Electric Sensor-Based Control of Underwater Robot Groups

Christine Chevallereau, Member, IEEE, Mohammed-Rédha Benachenhou, Vincent Lebastard, Member, IEEE, and Frédéric Boyer

Abstract —Some fish species use electric sense to navigate effi-

perceive their environment efficiently and can easily navigate

ciently in the turbid waters of confined spaces. This paper presents

in the turbid waters of tropical forests, which are their ecologi-

a first attempt to use this sense to control a group of nonholonomic rigid underwater vehicles navigating in a cooperative way.

cal niche. Named electrolocation, this sensorial ability has high

A leader whose motion is unknown to the others serves as an active

potential interest for underwater robotics applications such as

agent for its passive neighbor, which perceives the leader’s elec-

deep-sea exploration, rescue missions under catastrophic con-

tric field via current measurements and moves in order to follow

ditions or navigation in muddy waters, i.e., under conditions

a trajectory relative to it. Then, this passive agent, becomes in its

where neither vision nor sonar can work. Based on these obser-

turn the leader for the next agent and so on. Sufficient conditions of convergence of the control law are derived for electric current

vations, roboticists have recently proposed the first bioinspired

servoing. This is achieved without the explicit knowledge of the

electric field sensors with the aim of equipping a new gener-

location of the agents. Some limits on the possible motion of the

ation of underwater vehicles (UVs) able to navigate in turbid

leader along with the importance of the choice of controlled out-

waters and confined environments [2]–[5]. In [4], the sensor is

puts are demonstrated. Switching between different group config-

an insulating mobile body on the boundaries of which are fixed

urations by following a virtual agent is also described. Simulation

an arbitrary number of electrodes. A voltage generator imposes

and experimental results illustrate the theoretical study.

a potential on one of these electrodes (located in the tail), with

Index Terms —Biologically inspired robots, electric sense, marine

respect to all the others (which are set to a common ground).

robotics, nonholonomic agent, visual servoing.

Once immersed in a conductive fluid, this device produces a nearby dipolar electric field, and by Coulomb’s law, a continuous network of closed current lines linking the tail electrode

(emitter) to all the others (receivers) across which the currents LECTRIC sense is a mode of perception used by sharks, are measured. In this case, the sensor is said to work in active

NTRODUCTION I. I

rays [1], and several hundreds of fish species of the Gym- mode, since it produces the carrier whose modulations by the

notid and Mormyrid families, who live in the tropical forests environment, contain the measured information. of Africa and South America. While sharks and rays pas-

Electric fish also use a passive version of the electric sense in sively sense the electric fields generated in their surroundings, which the fields are not generated by themselves but by some

mormyrids and gymnotids have developed an active version of exogenous sources [6]. These exogenous fields can be produced this sense by generating their own electric field called a ”basal by prey or other nearby electric fish, which is a very usual field” or ”carrier.” While in the gymnotid fish, the carrier has a case, since these fish swim mostly in small social groups of a sine-wave nature, mormyrid fish produce a pulsed dipolar elec- few individuals [7]. Biological experiments [8]–[12] have re- tric field by polarizing on short durations with a specific organ vealed that active electric fish use specific strategies to organize called the electric organ discharge (EOD) located just anterior their collective electric activity. In particular, in order to avoid to the tail with respect to the rest of the body. In all cases, the jamming and overlapping between individual electric pulses, distortions of this basal field by the surrounding objects are several species of mormyrid order the electric activity of each measured by a dense distribution of electroreceptors distributed member of a group in a fixed sequence of individual pulses over the fish’s skin. By processing these measurements, the fish separated by “silent periods.” On the other hand, the wave elec-

tric fish of Gymnotide family uses different strategies based on frequency shifting [13]. In order to extend electric sense from the single-agent case to the multiagent case, we have proposed

Manuscript received July 1, 2013; accepted December 17, 2013. Date of pub-

in [4] an adaptation of our active sensor to the passive case. The-

lication January 13, 2014; date of current version June 3, 2014. This paper was recommended for publication by Associate Editor I. H. Suh and Editor B. J.

oretically, either currents or potentials can be measured by the

Nelson upon evaluation of the reviewers’ comments. This work was supported

receiving electrodes. 1 However, in practice, we have observed

by the ANGELS project funded by the European Commission, Information So-

that current-based sensors benefit from a higher sensing range

ciety and Media, Future and Emerging Technologies under Contract 231845. The authors are with the Centre national de la recherche scientifique, Ecole

(which is about one sensor length as in the fish). In [14] and [15],

des Mines de Nantes, Institut de Recherche en Communication et Cybern´etique

the collective navigation of a group of electric robots has been

de Nantes, Nantes 44321, France (e-mail: Christine.Chevallereau@irccyn. ec-nantes.fr; Mohammed-Redha.Benachenhou@irccyn.ec-nantes.fr; Vincent. Lebastard@irccyn.ec-nantes.fr; Frederic.Boyer@irccyn.ec-nantes.fr).

1 When the currents are measured, the receiving electrodes are grounded [4], Color versions of one or more of the figures in this paper are available online

while reciprocally, when the voltage is measured on the receivers, a very high at http://ieeexplore.ieee.org.

impedance on each of the receiving electrodes is imposed so forcing the currents Digital Object Identifier 10.1109/TRO.2013.2295890

to zero.

CHEVALLEREAU et al.: ELECTRIC SENSOR-BASED CONTROL OF UNDERWATER ROBOT GROUPS 605

Fig. 1. (Left) Photograph and (right) schematic view of a four-electrode sensor.

addressed but by using the voltage measurements. The voltage mode has been chosen for its simplicity and results were tested on simulations only. On the contrary, in the paper presented here, electronavigation in a group is addressed with current measurements, an extension which allows us to test our results both in simulation and experiments. The sensors are slender probes (see Fig. 1), whose technology and models are detailed in [4] and [16], respectively.

