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Barely three months into the new year and we are happy to announce a monumental milestone reached - 150 million downloads.
\n\nThis achievement solidifies IntechOpen’s place as a pioneer in Open Access publishing and the home to some of the most relevant scientific research available through Open Access.
\n\nWe are so proud to have worked with so many bright minds throughout the years who have helped us spread knowledge through the power of Open Access and we look forward to continuing to support some of the greatest thinkers of our day.
\n\nThank you for making IntechOpen your place of learning, sharing, and discovery, and here’s to 150 million more!
\n\n\n\n\n'}],latestNews:[{slug:"webinar-introduction-to-open-science-wednesday-18-may-1-pm-cest-20220518",title:"Webinar: Introduction to Open Science | Wednesday 18 May, 1 PM CEST"},{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"}]},book:{item:{type:"book",id:"3835",leadTitle:null,fullTitle:"Fecal Incontinence - Causes, Management and Outcome",title:"Fecal Incontinence",subtitle:"Causes, Management and Outcome",reviewType:"peer-reviewed",abstract:"Fecal incontinence is a common and disabling condition that unfortunately remains an “orphan” in terms of medical research and effective therapies. 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Dr. Kaykhaii has invented two new sample preparation techniques: salt-saturated single-drop microextraction and micro cloud point extraction. He is a member of the European Chemical Society Sample Preparation Study Group and Network as well as an affiliate member of the Royal Society of Chemistry.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"349151",title:"Prof.",name:"Massoud",middleName:null,surname:"Kaykhaii",slug:"massoud-kaykhaii",fullName:"Massoud Kaykhaii",profilePictureURL:"https://mts.intechopen.com/storage/users/349151/images/system/349151.png",biography:"Massoud Kaykhaii has been a Professor of Analytical Chemistry at the University of Sistan and Baluchestan (USB), Zahedan, Iran, since 1989. His research work focuses on modern sample preparation techniques including miniaturized solid and liquid phase microextraction and stir bar sorptive extraction. He has written 150 research articles, 5 books, and 22 national standard procedures of analysis. He has three patents to his credit. He has presented at 150 seminars/conferences and (co)supervised 106 MSc and Ph.D. theses. He is a member of the editorial advisory board of ninety-eight journals and acted as secretary of three national mirror committees of ISO/TC. Professor Kaykhaii was recognized as the best researcher at USB and as being in the top 1 percent of reviewers in chemistry by Publons.",institutionString:"University of Sistan and Baluchestan",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Sistan and Baluchestan",institutionURL:null,country:{name:"Iran"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"81",title:"Analytical Chemistry",slug:"chemistry-analytical-chemistry"}],chapters:[{id:"79484",title:"Introductory Chapter: Evolution of Sample Preparation",slug:"introductory-chapter-evolution-of-sample-preparation",totalDownloads:151,totalCrossrefCites:0,authors:[{id:"349151",title:"Prof.",name:"Massoud",surname:"Kaykhaii",slug:"massoud-kaykhaii",fullName:"Massoud Kaykhaii"}]},{id:"79034",title:"Modern Sample Preparation Techniques: A Brief Introduction",slug:"modern-sample-preparation-techniques-a-brief-introduction",totalDownloads:228,totalCrossrefCites:2,authors:[{id:"349151",title:"Prof.",name:"Massoud",surname:"Kaykhaii",slug:"massoud-kaykhaii",fullName:"Massoud Kaykhaii"},{id:"356813",title:"Dr.",name:"Mona",surname:"Sargazi",slug:"mona-sargazi",fullName:"Mona Sargazi"},{id:"426250",title:"Dr.",name:"Sayyed Hossein",surname:"Hashemi",slug:"sayyed-hossein-hashemi",fullName:"Sayyed Hossein Hashemi"}]},{id:"78687",title:"Advanced Sample Preparation Techniques for Surface Spectroscopy Analysis of Organic: Inorganic Hybrid Silica Particles",slug:"advanced-sample-preparation-techniques-for-surface-spectroscopy-analysis-of-organic-inorganic-hybrid",totalDownloads:174,totalCrossrefCites:0,authors:[{id:"419009",title:"Dr.",name:"Smrutirekha",surname:"Mishra",slug:"smrutirekha-mishra",fullName:"Smrutirekha Mishra"},{id:"419124",title:"Mr.",name:"Harekrishna",surname:"Panigrahi",slug:"harekrishna-panigrahi",fullName:"Harekrishna Panigrahi"},{id:"427165",title:"Dr.",name:"Suraj Kumar",surname:"Tripathy",slug:"suraj-kumar-tripathy",fullName:"Suraj Kumar Tripathy"}]},{id:"78769",title:"Preparation of Tissues and Heterogeneous Cellular Samples for Single-Cell Analysis",slug:"preparation-of-tissues-and-heterogeneous-cellular-samples-for-single-cell-analysis",totalDownloads:192,totalCrossrefCites:1,authors:[{id:"355954",title:"Prof.",name:"E.",surname:"Celeste Welch",slug:"e.-celeste-welch",fullName:"E. Celeste Welch"}]},{id:"78143",title:"Particle and Cell Separation",slug:"particle-and-cell-separation",totalDownloads:161,totalCrossrefCites:0,authors:[{id:"357422",title:"Prof.",name:"J. Paul",surname:"Robinson",slug:"j.-paul-robinson",fullName:"J. Paul Robinson"}]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@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. 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The modeling of robot manipulators with rotational joints has been extensively studied for decades. A robot manipulator is formed from the mechanical point of view by a kinematic chain of rigid bodies (links) which are joined together by linear or rotational joints. The modeling of the kinematics for a robot manipulator allows us to analyze the movements of the end-effector in order to perform a specific task.
