Different parameters of our physics-inspired feature decomposition method PAGE.
\r\n\t
",isbn:"978-1-83881-922-4",printIsbn:"978-1-83881-921-7",pdfIsbn:"978-1-83881-923-1",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,hash:"dcfc52d92f694b0848977a3c11c13d00",bookSignature:"Dr. Fiaz Ahmad and Prof. Muhammad Sultan",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10454.jpg",keywords:"Agricultural Engineering, Technologies, Application, Sustainable Agriculture, Information Technology in Agriculture, Food Security, Renewable Energies, Precision Farming, Smart Agriculture, Farm Mechanization, Robotics, Post Harvest Technologies",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 25th 2020",dateEndSecondStepPublish:"December 23rd 2020",dateEndThirdStepPublish:"February 21st 2021",dateEndFourthStepPublish:"May 12th 2021",dateEndFifthStepPublish:"July 11th 2021",remainingDaysToSecondStep:"25 days",secondStepPassed:!0,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Ahmad is a researcher in the field of agricultural mechanization and agricultural equipment engineering, in-charge of Farm Machinery Design Laboratory at Bahauddin Zakariya University, with expertise in modeling and simulation. He applied for two patents at the national level.",coeditorOneBiosketch:"Renowned researcher with a focus on developing energy-efficient heat- and/or water-driven temperature and humidity control systems for agricultural storage, greenhouse, agricultural livestock and poultry applications including HVAC, desiccant air-conditioning, adsorption, Maisotsenko cycle (M-cycle), and adsorption desalination.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"338219",title:"Dr.",name:"Fiaz",middleName:null,surname:"Ahmad",slug:"fiaz-ahmad",fullName:"Fiaz Ahmad",profilePictureURL:"https://mts.intechopen.com/storage/users/338219/images/system/338219.jpg",biography:"Fiaz Ahmad obtained his Ph.D. (2015) from Nanjing Agriculture University China in the field of Agricultural Bioenvironmental and Energy Engineering and Postdoc (2020) from Jiangsu University China in the field of Plant protection Engineering. He got the Higher Education Commission, Pakistan Scholarship for Ph.D. studies, and Post-Doctoral Fellowship from Jiangsu Government, China. During postdoctoral studies, he worked on the application of unmanned aerial vehicle sprayers for agrochemical applications to control pests and weeds. He passed the B.S. and M.S. degrees in agricultural engineering from the University of Agriculture Faisalabad, Pakistan in 2007. From 2007 to 2008, he was a Lecturer in the Department of Agricultural Engineering, Bahauddin Zakariya University, Multan-Pakistan. Since 2009, he has been an Assistant Professor in the Department of Agricultural Engineering, BZ University Multan, Pakistan. He is the author of 33 journal articles. He also supervised 6 master students and is currently supervising 5 master and 2 Ph.D. students. In addition, Dr. Ahmad completed three university-funded projects. His research interests include the design of agricultural machinery, artificial intelligence, and plant protection environment.",institutionString:"Bahauddin Zakariya University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Bahauddin Zakariya University",institutionURL:null,country:{name:"Pakistan"}}}],coeditorOne:{id:"199381",title:"Prof.",name:"Muhammad",middleName:null,surname:"Sultan",slug:"muhammad-sultan",fullName:"Muhammad Sultan",profilePictureURL:"https://mts.intechopen.com/storage/users/199381/images/system/199381.jpeg",biography:"Muhammad Sultan completed his Ph.D. (2015) and Postdoc (2017) from Kyushu University (Japan) in the field of Energy and Environmental Engineering. He was an awardee of MEXT and JASSO fellowships (from the Japanese Government) during Ph.D. and Postdoc studies, respectively. In 2019, he did Postdoc as a Canadian Queen Elizabeth Advanced Scholar at Simon Fraser University (Canada) in the field of Mechatronic Systems Engineering. He received his Master\\'s in Environmental Engineering (2010) and Bachelor in Agricultural Engineering (2008) with distinctions, from the University of Agriculture, Faisalabad. He worked for Kyushu University International Institute for Carbon-Neutral Energy Research (WPI-I2CNER) for two years. Currently, he is working as an Assistant Professor at the Department of Agricultural Engineering, Bahauddin Zakariya University (Pakistan). He has supervised 10+ M.Eng./Ph.D. students so far and 10+ M.Eng./Ph.D. students are currently working under his supervision. He has published more than 70+ journal articles, 70+ conference articles, and a few magazine articles, with the addition of 2 book chapters and 2 edited/co-edited books. Dr. Sultan is serving as a Leading Guest Editor of a special issue in the Sustainability (MDPI) journal (IF 2.58). In addition, he is appointed as a Regional Editor for the Evergreen Journal of Kyushu University. His research is focused on developing energy-efficient heat- and/or water-driven temperature and humidity control systems for agricultural storage, greenhouse, livestock, and poultry applications. His research keywords include HVAC, desiccant air-conditioning, evaporative cooling, adsorption cooling, energy recovery ventilator, adsorption heat pump, Maisotsenko cycle (M-cycle), wastewater, energy recovery ventilators; adsorption desalination; and agricultural, poultry and livestock applications.",institutionString:"Bahauddin Zakariya University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Bahauddin Zakariya University",institutionURL:null,country:{name:"Pakistan"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"8",title:"Chemistry",slug:"chemistry"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"56327",title:"Motion Coordination Problems with Collision Avoidance for Multi-Agent Systems",doi:"10.5772/intechopen.69845",slug:"motion-coordination-problems-with-collision-avoidance-for-multi-agent-systems",body:'\nMulti-agent systems are defined as bundles of multiple autonomous robots coordinated to accomplish cooperative tasks. In recent years, the study of multi-agent systems has gained special interest, because these systems can achieve tasks that would be hard or impossible to achieve by agents working individually. Multiple agents can solve tasks working cooperatively, making them more reliable, faster and cheaper than it is possible with a single agent [1].
