Multi-agent coverage control design with dynamic sensing regions

Longbiao MA,Fenghua HE,Long WANG,Yu YAO

Control and Simulation Center,Harbin Institute of Technology,Harbin Heilongjiang 150080,China

Abstract A cooperative region reconnaissance problem is considered in this paper where a group of agents are required to reconnoitre a region of interest.A main challenge of this problem is the sensing region of each agent varies with its altitude within an altitude constraint.Meanwhile,the reconnaissance ability of an agent is determined by its altitude and radial distance.First,the region reconnaissance is formulated as an effective coverage problem,which means that each point in the given region should be surveyed until a preset level is achieved.Then,an effective coverage control law is proposed to minimize coverage performance index by adjusting the altitude of an agent.Finally,the effectiveness of the proposed control law is verified through numerical simulations.

Keywords:Effective coverage,multi-agent systems,region reconnaissance,altitude constraint,dynamic sensing regions

1 Introduction

In recent years,there has been a growing interest in the area of coverage control of multi-agent systems.Coverage control can be broadly classified as static ones such as[1–4]and dynamic coverage ones such as sweep coverage problem[5]or effective coverage problems[6,7].The static coverage problem can be considered as a local optimization problems where the main objective is the optimal placement of sensors to cover a given region[8].The dynamic coverage problem is fundamentally different than the static one.The objective in dynamic coverage control is to develop control strategies for a group of mobile agents with limited range sensors such that the coverage level reaches a desired value at every point[8].We refer to it as dynamic because the agents move around to explore the given area instead of converging to a final configuration.In the field of dynamic coverage problems,mobility is the key feature.

Effective coverage control,which belongs to dynamic coverage control,over a planar region by ground agents has been studied extensively.A coverage error function was proposed to formulate the effective coverage problem in[9,10],and centralized control laws were designed to minimize the coverage error function.In[11],the authors provided more insight on some technical issues regarding the derivative of the coverage error function introduced in[9,10],and control laws were proposed to solve the effective coverage problem for agents with affine nonlinear dynamics.In[12],an awareness model was proposed to reformulate the effective coverage problem differently.It is assumed that the update of a discrete information takes place between agents whenever they get sufficiently close to each other.Inspired by the approach of[12],an adaptive coverage control design was proposed in[13]that the agents estimate the underlying density of a given region under the adaptive control law while pursuing the coverage objective.

Meanwhile,a few of effective coverage algorithms have been developed in applications involving unmanned aerial vehicle(UAV)and autonomous underwater vehicle(AUV).Extensive researches have been conducted in the field of area search by UAVs equipped with cameras[14–17].Also,the effective coverage control problem for underwater applications using a fleet of cooperative submarines with vision-based cameras was studied in[18]and the coverage goal was to collect a desired amount of satisfactory quality samples at every point in a given region.Based on assumption that the fleet was operating in a plane perpendicular to the direction of gravity(i.e.,no vertical motion),a gradient descent control law was designed so that the coverage goal is achieved.

However,in these pioneering research efforts,the sensing region for each agent is time invariant.Meanwhile,no constraints are imposed on the altitude of the UAV or AUV,and the overlap avoidance between the sensing region of different agent is not considered.

This paper studies the effective coverage problem for a group of UAVs,which is motivated by region reconnaissance application using visual sensors to reconnoitre a given region until each point in the given region is surveyed to a certain preset level.A main challenge of this problem is the sensing region of each agent varies with its altitude within an altitude constraint.Meanwhile,the reconnaissance ability of an agent is determined by its altitude and radial distance(the distance between the projection of the centre of UAV on the ground and a point in the sensing region).The altitude control of a UAV is considered within altitude constraints which is different from the work in[18].Each UAV is assumed to have downwards facing vision sensor with a conical field ofview,which creates a circle sensing region on the ground.The sensing region is dependent on the altitude of UAV.UAV at higher altitude can cover more region but the reconnaissance ability is lower compared to UAV at lower altitudes.In order to avoid overlap between the sensing regions of different agents,we give a novel coverage performance index to describe it.Then,based on partitioning scheme of the sensed region,we provide an effective coverage control law to drive the agent networks to minimize the coverage performance index and guarantee all the agents remaining within a predefined altitude range,and overlap between the sensing regions of different agents is avoided.To verify the effectiveness of the proposed control law,we compare the result in this paper with that of[18]through numerical simulations.

