Link parameter values.
\r\n\tBook, “Mites, Ticks and Humans", is written by keeping in vision non-availability of any standard text dealing in different aspects of acarology at one place. Separate chapters in this book are devoted to medical importance of mites and ticks; ectoparasites, endoparasites and disease transmitting mites; classification, biology and epidemiology of dust mites; manifestations, diagnostics and preventions of dust mites allergy; ticks transmission of disease causing pathogens; and measures to mitigate mites and ticks. Book will stimulate interest in the readers for more information about different mites and ticks affecting publics. The knowledge contained in the book may prove as best material for graduate and post-graduate level courses, teachers and researchers in entomology, pestss control advisors, professional entomologists, pesticide industry managers, policy planners, and other experts having interest in mites and ticks.
",isbn:"978-1-83880-186-1",printIsbn:"978-1-83880-185-4",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"1ab684433f948520e8e90a2e74e2801a",bookSignature:"Dr. Muhammad Sarwar",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8964.jpg",keywords:"Basic biology, Diversity of lifestyles, Scabies, Mange, Ecosystem, Soil mite, Dust mites allergy, Asthma, Vectors, Dispersal or spreading modes, Pest management, Detection and survey",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"August 26th 2019",dateEndSecondStepPublish:"September 16th 2019",dateEndThirdStepPublish:"November 15th 2019",dateEndFourthStepPublish:"February 3rd 2020",dateEndFifthStepPublish:"April 3rd 2020",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,editors:[{id:"272992",title:"Dr.",name:"Muhammad",middleName:null,surname:"Sarwar",slug:"muhammad-sarwar",fullName:"Muhammad Sarwar",profilePictureURL:"https://mts.intechopen.com/storage/users/272992/images/system/272992.jpeg",biography:"Dr. Muhammad Sarwar, Principal Scientist, completed his B. Sc. (Hons.), M. Sc. (Hons.) and Ph. D. in Entomology, during the years 1987, 1989 and 1998, respectively. He has years of Service experience in Research orientated Department of Agriculture (16-05-1991 to 31-05-2001, Government of Punjab) and Pakistan Atomic Energy Commission (01-06-2001 to date). Dr. Sarwar has 253 research work publications in National and Foreign Journals with suitable Impact Factor and Citations in his credit. He has been award by Higher Education Commission of Pakistan “Post-Doctoral Scholarship Phase II, 2006” on the basis of merit for research work at Institute of Plant Protection, Chinese Academy of Agricultural Sciences, Beijing, China. Shield award, Letters of Appreciation and Certificates of performance have been granted to him from faculty members of Chinese Academy of Agricultural Sciences, Beijing, China. He completed his Post Doctorate in 2008 from Institute of Plant Protection in Chinese Academy of Agricultural Sciences, Beijing, China. The Zoological Society of Pakistan in recognition of his research contributions presented him Prof. Dr. Mirza Azhar Beg Gold Medal-2010 during the 30th session of Pakistan Congress of Zoology hosted by University of Agriculture, Faisalabad on March 2, 2010.\r\nDr. Sarwar carried our Research on Integrated Pests Management of Rice, Cotton, Chickpea, Brassica crops, Fruit flies, and Stored grains. Undertaken research work on Predatory Mites, Ladybird Beetle, Chrysoperla, Trichogramma and parasitoids of Fruit flies culturing as Bio-control agents, Integrated Management of Fruit Fly and Mosquito, and other arthropod pests control methodology. He also conducted work on Vertebrate pests control especially controls of rodents in field crops and storage. He explored, hitherto the unexplored 36 new species of stored grain and stored products mites belonging to 8 genera viz., Forcellinia, Lackerbaueria, Acotyledon, Caloglyphus and Troupeauia of family Acaridae; and Capronomoia, Histiostoma, and Glyphanoetus in family Histiostomatidae. Identification keys, taxonomical observations, differentiation remarks, comparison of characters, similarity matrices, Phenograms and Geographical maps of new species along with 48 alien species have been prepared as well. He conducted research work on Integrated Management of Cotton Leaf Curl Virus (CLCV), Pest scouting, Pest monitoring and forecasting, planning, designing and layout of different research trials and data recording for integrated pest management on different crops, vegetables, and orchards. He conducted training of the farmers and Field Staff and provided advisory services to the farmers regarding plant protection practices. Training of the pesticide’s dealers for proper handling, distribution and storing of pesticides, their legal aspects, and sampling of pesticides for their quality control have been done by him.