This paper aims at exploring the potentialities and the diffi- culties of electric multiagent navigation. In this perspective, we will consider a set of rigid slender probes (or agents) able to move in the same horizontal plane. The motion of each agent is directly controlled using a kinematic model, here reduced to that of a nonholonomic unicycle. Thus, as this is the case in many UVs, the motion of each agent is controlled through the axial velocity

V (aligned with the sensor body axis) and the yaw angular velocity ω (orthogonal to the plane of the sensors motion).

Based on this framework, we will address the problem of the navigation in formation along with that of the formation shifting. In a given formation, the agents have to maintain a specified posture relative to each other. Drawing inspiration from the mormyrid fish, we will impose the following basic rule: two agents cannot emit an electric field at the same time. Based on this rule, we will see how we can elaborate successful control strategies where at each instant each of the passive agents takes

a prescribed position in the electric field of the active one, while the role of the active agent of the group rotates over time. It is worth noting here that in order to control the relative position of a passive agent with respect to the active one, we could first seek to invert the model of the passive agent’s electric measurements with respect to the position and orientation between the two agents. Unfortunately, this problem poses many difficul- ties, among them, it is nonlinear, sensitive to noise, and does not generally have a unique solution. However, several tech- niques based on optimization or observer control theory have been used with quite good success [17]–[21]. In this paper, we will follow a sensor-based control approach [22], [23]. The approach is based on a direct feedback of the measurements and not, as in the previous approach, on the state variables of a model that represents the world. This kind of approach is well known to the community working in vision. In this context, it is named “visual servo control” [24]. From that point of view, we here propose an “electric servo control,” in which the electric field of the active agent is used as an immaterial prolongation of its body in its surroundings, while the passive agent senses

this electric body in which it seeks to achieve some prescribed measurements preliminarily measured in a specified (desired) relative posture. Unfortunately, the set of poses ensuring the desired measurements cannot be reduced to the desired one, but rather spans a continuous set of poses, named “zero-dynamics” in nonlinear control theory [25]. As a result, this concept will play a crucial role in the subsequent analysis, in particular in all matters concerning closed-loop stability.

This paper is structured as follows. In Section II, we briefly present the model of the electric sensors as well as the available measurements. Section III deals with the locomotion model and in particular, with the effects of nonholonomy of agents (underactuated systems). In Section IV, the principles of group navigation in formation are explained. In particular, the convergence conditions of the measurements toward their desired values are listed. The zero dynamics, i.e., the internal dynamics with measurement variables forced to their desired values, are given as well as the convergence conditions toward the desired relative posture of the passive agent with respect to the active one. In Section V, one example illustrates in simulation the main properties of the control approach. The problem of switching between formations is addressed in Section VI. In Section VII, our testbed is described and first experiments with two kinds of formation are presented to prove the feasibility of the approach, while switching between formation is also illustrated. Finally, this paper ends with a few concluding remarks and perspectives in Section VIII.

EASUREMENT II. M M ODEL

As already discussed, the sensors (that we will also call “agents” or “robots”) are slender probes composed of n electrodes fixed on insulating axi-symmetric boundaries. The two- tip electrodes are hemispherical, while the others are ring shaped (see Fig. 1). For the sake of simplicity, all these electrodes will

be called “rings.” The two-tip rings are called the head and the tail electrodes. Except for the tail ring, each ring is divided into

a pair of symmetric left and right semiring electrodes, across which the 2(n − 1) currents flowing from outside can be measured independently. Each sensor can be active or passive depending on whether it measures its own electric field (in fact its perturbations), or if it measures the field of another agent, respectively. In the first case (it is active), a voltage generator sets

a controlled voltage between the tail electrode (emitter) and all the other rings (receivers). Once immersed in a conductive fluid, the probe so generates an electric field in its surroundings. In the second case, the probe is passive since all the electrodes are grounded, while the currents flowing across them are measured. In this case, the currents are produced by an exogenous electric field generated by an active agent. As proposed in [4], the control electronics of the agents enable instantaneous switching between active and passive modes. Subsequently, we consider

a group of such agents moving in a homogeneous conductive fluid. In order to avoid jamming between several agents’ electric fields, we require that at any instant, only one of the agents in the group is active while all the others are passive, the active role

606 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014

being shifted from agent to agent in a fixed ordered sequence repeated in a continuous loop.

Due to the short range of the sensor [between once (active mode) and twice (passive mode) of the sensor length], the mod- eling problem can be reduced to that of a single pair of neigh- boring agents one being active and the other being passive. To introduce the model of measurements in such a scenario, let us consider the succession of causes and effects, which relate the imposed voltages on the active agent to the measured currents

(b) on the passive one. 2 First, the active agent is set under voltage through a vector of potentials U = (U ,U 2 ,...U n ), imposed Fig. 2. 1 Active dipole generates an electric potential, which perturbs the electric currents measured by the electrodes of the passive agent. The differences on the rings by the voltage generator. Once U is imposed, the between the two figures illustrate the influence of the passive agent on the elec- sensor recovers its electric equilibrium by generating an (n × 1) tric field emitted by the active agent. (a) Electric potential φ (0 ) . (b) Electric

(a)

vector of currents I (0) flowing across the rings. 3 These currents potential φ.

are given by the intrinsic model of the sensor

(1) (2)] evaluated at the center of the rings. The second component represents the lateral response of each ring to the application where C (0) is the (n × n) conductivity matrix of the sensor of the electric field −∇φ (0) . It can be modeled by defining a