\nTo determine the kinematics model for a robot manipulator, conventionally there are two types of analysis, the first is called “Direct Kinematics Problem” and the second analysis is “Inverse Kinematics Problem.” Depending on what is required for a specific task in the manipulator, different mathematical models can be used to control the behavior of a wide variety of robot manipulators [1–3]. The Direct Kinematics Problem determines the position of the end-effector of the robot manipulator as a function of the degrees of freedom (joint), wherein each degree of freedom is an independent movement in the mechanical structure of the robot manipulator. In contrast, the Inverse Kinematics Problem is oriented to determine the variation of the degrees of freedom of the manipulator according to the kinematics of the end-effector of the robot manipulator. This last situation is usually more complex from the point of view of mathematical modeling and its solution, given the high nonlinearity of behavior of the robot manipulators. A further disadvantage of conventional models of the Inverse Kinematics Problem is called redundancy. This happens when more than one solution is obtained by the mathematical model of the robot manipulator, causing theoretical configurations that do not occur in practical movements of an industrial manipulator. Additionally, another problem that occurs in the analysis of manipulators is when matrices for translations and rotations are used in the modeling. There is the possibility that this inverse matrix shows singular points, causing undefined mathematical solutions. Translations and rotations are essentials to solve the Kinematics Inverse Problem, to do that, auxiliary reference systems are used to refer the kinematics of the end-effector in the reference inertial system. In this regard, this chapter shows a new strategy to solve the Inverse Kinematics Problem for manipulators with rotational joints by approximating with cubic polynomial functions that define the positions of the joints.
\nOne of the first methods developed to solve the inverse kinematics for manipulator is by "Algebraic Method". In general, the algebraic methods are obtained by vector equations that respond to the links geometry of the manipulator. By this way, trigonometric relationships and algebraic equations are obtained to get mathematical functions that determine the behavior of the degrees of freedom in the manipulator [4–6]. It might be supposed that this method is the most effective however; it presents mathematical uncertainties, inconsistencies, and considerable complexity.
\nAnother method developed to determine inverse kinematics is by spatial geometry. This method is based on the decomposition of the spatial geometry of the manipulator in planar geometric systems. For many serial or parallel manipulators, this decomposition is simple when the axes of consecutive degrees of freedom present changes as 0°, 90°, and 270°. However, it is not always the case, so this method cannot always be applied. The inverse kinematics of each degree of freedom usually has more than one solution. Also, given the trigonometric relationships of mathematical models obtained, they cannot always be solved because it is more complex to obtain an exact solution for a greater number of degrees of freedom [7, 8].
\nOne of the most used alternative to solve the inverse kinematics of robot manipulators is based on Newton methods. These algorithms seek target configurations which posed as solutions to minimization problem [9]. Due to extreme complexity, these methods are known to be less practical. There are also methods based on statistical filtering [10] and sequential iterative approaches [11]. However, these statistical methods suffer high computational cost.
\nResearch about the use of polynomial functions with some relationships with the inverse kinematics problem for manipulators is not new. However, some of the applications are oriented to trajectory planning [12], and also to solve the inverse kinematics problem by polynomial of n-degrees by using genetic algorithms [13, 14].
\nThis work presents a novel approach to the inverse kinematic problem by cubic polynomial functions which are built under the definition of new parameters and the divided differences recursive method. The present approach is different to other strategies to get a solution for inverse kinematics problem. The benefits of the proposed method are essentially the simplicity to obtain a polynomial function for rotational joints that solve the inverse kinematics problem (position, speed, and acceleration), and the warranty to get a unique solution with no singularities. However, as many of the mathematical models that attempt to describe a real physical phenomenon, the method has several restrictions: bi-dimensional work space, the end-effector paths are straight lines, two joints are free and a third joint have fixed orientation, and four points of the path trajectory are needed.