\r\nUnder the Coordinated Research Program, he collaborated with other Institutes to trace resistance sources for cotton, rice, gram, rapeseed, mustard plants, and stored cereals and pulses. Imparted training by him to field staff and the progressive farmers regarding plant protection practices in some Districts of Sindh and Punjab. Dr. Sarwar supervised Post-graduate research work and is acting as an external Examiner for Post-graduate studies. Various workshops have been organized by him, is Reviewer for Scientific Journals and joined different working Committees. A new avenue on the rearing of Predatory Mites as bio-control agents of insects & mites pests in greenhouse and field crops has been opened by him. \r\nDr. Sarwar has been granted with Research Productivity award in 2011, by Pakistan Council for Science and Technology. He is included in the list of Higher Education Commission (HEC) of Pakistan approved Supervisor. He acquired different National and International training in Beijing, Bangkok, Vienna, and Havana. Dr. Sarwar completed “Basic Management course” organized by Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, held from 31 January to 18 February 2011.",institutionString:"National Institute for Biotechnology & Genetic Engineering",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"5",title:"Agricultural and Biological Sciences",slug:"agricultural-and-biological-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"297737",firstName:"Mateo",lastName:"Pulko",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/297737/images/8492_n.png",email:"mateo.p@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:"6418",title:"Hyperspectral Imaging in Agriculture, Food and Environment",subtitle:null,isOpenForSubmission:!1,hash:"9005c36534a5dc065577a011aea13d4d",slug:"hyperspectral-imaging-in-agriculture-food-and-environment",bookSignature:"Alejandro Isabel Luna Maldonado, Humberto Rodríguez Fuentes and Juan Antonio Vidales Contreras",coverURL:"https://cdn.intechopen.com/books/images_new/6418.jpg",editedByType:"Edited by",editors:[{id:"105774",title:"Prof.",name:"Alejandro Isabel",surname:"Luna Maldonado",slug:"alejandro-isabel-luna-maldonado",fullName:"Alejandro Isabel Luna Maldonado"}],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:"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:"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:"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:"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:"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:"3794",title:"Swarm Intelligence",subtitle:"Focus on Ant and Particle Swarm Optimization",isOpenForSubmission:!1,hash:"5332a71035a274ecbf1c308df633a8ed",slug:"swarm_intelligence_focus_on_ant_and_particle_swarm_optimization",bookSignature:"Felix T.S. Chan and Manoj Kumar Tiwari",coverURL:"https://cdn.intechopen.com/books/images_new/3794.jpg",editedByType:"Edited by",editors:[{id:"252210",title:"Dr.",name:"Felix",surname:"Chan",slug:"felix-chan",fullName:"Felix Chan"}],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:"52880",title:"Stabilization Control for the Giant Swing Motion of the Horizontal Bar Gymnastic Robot Using Delayed Feedback Control",doi:"10.5772/65831",slug:"stabilization-control-for-the-giant-swing-motion-of-the-horizontal-bar-gymnastic-robot-using-delayed",body:'In the past decades, scientists have found that unusual and unexpected evolution patterns arise frequently in numerous natural and model nonlinear systems in physics, chemistry, biology, engineering, medicine, economics, and so forth, since Lorenz [1] in 1963, May [2] in 1976, and others reported chaotic behavior in very simple dynamical models. The most peculiar aspect of these patterns is their random-like behavior, although the systems are deterministic, in other words, the deterministic nature of these systems does not make them predictable [3]. This behavior is known as deterministic chaos, or simply chaos. It is considered that chaos is due to sensitive dependence on the initial conditions. This property implies that two trajectories emerging from two different close-by initial conditions separate exponentially in the course of time. Due to this property, and due to the fact that, in general, experimental initial conditions are never known perfectly, chaotic systems cannot be predicted by their long-term behaviors.
This feature of their critical sensitivity to initial conditions, often regarded as a troublesome property, made chaos undesirable in engineering control practice, and most experimentalists considered such characteristics as something to be strongly avoided since it restricts the operating range of many electronic and mechanic devices. Besides that, chaotic systems exhibit two other important properties. One is that there is an infinite number of unstable periodic orbits embedded in the underlying chaotic set. The other is that the dynamics in the chaotic attractor is ergodic; that is to say, the system visits ergodically small neighborhood of every unstable fixed point during its temporal evolution [4].