I (0) =C (0) U

when there is no object around it. Second, once the currents are dipolar tensor of lateral polarization for each ring [16]. Finally, generated on the active agent, they produce in the nearby sensor due to the symmetry of the electrodes distribution, the two com-

a basal field, which can be written [16] as ponents I ax and I lat can be simply deduced from the vector of

measured currents

I by taking for each ring (except the tail),

φ (x) = i (2) the half-sum and the half-difference of the left and right cur-

rents flowing across the symmetric electrodes that constitute this where γ is the conductivity of the (homogeneous) fluid, and ring. In the following, we will use the (n m × 1) vector of mea-

r i denotes the distance between the center of the ith ring and surements m = (I ax,1 ,...,I ax,n −1 ,I lat,1 ,...,I lat,n −1 ) T with the point x at which the potential is evaluated. Third, this basal n m = 2(n − 1), and I ax,i ,I lat,i are the axial and lateral currents

= I 1 +I 2 and I = ax,1 I 1 −I field polarizes its nearby surroundings and in particular any pas- 2 2 lat,1 2 . sive sensor close to the active one. This polarization generates These measurements obviously depend on the relative posture

of the ith ring. For example, I

the vector of 2(n − 1) perturbative currents measured by the between the passive and active agents. passive electrodes, 4 that we simply note

I = (I 1 ,I 2 ,...I 2n −2 )

and which is naturally a function of the pose between the two OCOMOTION III. L M ODEL OF A G ROUP agents. Due to the slender geometry of the sensor, the measured

Throughout this paper, the robots are assumed to move in currents can be written under the following form:

the same horizontal plane. Furthermore, the robots’ motions are I=I ax +I lat

(3) modeled using the nonholonomic kinematic model of the unicycle. The control inputs are the linear axial velocity

V and where I ax and I lat are called axial and lateral components of the angular vertical velocity ω (see Fig. 3). When they are con-

the currents. Physically, they correspond to the electric response trolled in formation, the control objective consists of keeping a of the passive agent to its lateral (from left to right) and axial constant position and orientation (or “relative posture”) between (from rear to front) polarization by the active agent field (2). As the robots. Following a usual control strategy in multirobot nav- shown in Fig. 2, this polarization takes into account the non- igation [27], one of the robots is distinguished as a “leader,” negligible interactions of the passive agent body with the active defining a reference motion for the group. This reference motion

agent field. 5 This contrasts with [14], where the voltages were is unknown to the other robots or “followers,” and is compatible measured while the passive agent body was neglected. The first with the nonholonomic constraints of the kinematic model of component I ax is modeled by distributing equally the (n − 1) locomotion. last components of (1) on the 2n − 2 half rings after having

In order to maintain a constant relative posture between the replaced the components of U by the values of −φ (0) [given by agents of a group, we combine the “follower-leader” and the

“active-passive” strategies. As a result, in a first step, the active

2 For the sake of clarity, we do not detail the model of electric interactions,

agent is the leader and its closest neighbor is passive and follows

which is the subject of other works progressing in parallel, and refer the reader to

it using the sensor-based control. In a second step, the follower

the preliminary results [26] where an analytical approach based on the method of refections is applied to the case of complex scenes (several agents, objects).

becomes active and its next neighbor follows it using a sensor-

3 The upper index (0) indicates that the quantity is considered without any

based control. This principle is then propagated from agent to

object around the sensor [16].

agent and repeats according to an ordered sequence. In such

5 That is the currents flowing across the 2(n − 1) half rings. Going further into the details, this polarization represents the instantaneous

a way, the reference motion is propagated to the whole group

reconfiguration of the electric charges in order to ensure the electric balance of

incrementally. For the sake of clarity, in the following, we will

the passive agent

only consider a pair of agents including the leader and its closest

CHEVALLEREAU et al.: ELECTRIC SENSOR-BASED CONTROL OF UNDERWATER ROBOT GROUPS 607

which takes the form of the following control system: ˙ X = Ju + P (t)

where

⎡ cos(θ) 0 ⎤ −V r + yω r J= ⎣ sin(θ) 0 ⎦ , and P (t) = ⎣ −xω r

0 1 −ω r is a perturbation due to the unknown motion of the active agent.

From these considerations, it appears that the formation motion can be perfectly tracked (with X=X d and ˙ X = 0) by the passive agent if, for any V r , ω r , there exist values of

V and ω

that satisfy the following constraints:

V cos(θ) − V r + yω r =0

V sin(θ) − xω r =0 ω−ω =0

Fig. 3. Passive agent (blue) and the active agent (red) are nonholonomic r systems. The red electrode is the emitter and blue ones are the receptors. The

the third equation allowing us to define

ω directly. The desired formation X d = (x d ,y d ,θ d ) T is feasible if there exists a value

control objective for the follower agent is to maintain constant values x, y, θ.