\nFrom the mathematics point of view, the divided differences is a recursive division process of increments. The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form when data points (xi, yi) are known, where i is an integer ranging from 0 to (n−1) and n is the number of points.
\nBy definition, it is called divided difference of
To properly apply the method, it is essential to order in a table the values corresponding to the points (xi, yi), so that the xi column is sorted from the highest to lowest number, or vice versa. Considering an approach with four known points, the divided differences are defined as shown in Figure 1:
\nSequence of Divided Differences to four points.
Such polynomial function f(x) is determined as:
\nor expressed in the reduced form:
\nA new feature in the mathematical development shown in this work is to adapt the method of divided differences to approximate the functions that define the changes of the rotational joints θ
Point | \nx | \ny | \nθ | \nθ | \nθ | \n
---|---|---|---|---|---|
θ11 | \nθ21 | \nθ31 | \n|||
θ12 | \nθ22 | \nθ32 | \n|||
θ13 | \nθ23 | \nθ33 | \n|||
θ14 | \nθ24 | \nθ34 | \n
Known values of the four positions of the manipulator.
Before showing how to construct polynomials, it is relevant to note that the movement of the end-effector of the robot manipulator performs a linear motion by keeping constant the orientation of the end-effector (θ3). In the case of analysis shown in the X-Y coordinate plane, the positions changes occur in both coordinates. In this case, proportionality of change of θ
Taking into account that the sum of the percentages of both coordinates is 100%, the incidence factor of y in θ (in percentage) is determined as:
\nFor example, for the line defined by the equation y = x, the proportionality of the change is the same for both coordinates, so the variation of the x coordinate has a 50% incidence in the changes of θ1 and θ2. Similarly, the variation of the y-coordinate has a 50% incidence in the changes of θ1 and θ2.
\nThe determination % of x is obtained by the value of the slope of the straight line m, the incidence factor of x in θ (percentage) is calculated using the following equation:
\nThe absolute value of m is obtained because it is only of our interest to obtain the proportionality of the changes of the angles θ1 and θ2 as a function of lineal movements. So that,
\nsimilarly,
\nThe tables to determine the polynomials that approximate the behavior of the angle θ1 are built based on the new approach strategy presented here, remaining as:
\nx vs. influence of x in
y vs. influence of y in
In a similar way,
\nx vs. influence of x in
y vs. influence of y in
One of the contributions of the strategy presented here is the composition of polynomial functions of the joints of the robot manipulator θ
where ai, bi are coefficients obtained from Tables 2 and 3, and the coefficients ci, di are obtained by Table 4 respectively. Thus, divided differences are shown in Figure 2.
\nThe Divided Differences of x vs. θ
where,
\ndeveloping the general equation,
\nThis equation allows us to obtain the model of the polynomial that approximates the angular position θ with respect to the variable x.
\nIn developing this equation and simplifying results,
\nSimilarly, the polynomial θ(y) is obtained, which has the following form:
\nClearly, the differences divided for this equation refers to the rate of change of the θ with respect to y, being defined by the polynomial approximation model of joint θ1 and θ2 of the robot manipulator, which in simplified form are expressed as:
\nIn this section, the strategy proposed to approach the Kinematics Inverse Problem by polynomial functions of the rotational joints of a robot manipulator is shown. The procedures to three different cases of lineal path in the end-effector are analysed: a) slope of a straight line, m = 1, b) slope of a straight line, m < 1 and c) slope of a straight line, m > 1. To show the procedures, it is important to describe the mechanical configuration of the robot manipulator that is shown in Figure 3.
\nConfiguration of the robot manipulator, Θ1=90°, Θ2=0°, Θ3=90°.
The three degrees of freedom that are considered known in the plane of the robot manipulator moves are (
Remembering that the procedure exposes that four positions of the configuration robot are known, the approach by polynomial function are described in the next lines:
\na) Case 1. Slope of a straight line, m = 1.
\nThe four Robot positions, slope = 1.
In order to show the effectiveness of the developed strategy, the movement of the end-effector of the robot manipulator was selected by considering that the maximum length of the lineal trajectory is close to the effective length of link 1. For this case, the orientation of the end-effector is 90° and is keeping constant during its moves. Under these considerations, Figure 4 shows the selected lineal trajectory.
\nThe values of the four known positions of the robot manipulator are shown in Table 6.