Although the existence of steady states and an infinity of different unstable periodic orbits embedded in chaotic motion is not usually obvious in free-running chaotic evolution, these orbits offer a great potential advantage if one wants to control a chaotic system. The presence of chaos may be a great advantage for control in a variety of situations. In a nonchaotic system, small controls typically can only change the system dynamics slightly. However, in a chaotic system, one can choose between a rich variety of dynamical behaviors. It is then not surprising that the matter of controlling chaotic systems has come under detailed investigation by several different scientific communities. Chaotic dynamics consists of a motion where the system moves in the neighborhood of one of the unstable periodic orbits (UPO) for a while, then falls close to a different UPO and remains for a limited time. Thus, it allows to exploit a single dynamical system for the production of a large number of different periodic behaviors so that the single system can carry out different performances with different yields [5]. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny, to avoid significant modification of the system’s natural dynamics.
Since Ott, Grebogi, and Yorke [6], in 1990, pointed out the existence of many unstable periodic orbits (UPOs) embedded in chaotic attractors that raise the possibility of using very small external forces to obtain various types of regular behavior, Pyragas [7] in 1992, proposed a so-called “Delayed Feedback Control (DFC)” idea that an appropriate continuous controlling signal formed from the difference between the current state and the delayed state is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit, several techniques were devised for controlling chaos [8–21] during the past years and applied to various systems [22–31]. It is worth noting that in spite of the enormous number of applications among the chaos control, very few rigorous results are so far available. Most results are justified by computer simulations rather than by analytical tools. Therefore, many problems remain unsolved.
On the other hand, recently in the field of mechanical and control engineering, the open-loop link mechanism, in terms of diversity and ease of movement, has been used in industrial robots, although a large control input is needed due to speeding up, and the opposing force to the joint and the base increases. Therefore, the actuator with a large driving performance is needed, and the mass of the entire mechanism is increased. It is thought that the control method using the opposing force and gravity is important to achieve the desired movement with high speed and high efficiency by the limited driving performance. The joint type movement robot is actively researched in recent years, and the achievement of dynamic walking and running with high speed and efficiency becomes a problem. Although living creatures, such as humans or animals, have achieved the high-speed movement by using the open-loop link mechanism, it is thought that this is because that a good movement possessing the energy efficiency and small joint drive power and small impact power, and so forth in the gravitational filed had been acquired adaptively during the process of evolution. For example, the human biped locomotion is performed by dynamic link mechanism coupled to the pendulum motion [32, 33], and brachiation (movement across branches) used by the monkeys is a specialized form of arboreal locomotion that can be seen as a continuous pendulum motion [34, 35]. In addition, humans have competed for the manipulation of their own multi-degree of freedom link mechanism through sports or physical exercise, and so forth, and the achievement of a complex movement meeting a specific initial terminal requirement as a technique.
The systems, such as humans and robots, which control the internal force to achieve motions by using reaction force from the outside world, can be modeled as an underactuated link mechanism. These systems usually have nonholonomic second-order constraints due to the presence of passive joints, which makes development of control methods a difficult task. In recent years, studies on underactuated mechanical systems that possess fewer actuators than degrees of freedom have received increased interest [36–40].
Open-loop dynamic characteristics of an underactuated system with nonholonomic constraints show the chaotic nature that even though there are small differences in the initial conditions, the amplitude grows to be a completely different movement for its nonlinearity due to centrifugal force, Coriolis force, and gravity. Motions control of underactuated systems known also as nonholonomic control, such as wheeled robot positioning control [41], aerial posture control of space robots [42], and positioning control of underactuated link mechanism robot [43, 44], have been studied. However, a generalized control method has not yet been established to this kind of system due to the difficulties in analysis, and the fact that for an underactuated system, it cannot be directly controlled because of the generalized coordinates of its passive parts. Control methods, which utilize empirical skills catching the motion characteristics and proper motion of the system, are desirable. From the above-mentioned viewpoint, in order to achieve a robot with a high-speed, highly effective optimal motion, and the skill in movement seen in living creatures, fundamental researches on the control method and the trajectory planning method using periodic free movement for the linkage with a passive joint have been done [45–51].
The remaining parts of this chapter are organized as follows: Section 2 discusses a kind of DFC method, PDFC, for a two-link gymnastic robot by using of a Poincaré section. In the Section 3, an improved DFC method, Multiprediction Delayed Feedback Control, is extended to a four-link gymnastic robot. Section 4 gives some numerical simulations to show the effectiveness of the proposed method. Section 5 summarizes the chapter.