V that satisfies the two first equations for any V r , ω r . In passive neighbor. The case of any agent is easily deduced by particular, a motion along a straight line ( ω r = 0, V r propagating this basic scenario according to the previous rules.

be tracked only if θ = 0. To achieve a satisfactory behavior Considering this isolated pair of agents, the relative posture of when the leader moves along a straight line, we limit our study the passive agent with respect to the active one (here the leader) to the case where θ d = 0, i.e., to the case where the agents are can be defined in terms of the measurements of the passive parallel. Under these conditions, (8) can be satisfied for any agent.

motion of the leader agent (i.e., with any V r and ω r ) only when

In an absolute frame, the leader posture is (x r ,y r ,θ r ), while θ d = 0 and x d = 0. If x = x d

its velocity is defined by (V r ,ω r ). Similarly, the posture of of the leader ( ω r the passive agent in the absolute frame is (x p ,y p ,θ p ), and its relative posture of the passive agent with respect to the active corresponding velocity (V, ω) defines the two control inputs of leader. This point will be further discussed since the formation

the problem: u = (V, ω) T (see Fig. 3). Since these two agents that we will consider will not satisfy x d = 0. are nonholonomic unicycle systems, we have

Since the motion of the passive agent is controlled by two ⎧

V and ω while three parameters define its evolution, ⎨ ˙x r =V r cos(θ r )

⎨ ˙x p = V cos(θ p )

inputs

˙y its relative evolution with respect to the active agent can be r =V r sin(θ r )

˙y p = V sin(θ p )

⎩ ˙θ expressed by the following equation, which is independent of

the control

The relative posture of the passive agent in the active agent’s ˙x sin(θ) − ˙y cos(θ) = −V r sin(θ) + yω r sin(θ) − xω cos(θ). frame is defined by the state vector

X = [x, y, θ] T . With this

(9) notation, the objective of the control strategy, defined in the This equation is obtained by combining the two first equations frame of the active agent is to maintain

X at a desired value

X of (6). It represents the nonholonomic constraint imposed to d = [x d ,y d ,θ ] X expressed as a function of the control inputs

d T . Our control model is defined by ˙

V and ω of the the leader tracking. The control law will not affect this equa- passive agent. In order to obtain it, let us first write the relation tion, which is independent of the two control inputs

V and ω. between the relative and absolute postures as

However, it will impose via

V and ω, two relations on the three variables x, y, and θ and their derivatives. The control law must

x = (x p −x r ) cos(θ r ) + (y p −y r ) sin(θ r )

be such that these relations, once combined with (9), ensure a stable behavior of the passive agent.

y = −(x p −x r ) sin(θ r ) + (y p −y r ) cos(θ r ) θ=θ p −θ r .

ENSOR IV. S -B ASED C ONTROL S CHEMES FOR M OTION IN F ORMATION

Time-differentiating these equations and combining them with (4), we obtain

Because the motion of the passive agent is defined by two inputs

V and ω, only two outputs can be tracked in a closed loop. ˙x = V cos(θ) − V r + yω r

Various control strategies have been proposed for the control in ˙y = V sin(θ) − xω formation of nonholonomic vehicles when their relative posture

is known [28]–[31]. However, since we have chosen to base ˙θ = ω − ω r

(6) the control law on a direct feedback of the electric measure-

608 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014

ments, with no calculation of the relative posture of the leader which ensures that the error decreases if the matrix (C ∂m ∂X JL) and follower agents, these methods cannot be directly applied. is positive definite, i.e., if it satisfies the condition [23] Nonetheless, they reveal the importance of the choice of the outputs for this kind of system. In our case, the two controlled

∂m

C J L > 0. (15) outputs being two functions of the electric measurements, and

∂X

since the relative posture of the agents is defined by the three To ensure positive definiteness, we propose to take variables x, y, and θ: an infinite set of relative postures ensuring

C ∂m ∂X J

the desired outputs exists. As a result, the closed-loop dynamics to the relation of the system when the outputs are forced to follow their de-

sired evolution, i.e., the so called “zero dynamics” of [25], must

be studied. Generically, these dynamics are determined by the

∂X

choice of the outputs, by X d , and by the motion of the leader. where L is an estimation of the inverse of C ∂m In particular, the control outputs must be chosen in order to

∂X J. If the robot

X would be known, the classical solution would consist of produce, at least locally, a stable zero dynamics. In our case, de- using the model of measurements to calculate the current value pending on the curvature of the motion of the active agent, some of (C ∂m (X)J(X)) −1 . However, since the state is unknown (no errors on the relative posture of the controlled passive agent can

state

∂X

observer will be used), this approach is not feasible. Nonethe- occur. These points will be detailed in the Section IV-C. Before less, since a constant relative posture of the passive agent with entering into these details, let us remark that the control schema respect to the active one is desired ( X=X d ), one can choose

derived hereafter address the general measurement-based ser- L as (C ∂m (X d )J(X d )) −1

, whose the value can be calculated by using the model, or directly measured in a preliminary ex-

voing problem of an unicycle plant model, and as such, could

∂X

be extended to other sensors.

perimental phase 6

A. Principle ∂m −1 L= C (X d )J(X d )

. (17) As in the case of vision-based control schemes of [24], the

∂X

control aims at minimizing an error vector e(t) defined as a If C ∂m ∂X J or L are singular, then the convergence condi- function of the measurements. Such a vector is here restricted tions fail and the desired behavior can no longer be guaran-

to a linear combination of the measurements: s = Cm(X(t)), teed. As a result, the choice of

C is critical to ensure that where

C is a constant (2 × n m ) matrix, and

C ∂m

)J(X ∂X d (X d )

e(t) = Cm(X(t)) − s d (10) since we have for any (2 × 2) invertible matrix C 0 , the invari-

ance property

where the vector m(X(t)) denotes the vector of measurements (e.g., the electric currents). Once

C is selected, the definition of

∂m

C 0 e the control law is straightforward. Indeed, invoking the relation

u = −λ C (X d 0 d C )J(X )

∂X

between the state and the control inputs (7) allows one to express

∂m

the relation between the error and the control inputs as

= −λ C (X d )J(X d )

e. (18)

∂X

∂m ˙e(t) = C

∂X the control law is unchanged by replacing

(Ju + P (t)) − ˙s d (t).

C by C 0 C, and we can for instance choose a normalized matrix C.