\nx | \ny | \nΘ1 [°] | \nΘ2 [°] | \nΘ3 [°] | \n
---|---|---|---|---|
−15 | \n25 | \n193.51° | \n50.58° | \n90° | \n
−5 | \n35 | \n158.15° | \n23.53° | \n90° | \n
0 | \n40 | \n144.28° | \n18.59° | \n90° | \n
10 | \n50 | \n121.37° | \n19.71° | \n90° | \n
The four known positions of the lineal trajectory.
The incidence factor of x in θ, according to Eq. (9) and the value of the slope of a straight line is calculated as:
\nThen, from Eq. (8) the incidence factor of y in θ takes the next value:
\nConsidering both factors, the tables to obtain the polynomial functions for the joint θ1 are defined as:
\n96.755 | \n||
79.075 | \n||
72.14 | \n||
60.685 | \n
x vs. incidence of x in
96.755 | \n||
79.075 | \n||
72.14 | \n||
60.685 | \n
y vs. incidence of y in
The polynomial approach of Table 7 is determined by the divided differences according to Eqs. (14) to (19). In this way, next table shows the values obtained:
\nX | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
79.075 | \n−1.387 | \n0.0161 | \n\n | |
72.14 | \n−1.1455 | \n\n | \n | |
60.685 | \n\n | \n | \n |
Values of the divided differences for θ1(x).
Applying Eq. (22) and simplifying results:
\nThis polynomial function represents the incidence of x in joint of θ1. To complete the approach, it is also necessary to add the incidence of y in the same joint of θ1. So the polynomial approach from Table 8 is obtained by applying Eqs. (14) to (19).
\nY | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
79.075 | \n−1.387 | \n0.0161 | \n\n | |
72.14 | \n−1.1455 | \n\n | \n | |
60.685 | \n\n | \n | \n |
Values of the divided differences for θ1(y).
Applying Eq. (23) and simplifying it gives:
\nAccording to Eq. (24) the complete function approximation of θ1 becomes:
\nThe next step is to obtain the function approximation of θ2(x). According to the above procedure, the differences divided for Table 8 to get the polynomial approximation with respect of x takes the form:
\nX | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
11.765 | \n−0.494 | \n0.0366 | \n\n | |
9.295 | \n0.056 | \n\n | \n | |
9.855 | \n\n | \n | \n |
Values of the divided differences for θ2(x).
Applying Eq. (22) and simplifying it gives:
\nSimilarly, the divided differences of Table 8 to get the polynomial approximation for θ2(y) takes the form:
\nY | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
11.765 | \n−0.494 | \n0.0366 | \n\n | |
9.295 | \n0.056 | \n\n | \n | |
9.855 | \n\n | \n | \n |
The divided differences values for θ2(y).
Applying Eq. (23) and simplifying it gives:
\nAccording to Eq. (25) the complete function approximation of θ2 becomes:
\nObtaining result from the complete approximation by polynomials θ1 and θ2:
\nAfter obtaining the desired approach, the next step is to check the positions of the end-effector of the robot manipulator in order to verify that a lineal trajectory is made. According to the Direct Kinematics Model of the robot manipulator, the approximate position of the end-effector is determined by:
\nRemembering that L1 = 35, L2 = 30, L3 = 10 y θ3 = 90°, the Eq. (36) is evaluated in the range
−15 | \n25 | \n193.497 | \n50.580 | \n−14.983 | \n25.007 | \n
−12.5 | \n27.5 | \n183.534 | \n41.286 | \n−12.391 | \n27.637 | \n
−10 | \n30 | \n174.368 | \n33.782 | \n−9.896 | \n30.116 | \n
−7.5 | \n32.5 | \n165.930 | \n27.915 | \n−7.441 | \n32.554 | \n
−5 | \n35 | \n158.149 | \n23.530 | \n−4.980 | \n35.004 | \n
−2.5 | \n37.5 | \n150.955 | \n20.473 | \n−2.493 | \n37.485 | \n
0 | \n40 | \n144.280 | \n18.590 | \n0.019 | \n39.998 | \n
2.5 | \n42.5 | \n138.053 | \n17.726 | \n2.544 | \n42.530 | \n
5 | \n45 | \n132.205 | \n17.728 | \n5.063 | \n45.061 | \n
7.5 | \n47.5 | \n126.665 | \n18.440 | \n7.560 | \n47.565 | \n
10 | \n50 | \n121.364 | \n19.710 | \n10.026 | \n50.003 | \n
Values of θ1, θ2, xapprox and yapprox.