\nThe main applications of multi-agent systems include the transport and manipulation of objects, localization, exploration and motion coordination [1, 2]. The main idea of motion coordination is the strategic navigation of a group of agents. Some of the main areas of research in the motion coordination are the formation control, where the goal is to achieve a desired pattern defined by relative position vectors, the time-varying formation tracking control, where the goal is to track a pre-established trajectory while the agents maintain a time-varying desired formation and the time-varying containment control, which consists in a group of mobile agents (called leaders) that track a predetermined trajectory, while another group of agents (called followers) remain within the region determined by the leaders [3].
\nThe time-varying formation problem has been scarcely studied and some examples can be found in [4–7]. The time-varying formation control can be applied as the solution to complex motion coordination problems. In our case, the time-varying formation allows trajectory tracking with formations oriented to the heading angle of a leader robot, as well as changes in the physical dimensions of the formations. More specifically, the time-varying formation is composed of a predefined static formation which is transformed by a rotation matrix, which depends on the orientation of a specific leader robot and a scaling matrix, which depends on a factor that varies with respect to time. This time-varying formation allows the group of agents to behave as a rigid body which can be translated, rotated and scaled in the plane.
\nAnother ubiquitous problem in all areas of motion coordination is the possible collision between agents when they try to achieve a desired position into a formation or during the trajectory tracking. In the literature, we can find different methods to predict/avoid collisions. In Ref. [8], a mechanism for collision avoidance under central control mode (traffic control type) is presented. In Refs. [9–11], navigation functions and artificial potential functions are used to avoid collisions between agents. These non-collision strategies are developed based on a combination of attractive potential functions (APFs) and repulsive potential functions (RPFs). Works [12–15] address the formation control problem without collisions using discontinuous vector fields.
\nThe interaction topology between agents is modelled by formation graphs, where each agent is represented by a vertex, and the sharing of information between agents is represented by an edge. The control strategies designed in this work are presented for differential-drive mobile robots. This kind of mobile robots is commonly chosen as test bed because of simplicity and commercial availability. Differential-drive mobile robots present interesting challenges because they possess non-holonomic restrictions and even though have a simple kinematic model, it presents singularities. For this reason, the stabilization of such kind of mobile robots has been studied for several years by researches from diverse viewpoints.
\nThe goal of this chapter is to design decentralised control strategies that allow motion coordination for multi-agent systems avoiding collisions between agents. The non-collision strategy is based on previous works [16, 17]. We use bounded control strategies based on sigmoid functions adding a repulsive vector field.
\nLet \n
Kinematic model of the differential-drive mobile robot.
where v\n\ni\n is the longitudinal velocity of the middle point of wheels axis of the ith robot, w\n\ni\n its angular velocity and θ\n\ni\n the orientation with respect to the X axis. It is known that systems like Eq. (1) cannot be stabilised by any continuous and time-invariant control law [18]. Moreover, if the position ξ\n\ni\n is taken as output of the system Eq. (1), the so-called decoupling matrix becomes singular. For this reason, to avoid singularities in the control law, it is common to study the kinematics of a point α\n\ni\n off the wheels axis. The coordinates of point α\n\ni\n are given by
\nThe kinematics of point α\n\ni\n is given by
\nwhere A\n\ni\n(θ\n\ni\n) is the decoupling matrix for each robot R\n\ni\n. The decoupling matrix is non-singular since \n
\nDefinition 1. (Formation Graph). Let \n
\nA set of vertices \n
\nA set of edges \n
\nA set of labels \n
\nDefinition 2. (Laplacian). Let us have a formation graph G, the Laplacian associated with G is given by\n
\n\nWith ∆ the degree matrix defined by\n
\n\nwhere \n\n\n\ng\ni\n\n=\ncard\n\n{\n\n\nN\ni\n\n\n}\n\n,\ni\n=\n1\n,\n…\n,\nn\n\n\n\n and \n\n\n\nA\nd\n\n\n\n is the adjacency matrix of G defined by\n
\nGiven a formation graph G, there exist a path in this graph if between the vertices R\n\ni\n and R\n\nj, there is a sequence of edges \n
For further details about formation graphs, Laplacian and its properties and algebraic graph theory, the reader is referred to Refs. [19–21].