The two-link horizontal bar gymnastic robot is a highly simplified model of a human gymnast on a high bar, where the underactuated first joint models the gymnast’s hands on the bar and the actuated second joint models the gymnast’s waist [52]. Studies on such underactuated mechanical systems that possess fewer actuators than degrees of freedom, as gymnastic robot with the underactuated first joint, have received considerable interest in recent years. Open-loop dynamic characteristics of such a linkage as this kind of system, which is classified as an underactuated systems with nonholonomic constraints, show the chaotic nature that even though there are small differences in initial conditions, the amplitude grows to be a completely different movement for its nonlinearity due to centrifugal force, Coriolis force, and gravity. Since a generalized control method has not yet been established to this kind of system for the difficulties in analysis, their control problems are challenging.
While applying the original DFC to the continuous system, such as gymnastic robot, stability analysis of the closed-loop system becomes a very difficult task because of the time-delay dynamics described by a difference-differential equation. A modified delayed feedback control method, called Prediction-based DFC (PDFC), is proposed to stabilize the giant swing motion of two-link horizontal bar gymnastic robot.
The two-link horizontal bar gymnastic robot is a simplified model of a human gymnast on a high bar, where the first joint being passive models the gymnast’s hands on the bar and the second joint comprising of an actuator models the gymnast’s hips. The formula mi, li, ai, Ii(i = 1, 2) in Figure 1 denotes the ith link’s mass, length, distance from joint to its center of mass, inertia moment around its center of mass, respectively.
Two-link gymnastic robot model.
The equation of motion of the gymnastic robot is as follows:
where x = (x1, x2)T ∈ R2 is the generalized coordinate vector, u = (u1, u2)T ∈ R2 is the joint torque vector, and M ∈ R2×2, C ∈ R2×1, G ∈ R2×1 denote the inertia matrix, Coriolis matrix, and gravitational matrix, respectively.
In addition, since the first joint cannot generate active torque, the following constraint must be satisfied.
In order to facilitate the analysis, we rewrite the motion Eq. (1) into the following equation of state:
where
Here,
As is known, the stability of a periodic orbit of the original continuous-time system is closely related to the stability of the fixed point of the corresponding Poincaré map. Since the stability of the periodic orbit means that the sequence of points converges to a fixed point in the phase plane, the objective system can be expressed as the following difference equation for the discrete-time systems.
where k,
Here, let
As shown in Figure 2, the (k + 1)th state vector xp,k + 1 can be calculated by one-cycle integration in accordance with the control law by defining the kth state vector passing through the section P as xp,k.
Poincaré map of a closed periodic trajectory.
This relationship representing this state transfer map is defined as ϕ as follows:
If given an initial condition from which a periodic orbit can be formed, it will return to the same point after one period. Hence, the following equation holds:
While a difference occurs in the initial state, the motion will stray away from the periodic orbit. Suppose that there was an error corresponding to ep,k in kth state vector, the (k + 1)th state xp,k + 1 can be described as
Using the above equation’s Taylor series at ep,k and neglecting the higher-order terms can yield
It is obvious that ∂ϕ(xp,ref)/∂xp is equivalent to the ETM A defined in Eq. (5). It can be judged from this equation that if the absolute value of the ETM’s eigenvalue >1, then the error increases with each cycle, and if <1, the motion will approach asymptotically the periodic orbit. However, because the state transfer mapping cannot be stated explicitly as a function, in previous studies, this ETM had been computed based on numerical differentiation. In this work, we obtained ETM with an analytical method by using variational equation.
Now consider the objective system given by the equation of motion in Eq. (3). First, given the case of u = 0, the objective system becomes
Assume the state x to be the following vector function
where fi, φi (i = 1 ∼ 4) are the functions of x. The solution of equation passing through x(0) at t = 0 is defined as follows:
Moreover, the periodic solution satisfies the following equation.
Here, a Poincaré map is defined as follows:
Then, the T-period sequence of points obtained by sampling the trajectory from the initial value x0 is
Thus, it is obvious that the periodic solution is shown as a point passing through the same position. Substituting x = φ(t, x0) into Eq. (11), we obtain an equation for the variable φ(t, x0), which can be written as
Next, taking differentiation for Eq. (16) at x0, we can obtain
The differential order of the left-hand side in the above equation can be modified as
Thus, it becomes a time-varying linear matrix differential equation with the form of
which is called the Variational Equation if defining
where xi,0 refers to the solution at t = 0 of the ith state corresponding to the state vector x defined in Eq. (12).
Note that X(0, x0) = I. Carrying out integrals in numerical integration to the above variational equations over the interval t ∈ [0, T], one column of the matrix X(T, x0) can be obtained. Hence, repeating it four times, all of its values can be calculated. It should be emphasized here that to solve these equations since the value of Ã(t) is required, however, Ã(t) contains x(t) in terms of the relevant time, they have to be solved together with Eq. (11) simultaneously. From Eqs. (10) and (15), it is not difficult to be verified that X(T, x0) can be regarded as ETM.