The rest of this paper is organized as follows.In Section 2,the region reconnaissance is formulated as an effective coverage problem.In Section 3,an effective coverage control law is designed and its convergence is proved.Simulation results are provided to illustrate the performance of the proposed control law in Section 4.Section 5 concludes the paper and describes directions for future work.

2 Problem formulation

In this section,the region reconnaissance is formulated as an effective coverage problem.First,a simplified dynamics model of an agent is given.Then,the reconnaissance ability of an agent is described by a function with respect to its altitude.Finally,an effective coverage performance index is constructed.

2.1 Dynamics model of agent

In this paper,we consider a scenario in which a group of agents cooperatively reconnoitre a region of interest using their vision sensors.Before our problem formulation,a global coordinate frame OX Y Z is defined as follows:the origin point of the coordinate OX Y Z is set to be center of the region of interest.OX and OY axis are in the horizon,pointing to the east and north direction,respectively;OZ axis is determined by the right-hand rule.

For simplicity,each agent is considered to be a point mass.We consider a group of n agents,and the i th agent is denoted by Ai.The dynamic model of Aiis described by

where i∈In={1,...,n},Xi=[xi,yi,zi]Tis the position vector of agent Ai,ui,x,ui,yand ui,zare the control inputs of Aialong the axis of OX,OY and OZ,which are assumed to be bounded,i.e.,denotes the altitude of Aiwhere the altitude is within a constraint.The minimum and maximum altitude of Aiare denoted byrespectively,i.e.,It is assumed thatare same for all agents.Let zminand zmaxdenote the minimum altitude and maximum altitude for all agents,respectively.The minimum altitude zminis assumed zmin＞0 to ensure all the agents will fly above ground obstacles,whereas the maximum altitude zmaxis determined by the reconnaissance ability of each agent.

2.2 Reconnaissance ability function

Let qi=[xi,yi]Tbe the projection position of the center of Aiin the OX Y plane.An illustration of cooperative reconnaissance of n agents is shown in Fig.1,where Xiis the position vector of Ai,Ω denotes the given region under reconnaissance in the plane OXY,Siis sensing region of Aiin the OX Y plane.Assume each agent is equipped with a downwards facing vision sensor with the angle-of-view αi.In the sequel,we assume that all the agents have the same angle-of-view,i.e.αi= α.The sensing region Siin the OXY plane is a circular area,which can be expressed by

where q∈Ω.

Fig.1 Reconnaissance coverage concept.

Now,let di= ‖qi? q‖denote the radial distance between the projection of Xion the ground and a point in the sensing region Si.Then,a function f(zi,di)∶[zmin,zmax]×[0,zitanα]→ [0,1]is defined to describe reconnaissance ability of Ai,which is dependent on the altitude constraints zmin,zmaxof Aiand di.The function f(zi,di)is required to have the following properties:

i)The reconnaissance ability with respect to Aishould be zero outside Si(Xi,α),i.e.,f(zi,di)=0 if q? Si.

ii)Due to the decrease of reconnaissance ability while the increase of its altitude and di,f(zi,di)should be a decreasing function of ziand di.

iii)Since the altitude ziis within a constraint zmin≤zi≤zmax,thus,f(zi,di)=1 when zi=zminand di=0,and f(zi,di)=0 when zi=zmax.

iv)f(zi,di)is first order differentiable with respect toexists within Si(Xi,α).The integrals over some part of?Si(Xi,α)are not zero.

v)Since the sensing patterns are rotation invariant,f(zi,di)is symmetric around the OZ-axis.

The definition of the reconnaissance ability function is not unique.In this paper,the reconnaissance ability function is chosen as follows:

where 0≤a≤1,

and Ez=zmax?zmin.

where

The function f(zi,di)is shown in Fig.2(a=0.3).As shown in Fig.2,this function has the property that f(zmin,0)=1 and f(zmax,di)=0.In addition,f(zi,di)is first order differentiable with respect to ziand qi,exist within Si,which is required in the control law design in Section 3.

Fig.2 Reconnaissance ability function.

2.3 Coverage performance index

The aim of the effective coverage problem under consideration is to cover a given region using the sensing regions of multi-agent such that all the points in the given region are surveyed until a preset level is achieved.