\n\nDefinition 3. [22, 23] Let \n
\n\n
\nThe real part of each eigenvalue of A is positive.
\nAll principal minors of A are positive.
\nA\n−1 exists and the elements of A\n−1 are non-negative.
\nDefinition 4. [24] The convex hull of a set of vectors \n
\nDefinition 5. Given a point \n
\nDefinition 6. Given a vector \n
\nDefinition 7. Given a matrix \n
\nDefinition 8. Let \n
\nWith \n
\nDefinition 9. [17] Consider the dynamical system \n
Let \n
where d is the minimum allowed distance between the agents. To avoid collisions between agents, we propose repulsive vector fields given by
\nwhere ϵ > 0 and the parameter δ\n\nij\n is given as follows
\nThe repulsive vector fields are proposed in such a way that there is an unstable focus that rotates anticlockwise as shown in \nFigure 2\n, centred on the position of the other agents that are in risk of collision.
\nPhase plane of the repulsive vector field β\n\nij\n.
For the control strategies designed in this chapter, we will take into account the following assumptions:
\n\nAssumption 1. The initial conditions of all agents satisfy \n
\nAssumption 2. The ith agent, besides knowing the position of the agents of the set N\n\ni\n, it can detect the presence of any other agent that is within the circle of radius d.
\nAlso, consider the following:
\n\nRemark 1. It should be clear that the minimum allowed distance between agents d must be less than the minimum distance between agents within a desired formation, i.e. \n
The desired relative position of the ith agent in a desired formation is given by
\nwhere c\n\nki\n is the position vector between agents R\n\ni\n and R\n\nk\n. The goal is to design a decentralised control law \n
The agents achieve a desired formation, i.e.
Collision avoidance among agents is achieved. In addition, for all time t, the agents remain at a distance greater than or equal to a predefined minimum distance d between them, i.e.
A control law to achieve a desired formation is given by
\nwhere \n
where \n
where μ is the constant velocity of all agents.
\n\nProposition 1. Consider the system Eq. (3) and the control law Eq. (17) with γ\n\ni\n given by (18) and a connected formation graph composed entirety by the superposition of different cycles. Then, in the closed-loop system Eqs. (3)–(17), we have finite time convergence of the agents to the desired formation.
\n\nProof. The proof of this Proposition is detailed in [17].▪
\nTo achieve formation with collision avoidance between agents, we propose a control law given by
\nWith γ\n\ni\n given by Eq. (18) and β\n\ni\n the repulsive vector field given by Eq. (11).
\n\nProposition 2. Consider the system Eq. (3) and the control law Eq. (19) with Eqs. (18) and (11). Also consider a connected formation graph composed entirety by the superposition of different cycles. Suppose that there exist risk of collision between n agents at time instant t and \n
\nProof. For the proof of this Proposition, mathematical induction is performed, first showing the cases of risk of collision between two agents and between three agents, applying induction to arrive at the general solution of n agents. This proof is detailed in Ref. [17]. It is worth mentioning that, geometrically, the worst case occurs when an agent is surrounded by other six agents. Also, the value of \n
The results obtained from a numerical simulation using the control strategy given by Eq. (19) are shown below. For the simulation, three differential-drive mobile robots are considered, where the point α\n\ni\n to be controlled is located at 0.045 m in front of the mid-point of wheels axis. The formation graph using in the simulation is shown in \nFigure 3\n.
\nFormation graph for the simulation (formation with collision avoidance problem).
The parameters used in the simulation are \n
\n\nFigure 4\n shows the motion of the agents in the plane. It is observed how the agents achieve the desired formation avoiding collisions. The effect of the repulsive vector fields can be seen when modified the trajectories of the agents to avoid collisions. In \nFigure 5\n, the distances between agents are shown, we can see that the minimum distance between agents is always greater than or equal to the predefined distance d = 0.2. \nFigure 6\n shows the position errors of the agents. Such errors converge to zero.
\nTrajectories of the agents in the plane (formation with collision avoidance problem).
Distances between agents (formation with collision avoidance problem).
Position errors of the agents (formation with collision avoidance problem).
In order to maintain a formation oriented to the direction of a leader agent R\n\nn\n and resize the formation, we use a time-varying position vector given by
\nwhere c\n\nji\n is a position vector corresponding to the static desired formation, R(θ\n\nn\n) is a rotation matrix given by
\nand δ(t) is a scaling factor. The time derivative of Eq. (14) is given by
\nwhere
\nIn the time-varying formation tracking problem presented in this subsection, the agent R\n\nn\n is the leader, responsible for tracking a desired trajectory. The n–1 remaining agents are follower, responsible for performing a time-varying formation with respect to the leader. The leader agent does not know the position and velocities of the followers agents but only knows the desired trajectory and velocity. The followers do not know the desired trajectory and velocity but only knows the positions and velocities of others agents in the system.