Then, if we want to assign the following desired behavior to the error e(t):

C. Convergence Condition for the Relative Position

˙e(t) = −λe

(12) Between the Agents

with a chosen parameter λ > 0, we can use the following linear The control law is designed to ensure the convergence of the feedback control:

output e(t) to zero. Because the dimension of the state X is

3, while the relative degree of the two independent controlled

(13) outputs is 1, the dimension of the zero dynamics is 1. In general, where L is a (2 × 2) matrix of gains.

u = −λLe

zero dynamics characterize the closed-loop behavior of a controlled system [25]. Geometrically, they define a submanifold

B. Convergence Condition for the Error e Z of the state space, which contains all the states of the system such that the controlled outputs are zero, and in particular, the

d . Thus, while the convergence of the control law entails law. To that end, let us remark that when the vector of desired

We now study the convergence of the aforedefined control state X

outputs s d is constant, ˙s d that the state tends to reach = 0, and the active robot is at rest Z, that does not mean that X converges to

X d since the system is underactuated, and Z is not ( P (t) = 0), the behavior of the error in a closed loop is then reduced to X

d . Therefore, zero dynamics Z must be studied in governed by

∂m

˙e = −λ 6 C J Le (14) By imposing small displacements as inputs and measuring the corresponding

∂X

variations of the outputs.

CHEVALLEREAU et al.: ELECTRIC SENSOR-BASED CONTROL OF UNDERWATER ROBOT GROUPS 609

in a closed loop (the state of the robot remains in Z) the relation

∂m(X)

C X=C ˙ m X=0 ˙ (19)

∂X

where ∂ m (X ) C

m =C ∂X will denote a (2 × 3) matrix of rank =

2. Assuming that the left (2 × 2) submatrix C m1 of C m is invertible, and denoting the right (2 × 1) submatrix of C m by C m2 , the velocities will verify in a closed loop the simple relation

which can be rewritten as 8

˙x = Z x (θ) ˙θ

Fig. 4. Different configurations of the passive agent belonging to Z , i.e.,

˙y = Z

y (θ) ˙θ. (21)

corresponding to desired output vector

C m(X ). The set of the sensor center

positions x, y is represented by the red line. Several corresponding configura-

The control law allowing the maintenance of the state of the

tions of the passive agent are drawn in blue, the desired configuration of the follower agent is in black, and the active agent is in red with a contrasted emitter.

robot in Z, its velocity ˙ X will necessarily satisfy (21). Since the two agents are nonholonomic, depending on the motion of the leader, it is possible that the state of the follower does not tend toward the desired state. In fact, as already discussed in Section III, the behavior of the follower robot is governed by (9), which once combined with (21), gives

Z x (θ) ˙θ sin(θ) − Z y (θ) ˙θ cos(θ)

r sin(θ) + y Z (θ)ω r sin(θ) + x (θ)ω r cos(θ). (22) This equation describes the closed-loop behavior of the follower

= −V

agent as a function of θ and ˙θ only. It plays a crucial role in all the following. In particular, it will reveal the motions of the follower robot in a closed loop. Using this result, we are now able to address the question of whether the agent will be steered

to the desired state X d (or equivalently θ d ) or not, and if this is

Fig. 5. Relative position of the passive agent x and y belonging to Z

( C m(X ) = s d ω ) as a function of r θ (top and middle), and the corresponding ω r

the case, will it remain in that desired state? In order to address

value of V r (bottom). When the active agent moves in a straight line V r = 0,

this question, we will consider (22) in two cases. First, when

the equilibrium position is the desired one x = −0.22 m, y = −0.10 m, and θ = 0. When the active agent turns with ω r

= 2.61 rd/m, the equilibrium

the leader is moving in a straight line (see Section IV-C1) and

position is defined by x = −0.2532 m, y = −0.041 m, and θ = −0.539 rd

second, when the leader is turning (see Section IV-C2).

so generating a vector of error:

d x−x d = −0.0332 m, y − y d = 0.059 m,

1) Leader Moves in a Straight Line ( ω r = 0): Equation (22)

θ−θ = −0.539 rd.

then becomes

detail in order to find out whether the convergence of the outputs Z x (θ) ˙θ sin(θ) − Z y (θ) ˙θ cos(θ) = −V r sin(θ)

(23) to zero does imply the convergence of the state towards X d , at whose obvious solution is given by ( ˙θ = θ = 0). Thus, θ =

least locally. θ d = 0 (and so any X = X d ) is an equilibrium state. Further- The zero dynamics being spanned by all the possible evolu- more, the first-order linearization of (23) around the desired

tions of the robot when the controlled outputs are identically state θ d zero, it is in our case given by Z=

manifold whose analytical construction would require the inver-

d θ=0 (24) sion of the model of electric measurements. To circumvent this

Z y (θ )

difficulty, and since the dimension of Z is 1, the state X ∈ Z defines the local behavior of the follower agent. A simple anal- can be parametrized by θ as: X = [x Z (θ), y Z (θ), θ] T . In doing ysis of (24) leads to the following four conditional conclusions. so, the functions x Z and y Z can be constructed numerically

a) If V Z r y (θ d ) < 0, then θ will converge toward zero. Therefore,

x and y will converge toward x d and y d if X=X d is the Time differentiating this context, the robot velocity ˙ X satisfies

around the desired state X d7 as illustrated in Figs. 4 and 5.

unique state belonging to Z with θ = 0.

∂y that Z θ is a convenient choice. A parametrization by x or y or any function of

7 More generally Z can be parametrized by a single variable. We assume here

, but the analytical expression of the state can be used in a similar way. Parametrization by θ implies that θ is

By definition Z x =

are unknown; thus, these terms are calculated numerically using

monotonic in the studied subspace of Z around θ d (20).