In order to ensure the validity of the polynomial function, the Error of x, Error of y, and Error(x, y) are calculated by the next equations:
\nGetting the following table for the values of Table 13:
\nError of x | \nError of y | \nError (x, y) | \n||
---|---|---|---|---|
−15 | \n25 | \n−0.017 | \n−0.007 | \n0.0179 | \n
−10 | \n30 | \n−0.104 | \n−0.116 | \n0.1555 | \n
−7.5 | \n32.5 | \n−0.059 | \n−0.054 | \n0.0801 | \n
−5 | \n35 | \n−0.020 | \n−0.004 | \n0.0206 | \n
−2.5 | \n37.5 | \n−0.007 | \n0.015 | \n0.0161 | \n
0 | \n40 | \n−0.019 | \n0.002 | \n0.0190 | \n
2.5 | \n42.5 | \n−0.044 | \n−0.030 | \n0.0529 | \n
5 | \n45 | \n−0.063 | \n−0.061 | \n0.0879 | \n
7.5 | \n47.5 | \n−0.060 | \n−0.065 | \n0.0881 | \n
Errors of the polynomial approach.
The error of interest is the maximum error, in this case the maximum error is 0.1754 units and occurs in the point (−12.5, 27.5). The percentage of maximum error is determined by the following equation:
\nwhere (xi, yi) are the coordinates of the starting point and (xf, yf) are the coordinates of the endpoint. That is, for an error rate, the maximum error is compared with the path length. Substituting values in Eq. (40), it gives:
\nConsidering that this value is acceptable, Eqs. (34) and (35) can be used to do a lineal trajectory in the end-effector of the robot manipulator.
\nb) Case 2. Slope of a straight line, m < 1.
\nSimilarly, as in the previous case, the end-effector of the robot manipulator makes a lineal path, which is shown in Figure 5. In this case it is considered θ3 as a constant, with the value of 45°.
\nThe four Robot positions, negative slope.
For this case, the values of four known positions of the manipulator robot are shown in the following table.
\nΘ1[°] | \nΘ2[°] | \nΘ3[°] | \n||
---|---|---|---|---|
−10 | \n45 | \n159.34 | \n58.46 | \n45 | \n
0 | \n40 | \n153.87 | \n35.69 | \n45 | \n
10 | \n35 | \n139.48 | \n9.94 | \n45 | \n
20 | \n30 | \n117.01 | \n-15.99 | \n45 | \n
Four known positions of the manipulator, negative slope.
The four known positions correspond to the linear relationship y = −0.5 x + 40. Considering that slope: m = −0.5, the incidence factors can be calculated by the (8) and (9) equations, like this:
\nAccording to the presented strategy, the incidence table to approximate polynomial functions θ1 takes the next form:
\nθ1i(x) [°] | \n||
---|---|---|
−10 | \n106.216 | \n|
0 | \n102.569 | \n|
10 | \n92.977 | \n|
20 | \n77.998 | \n
x vs. incidence of x in
45 | \n53.108 | \n|
40 | \n51.284 | \n|
35 | \n46.488 | \n|
30 | \n38.999 | \n
y vs. incidence of y in
Calculating the polynomial approximation to θ1(x), the next table is obtained:
\nX | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
−10 | \n||||
0 | \n102.569 | \n−0.9592 | \n−0.026935 | \n\n |
10 | \n92.977 | \n−1.4979 | \n\n | \n |
20 | \n77.998 | \n\n | \n | \n |
Divided differences values for θ1(x).
Applying Eq. (22) and simplifying it gives:
\nLikewise,
\nY | \nΔ1i | \nΔ2i | \nΔ3i | \n|
---|---|---|---|---|
45 | \n53.108 | \n|||
40 | \n51.284 | \n0.9592 | \n−0.05386 | \n\n |
35 | \n46.488 | \n1.4978 | \n\n | \n |
30 | \n38.999 | \n\n | \n | \n |
Divided differences values for θ1(y).