\nWe make the following standing assumption
\n\nAssumption 3. For each follower agent, there is a path to the leader agent, i.e., for all \n\n\n\nR\ni\n\n,\ni\n=\n1\n,\n…\n,\nn\n−\n1\n\n\n, there are edges \n\n\n(\n\nR\nn\n\n,\n\nR\n\n\nm\n1\n\n\n\n)\n,\n(\n\nR\n\n\nm\n1\n\n\n\n,\n\nR\n\n\nm\n2\n\n\n\n)\n,\n…\n,\n(\n\nR\n\n\nm\nr\n\n\n\n,\n\nR\ni\n\n)\n∈\nE\n\n\n\n.
\nLet \n
The desired relative position of the ith follower within the desired time-varying formation is given by
\nwhere C\n\nji\n(t) is a time-varying position vector between the agents R\n\ni\n and R\n\nj\n. The time derivative of C\n\nji\n(t) satisfies \n
The goal is to design a decentralised control law \n
Asymptotic tracking of a prescribed trajectory by the leader agent, i.e.
Asymptotic time-varying formation by the follower agents, i.e. for \n
Collision avoidance between agents, that is, all agents in the system remain at some distance greater than or equal to a predefined minimum distance d from each other, i.e.
To achieve time-varying formation tracking, we propose a control law defined by
\nwhere \n
The first main result of this subsection is the following:
\n\nProposition 3. Consider the system Eq. (3) and the control law Eq. (28). Suppose that \n\n\n\nk\nm\n\n,\n\nk\nf\n\n>\n0\n\n\n. Then, in the closed-loop system defined by Eqs. (3)–(28), it follows that the leader agent R\n\nn\n converge to the desired trajectory, i.e.\n\n\n\n\n\nlim\n\n\nt\n→\n∞\n\n\n\n\n(\n\n\nα\nn\n\n\n(\nt\n)\n\n−\nm\n\n(\nt\n)\n\n\n)\n\n=\n0\n\n\n, whereas the follower agents converge to the desired formation, i.e. \n\n\n\n\n\nlim\n\n\nt\n→\n∞\n\n\n\n\n(\n\n\nα\ni\n\n\n(\nt\n)\n\n−\n\nα\ni\n*\n\n(\nt\n)\n\n)\n\n=\n0\n\n\n, for \n\n\ni\n=\n1\n,\n…\n,\nn\n−\n1\n\n\n\n.
\n\nProof. The closed-loop system Eqs. (3)–(28) is given by
\nwhere \n
\n\n
At this point, we have to show that \n
Now define the errors of the system as
\nThe system errors in matrix form are given by
\nwhere \n
We propose a Lyapunov function candidate given by
\nand evaluating its time derivative along the trajectories of the system, we have
\nthen the errors converge asymptotically to zero.▪
\nModifying the previous control law Eq. (28) by adding the repulsive vector field Eq. (11), finally, we have the strategy to achieve time-varying formation tracking with collision avoidance given by
\nTo analyse the relative distance among the jth and ith agents, we define the variables \n
In order to present our second main result, we need to establish the following Technical Lemma.
\n\nLemma 1. Consider the system Eq. (3) and the control law Eq. (28) along with definitions \n
\nProof. Taking the norm of the system Eq. (29), we get
\nwhere \n
This concludes the proof.▪
\nNow, we can state our second main result. First, we consider the case when only two agents are in risk of collision. From this simplest case, we state a series of theorems leading to the general case.
\n\nProposition 4. Consider the system Eq. (3) and the control law Eq. (37). Suppose that there exists risk of collision between only two agents at time instant t and the parameter ϵ satisfies \n\n\nϵ\n>\n\n\nη\n^\n\n\n/\nd\n\n\n. Then, in the closed-loop system Eqs. (3)–(37), the agents tend asymptotically to their desired positions, and they stay at a distance greater than or equal to d, \n
\nProof. We show that the rth and sth agents will avoid collision between them, and they stay at some minimum distance from each other. Define a surface given by
\nTo determine the behaviour under the action of the repulsive vector fields, we use the positive definite function \n
Therefore, \n
Solving for ϵ, we have that, if \n
Now, we consider the case when three agents are in risk of collision, that is, agent R\n\nr\n is in risk of collision against agents R\n\ns1 and R\n\ns2.▪
\n\nProposition 5. Consider the system Eq. (3) and the control law Eq. (37). Suppose that there exists risk of collision between three agents and the parameter ϵ satisfies \n\n\nϵ\n>\n2\n(\n\n\nη\n^\n\n\n/\nd\n)\n\n\n. Then, in the closed-loop system, Eqs. (3)–(37), the agents converge asymptotically to their desired positions, and they stay at a distance greater than or equal to d, \n
\nProof. We define a surface composed of two components given by
\nWe use the positive definite function \n
In this scenario, agents R\n\ns1 and R\n\ns2 can be positioned at any point of the circumference of radius d around the agent R\n\nr\n, considering that, from Proposition 4, they must remain at a distance greater than or equal to d between them. The worst case occurs when the agents R\n\ns1 and R\n\ns2 are uniformly distributed over the circumference of radius d. Thus, \n
Geometrically, the most general case occurs when the rth agent is surrounded by six agents, i.e. seven agents are in danger of collision.