The effective coverage achieved by Aisurveying one point in the given region Ω from an initial time t0=0 to time t is defined as follows:

The effective coverage by a group of n agents surveying one point in the given region Ω is given by

Let C＊(q)be the desired effective coverage level of each point in region Ω.The definition of effective coverage of all points in the region Ω by n agents is given as follows:

Definition 1We say that the effective coverage of all the points in region Ω can be achieved by the multiagent networks,if ΥIn(q,t)=C＊(q)at some time t for?q∈Ω.

Remark 1The effective coverage is a kind of sweep coverage.The objective of sweep coverage is to move a group of agents across a coverage area in a manner which addresses a specified balance between maximizing the number of detections per time and minimizing the number of missed detections per area.The purpose of effective coverage is to design control law to address a specified balance between maximizing the area in which the points are effective covered with quantative performance measurement C＊(q)and minimizing the area in which the points are not effective covered.

Let q denote one point in Ω.A space density function Φ∶Ω→ R+is assigned to describe the probability that some event takes place over Ω.Then,the coverage performance index for effective coverage by multi-agent networks can be described as

where h(x)is a penalty function that is positive definite,twice differentiable,strictly convex on(0,C＊(q)]and satisfies h(x)=h′(x)=0,for all x ≤ 0.Positive definite and strict convexity here refer to that h(x)=h′(x)=h′′(x) ＞ 0,for all x ∈ (0,C＊(q)].This definition for h(x)is chosen to prevent a negative contribution to the coverage performance index at the points q|ΥIn(q,t) ＞ C＊(q).H(t)can be regarded as a measure of how better the coverage provided by the multi-agent networks is.As H(t)→0 means that each point in region Ω has been covered effectively.Then H(t)=0,means that the mission is accomplished.Thus we are interested in minimizing H(t)by designing control law of each agent in the next section.

3 Control law design

In this section,an effective coverage control law is proposed which consists of a nominal control law and a perturbation control law.Under the nominal control law,Aican be driven to a condition where all points in its sensing region are fully covered.However,the coverage performance index function H(t)is not driven to a neighborhood of zero.Hence,the perturbation control law is designed which can drive Aito a desired destination point that is not covered satisfactorily.

Before designing the control law,a region partitioning method is developed.

3.1 Sensed region partitioning

In order to avoid overlapping between sensing regions of different agents,a partitioning scheme is given based on the reconnaissance ability.The partitioning scheme is achieved in a manner similar to[19],where only the subset of Ω sensed by all the agents is partitioned.It means thatis partitioned by(11)and Aiis assigned a cell

with the equality holding true only zi=zj,so that the cells Wicomprise a complete tessellation of the sensed region

For the region partitioning,it is necessary to solve f(zi,di)≥ f(zj,dj)in order to find part of?Wj∩Wi.The resulting solution on the OX Y plane is a diskwhen zi≠ zj.The cell of Aiwith respect to Ajis Wi=Si∩if zi＜zjand Wi=Si(Sj∩if zi＞zj.If zi=zj,the halfplane defined by the perpendicular bisector of qiand qj.If the sensing region Siis contained within the sensing region Si,then Wi?Siand Wj=SjWi.

In the general case,it should be noted that?Wi∩?Wjconsists of circular arcs from bothand ?Sjor?Si.An example of partitioning with all of the aforementioned cases illustrated can be seen in Fig.3,where the boundaries of the sensing regions?Siare in dashed and the boundaries of the cells?Wiare in solid black.

Fig.3 Region partitioning examples.

A1,A2and A3illustrate the general case.A4and A5are at the same altitude so the arbitrary partitioning scheme is used.The sensing region of A6contains the sensing region of A7.

Remark 2The aforementioned partitioning is a complete tessellation of the sensed region.However it is not a complete tessellation of Ω.The neutral region witch is not assigned by the partitioning schemeis denoted as Θ=ΩThe cells Wiare compact but they are not always convex.It is also possible that a cell Wiconsists of multiple disjoint regions,such as the cell of A2which is shown in Fig.3.

3.2 Nominal control law

Let ui,q=[ui,x,ui,y]T∈R2.Then,ui,qand ui,zare the corresponding control inputs for Ai.The effective coverage control law consists of a nominal control law and a perturbation control law.In this subsection,the nominal control law of Aiis designed.Letˉui,qandˉui,zdenote the nominal control law of Ai.Based on the agent’s dynamics(1),the reconnaissance ability(3)and the coverage performance index(10),the nominal control laws are presented as follows:

Before proceeding,the following condition is introduced.