Applying Eq. (23) and simplifying it gives:
\nThe complete function approximation of θ1 becomes:
\nAlso, the tables to approximate polynomial functions θ2 are:
\nX | \n||
---|---|---|
−10 | \n38.969 | \n|
0 | \n23.790 | \n|
10 | \n6.626 | \n|
20 | \n−10.658 | \n
x vs. Incidence of x in
Y | \n||
---|---|---|
45 | \n19.484 | \n|
40 | \n11.895 | \n|
35 | \n3.313 | \n|
30 | \n−5.329 | \n
y vs. incidence of y in
Calculating the polynomial approximation for the Tables 20 and 21, we obtain:
\nAccording to Eq. (25) the function approximation of θ2 becomes:
\nDetermining the complete polynomial approximation of θ1 and θ2 by Eqs. (46) and (47) respectively, it results to:
\nApplying the direct kinematic model of the robot manipulator defined by Eq. (36), it proceeds to verify polynomials approach θ1 and θ2. In this case, θ3 is a constant with the value of 45°, where y = −0.5 x + 40, with x ∈ (−10.20). Considering an increase Δx = 3 units, the following table is obtained:
\n−10 | \n45 | \n159.324 | \n58.453 | \n−9.979 | \n44.995 | \n
−7 | \n43.5 | \n158.669 | \n52.102 | \n−7.103 | \n43.476 | \n
−4 | \n42 | \n157.158 | \n45.306 | \n−4.085 | \n41.984 | \n
−1 | \n40.5 | \n154.815 | \n38.142 | \n−1.007 | \n40.493 | \n
2 | \n39 | \n151.662 | \n30.685 | \n2.065 | \n38.994 | \n
5 | \n37.5 | \n147.721 | \n23.009 | \n5.093 | \n37.489 | \n
8 | \n36 | \n143.016 | \n15.192 | \n8.065 | \n35.989 | \n
11 | \n34.5 | \n137.568 | \n7.308 | \n10.995 | \n34.502 | \n
14 | \n33 | \n131.401 | \n−0.566 | \n13.923 | \n33.028 | \n
17 | \n31.5 | \n124.536 | \n−8.357 | \n16.910 | \n31.543 | \n
20 | \n30 | \n116.997 | \n−15.987 | \n20.023 | \n29.995 | \n
Values θ1, θ2, xapprox and yapprox.
Applying Eqs. (37), (38), and (39) the table of the errors become:
\nError of x | \nError of y | \nError(x, y) | \n||
---|---|---|---|---|
−10 | \n45 | \n−0.021 | \n0.005 | \n0.0218 | \n
−4 | \n42 | \n0.085 | \n0.016 | \n0.0863 | \n
−1 | \n40.5 | \n0.007 | \n0.007 | \n0.0099 | \n
2 | \n39 | \n−0.065 | \n0.006 | \n0.0654 | \n
5 | \n37.5 | \n−0.093 | \n0.011 | \n0.0938 | \n
8 | \n36 | \n−0.065 | \n0.011 | \n0.0656 | \n
11 | \n34.5 | \n0.005 | \n−0.002 | \n0.0059 | \n
14 | \n33 | \n0.077 | \n−0.028 | \n0.0816 | \n
17 | \n31.5 | \n0.090 | \n−0.043 | \n0.0995 | \n
Errors of the polynomial approach.
The four Robot positions, infinite slope.
The maximum error occurs at (−7, 43.5). From Eq. (40) the maximum percentage of error becomes:
\nSimilarly as in the previous case, this value is reasonably acceptable. So for this case, Eqs. (48) and (49) can be used to do the lineal trajectory in the end-effector of the robot manipulator.
\nc) Case 3. Slope of a straight line, m > 1.
\nAs in the two previous cases, the end-effector of the robot performs a linear path. In this case, the geometric robot positions were made considering an infinite slope (x = constant). Figure 6 shows these positions, and the angle of the end-effector is considered constant with a value of 180°.
\nThe values of the known positions are shown in the Table 24:
\nΘ1 [°] | \nΘ2 [°] | \nΘ3 [°] | \n||
---|---|---|---|---|
−50 | \n−20 | \n248.6 | \n155.16 | \n180 | \n
−50 | \n−10 | \n239.46 | \n137.78 | \n180 | \n
−50 | \n0 | \n226.55 | \n122.06 | \n180 | \n
−50 | \n15 | \n203.45 | \n105.24 | \n180 | \n
Four know positions of the manipulator, infinite slope.
Considering that slope: m = ∞, the incidence factors can be calculated by the (8) and (9) equations like this:
\nAs expected, the change in x has no any incidence on changes in θ1 and θ2, it is in this case that the change of y causes the change in both degrees of freedom of the robot manipulator.
\nThe incidence tables to approximate polynomial functions θ1 and θ2 take the next form:
\n-20 | \n248.6 | \n|
-10 | \n239.46 | \n|
0 | \n226.55 | \n|
4 | \n15 | \n203.45 | \n
y vs. incidence of y in
-20 | \n155.16 | \n|
-10 | \n137.78 | \n|
0 | \n122.06 | \n|
4 | \n15 | \n105.24 | \n
y vs. incidence of y in
Performing the same procedure as in the previous cases, the desired functions of polynomial approximation are obtained for both Tables 25 and 26:
\nIn this case θ3 = 180°, with
−20 | \n248.600 | \n155.16 | \n−19.983 | \n−19.983 | \n
−16.5 | \n245.925 | \n148.990 | \n−16.500 | \n0.000 | \n
−13 | \n242.667 | \n142.894 | \n−12.994 | \n−0.006 | \n
−9.5 | \n238.891 | \n136.940 | \n−9.484 | \n−0.016 | \n
−6 | \n234.663 | \n131.199 | \n−5.979 | \n−0.021 | \n
−2.5 | \n230.048 | \n125.743 | \n−2.481 | \n−0.019 | \n
1 | \n225.110 | \n120.644 | \n1.014 | \n−0.014 | \n
4.5 | \n219.917 | \n115.972 | \n4.512 | \n−0.012 | \n
8 | \n214.532 | \n111.799 | \n8.015 | \n−0.015 | \n
11.5 | \n209.021 | \n108.198 | \n11.520 | \n−0.020 | \n
15 | \n203.450 | \n105.239 | \n15.017 | \n−0.017 | \n
Values θ1, θ2, yaprox, and Error of y.