Note that Eq. (3) can be rewritten into
where
The input torque u can be described as
where
Here, u0 is the control input during (t0, τ), while x = x0 and τ are control parameters.
Since the solution of equation at x = x(0) can be stated as
Define a Poincaré map as follows:
The continuous system described by Eq. (21) will change to be a discrete system by mapping of
It is obvious here that by taking differentiation at x0, its result will be equivalent to Eq. (18). Consider here the case of differentiation at u0, which can yield
The differential order of left-hand side in the above equation can also be changed.
Let X(t, x0, u0) = ∂φ/∂x0(t, x0, u0). Similar to previous subsection, the above equation is equivalent to the following equation.
Note that X(0, x0, u0) = 0. Carrying out integrals in numerical integration to Eq. (29) over the interval t ∈ [0, T], it is not difficult to obtain the value of the matrix X(T, x0, u0) which can be regarded as the input matrix B defined in Eq. (5).
Since the value of both the matrices A and B can be calculated analytically, i.e., the continuous system of two-link gymnastic robot can be expressed in a discrete model described in Eq. (5). In order to study the stabilization of gymnastic robot’s continuous system by delayed feedback control, it is sufficient to consider the discrete model.
Consider the following Prediction-based feedback control:
where K ∈ R1×4 is a feedback gain, x(k) is the state vector at k-step, xP(k) denotes one period future states of uncontrolled system which can be obtained from Eq. (3). Moreover, notice that the following almost equality holds.
Here
Therefore, the closed-loop system can be described by
Let
then the stabilization problem is reduced as follows:
Given a system Eq. (5), find a feedback gain
By using the pole placement technique, it is not difficult to obtain the value of
Thus, the above design procedure can be applied to analyze the stability of two-link gymnastic robot system.
As is known, the simplified two-link robot shown in the previous section is not an ideal physical model of a human gymnast on a high bar. In order to mimic gymnastic routine more realistically, the complicated robot model with higher degrees of freedom (DOF) needs to be considered. An improved method based on PDFC which control a three-link gymnastic robot via a periodic gain has been proposed [51]. In this section, the Multiprediction Delayed Feedback Control (MDFC), is extended to a more complicated gymnastic robot with four DOF.
Figure 3 shows a four-link horizontal bar gymnastic robot model, which consists of four links and four joints. Shoulder, hips, and knees are active, while the pivot connecting the hand and bar is a passive joint. Assume mi, li, ai, Ii (i = 1, 2, 3, 4) to be the link mass, length, distance from joint to its center of mass, and inertia moment around its center of mass, respectively.
Four-link gymnastic robot model.
The motion equation of the robot is
where x = (x1, x2, x3, x4)T ∈ R4 is the generalized coordinate vector, (u1, u2, u3, u4)T ∈ R4 is the joint torque vector, and Mij, Ci, Gi (i, j = 1, 2, 3, 4) are, respectively, the terms of inertia matrix, Coriolis matrix, and gravitational matrix. Each item of Mij, Ci, Gi is as follows. Note that Mij = Mji, (i ≠ j).
Similar to the motion equation of the two-link robot, Eq. (36) can be rewritten as follows.
As is discussed in the reference [51], multiprediction delayed feedback control is based on a new discretization method to increase the control performance by introducing a notion of plural Poincaré maps, (Pi, P2,..., PN) which divide the first link angle x1 ∈ (−π, π) into N sections as shown in Figure 4.
Image of Poincaré maps [51].
By the ith (i = 1, 2,..., N) Poincaré map, the four-link horizontal bar gymnastic robot system can be discretized into the following
where k is the discrete time and x(k, i)(i = 1, 2,..., N) ∈ Rn, Ai ∈ Rn×n(n = 8), Bi ∈ Rn×3 denote, respectively, the state error, the error transfer matrix, and the input matrix in terms with the ith Poincaré map. Furthermore, u(k, i) is defined as
Here, the equivalence of
where rect(t) is a rectangular function as Eq. (41).
Here, ti is the time at ith Poincaré map Pi(i = 1, 2,..., N).
Summarize the state variables at each one of the ith Poincaré section into one vector as follows:
Therefore, the close-loop system relating to X(k, i) can be stated as the following discrete-time system with periodic N.
where,
As it is known, the local stability of such a system can be determined by the eigenvalue of the monodromy matrix as follows:
The solution of the error transfer matrix Ai and the input matrix Bi is similar to the determination of the two-link gymnastic robot, so only the difference is shown below.