Condition 3.1C＊(q)= ΥIn(q,t),?q∈ Wi,i∈ In.

This condition describes a coverage condition where all points in the sensing region of Aiare satisfactorily covered.The nominal control laws(12)and(13)result in Condition 3.1 satisfaction.Then the lemma is given as follows.

Lemma1Consider the agent’s dynamics(1)and the reconnaissance ability(3),the nominal control laws(12)and(13)drive each agent converge to Condition 3.1.

ProofSelect a function

where

In view of the properties of f(zi,di),we have f(zi,di)≠0 if q∈Siand f(zi,di)=0 if q?Si.Thus,V(t)can be rewritten as

Based on the partitioning scheme(11),the function V(t)can be rewritten as

Then,the derivative of V(t)with respect to t can be obtained as follows:

where h′′=h′′(C＊(q)? ΥIn(q,t)).

According to the Leibniz integral rule[20],we obtain

in which the two previous terms indicate how the movement of Aiaffects the boundary of its cell and the boundaries of the cells of other agents.It is clear that only the cell Wjwhich has a common boundary with Wiwill be affected.Thus,the boundary?Wican be decomposed to disjoint sets shown as follows:

These sets represent the parts of?Withat lie on the boundary of Ω,the boundary of the sensing regions which have common boundary between the boundary of Wiand those of other cells.An example of the decomposition of?Wican be seen in Fig.4.In Fig.4,the sets?Wi∩ ?Ω,?Wi∩ ?Θ and?Wi∩ ?Wj)appear in solid green line.

Fig.4 Boundary decomposition into disjoint sets.

Since the boundary ?Wi∩?Wjbelongs to both Aiand Aj,it is true that=and nj= ?ni.Thus,the sums and the integrals within them can be combined.Then,can be rewritten as

where ef=f(zi,di)?f(zj,dj).

Similarly,by using the decomposition of?Wiand defining

Compare the forms of the nominal control law of Aiwith(24)and(27),we have

Substituting(28)into(18),we can obtain

It is clear that˙V(t)≤0.The equality holding true only C＊(q)= ΥIn(q,t),?q ∈ Wi,i ∈ Inand V(t) ≥ 0 and V(t)=0 if and only if C＊(q)= ΥIn(q,t),?q ∈ Wi,i∈ In.Thus,Condition 3.1 is achieved under the control laws(12)and(13). □

Remark 3For the initial locations of Ai,one of the following statements holds:

i)the interior of sensing regions of different agents intersects with the boundary of?Ω,i.e.,?Si∩ ?Ω ≠ ?,?i∈ In;

ii)or interior of the sensing region is inside the given region Ω.

Note that if 1)and 2)are violated,the control law(12)and(13)are guaranteed to converge to zero.

3.3 Perturbation control law

Under the nominal control law,the agents can be driven to Condition 3.1 where all points in its sensing region are fully covered.However,the satisfaction of Condition 3.1 does not imply that the coverage performance index function H(t)is driven to zero.Meanwhile,when Condition 3.1 holds,the nominal control law(12)and(13)is zero.Hence,other control law is needed to be designed which can drive Aito a desired destination point that is not covered satisfactorily so that the Condition 3.1 is violated.Once away from Condition 3.1,the nominal control law is not zero and the controller is switched back to the nominal control laws.This procedure is repeated until the coverage performance index function H(t)is zero.

Under the control law(12)and(13),all agents are continuous moving as long as Condition 3.1 is avoided.Whenever Condition 3.1 holds with H(t)≠0,Aihas to be perturbed by switching to some other control law that ensures violating Condition 3.1.Once away from Condition 3.1,the controller is switched back to the nominal control in(12)and(13).When both Condition 3.1 and H(t)=0 are satisfied,there is no need to switch to(12)and(13).Thus,the goal is to propose a simple linear feedback controller which guarantees driving the system away from Condition 3.1.

Define the time varying set:

This choice is efficient since the perturbation maneuver seeks the minimum-distance for redeployment.