\n\nProposition 6. Consider the system Eq. (3) and the control law Eq. (37). Suppose that there exists risk of collision between n ≥ 3 agents and the parameter ϵ satisfies \n
\nProof. We follow a similar procedure to that presented in the proof of Proposition 5, considering a surface with n – 1 components and showing that, if \n
The results of a numerical simulation using the control strategy given by Eq. (37) are shown below. For the simulation, we considered five differential-drive mobile robots, where the point α\n\ni\n to control is located 0.15 m ahead the mid-point of the wheel axis. The formation graph employed in the simulation is shown in \nFigure 7\n.
\nFormation graph for the simulation (formation tracking with collision avoidance problem).
The control gains used in the simulation are k\n\nm\n = 2 and k\n\nf\n = 3. The desired marching trajectory is a quadrifolium curve given by \n
The minimum allowed distance between agents is d = 0.3 m and the parameter ϵ was set to \n
\n\nFigure 8\n shows the motion of the agents in the plane. The initial position of the agents is indicated with an ‘x’ and positions in different time instants are represented with a circle ‘o’. Is observed how the leader follows the desired trajectory while the followers achieve a time-varying formation. Furthermore, the minimum distance requirement is satisfied as shown in \nFigure 9\n, which depicts all the possible distances between agents. The distances between any pair of agents are always greater than or equal to the predefined distance d = 0.3. \nFigure 10\n shows the position errors of the agents. Such errors converge to zero.
\nTrajectories of the agents in the plane (formation tracking with collision avoidance problem).
Distances among agents (formation tracking with collision avoidance problem).
Position Errors of the agents (formation tracking with collision avoidance problem).
Let \n
In this subsection, we make the following standings assumptions
\n\nAssumption 4. For each follower, there is a path to at least one leader agent, i.e. for all \n
\nAssumption 5. For each secondary leader, there is a path to the main leader, i.e. for all \n
In order to define the problem statement, let us introduce some notation. Let \n
where \n
The goal of this work is to design a decentralised control law \n
Asymptotic tracking of a prescribed trajectory by the main leader agent, i.e.
Asymptotic time-varying formation by the secondary leader agents, i.e.
Convergence of the follower agents to the convex hull formed by the leaders, i.e.
Collision avoidance among all agents, that is, all agents in the system remain at a distance greater than or equal to a predefined minimum distance d from each other, i.e.
To achieve time-varying containment, we propose a bounded control law given by
\nwhere k\n\nm\n, k\n\nf\n and k\n\nc\n are control gains. Note that for each secondary leader and each follower, the control input depends on the position and velocity of the agents with which has a communication. In practical implementations, these velocities can be calculated by numerical differentiation.
\nThe first main result of this subsection is the following.
\n\nProposition 7. Consider the system Eq. (3) and the control law Eq. (51). Suppose that \n
\nThe main leader R\n\nn\n converges to the desired marching trajectory, i.e. \n
\nThe followers converge to the convex hull formed by the leaders, i.e. \n
\nProof. For part 1, the proof has a procedure similar to that performed in the Proposition 3.
\nFor part 2, the system errors are given by
\nwhere \n
where \n
Solving for the position of follower agents α\n\nF\n(t) of Eq. (52), we have
Since \n
where η = 1, \n
Modifying the previous control law Eq. (51) by adding the repulsive vector field Eq. (11), finally, we have the strategy to achieve time-varying containment with collision avoidance given by
\nThe second main result in this subsection is very similar to the second presented in the previous subsection, which consists of a series of three propositions, considering the simplest case, when there is risk of collision between two agents, then the case when there is risk of collision between three agents and, finally, the general case.
\nThe results of a numerical simulation using the control strategy given by (56) are shown below. For the simulation, we considered eight differential-drive mobile robots, where the point α\n\ni\n to control is located 0.15 m ahead the mid-point of the wheels axis. The formation graph employed in the simulation is shown in \nFigure 11\n.
\nFormation graph (time-varying containment with collision avoidance problem).
The parameters used in the simulation are \n
The minimum allowed distance between agents is d = 0.2 m, and the parameter ϵ was set to \n
\n\nFigure 12\n shows the motion of the agents in the plane. The initial positions of leader and follower agents are indicated with an ‘x’ ‘*’ and positions in times \n
Trajectories of the agents (time-varying containment with collision avoidance problem).