610 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014

y (θ d ) > 0, then θ will diverge and consequently x and y will diverge too.

b) If V Z r

a perturbation

surement error. Moreover, in a transient maneuver where the

c) If V y r (θ d ) = 0, then neither θ changes nor do x or y. The Z robots move in order to configure the group in a given forma- passive agent does not move. The error if it exists, remains tion (see Section VI), ˙s d unchanged.

governed by

d) Since V r appears in V Z r y (θ d ) , if the chosen formation and

∂m

outputs are such that the control law is stable for V > 0,

˙e = −λ C J Le + C

P (t) − ˙s (t). (29)

∂X

then the same control law will be unstable for V r < 0.

As a result, a small error in position will be induced too. Finally, Consequently, in order to ensure that a formation can go backward, a new set of controlled outputs has to be defined. the proposed control strategy will be tested in simulation in

2) Leader Turns

(ω Section V and in experiments in Section VII.

two subcases depending on whether x d is equal to zero or not:

a) If

OTION IN x ORMATION d = 0, all the desired motions are feasible (indepen- V. E M F

XAMPLE OF

For the purpose of illustration, a formation of four identical and x=x d = 0) is an equilibrium state for (22). More- agents is considered. The relative posture between two neigh- over, the local behavior of the passive agent is governed by bors is X d = (x d ,y d ,θ d ) = (−0.22, −0.10, 0) (in meters and the first-order expansion of (22) around the desired state radians).

dently of ω r and V r ). Thus, any X=X d (with θ=θ d =0

V r −Z x (θ d )ω r −y d ω r

A. Choice of Control Outputs

In this desired configuration X d , considering only one active Thus, similar conditions to those met in the previous case agent and its passive neighbor, the vector of measurements of the

Z y (θ d )

can be stated, except that the critical velocity is no longer passive agent is m = [I ax,1 , I ax,2 , I ax,3 , I lat,1 , I lat,2 , I lat,3 ] −4 T

V but rather = 1.10 [0.315, −0.357, −0.645, 0.106, 0.317, 0.499], while

its Jacobian matrix is

V ⎤ r 1 − (y d +Z x (θ d )) . (26) −0.4132 0.1297

−0.3053 ⎢ ⎥ 1.3258 ⎥ forward velocity now depends on r , y d

Consequently, to ensure stability, the critical value of the

but a different state denoted θ e , and such that

(θ r e sin(θ e )+y Z )ω r sin(θ e )+x Z (θ e )ω

e r cos(θ )=0

Examining this matrix, it appears that many combinations of (27) ∂m the outputs can be chosen to obtain a nonsingular matrix C ∂X J.

when θ e exists. If for a given value of the turning curvature In the following, we choose as control outputs I ax,3 and I ax,1 −

0.4336 ). According 0.0794 the other hand, when (27) has a solution θ e , the first-order to the conditional conclusions of Section IV-C, we have two

V r , (27) has no solution θ , then the motion of the passive I (i.e., C ax,2 ∂m J = 1.10 −3

agent cannot be stabilized and there is no equilibrium. On

∂X

expansion of (22) around this equilibrium state can be cases depending on wether the leader moves in a straight line written as

with V r > 0, or turns along a path of curvature ω r /V r . In the

e first case, the stability is ensured if Z y (θ d α ˙θ + V ) < 0 [see (22), with

r β(θ − θ )=0

(28) Z y defined by (21)], a condition which is verified by our chosen with α=Z x (θ e ) sin(θ e )−Z y (θ e ) cos(θ e ) and β = cos outputs, since in this case Z y (θ d ) = −0.1370 < 0. In the second

(θ e )−

ω r (y Z (θ e ) cos(θ e )+Z x e (θ d ) cos(θ e )−x Z (θ e ) case, since x

sin(θ e V )+Z r y (θ e ) sin(θ e )).

of the follower (see Section IV-C2).

Finally, three conditional conclusions arise. Then, the control will force the follower agent to reach the

and x zero dynamics manifold Z(= and y will converge toward x e and y e . (θ e ,x e ,y e ) defines the knowledge of Z is essential. In particular, imposing a ro- the posture of the passive agent with respect to the active tation of the leader ω r one when ω r

1) If V r > 0 and β > 0, then θ will converge toward θ e

Z. Fig. 4 illustrates the states of the follower agent belonging

2) For ω r = 0, (28) is reduced to (24). Since θ e V r = 0, we have to Z. Another representation of the zero dynamics manifold is α = −Z (θ e ) and β = 1.

given in Fig. 5. In this case, the two upper plots numerically

define the functions x Z (θ) and y 3) The condition of stability is based on Z | (θ). The lower plot defines

V r | and on the

property of the equilibrium point defined by the choice of the equilibrium state on Z for different curvature of the leader ω r the output (which affects Z).

motion. It shows V r as a function of θ deduced from (27) (i.e., Note that we have hitherto assumed that ω P (t) = 0 in (11). r

V r = y Z (θ ) sin(θ )+x Z (θ ) cos(θ ) ). For a motion of the leader turn- However, in reality, the motion of the active agent will produce ing on a path of constant curvature, these three numerical plots

sin(θ )

CHEVALLEREAU et al.: ELECTRIC SENSOR-BASED CONTROL OF UNDERWATER ROBOT GROUPS 611

Fig. 7. Paths followed by four agents illustrate the stability of the formation,

Fig. 6. Two eigenvalues of C

even if the leader (red) has a complex motion with various curvatures. (15) (top), and β

∂X ∂m J L involved in the convergence condition

α involved in (28) as a function of V r r for configurations

belonging to Z (bottom).