Let tsbe the time at which Condition 3.1 holds and H(ts)＞0.That is,tsis the time of entering into Condition 3.1 with H(ts)≠0.At ts,for Ai,consider a pointNote that the setmay include more than a single point.In this case,any rule can be assigned to pick up a point(ts).This rule is immaterial in the present discussion,and we will simply assume there is a single such point.Consider a simple perturbation control law shown as

where

which denotes the attraction potential field function,is attraction potential force driving the Aito the destination point

When Condition 3.1 holds and H(t)＞0,this control law is a simple linear feedback controller and will drive Aitowards its associated pointAccording to linear systems theory,the feedback control law(32)will result in havingfor some i∈In,being inside a circle of radiustanα at some timeHence the pointis guaranteed to lie strictly inside the sensing range of Ai.

3.4 Overall control law

Now,we can present the main result.

Theorem 1Consider the agent’s dynamics(1)and their reconnaissance ability(3),the control laws

drive the H(t)→ 0 as t→ ∞.

ProofUnder the nominal control law,Aiis driven to the state described in Condition 3.1.Whenever Condition 3.1 holds and the coverage performance index function H(t)is not driven to a neighborhood of zero,Aiis perturbed from Condition 3.1 by switching to the perturbation control law(32).Once away from Condition 3.1,the controller is switched back to the nominal control laws(12)and(13).This procedure is repeated until the set Γ(t)is empty.When the set Γ(t)is empty,via the definition of Γ(t),one has

It implies that?q∈ Ω,h(C＊(q)?ΥIn(q,t))=0.Then,we have H(t)→0 as t→ ∞.

Next,we prove that switching between the nominal control law and the perturbation control law is finite.Consider the following function:

When the nominal control lawsandare employed,(t)＜0 always holds since Condition 3.1 is not satisfied.Thus,the function H(t)decreases by an amount of non-zero value during the applications ofandWhen Condition 3.1 is satisfied and the set Γ(t)is not empty,the perturbation control lawis used.Then,H(t)is non-increasing.It implies that a region of measure larger than zero is covered after every single switch(fromtoand since the region Ω is compact,there will be no infinite switch.Thus,finite switching will be performed to guarantee that the set Γ(t)is empty.Clearly,if Ω is open or unbounded,there is no guarantee that after each switch the nonzero measure region covered by the formation will eventually cover the entire region.This is the main reason for requiring that Ω be compact.

Remark 4Note that if Condition 3.1 holds,the control lawis in effect.Once it converges toand H(t)=0,then the goal is met and the control converges to=0.If H(t)≠0,the controller switches back toSwitching will recur until H(t)=0.

4 Numerical simulations

In this section,we will give some numerical examples to illustrate the performance of the control laws(33)and(34).First,the effectiveness of the proposed coverage control law is verified when the space density function Φ(q)is equal to unity.Moreover,the proposed control law has better convergence rate compared with the control law in[18].Then,the effectiveness of the proposed coverage control law is verified when the space density function Φ(q)is set to a two dimensional Gaussian function.Finally,the effect of initial position of agent on the convergence rate of coverage performance index is revealed via simulations.

The given region Ω is a square region of side length d=1.7 units length,the desired effective coverage level C＊=0.3,the control gains in the control laws(33)and(34)are set to be kqi=0.8,kzi=1.2 and kqi=0.22.It is assumed thatfor all agents.A simple trapezoidal method is used to compute integration over Wi,?Wi∩?Wjand ?Wi∩Θ and a simple first order Euler scheme is used to integrate with respect to time.

4.1 Case study I

In this case,the number of agent is selected identical to that used in[18]for consistency,while the space density function Φ(q)is set equal to unity for visualisation purposes.This numerical example is to illustrate the performance of the control law(33)with the exception of the space density function Φ(q)=1.There are 4 agents(n=4)with a randomly selected initial deployment as shown in Fig.5(b).Fig.5(a)shows the 3D trajectories of four agents and their projections on Ω is denoted by black lines.The initial positions of the agents are marked by squares and their final positions by circles.It can be seen that guarantees all the agents remain within a predefined altitude range[0.2,2.0].Initial and final network configuration in Ω can be seen in Fig.5(b)and Fig.5(c).Fig.5(d)shows the H(t)with switching control converge to zero.Compare to the control law developed in[18],under the control law proposed in this paper,H(t)convergence to the zero faster.