Distances among agents (time-varying containment with collision avoidance problem).
Position errors of the agents (time-varying containment with collision avoidance problem).
This chapter presents motion coordination problems with collision avoidance for multi-agent systems, where the agents are differential-drive mobile robots. We propose decentralised control strategies which ensure formation, time-varying formation tracking and time-varying containment. Furthermore, collision avoidance between agents is achieved. We use formation graphs to represent interactions between agents. As shown in numerical simulations, the goals are achieved and system errors converge to zero.
\nAs future work, it is proposed to control the mid-point of wheel axis of the differential-drive mobile robots and include a strategy for obstacle avoidance. It is also intended to validate the theoretical results obtained through real-time experiments.
\nPhysical phenomena described by partial differential equations (PDE) have inspired a new field in computational imaging and computer vision [1]. Such physics-inspired algorithms based on PDEs have been successful for image smoothening and restoration. Image restoration can be viewed as obtaining the solution to evolution equations by minimizing an energy function. The most popular PDE technique for image smoothening treats the original image as the initial state of a diffusion process and extracts filtered versions from its evolution at different times. This embeds the original image into a family of simpler images at a hierarchical scale. Such a scale-space representation is useful for extracting semantically important information [2]. Physics based algorithms not only outperform their conventional counterparts, but also have enabled new applications. Usage of these algorithms range from feature detection in digital images [3, 4, 5], to 3D modeling of objects from 2D images [6, 7], to optical character recognition [8] as well as for restoring audio quality [9].
Phase Stretch Transform (PST) is a physics inspired algorithm that emulates 2D propagation through a medium with group velocity dispersion, followed by coherent (phase) detection [10, 11]. The algorithm performs exceptionally well as edge and texture extractor, in particular in visually impaired images [12]. This transform has an inherent equalization ability that supports wide dynamic range of operation for feature detection [12, 13, 14]. It also exhibits superior properties over conventional derivative operators, particularly in terms of feature enhancement in noisy low contrast images. These properties have been exploited to develop image processing tools for clinical needs such as a decision support system for radiologists to diagnose pneumothorax [15, 16], for resolution enhancement in brain MRI images [17], single molecule imaging [18], and image segmentation [19].
PST emulates the physics of photonic time stretch [20], a real time measurement technology that has enabled observation as well as detection of ultrafast, non-repetitive events like optical rogue waves [21], optical fiber soliton explosions [22] and birth of mode locking in laser [23]. Further, by combining photonic time stretch technology with machine learning algorithms, a world record accuracy has been achieved for classification of cancer cells in blood stream [24, 25].
The photonic time stretch employs group-velocity dispersion (GVD) in an optical fiber to slow down an analog signal in time by propagating a modulated optical pulse through the time stretch system which is governed by the following equation:
where,
In the above equations,
PST utilizes the GVD dispersion to convert a real image to a complex quantity such that the spatial phase after the
In this chapter, we introduce Phase-stretch Adaptive Gradient-field Extractor (PAGE), a new physics inspired feature engineering algorithm that computes a feature set comprising of edges at different spatial frequencies, at different orientations, and at different scales. These filters metaphorically emulate the physics of birefringent (orientation-dependent) diffractive propagation through a physical medium with a specific diffractive property. In such a medium, the dielectric constant of the medium and hence, its refractive index is a function of spatial frequency and the polarization in the transverse plane. To understand this metaphoric analogy, we consider an optical pulse with two linearly orthogonal polarizations,
As the propagation constant
By controlling the value of
Further, we address the dual problem of spatial resolution and dynamic range limitations in an imaging system. In an ideal imaging system, the numerical aperture and the wavelength of an optical set up are the only factors that determine the spatial resolution offered by the modality. But under non-ideal conditions, the number of photons collected from a specimen control its dynamic range (the ratio between the largest and the smallest value of a variable quantity) which in turn also limits the spatial resolution. This leads to the fundamental dual-problem of spatial resolution and dynamic range limitations in an imaging modality [26].
Certain approaches to improve the resolution of the imaging system include use of wide-field fluorescence microscopy [27, 28] which offers better resolution than confocal fluorescence microscopy [29], multiple fluorophores [30, 31]. Also, various image processing techniques such as multi-scale analysis using wavelets [32, 33] have been proposed for improving the resolution while retaining important visual information post the image acquisition. We show later in the chapter that we are able to alleviate this dual-problem by incorporating, in our algorithm, a local adaptive contrast enhancement operator, also known as Tone Mapping Operator (TMO) which leads to excellent dynamic range.
Other steps of the proposed decomposition method are discussed at length in the next section. The organization of the chapter is as follows. In Section 2, we describe the details of the proposed decomposition method. Experimental results and conclusions are presented in Sections 3 and 4, respectively.