allow us to fully determine the relative position of the two agents

when the control law is stable. 9 As regards the stability of the

closed-loop dynamics, it is conditioned by the sufficient condi-

tion (15) along with the stabilization of (28) on θ e . In the upper part of Fig. 6, the two eigenvalues of C ∂m ∂X J

the equilibrium configuration corresponding to the curvature of

the leader path defined by ω r V r . On the upper part of Fig. 6, the

term β α involved in (28) is plotted under the same conditions. β α has to be positive to ensure that the passive agent does follow

the active one with V r > 0. Based on these plots, a stable be- Fig. 8. Results for the first follower agent are presented in the first column, havior is obtained when the leader agent follows a path with a while for the last one the results are in the second column. All the curves are curvature such that −5 rd/m< ω r V r presented as a function of time. In the first line, the error on the controlled < 5.35 rd/m.

measurements is shown. In the second line the error on the relative position are presented and in the last line the control input

V ω (solid line) and the velocity

B. Simulation

of the active agent V r ω r (dotted line) are drawn.

The leader agent is subject to a motion composed of different phases. The leader is initially motionless (phase A, from t = 0 to

leader moves along a straight line with V r > 0, one observes

5 s), then it follows a straight line with a linear velocity

V = 0.05

the convergence of the robot posture toward the desired value m/s (phase B, from t = 5 to 20 s). During the phase C (from

x−x d = 0, y − y d = 0, and θ = 0.

t = 20 to 30 s), the angular velocity smoothly increases up to Due to the nonzero distance between agents along the x-axis, reach a constant value of 0.13 rd/s during phase D (from t = 30

to 60 s), where the leader follows a circle of curvature ω r

a nonzero error on the relative position of the passive agent

V r = necessarily occurs for a leader following a curved motion [see

2.61 rd/m. A symmetric evolution of the velocity of the leader is equation (8)]. This is confirmed in Fig. 8 during phases C, D, then imposed. In this case, the angular velocity decreases during and E, where nonnegligible posture errors can be observed, phase E ( t = 60 to 70 s), then a straight line is followed during while the measurement errors are very small. During phase D, phase F ( t = 70 to 95 s) and the leader stops in the final phase

the velocity of the leader is defined by G(

V r = 0.05 m/s, and t = 95 to 100 s). As illustrated in Fig. 7, the formation is

stably maintained throughout the entire motion. However, as the ω r = 0.13 rd/s; thus, its curvature is V r = 2.61 rd/m, and its follower converges toward the equilibrium posture defined in curvature increases, errors with respect to the relative positions Fig. 5. In Fig. 5, the red dotted line defines the equilibrium increase. point corresponding to this curvature of the leader path. As The initial posture of the passive agent being biased by errors indicated on these plots, when the second agent is the follower, (see Fig. 8), the initial measurements do not coincide with the the equilibrium point corresponding to ω r = 2.61 rd/m is x = desired ones

−0.2532 m, y = −0.041 m, and θ = −0.539 rd, thus an error measurement error during the phase A so ensuring the passive x−x d = −0.0332 m, y − y d = 0.059 m, θ − θ d = −0.539 rd

agent to reach the zero dynamics Z. However, since V r = 0, exist and correspond to the final configuration of phase D in an error on the position remains. In phases B and F, while the Fig. 8. Numbering the agents from left to right starting from the

9 Indeed, for any value of

leader, the farthest agent has the third agent for leader. Since it

ω r /V r , the lowest plot of Fig. 5 gives the corresponding value of θ, while the two others give x(θ) and y(θ) and finally define

is on the external side of the curve, the leader angular velocity

the point on Z toward which the state converges.

is the same but its linear velocity increases ( V r = 0.079 m/s,

612 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 3, JUNE 2014

ω r = 0.131 rd/s). Therefore, its curvature is ω r V r = 1.66 rd/m and which generalizes (13) from formation maintaining to forma- the position error of the last agent with respect to its neighbor tion tracking. Thus, as in Section IV, a condition of conver- is reduced (see Fig. 5).

gence on the error measurement is given by (15) with L v = Due to the leader motion (

∂ m (X (C ) J(X v )) . This condition implies that (C ∂m position and on the measurements persist when the leader is not

∂X J) is stopped.

∂X

not singular throughout the entire maneuver. Since the passive agent moves with respect to the active one ( X v (t) is not constant), this condition is more difficult to satisfy than in the case

ENSOR VI. S -B ASED C ONTROL S CHEMES FOR

of formation maintaining. Finally, even if the proposed control

F ORMATION S WITCHING law allows us to deduce the desired value for the measurement, Since different formations can be used, it is useful to be able to the motion of the robot remains unknown. switch between them. This is achieved by changing the relative desired position of the passive agent with respect to the active

C. Condition of Stability

one. Both the virtual and real agents are nonholonomic systems

(9), whose movements satisfy the constraints

A. Statement of the Problem To tackle the difficulty of the agents, nonholonomy, a refer-

(36) ence motion (or “maneuver”) is first defined in order to steer

˙x v sin(θ ) − ˙y v cos(θ v ) = 0.

a virtual agent from the initial desired posture X d1 to the final which correspond to (9) with a leader kept fixed ( V r =ω r = 0). one X d2 . To that end, the active agent is kept fixed, while in Being interested in the difference between the motion of the its frame, the motion of the virtual passive agent is defined by virtual and the real agent, we define the vector dX = X − X v .

X v = [x v (t), y v (t), θ v (t)] T , with

A first-order expansion of (36) around dX = 0, combined with ˙x v =V

(31), gives the following constraint on tracking errors (between

v cos(θ )

the virtual and real agents):

˙y v =V

d ˙x sin(θ v ) − d ˙y cos(θ v )+V v dθ = 0 (37) ˙θ v =ω v

v sin(θ )

which models the nonholonomy of the agents. As expected, the and X d1 =X v (0), X d2 =X v (T ). The control of the virtual control law is designed to ensure that the controlled measure- agent is u v = [V v ,ω v ] T . At any time, the electrodes of the virtual ments of the real agent match those of the virtual one. Thus, agent measure the currents produced by the active agent, which with

C ∂ m (X )

m =C

, we have

are gathered into m (X v

∂X

(t)).

v (t))

C m (X(t)) ˙ X=C m (X (t)) ˙ X v

(t). (38)

B. Control Law

Based on direct electric measurements, the control law aims Since our objective is to satisfy dX = 0 (i.e., the real agent at making the real passive agent track the maneuver of the virtual tracks the virtual one), we study the linearized evolution of dX passive agent. To that end, the choice has been made to control around dX = 0. To that end, we linearize (38) around dX = 0 the measurement error

and find

⎨ C m (X v e(t) = Cm(X(t)) − Cm (t))dX = 0

v (X (t)).

C m v (t)) ˙ X m v (X (t) dX.

Differentiating the above relation with respect to time and com- ⎩ C (X (t))d ˙ X=−

∂X

bining it with the model (7) (with V r =ω r = 0 since the active Starting from (39) and using similar notation and calculation agent is fixed) gives

to those used in (20) and (21), allows one to parameterize the ∂m(X)

evolution of the tracking error as a function of (dθ, d ˙θ) only ˙e(t) = C

∂m

J(X)u − C

(X v (t))J(X v

(t))u v (t).

To obtain the desired behavior in a closed loop ˙e = −λe, we

∂m(X)

∂m

u= C J(X)

−λe(t) + C

(X v )J(X v )u

C m (X v )˙ X v ⎣ Z y (X v ) ⎦ . Since the expected value for

= −C m (X v ) −1

X is X = X (t), the estimation of

∂X

∂ m (X )

∂ m (X (t))

C ∂X J(X) can be replaced by C ∂X v J(X (t))

which is known. As a result, the control law becomes Extending the procedure of Section IV-C from regulation to tracking, we combine (37), which expresses the effect of non-

u = −λL v e(t) + u v (t) (35) holonomy, with (40), which models the effect of the control law,

CHEVALLEREAU et al.: ELECTRIC SENSOR-BASED CONTROL OF UNDERWATER ROBOT GROUPS 613

Fig. 9. Electrolocation test bench.

(b) and obtain the evolution equation of dθ

(a)

Fig. 10. Experimental test bench, the allowed motion of the active sensor is

(Z x (X v ) sin(θ v )−Z y (X v ) cos(θ v )) d ˙θ

the straight line and the motion of passive sensor is nonholonomic. (a) Test bench top view. (b) Test bench side view.

+ (V v +v x sin(θ v )−v y cos(θ v ))dθ = 0

via the control inputs u = [V, ω] from which we deduce the following three cases. T based on the electric currents

1) Starting from dθ = d ˙θ = 0 ensures a perfect tracking of feedback. Both agents are in the same horizontal plane. The the virtual agent motion by the real agent.

sampling period of the bench is T e = 0.015 s.

To illustrate the control laws in our tank, we now report the which occurs, for example, when the virtual agent stops, results of three experiments. In the first two experiments, the then the error will be kept constant.

2) If V v +v x (X v ,˙ X v ) sin(θ v )−v y (X v ,˙ X v ) cos(θ v ) = 0,

active agent moves in a straight line, while the passive one

V v +v x (X v ,˙ X v ) sin(θ v )−v y (X v ,˙ X v ) cos(θ v 3) If ) γ=

follows it in order to maintain a given relative configuration

Z x (X v

) sin(θ )−Z y (X v ) cos(θ v )

> 0, the

with y

d d = 0 in the second (in the second case the two agents are aligned). The third experiment

motion of the real agent will converge to the motion of

the virtual agent. consists of switching between the two previous configurations For a desired motion of the virtual agent, the controlled (from a formation where the two probes are aligned to another measurement Cm(t) will be chosen to avoid singularity of

C ∂m (X v (t))J(X v

where they are not), while the active agent is kept fixed. For all

(t)) and to ensure γ > 0. experiments, the speed of convergence of the error dynamics

∂X

∂m ( X d )

XPERIMENTS VII. E

is such that λ = 0.7, while the matrices C, ∂X and L are precomputed with simulations on our fast analytic simulator. All

A. Electrolocation Testbed

the desired measurements m X d

In order to test our sensors [4] and algorithms under controlled in a preliminary phase that allows emancipating ourselves from and repeatable conditions, an automated test bench consisting the influence of the conductivity fluctuations. of a cubic tank of 1 m side with insulating walls, filled with fresh water, and a three-axis cartesian robot has been built (see

B. First Motion in Formation

Fig. 9). The robot fixed on top of the aquarium allows probes The desired posture of the follower in the leader frame is fixed positioning in translation along

X and Y with a precision of to x d = −0.22 m, y d = −0.10 m, and θ d = 0 as in Section V. 1/10 mm and the orientation in the (X, Y ) plane is adjusted

We define a motion of the leader in three phases. In Phase A, the in 0.1 using an absolute yaw-rotation stage. The two agents leader is at rest and the follower is positioned in the leader frame

tested are positioned in the aquarium at adjustable height using with an initial error. In Phase B, we move the leader ( V r > 0,

a rigid perch. This vertical insulating tube allows the passage ω r ≃ 0). In Phase C, the leader is stopped. of electrical cables dedicated to the signals coming from the

Electric Sensor-Based Control of Underwater Robot Groups