Here, let
Consider the objective system given by the equation of motion in Eq. (37). First, given the case of u = 0, the objective system becomes
Assume the state x to be the following vector function
where fĩ, φĩ(ĩ = 1 ∼ 8) are the functions of x. Introducing the following definition,
the variational equation becomes
where xĩ,0 refers to the solution at t = 0 of the ĩth state corresponding to the state vector x defined in Eq. (48).
Note that the following equation holds.
Carrying out integrals in numerical integration to the variational equations of Eq. (50) over the interval t ∈ [0, T] by eight times, each item of the error transfer matrix, Ai(i = 1, 2,..., N), can be calculated.
Note that Eq. (37) can be rewritten as
Here,
Since the solution of equation at x = x(0) can be written as
The continuous system described by Eq. (53) will change to be a discrete system by mapping of
Here, consider the case of differentiation at u0, which can yield
Let
Similar to previous subsection, Eq. (53) is equivalent to the following equation.
Note that the following equation holds.
Carrying out integrals in numerical integration to Eq. (58) over the interval t ∈ [0, T] by three times, it is not difficult to obtain the value of the input matrix Bi(i = 1, 2,..., N).
This section shows some simulation results to verify the validity of the proposed prediction-based delayed feedback control for a two-link horizontal bar gymnastic robot. The parameters of the robot model are shown in Table 1. The stability of giant swing motion with period T = 0.7(s) was examined.
1st link | 2nd link | ||
---|---|---|---|
Mass | mi (kg) | 39.5 | 20.7 |
Moment of inertia | Ii (kgm2) | 3.25 | 1.56 |
Link length | li (m) | 1.2 | 0.879 |
Offset of mass center | ai (m) | 0.756 | 0.377 |
Link parameter values.
By means of the shooting method, an initial condition was obtained such as x(0) = (π, 0.0, 8.352, − 1.19) that can achieve a free giant swing motion with a period of 0.7(s). During the first 30 s starting from this initial condition, the phase portrait of the uncontrolled system is depicted in Figure 5 in the phase plane (X,
Phase portrait of X−x. (no error).
To see the stability of periodic orbits via delayed feedback control, numerical simulation was exercised from the initial condition x(0) = (π, 0.1, 9.252, − 2.39) that contained errors. First, consider the uncontrolled case, the 30 s trajectory of the orbits in the phase plane filled up a section of the phase space as in Figure 6. It can be confirmed that there was no periodicity without control.
Phase portrait of X−x. (with error).
Next, consider the system controlled by the proposed method, with which the error transfer matrix A was obtained as follows:
The input matrix B in the case of τ = 10(ms) is obtained as follows:
Based on the value of the matrix A and B shown in the above, the feedback gain K was obtained by assigning all of its poles of the system matrix
In Case I, the numerical simulation results via calculated feedback gain K are shown in Figure 7. Figure 7(a) and (b) show the trajectory of the orbits in the phase plane (x,
Phase portrait and joint torque (Case I): (a) Phase portrait of X−x.(t=0~50s); (b) Phase portrait of X−x. (t=50~60s); (c) Joint torque.
However, on further analysis conducted, it was found that one can change the type of giant swing type by changing the pole placement of the matrix
Phase portrait and joint torque (Case II): (a) Phase portrait of X−x. (t=0~150s); (b) Phase portrait of X−x. (t=150~160s); (c) Joint torque.
It could be said that in Case II, the proposed method showed the ability of controlling the giant-swing motion to an unknown periodic orbit, which is known as one of the advantages of delayed feedback control.
Moreover, Figures 9 and 10 depict the stick diagrams based on one period data after the giant swing motion converging to a stable orbit in Case I and Case II, in which the figures of (a), (b), and (c) plot, respectively, the angle, angular velocity, and stick figure. From the above two figures, it can be seen that the difference between the assigned poles caused periodic giant swing motion to change to a different type.
Angle, angular velocity, and stick (Case I).
Angle, angular velocity, and stick (Case II).
This chapter studied the behavior of the giant swing motions of a two-link and a four-link horizontal bar gymnastic robot systems via delayed feedback control. First, it has been discussed as a solution to calculate analytically the error transfer matrix and the input matrix by which the stability of DFC for the original two-link gymnastic robot system is equivalent to that of the corresponding discrete system. Moreover, it was introduced as a method by which a feedback gain to ensure the stability of original system can be determined. Second, a modified DFC method, MDFC, has been extended to a four-link gymnastic robot. Plural Poincaré maps were defined so that the stability of the close-loop system can be evaluated based on the theory of monodromy matrix. Finally, the simulation results showed its effectiveness.
This work is supported in part by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and in part by the Scientific Research Foundation for Young Teachers, Shanghai JiaoTong University.
The major source of energy for mammalian cells is glucose. Glucose derived from the diet and synthesized within the body is transported from the circulation into target cells. The transfer of glucose across the plasma membrane is necessary. Cell membrane is composed by lipid bilayer, which is hydrophobic. Glucose has hydrophilic nature. Therefore, cell membranes act as barriers to most molecules. For water molecules and a few other small molecules, such as oxygen and carbon dioxide, the lipid bilayer is permeable. These molecules move spontaneously down their concentration gradient by diffusion. For cations such as K+, Na+, and Ca2+; anions such as Cl− and HCO3−; and hydrophilic molecules and macromolecules such as proteins and RNA, lipid bilayer is impermeable. Therefore, these molecules and ions need specific transport system. There are two general classes of membrane transporters: channels and carriers.
\nGlucose transporters belong to the major facilitator superfamily (MFS). MFS contains 74 families of membrane transporters including more than 10,000 members. These transporters transport variety of molecules.
\nGlucose as well as other monosaccharides cannot penetrate the lipid bilayer because they are hydrophilic in nature; therefore, they require specific carrier proteins to undergo diffusion through the bilayer. In humans, there are three families of genes that encode for glucose transporters: SLC2A, SLC5A, and SLC50A [1].
\nGlucose is transported across the cell membranes and tissue barriers by a sodium-independent glucose transporter (facilitated transport, GLUT proteins, and SLC2 genes), sodium-dependent glucose symporters (secondary active transport, SGLT proteins, and SLC5 genes), and glucose uniporter—SWEET protein (SLC50 genes). Most cells express more than one kind of glucose transporters. However, these membrane carrier proteins are called glucose transporters; they are involved in the transport of several different molecules, not just glucose.
\nIn humans, 14 members of GLUT proteins have been identified. They are encoded by the solute-linked carrier family 2, subfamily A gene family, and SLC2A [2, 3]. All GLUT proteins are predicted to contain 12 hydrophobic membrane spanning, α-helical transmembrane (TM) domains. These domains are connected by hydrophilic loop between TM6 and TM7 of the protein [4, 5, 6]. GLUTs contain a site for single glycosylation on the exofacial end, either in the large loop between TM1 and TM2 (first extracellular loop) or between TM9 and TM10 (fifth extracellular loop) [7]. As was proposed for GLUT1, helices 1, 2, 4, 5, 7, 8, 10, and 11 form an inner bundle that is stabilized by the outer helices 3, 6, 9, and 12 [8].
\nBased on the phylogenetic analysis of sequence similarity and characteristic elements, the GLUT family of sugar transporters is divided into three classes [4, 5, 9, 10]: an N-linked glycosylation site for GLUTs of class I and II is positioned in the first exofacial loop between TM1 and TM2, and family members of class III contain the glycosylation site between TM9 and TM10 [5].
\nClass I GLUTs include GLUT1–GLUT4 and GLUT14, which are 48–63% identical in humans. Class II GLUTs comprise of GLUT5, GLUT7, GLUT9, and GLUT11. These transporters are 36–40% identical. Class III GLUTs include GLUT6, GLUT8, GLUT10, GLUT12, and GLUT13 (HMIT). GLUTs in this class are only 19–41% identical.
\nThe human GLUTs are involved in the transport of the several hexoses in addition to myoinositol, urate, glucosamine, and ascorbate [7]. All the members of the GLUT family are facilitative transporters except for GLUT13 (HMIT), which is an H+/myoinositol symporter [11].
\nTo date, four pseudogenes of SLC2A family were described [5, 7]:
SLC2A3P1 (alias GLUT6 or GLUT3 pseudogene) is located on chromosome 5q35.1 and is a retroposon of SLC2A3.
SLC2A3P2 (alias GLUT3 pseudogene 2) is located on chromosome 1p31.3 and is a retroposon of SLC2A3.
SLC2A3P4 (alias GLUT3 pseudogene 4) is located on chromosome 8q21.3 and is a retroposon of SLC2A3.
SLC2AXP1 is located on chromosome 2q11.2 and contains internal stop sequences.