Different steps of our proposed decomposition method Phase-stretch Gradient-field Extractor (PAGE) for feature engineering are shown in Figure 1. The first step is to apply an adaptive tone mapping operator (TMO) to enhance the local contrast. Next, we reduce the noise by applying a smoothening kernel in frequency domain (this operation can also be done in spatial domain). We then apply a spectral phase kernel that emulates the birefringence and frequency channelized diffractive propagation. The final step of PAGE is to apply thresholding and morphological operations on the generated feature vectors in spatial domain to produce the final output. The PAGE output embeds the original image into a set of feature maps that select semantic information at different scale, orientation, and spatial frequency. We show in Figure 2 how PAGE embeds semantic information at different orientations for an X-ray image of a flower.
Different steps of the phase-stretch gradient-field extractor (PAGE) algorithm. The pipeline starts with application of tone mapping in the spatial domain. This is followed by a smoothening and a spectral phase operation in the frequency domain. The spectral phase operation is the main component of the PAGE algorithm. The generated hyper-dimensional feature vector is thresholded and post-processed by morphological operations. PAGE embeds the original image into a set of feature maps that select semantic information at different scale, orientation, and spatial frequency.
The phase-stretch gradient-field extractor (PAGE) feature map of an X-ray image. The original image is shown on the left (A). PAGE embeds the original image into a feature map that selects semantic information at different orientations as shown in (B). The orientation of the edges is encoded into various color values here.
The sequence of steps of our physics-inspired feature extraction method, PAGE, can be represented by the following equations. We first define the birefringent stretch operator
where
In the above equations,
The PAGE operator
where
In the next subsections, we discuss each of the above mentioned kernels in detail and demonstrate the operation of each step using simulation results.
A tone mapping operator (
We operate on the input image using a
Phase-stretch adaptive gradient-field extractor (PAGE) filter banks are defined by the PAGE kernel
such that the frequency vector rotates along the origin with
We then define the PAGE kernel
where
There are two important things that should be noted here. First, we consider the modulus of our translated frequency variable
These filter banks can detect features at a particular frequency and/or in a particular direction. Therefore, by selecting a desired direction and/or frequency, a hyper-dimensional feature map can be constructed. We list all parameters in Table 1 that control different functionalities of our proposed decomposition method PAGE.
Notation | Variable |
---|---|
Spatial frequency | |
Steerable angle | |
Translated spatial frequency | |
Normal filter | |
Log normal filter | |
Strength of | |
Strength of | |
Mean of normal distribution for | |
Mean of log-normal distribution for | |
Sigma of normal distribution for | |
Sigma of log-normal distribution for | |
Sigma of Gaussian distribution for | |
Bi-level feature thresholding for morphological operations |
Different parameters of our physics-inspired feature decomposition method PAGE.
The values of these parameters for Figure 2 simulation result are:
Figure 3A–P show the generated phase profiles for
Phase-stretch gradient-field extractor (PAGE) filter banks (A)–(P) phase filter banks as defined in Eqs. (8)–(13) for various frequencies and directions. The frequency variables u and v are normalized from −ωu to +ωu and −ωv to +ωv, respectively. The center μv′ of the phase kernel Sv′ is gradually increased for control over the frequency distribution. The values for steerable angle θ considered here are 0, π/4,π/2, 3π/4.
In order to detect features in a particular direction spread over the all the frequency components in the spectrum, we construct the PAGE filter banks by using Eqs. (9)–(13) for
Phase-stretch gradient field extractor (PAGE) directional filter banks (A)–(D) the directional filter banks of PAGE computed using the definition in Eqs. (9)–(13) for steerable angle θ = 0, π/4, π/2 and 3π/4, respectively. By monitoring the value of sigma σu′ of the normal filter ϕ1u′, the angular spread of kernel K˜uvθ can be controlled to avoid any overlapping of directional filters.
We first evaluate the performance of these kernel by qualitatively comparing the feature detection of PAGE with PST. The image under analysis is a gray-scale image of a rose. For a better visual understanding of our method, we first compute orthogonal directional responses as shown in Figure 5. We then show results of edge detection using PST and PAGE in Figure 6. The values for the parameters strength
Phase-stretch gradient-field extractor (PAGE) directional filter banks response the original image is shown in (A). We design two directional PAGE filters here to detect vertical (θ=π/2) and horizontal (θ=0) edges as shown in (B) and (C) respectively.
Comparison of feature detection using phase stretch transform (PST) and phase-stretch gradient-field extractor (PAGE) the original image is shown in (A). The output edge image obtained using PST without the support of directional response is shown in (B). The edge map obtained using PAGE filter banks that support edge detection at all frequencies is shown in (C). Different color values are used to show the orientation of the edges.
The PAGE filter banks can also be designed to detect edges at a particular frequency by controlling the spread of log normal distribution. To demonstrate this functionality, we show the features detected at low and high frequency using the rose image as an example in the Figure 7. As seen in the figure, the features detected at low frequency are smoother and at high frequency are sharper.
Feature detection using phase-stretch gradient field extractor (PAGE) at low and high frequency: Features detected at low frequency are much smoother whereas for high frequency, the features are sharper. This demonstrates the frequency selectivity for feature detection using PAGE.
We demonstrate the effectiveness of our decomposition method by comparing the directional edge response obtained by applying Gabor filter banks to an optical character image. We design 24 Gabor directional filters and augment the response from each of the filters to generate the image in Figure 8B. As seen in Figure 8C, with PAGE we have a better spatial localization of the edge response. By spatial localization, we mean that inherently PAGE has a sharper edge response, as seen in the figure. This is because, unlike the Gabor filters whose bandwidth is determined by the sigma parameter of the filter, in PAGE, the bandwidth of the response is determined by the input image dimension. Therefore, there is better localization of edge with PAGE. The parameters values are strength
Comparison to Gabor feature extractors: Features detected using Gabor do not have inherent spatial feature localization. With PAGE, the features are more sharper as the bandwidth of the response is determined by the input image dimension.
To demonstrate the superiority of our decomposition method, we compare the edge response obtained by applying derivative based operators to a test image shown in Figure 9A. The response to a derivative based operator is computed by using the edge function of Matlab software (canny) and is shown in Figure 9B. As seen in Figure 9C, PAGE outperforms derivative based operators by producing the orientation information and low contrast details. The parameters values are strength
Comparison to derivative feature extractors: Features detected with derivative based edge operators calculate the directionality based on the horizontal and vertical gradients and do not provide information about the spatial frequency of the edges. PAGE provides both the orientation as well as the spatial frequency selectivity in the output response.
We apply our decomposition method to different types of images to show that the directional edge response obtained by PAGE can be used for various computer vision applications. For example, in Figure 10, we show application of PAGE to a Single Electron Microscope (SEM) image of an integrated circuit chip. As seen, the PAGE feature response is able to capture the edges corresponding to the chip layout (even the low contrast details). Based on the viewing angle (camera position), the layout edges should appropriately be rendered in the image as well as in the edge map. This can be used to identify any chip artifacts during the fabrication process. The parameters values for generating the feature map shown in Figure 10 are strength
Fabrication artifact detection using phase-stretch gradient-field extractor (PAGE) on a single Electron microscope (SEM) image of integrated circuit chip. The original image is shown in (A). The output edge image obtained using PAGE filter banks that support edge detection at all frequencies is shown in (B). Different color values are used to show the orientation of the edges that correspond to the chip layout and can be used to detect fabrication artifacts.
We also apply PAGE to detect directional edge response to an image of a finger print as shown in Figure 11. Not only does PAGE detects a directional edge response, but also has an inherent equalization property to detect low contrast edges. The parameters values are strength
Fingerprint feature map using phase-stretch gradient-field extractor (PAGE). The original image is shown in (A). The output edge image obtained using PAGE filter banks that support edge detection at all frequencies is shown in (B). As the edges of the fingerprint rotate, the response value changes (shown here with different color value).
Next, we show application of our decomposition method PAGE to extract edges of vessels from a retinal image in Figure 12. The distribution of vessels based on the orientation of the edges can be used as an important feature to detect any abnormalities in the eye structure. As seen, the PAGE feature response is able to capture both the low contrast details as well as information about the directionality of the vessel edges which is coded in form of the color value in RGB space. The parameters values are strength
Vessel detection using phase-stretch gradient-field extractor (PAGE) on an image of a retina. The original image is shown in (A). The output edge image obtained using PAGE filter banks that support edge detection at all frequencies is shown in (B). Different color values are used to show the orientation of the edges. The low contrast vessels are not only detected using PAGE but also information on how the direction of the blood flow changes across the eye based on the vessel distribution is extracted.
In this chapter, a presentation is made on a new feature engineering method that takes inspiration from the physical phenomenon of birefringence in an optical system. The introduced method called Phase-stretch Adaptive Gradient-field Extractor (PAGE) controls the diffractive properties of the simulated medium as a function of spatial location and channelized frequency. This method when applied to 2D digital images extracts semantic information from the input image at different orientation, scale and frequency and embeds this information into a hyper-dimensional feature map. The computed response is compared to other directional filters such as Gabor to demonstrate superior performance of PAGE. Applications of the algorithm for edge detection and extraction of semantic information from medical images, electron microscopy images of semiconductor circuits, optical character and finger print images is also shown.
The authors would like to thank Dr. Ata Mahjoubfar for his helpful comments on this work during his post-doctoral studies in Jalali Lab at UCLA. This work was partially supported by the National Institutes of Health (NIH) Grant No. 5R21 GM107924-03 and the Office of Naval Research (ONR) Multi-disciplinary University Research Initiatives (MURI) program on Optical Computing.
The authors declare no conflict of interest.
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