4.2 Case study II

This simulation is identical to the previous one with the exception of the space density function Φ(q)which is set to a two dimensional Gaussian function.In this case the agents minimize the coverage performance index while also taking into account the importance of each point in the region as expressed by the space density function.The trajectories of the agents in three dimensional space and their projections on the given region can be seen in Fig.6(a).The initial positions of the agents are marked by squares and their final positions by circles.Fig.6(b)and Fig.6(c)show the initial and final configurations of the network,as well as the value of the function Φ(q)using a color gradient.It can be seen that the sensing region of each agent varies with its altitude and agents converge around the peak of the Gaussian function.Fig.6(d)shows the evolution of the coverage performance index.It can be seen that H(t)converges to zero under the switching control law.

Fig.5 (a)The trajectories of four agents and their projections on Ω.(b)Initial configuration of four agents.(c)Final network configuration of four agents.(d)The evolution of the coverage performance index H(t).

Fig.6 (a)The trajectories of four agents and their projections on Ω.(b)Initial network configuration of four agents.(c)Final network configuration of four agents.(d)The evolution of the coverage performance index H(t).

4.3 Case study III

In order to illustrate the effect of initial position of agent on the convergence rate of coverage performance index,four cases of initial configuration of five agents are considered as follows:

i)the projection positions of five agents are grouped inside the given region,which is shown in Fig.7(a);

ii)the projection positions of two agents are outside of the given region,and that of three a gens are scattered inside the given region,which is shown in Fig.7(b);

iii)the projection positions of five agents are scattered outside the given region,which is shown in Fig.7(c);and

iv)the projection positions of five agents are scattered inside the given region,which is shown in Fig.7(d).

The convergence trajectory of H(t)in different cases are shown in Fig.8.

Fig.8 Convergence rate of H(t)with different initial configuration.

From the results,it can be seen that the convergence rate in iv)is fastest and that in i is slowest.From Figs.7(a)–(d)and 8,it can be seen that the more scattered initial configuration of agents are,the faster the convergence rate of H(t)is.

5 Conclusions and future works

A region reconnaissance problem of multi-agent systems is considered in this paper in which a group of agents reconnoitre a given region until each point in the given region is surveyed to a certain preset level.The region reconnaissance of multi-agent systems is formulated as an effective coverage problem.The sensing region is described as a circle which varies with its altitude,and the reconnaissance ability function of each agent is expressed by a function with respect to the altitude of agent and its radial distance.A novel coverage performance index in given to describe the effective coverage problem.Based on region partitioning scheme,an effective control law is proposed to minimize the coverage performance index and guarantee all the agents remaining within a predefined altitude range,and overlap between the sensing regions of different agents is avoided.Numerical simulations are carried out to validate the efficiency of the proposed control law.

For future investigation,there are still some interesting questions:

i)How to design effective control law when the agents are deploying for reconnaissance in geometrically complex environment,such as the urban areas and mountainous?

ii)Since the part that was covered should be avoided when the agent moves,it may be extended to some regions with obstacles(the obstacle part can be viewed as a part that was covered).

iii)Theoretical analysis on the relationship between initial position of agent and the convergence rate of coverage performance index is needed.

First,ni,andon ?Wi∩Θ are considered,which is always an arc of the circle ρi(k)because of the partitioning scheme(11).The normal vector niis given by

It can be shown that

and similarly that

resulting in·ni=tanα,q ∈ ?Wi.

Then,ni,andon ?Wi∩ ?Wjneed to be calculated.Note that?Wi∩ ?Wjconsists of circular arcs from both ?and ?Sjor?Si.Since q∈ ?means that ef=0,the integral over?is zero.Thus,the remaining part of?Wi∩ ?Wjare circular arcs ?Sjor?Si.If ef=0,the value of ni,anddose not need to be considered since the corresponding integral will be 0 due to the ef=0 term.If ef＞0,according to the partitioning scheme(11),?Wi∩ ?Wjwill be an arc of ρ(k).Thus,ni,andis the same as it is over?Wi∩ Θ.If ef＜ 0,according to the partitioning scheme(11),?Wi∩?Wjwill be also an arc of ρj(k).Thus bothandwill be 0,since Sj(Xi,α)is not dependent on Xi.From the above,ni,andover?Wi∩?Wjcan be given as follows: