (Virtual)

978-1-7281-6211-9/20/$31.00 ©2020 IEEE 2496

0 2 4 6 8 10X-position (Km)

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)An isoline

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(a)

𝜙𝜙

Source

s = 𝑠𝑠𝑑𝑑

𝑑𝑑𝜃𝜃

𝑋𝑋

𝑌𝑌

𝑂𝑂

𝜑𝜑𝑠𝑠

s > 𝑠𝑠𝑑𝑑

s < 𝑠𝑠𝑑𝑑

Robot

ℎ

−𝑛𝑛

𝑛𝑛

𝜏𝜏

(b)

𝜙𝜙

s = 𝑠𝑠𝑑𝑑

𝜃𝜃

𝑋𝑋

𝑌𝑌

𝑂𝑂

𝜑𝜑𝜏𝜏

s > 𝑠𝑠𝑑𝑑 s < 𝑠𝑠𝑑𝑑

Robot

𝑠𝑠

𝑛𝑛

−𝑛𝑛

ℎ

(c)Fig. 1. (a) PM2.5 concentration observed in in an area of China. (b) Coordinates of the Dubins robot in circular fields. (c) Coordinates of the Dubinsrobot in scalar fields.

The rest of this paper is organized as follows. In SectionII, the problem under consideration is formulated in details.Particularly, we clearly describe the desired isoline trackingpattern. To achieve the objective, we propose a PI-likecontroller for a Dubins robot in Section III. In Section IV weshow that the isoline tracking system is locally exponentiallystable. Moreover, we explicitly show the upper bound ofthe steady-state error in scalar fields in V. Simulations areperformed in Section VI, and some concluding remarks aredrawn in Section VII.

II. PROBLEM FORMULATION

In Fig. 1(a), we provide a 2-D example of the concen-tration distribution of PM2.5 in an area of China. In theenvironmental monitoring, it is fundamentally important toinvestigate the concentration distribution of air pollutants. Toachieve it, we design a sensing robot to track an isoline of itsdistribution function. Mathematically, the concentration of a2-D scalar field can be described by

F (p) : R2 → R, (1)

where p ∈ R2 is the position. Given a concentration levelsd, an isoline L(sd) is defined as

L(sd) = {p|F (p) = sd}. (2)

The isoline tracking problem is on the design of a con-troller for a sensing robot to reach a given isoline andmaintain on the isoline with a constant speed. That is, theobjective is to asymptotically steer a sensing robot such that

limt→∞

|s(t)− sd| → 0 & ‖p(t)‖ = v, (3)

where s(t) = F (p(t)) is the concentration measurement ofthe scalar field at the GPS position p(t) of the robot and vis its constant linear speed. For a circular field, e.g., acousticfield, then

F (p) = I0 exp(−ς‖p− po‖2), (4)

where po is the source position of the field and I0, ς areunknown parameters. The isoline tracking in (3) is exactlyreduced to the celebrated circumnavigation problem [5]–[7].

In this work, we are interested in the scenario that boththe concentration distribution F (p) and the GPS positionof the sensing robot are unavailable. Moreover, we cannotmeasure a continuum of the scalar field, which implies thatthe gradient-based methods [1], [9], [16] cannot be appliedhere.

III. CONTROLLER DESIGN

In this section, we design a coordinate-free controller ina PI (proportional integral)-like form for a Dubins robot tocomplete the isoline tracking problem. The key idea lies inthe novel design of a sliding surface based error term in thestandard PI controller.

A. The PI-like controller for a Dubins Robot

Consider a Dubins robot on a 2-D plane

p(t) = v

[cos θ(t)sin θ(t)

], θ(t) = ω(t), (5)

where p(t) = [x(t), y(t)]′, θ(t), ω(t) and v are the position,heading course, tunable angular speed and constant linearspeed, respectively.