The maximum error in this case is in y =−6. From Eq. (40) the maximum percentage of error becomes:
\nThis value is so small that it can provide security to use the approximation obtained by Eqs. (53) and (54).
\nUntil now, there have been three different cases that demonstrate an acceptable approximation in the position of θ1 and θ2 by polynomial functions. Given that the position of the end-effector of the robot is known, it is considered in this work that the speed and acceleration of the end-effector are well-known. Considering that the polynomial approximation for any degree of freedom has the following form:
\nDeriving this equation, the speed for any degree of freedom is defined as:
\nDeriving newly, the acceleration for any degree of freedom becomes:
\nBy applying Eqs. (57) and (58), speed and acceleration for any degrees of freedom of the manipulator can be obtained respectively.
\nOne of the basic activities related to the movement of industrial robots is the solution of the Inverse Kinematics Problem. This chapter has presented a new strategy that allows for an approximate solution without the use of conventional methods. However, the approach method has several restrictions. Thus, it is possible to achieve an alternative solution to the Inverse Kinematics Problem by polynomial functions that define the behaviour of the rotational joints. An advantage of this strategy is that it is easy to implement, and after getting the polynomial function for the position of the joints, the speed and acceleration can be obtained by conventional derivation.
\nMoreover, because this approach is made from four known configurations of the robot, it can be considered that the methodology described is a way of discretizing the continuous kinematics of the robot, which is an interpolation technique to the inverse kinematic problem.
\nGiven the restrictions described in the analysed cases, further research is needed to determine if this strategy could be used for nonlinear paths; or for a kind of paths that require changes in the orientation of the end-effector, and also explore the possibility to extend the method for manipulator with more degrees of freedom.
\nThe c-myc oncoprotein is a well-studied, multifunctional transcription factor that controls cell proliferation, metabolism, and stress responses [1, 2]. Deregulation (amplification/overexpression) of c-myc occurs in many human cancers and is considered a “transition gatekeeper” for tumorigenesis. c-Myc controls metabolic reprogramming, a key event in tumorigenesis that provides gain-of-survival adaptive advantage, enabling cancers to thrive in tumor microenvironments with limited nutrient resources [3, 4, 5]. This rewiring of metabolic circuitry is characterized by (i) preferred use of glucose to produce ATP by aerobic glycolysis combined with increased lactate production even in oxygen abundance and (ii) dependence on glutamine for conversion to glutamate and subsequent entry into the tricarboxylic acid cycle by anaplerosis [6, 7]. The metabolic calibration by c-myc in cancer cells is orchestrated to adequately meet the demands of energy expenditure, macromolecular syntheses, and disposal of catabolic wastes, in concordance with robust cell growth. For example, the capacity for aerobic glycolysis in highly proliferative cancer cells is controlled by c-myc via an increase in glucose uptake as a result of induced expression of glucose transporter 1 (Glut1) and lactate dehydrogenase (LDH) to stimulate glycolytic flux [8, 9, 10, 11, 12]. Similarly, to accommodate the utilization of glutamine, c-myc upregulates the level of expression of sodium-dependent neutral amino acid transporter (SLC1A5), thereby facilitating the transport of glutamate and uptake of glutamine. c-Myc also induces glutaminase GLS which catalyzes the conversion of glutamine to glutamate [8, 9, 10]. We hypothesize that the inhibition of c-myc expression and function, in addition to restriction of access to glucose/glutamine are both bona fide anti-carcinogenic approaches.