Crane [12] showed that active glucose absorption by hamster’s small intestine required sodium ions in the bathing medium. He proposed that these symporters have two binding sites: one for glucose and one for sodium [13].
\nThe sodium-dependent glucose cotransporters belong to the gene family (SLC5A), the SGLTs, or sodium/substrate symporters family (SSSF), containing over 450 members [14, 15, 16]. In humans, 12 members of sodium-dependent glucose cotransporters have been identified. Amino acid comparison of the human sodium-dependent glucose cotransporters shows the range of identity from 57 to 71% [17]. The members of the SGLT family also share considerable homology among the proteins (21–70% amino acid identity with SGLT1) [10, 16]. These proteins contain of 580–718 amino acid residues, with a predicted mass of 60–80 kDa. There is a diversity in gene structure. In eight genes, the coding sequences are found in 14–15 exons (SLC5A1, SLC5A2, SLC5A4–SLC5A6, and SLC5A9–SLC5A11), and the coding sequence for SLC5A7 and SLC5A3 are present in exons 8 and 1, respectively. In SLC5A9–SLC5A11 and SLC5A3, there is evidence for alternative splicing. These proteins contain 14 TM α-helices (TMHs) in all but not in sodium-iodide symporter (NIS) and SMCT1, which lack TMH14 [18]. Both the hydrophilic N- and C-termini are located on the extracellular side of the cell membrane [1]. SGLTs are highly glycosylated membrane proteins; however, glycosylation is not required in the functioning of the protein. The human SLC5A genes are expressed in different tissues, and all of them code for sodium-dependent glucose cotransporter proteins, except for SGLT3 (SLC5A4), which acts as a glucose sensor [19]. These carrier proteins transport substrates such as glucose, myoinositol, and iodide; one is a Na+/Cl−/choline cotransporter, and another is a glucose-activated ion channel [16].
\nSWEETs transport mono- and disaccharides across vacuolar and plasma membranes. A new class of glucose transporters, SWEET, was first identified by expressing candidate Arabidopsis genes coding for polytopic membrane proteins in HEK293T cells [20]. SWEETs are ubiquitously expressed in plants. In contrast to Arabidopsis thaliana, in which up to two dozen SWEETs have been identified, animals usually have only one SWEET, except for Caenorhabditis elegans, where seven SWEET-encoding genes have been found. Homologs of the SWEETs are widespread in metazoan genomes, and there is a single homolog in human genome (SWEET1) encoded by the gene SLC50A1 [1].
\nHuman SWEET1 (RAG1AP1), encoded by SLC50A1, comprises 221 amino acids with a molecular weight of 25 kDa. Human SWEET1 did not promote glucose uptake but instead mediated a weak efflux. Human SWEET1 when expressed in HEK293T cells was predominantly found to be localized in the Golgi with minimum expression also found in the plasma membrane. Chen et al. [20] discovered the highest level of expression in the oviduct, epididymis, and intestine, and its expression was induced in mouse mammary gland during lactation. The authors suggest that the human SWEET1 serves to supply glucose for lactose synthesis in the mammary gland. Human SWEET1 glucose transporter is the missing glucose transporter in the basolateral membrane of enterocytes where it may account for the exit of glucose from the cell into the blood in patients with Fanconi-Bickel syndrome and in mice missing the GLUT2 transporter [21, 22].
\nSupporting women in scientific research and encouraging more women to pursue careers in STEM fields has been an issue on the global agenda for many years. But there is still much to be done. And IntechOpen wants to help.
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\\n\\nWe aim to publish 100 books in our Women in Science program over the next three years. We are looking for books written, edited, or co-edited by women. Contributing chapters by men are welcome. As always, the quality of the research we publish is paramount.
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\\n\\nAdvantages of Publishing with IntechOpen
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\n\nWe aim to publish 100 books in our Women in Science program over the next three years. We are looking for books written, edited, or co-edited by women. Contributing chapters by men are welcome. As always, the quality of the research we publish is paramount.
\n\nAll project proposals go through a two-stage peer review process and are selected based on the following criteria:
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\n\n“My scientific path has given me the opportunity to work with colleagues all over Europe, including Germany, France, and Norway. Editing the book Graph Theory: Advanced Algorithms and Applications with IntechOpen emphasized for me the importance of providing valuable, Open Access literature to our scientific colleagues around the world. So I am highly enthusiastic about the Women in Science book collection, which will highlight the outstanding accomplishments of women scientists and encourage others to walk the challenging path to becoming a recognized scientist." Beril Sirmacek, TU Delft, The Netherlands
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