To achieve the objective in (3) by the Dubins robot (5),we propose a novel PI-like controller

ω(t) = kpe(t) + kiσ(t), (6)

where σ(t) = e(t), kp > 0 and ki ≥ 0 are the controlparameters to be designed.

Let the tracking error be ε(t) = s(t) − sd. The majordifference of (6) from the standard PI controller lies in thenovel design of the following error term

e(t) = ε(t) + c1 tanh (ε(t)/c2) , (7)

where c1,2 > 0 are constant parameters, and tanh(·) isthe standard hyperbolic tangent function to ensure that theselection of the control parameters is independent of themaximum range of the operating space of the controller. Infact, the error term e(t) in (7) can also be regarded as asliding surface. For example, once reaching the surface, i.e.,e(t) = 0, it follows that ε(t) = −c1 tanh (ε(t)/c2) , which

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further implies that ε(t) will tend to zero with an exponentialconvergence speed, i.e., the robot will eventually reach theisoline L(sd).

Intuitively, the PI-like controller (6) consists of two terms:(i) the proportional term for global stability, and (ii) theintegral term to eliminate the steady-state error. Similar to thestandard PI controller, the integral coefficient ki is generallymuch smaller than the proportional coefficient kp. It is worthmentioning that c1 affects the convergence speed and c2affects the sensitivity to the tracking error ε(t).

Clearly, the PI-like controller (6) of this work only usesthe concentration measurement s(t) of the scalar field, andis particularly useful in GPS-denied environments.

B. Comparison with the existing methods

Some related methods to our proposed control laws are(i) the sliding mode controller in [4], (ii) the PD controllerin [19], (iii) the sliding mode controller with two-slidingmotions in [3], and (iv) our backstepping controller in [23].The sliding mode approach in [4] is originally designed forthe problem of target circumnavigation [17] with range-basedmeasurements, and then is adopted to isoline tracking in[4]. Besides the existence of the chattering phenomenon,their method cannot achieve zero steady-state error evenfor the task of circumnavigation. In contrast, our PI-likecontroller (6) is continuous and particularly useful to isolinetracking in circular fields, since the integral part can exactlyeliminate the steady-state error. Moreover, the PD feedbackcontroller in [19] is devised for a double-integrator robot,and their control parameters depend on maximum rangeof the controller operating space. We address this issueby introducing a hyperbolic tangent function tanh(·). Thecontroller in [3] needs two-sliding motions. They validatetheir controller by both simulations in a synthetic data-based environment and sea-trials by a C-Enduro ASV inArdmucknish Bay off Dunstaffnage in Scotland. However,their method is heuristic and in fact only offers uncompletedjustification. Furthermore, the backstepping controller in [23]is to steer the robot to follow a smooth reference command.

IV. ISOLINE TRACKING IN CIRCULAR FIELDS

In this section, we first consider the case of a circular fieldin (4). Taking logarithmic function on both sides of (4), thereis no loss of generality to write it in the following form

F (p) = sd − α(d(t)− rd), (8)

where sd is the desired concentration, α ≥ α is an unknownpositive constant, d(t) = ‖p(t) − po‖2 is the distance fromthe robot to the position po of the source, and rd denotesthe unknown radius when the robot travels on the desiredisoline, i.e., s(t) = sd.

Let n = ∇F (p) denote the gradient vector of F (p), seeFig. 1(b), and h = [cos θ, sin θ]′ represent the course vectorof the Dubins robot and τ represent the tangent vector ofh. By convention, h and τ form a right-handed coordinateframe with h× τ pointing to the reader.

After converting the coordinates of the robot from theCartesian frame into the polar frame, we use the concen-tration s(t) and angle φ(t) to describe the tracking system.See Fig. 1(c) for illustrations, where n exactly points to thesource and φ(t) is formed by the negative gradient vector −nand the heading vector h. The counter-clockwise directionis set to be positive.

By definitions of s(t) and φ(t), we have that

s(t) = −αd(t) = −αv cosφ(t),

φ(t) = ω(t)− v

d(t)sinφ(t).

(9)

If s(t) converges to sd, then d(t) also converges to rd.However, rd is unknown to the sensing robot, which is sub-stantially different from the target circumnavigation problem[5], [6], and we cannot use the control bias ωc = v/rd toeliminate the tracking error as in [7]. To solve it, we designan integral term kiσ(t) in (6).

Proposition 1: Consider the tracking system in (9) underthe PI-like controller in (6). Define x(t) = [s(t), φ(t)]′ andxe = [sd, − π/2]′. If the control parameters are selected tosatisfy that

kp(kp − 2)vα > ki and vα > c1 > 0, (10)

then xe is a locally exponentially stable equilibrium of thetracking system (9).

Proof: By (9), the tracking system under the PI-likecontroller (6) is written as

d(t) = v cosφ(t),

φ(t) = − kp(αd(t) + c1 tanh (α/c2 · (d(t)− rd))

)+ kiσ(t)− v sinφ(t)/d(t),

σ(t) =− αd(t) + c1 tanh (−α/c2 · (d(t)− rd)) .

(11)

Then, we define an error vector

z(t) = [z1(t), z2(t), z3(t)]′

= [d(t)− rd, φ(t) + π/2, σ(t) + v/kird]′,

and linearize (11) around [rd, −π/2, −v/kird]′ as follows

z(t) = Az(t), where A =

0 v 0

−kpc1αc2− v

r2d−kpvα ki

−c1α/c2 −vα 0

.(12)

Consider a Lyapunov function candidate as

V (z) = µ2z21(t) + µ3z

22(t) + µ4z

33(t) (13)

+1

2

(−µ1z

21(t)− c1vαz2(t) + c2z3(t)

)2,

where µ1 = kpc1α/c2 + v/r2d, µ2 = kpα(kpαvµ1 −kic1α/(2c2)), µ3 = µ1v/2 + kiαv/2, and µ4 =kpkic2vµ1/c1 − k2i /2. It is clear that the conditions in (10)ensure that V (z) is nonnegative for all z(t) 6= 0.

Then, we write (13) as the following form

V (z) = z′Pz, (14)

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Q =

kpαvµ

21 −

kic1αµ1

c20 0

0 (kpαv)3 ki(kpαv)

2 − k2i αv/2 +kpkic2(αv)

2µ1

c1α

0 ki(kpαv)2 − k2i αv/2 +

kpkic2(αv)2µ1

c1αkpk

2i αv

(17)

where

P =1

2

2µ2 + µ21 kpαvµ1 −kiµ1

kpαvµ1 2µ3 + (kpαv)2 −kpkiαv

−kiµ1 −kpkiαv 2µ4 + k2i

,which leads to that

λmin(P )‖z‖22 ≤ V (z) ≤ λmax(P )‖z‖22, (15)

where λmin(P ) and λmax(P ) denote the minimum andmaximum eigenvalues of P .

Taking the derivative of V (z) along with (12) leads to that

V (z) = −z′Qz, (16)

where Q is shown in (17) and is positive definite by theconditions in (10).

Then, it follows from (15) and (16) that

V (z) ≤ −λmin(Q)‖z‖22 ≤ −λmin(Q)

λmax(P )V (z). (18)

By the comparison principle [24], the tracking system (9) islocally exponentially stable under the PI-like controller (6).

V. ISOLINE TRACKING IN SCALAR FIELDS

In this section, we consider a scalar field in (1) under theassumption that F (p) is twice differentiable and satisfies

γ1 ≤ ‖∇F (p)‖ ≤ γ2, ‖∇2F (p)‖ ≤ γ3, ∀p ∈ R2 (19)

where γi is a positive constant. Note from (19) that|h′∇2F (p)h| ≤ γ3 for any h = [cos θ, sin θ]′.

To this end, we follow from Fig. 1(c) that

s(t) = vn′h = −v‖∇F (p)‖ cosφ(t). (20)

Then, taking the derivative of s(t) leads to that

s(t) = ω(t)vn′τ + v2h′∇2F (p)h (21)

= ω(t)v‖∇F (p)‖ sinφ(t) + v2h′∇2F (p)h.

Proposition 2: Consider the isoline tracking system in(20) and (21) under the PI-like controller in (6) and (19).If φ(t0) ∈ [−ε,−π + ε] where ε ∈ (0, π/2) and the controlparameters are selected to satisfy that

kp > max

(γ3v

γ1 sin ε (vγ1 cos ε− c1),c2γ3v + c1γ2c1γ1 sin ε

),

and ki = 0, then

lim supt→∞

|s(t)− sd| ≤ tanh−1(c2γ3v + c1γ2kpc1γ1 sin ε

).

The proof depends on the following technical result.

Lemma 1: Consider the following system

z(t) = −k tanh(z(t)) + b. (22)

If k > b > 0, then lim supt→∞ |z(t)| ≤ tanh−1 (b/k) .Proof: Consider a Lyapunov function candidate as

Vz(z) = 1/2 · z2(t).

Taking the derivative of Vz(z) along with (22) leads to that

Vz(z) = z(t) (−k tanh(z(t)) + b)

≤ −kz(t) tanh(z(t)) + b|z(t)|.

By k > b > 0, it holds that Vz(z) ≤ 0 forall |z(t)| ≥ tanh−1 (b/k). Furthermore, it follows thatlim supt→∞ |z(t)| ≤ tanh−1 (b/k) .

Remark 1: Given a specific b in (22), we can reduce theupper bound by increasing the gain k. Similarly, Proposition2 implies that increasing kp can reduce the upper bound ofthe steady-state tracking error.

Proof: [of Proposition 2] Firstly, we show that φ(t) cannot escape from the region [−ε,−π + ε]. Substituting thePI-like controller (6) into (21) yields that

s(t) = kpvn′τ (ε(t) + c1 tanh (ε(t)/c2)) + v2h′∇2F (p)h.

(23)

Since s(t) and φ(t) are continuous with respect to time t by(20) and (23), we only need to verify the sign of s(t) whenφ(t) = −ε and −π + ε. When φ(t) = −ε, it follows from(23) that

s(t) = v2h′∇2F (p)h− kpv‖∇F (p)‖ sin ε×(v‖∇F (p)‖ cos ε+ c1 tanh (ε(t)/c2))

≤− kpvγ1 sin ε (vγ1 cos ε− c1) + γ3v2 < 0. (24)

Similarly, φ(t) = −π + ε yields that

s(t) ≥− kpvγ1 sin ε (−vγ1 cos ε+ c1)− γ3v2 > 0. (25)

Thus, φ(t) stays in the region [−ε,−π + ε] for all t ≥ t0 ifφ(t0) ∈ [−ε,−π + ε].

Consider a Lyapunov function candidate as

Ve(e) = 1/2 · e2(t).

Its derivative along with (20) and (23) is obtained as

Ve(e) = e(t)(s(t) + c1/c2 ·

(1− tanh2 (ε(t)/c2)

)s(t)

)= kpvn

′τe2(t) + e(t)×(v2h′∇2F (p)h+ c1/c2 ·

(1− tanh2 (ε(t)/c2)

)s(t)

)≤ kpvn′τe2(t) +

(γ3v

2 + c1/c2 · γ2v)|e(t)|

≤ − (kpvγ1 sin ε) e2(t) +

(γ3v

2 + c1/c2 · γ2v)|e(t)|.

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P control kp = 1

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(c)Fig. 2. (a) Fields distribution and trajectory of the Dubins robot. (b) Trajectories of the Dubins robot with different initial states. (c) Tracking errors ofthe Dubins robot under different control parameters.

i

(north)ii

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v

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( , , )n e dp p p

(east)ij

(down)ik

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bj

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rate

(c)Fig. 3. (a) Coordinates of a fixed-wing UAV. (b) Trajectory of the fixed-wing UAV in the field of PM2.5. (c) Tracking errors and concentration rate ofthe fixed-wing UAV.

Thus, Ve(e) ≤ 0 holds for all

|e(t)| ≥ ρ =γ3v + c1γ2/c2kpγ1 sin ε

.

This implies that |e(t)| will be eventually bounded by ρ, i.e.,

lim supt→∞

|ε(t) + c1 tanh (ε(t)/c2)| ≤ ρ.

By Lemma 1, it holds that

lim supt→∞

|s(t)− sd| ≤ tanh−1(c2γ3v + c1γ2kpc1γ1 sin ε

).

VI. SIMULATIONS

The effectiveness and advantages of the PI-like controllerare validated by simulations in this section. Particularly, thePI-like controller (6) is performed on a realistic simulator ofa 6-DOF fixed-wing UAV [25].

A. Isoline Tracking in Scalar Fields

Consider a Dubins robot in (5), and let q(t) =[p′(t), θ(t)]′ denote its state. The linear speed of the robotis set as v = 0.5 m/s. Let the Dubins robot travel in a scalarfield of Fig. 1(a), under the PI-like controller (6) with the

TABLE IPARAMETERS OF THE CONTROLLER (6) IN SECTION VI-A

Parameter kp ki c1 c2

Value 10 0 0.1 1

TABLE IIPARAMETERS OF THE CONTROLLER (6) IN SECTION VI-B

Parameter kp ki c1 c2

Value 10 1 0.2 1

parameters shown in Table I. The field distribution and thetrajectory of the Dubins robot are depicted in Fig. 2(a) withsd = 10 and q(t0) = [0, 20, − π/2]. It is clear that theobjective (3) is eventually achieved.

B. Isoline Tracking in Circular Fields

In this subsection, we validate the performance of the PI-like controller (6) in a circular field

F (p) = 20 exp(−0.1

√x2 + y2

)(26)

where the source position is set to the origin. The controlparameters are selected as Table II. Fig. 2(b) illustrates the

2500

field distribution and trajectories of the Dubins robot withdifferent initial states. Furthermore, Fig. 2(c) portrays thetracking errors with different control parameters. It can beobserved that increasing kp can exactly enforce the steady-state error to approach zero, however only the controller (6)with ki = 1 eventually achieves the objective in (3) withzero steady-state error.

C. Isoline Tracking in a field of PM2.5

In this subsection, a 6-DOF fixed-wing UAV [25] isadopted to test the effectiveness of the PI-like controller (6)in the field of PM2.5, see Fig. 1(a) and Fig. 3(a). To beconsistent with the notions in [7], [25], [26], we also adopt[pn, pe, pd]

′ and [φ, θ, ψ]′ to denote the position and orienta-tion of the UAV in the inertial coordinate frame, respectively.Moreover, we use [u, v, w]′ and [p, q, r]′ to denote the linearvelocities and angular rates in the body frame. Due to pagelimitation, we omit details of the mathematical model of theUAV, which can be found in [25], and adopt codes from [27]for the model. Moreover, Fig. 3(b) depicts the distribution ofthe PM2.5 and the trajectory of the UAV, where the squareand arrow denote its initial position and course. Furthermore,the tracking error and the concentration measurement rate ofthe sensing robot versus time are illustrated in Fig. 3(c). Indetails, the sampling frequency for the PM2.5 is set as 1 Hzand the linear speed of the UAV is maintained as 30 m/s byits original controller.

Overall, the objective (3) is eventually achieved by theDubins robot (5) under the proposed PI-like controllers (6).

VII. CONCLUSION

To track a desired isoline of a scalar field, we havedesigned a coordinate-free controller in a simple PI-like formfor a Dubins robot by using concentration-based measure-ments in this work. A novel idea lies in the design of asliding surface based error term, which renders our PI-likecontroller different from the standard PI controller. Moreover,the simulation results validated our theoretical finding.

REFERENCES

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