\nCancer biology studies, for the past several decades, have focused on the identification and defining of DNA mutations, the narrowly centered focus being underpinned by the viewpoint that cancer is largely a genetic disease. However, cancer cells are also endowed by the acquisition of several hallmarks that enable them to become tumorigenic; among them is the apparent rewiring of gene:metabolite circuitry coordinating an altered metabolic adaptation with dysregulated cell proliferation in cancer cells [13]. Thus, the ability of cancer cells to preferentially utilize glucose and aerobic glycolysis as a paradoxical coincidence with ample oxygen supply was historically made by Warburg in the 1920s [14]. The so-called Warburg phenomenon amply illustrates the oncogenic addiction for glucose as a rapidly established metabolic adaptation that increases the capacity for ATP generation to meet the energy demands for unrestricted tumor cell proliferation and metastasis [7, 13, 15, 16, 17]. A similarly disproportionate reliance on amino acid glutamine is found in various cancer types [3, 4], compared to normal cells [18, 19]. The elevated levels of glutamine metabolism in cancer provide nitrogen for nucleotide and amino acid biosynthesis, and additionally, direct citrate and isocitrate in the TCA cycle for use in lipid synthesis, namely, reprogramming glutamine for energy production and for biosynthetic reactions via anaplerosis [6, 7, 20]. This metabolic rewiring is now recognized as an “Achilles heel” for cancer therapy.
\nc-Myc is a transcription factor that controls the expression of a large number of genes including those involved in ribosome and mitochondria biogenesis, glucose and glutamine metabolism, and lipid biosynthesis [8, 11, 12, 21]. c-Myc controlled gene sets underpin bioenergetics and synthesis of building blocks required for macromolecular assembly, transformation, proliferation and tumorigenesis. As a proto-oncogene, c-myc is frequently amplified/overexpressed in human cancers. Among a myriad of tumorigenic changes, c-myc controls are metabolic reprogramming—preferred use of glucose and glutamine, and sequential regulation of downstream targets/effectors [8, 11, 12, 21, 22]. c-Myc has been shown to induce the expression of glucose transporter Glut1 and glycolytic enzyme LDH to increase glycolytic flux. Similarly, c-myc also activates glutamine transporters SLC1A5 (also known as solute carrier family 1, member 5 and SLC38A5 - solute carrier family 38, member 5) that acts to increase uptake/transport of glutamine [12]. The rewiring of metabolic programming by c-myc provides cancer cells with survival advantage by meeting the demands of energy expenditure, macromolecular syntheses, and catabolic waste disposal. Studies have also shown that upregulation of c-myc is associated with an increase in mitochondrial glutaminolysis, which plays an overarching role on glutamine addiction in cancer cells. c-Myc also induces GLS expression through a dual negative mechanism – suppression of miR-23a/b by c-myc, and inhibition of GLS expression by miR-23a/b [21, 23]. The upregulation of GLS by c-myc facilitates the coordination of metabolic addiction with oncogenesis and the coalescence of metabolic calibration for survival with cellular transformation, proliferation, and cancer-related gene mutations. While metabolic addiction is clinically viewed as favoring carcinogenesis, it is equally plausible that the same nutrient dependence may become a restriction point for designing cancer therapy. Specifically, by limiting access to nutrients within the tumor microenvironments, novel oncogenic targets could become evident and are amenable for rational design of countermeasure strategies. A notable example can be found in the control of c-myc as a “master driver” for cancer; an equally appealing consideration likely involves the downstream targets regulated by c-myc [24, 25].
\nResveratrol is a grape polyphenol whose efficacy has been amply supported by tissue culture and animal studies, and in limited clinical trials [26, 27, 28, 29, 30]. Resveratrol has been reported to inhibit c-myc in multiple cancer types including medulloblastoma cells, breast cancer cells and osteosarcoma [31, 32, 33]. How resveratrol inhibits c-myc via rewiring of metabolite:gene circuitry in tumorigenesis remains largely unknown and will be discussed in the next section. Greater understanding of this will add yet another dimension to the cancer preventive and therapeutic efficacy of resveratrol.
\nSince c-myc is involved in control of metabolic processing and resveratrol can inhibit c-myc expression, we propose that the c-myc-mediated metabolic dysregulation of cancer can be countered using grape-derived resveratrol. It is our hypothesis that resveratrol exerts a dual role in disrupting c-myc-mediated cancer metabolic reprogramming—a
As a powerful factor governing the transcription of large gene sets that encode proteins playing critical roles in numerous cellular processes, both in normal and diseased states, the level of expression of c-myc is under stringent control. Ample data point to c-myc degradation being regulated by sequential phosphorylation of S62 and T58, by two external signal activated kinase cascades, respectively, the RAF/MEK/ERK and PI3K/AKT/GSK3β signaling pathways [46, 47, 48, 49, 50, 51]. T58 phosphorylation of c-myc promotes its interaction with the ubiquitin ligases Fbw7 and Skp2, ubiquitination and degradation by the proteasome [52, 53, 54]. Additionally, deubiquitinating enzymes, USP28 and USP36, also contribute to c-myc degradation [54, 55, 56]. Of note, the reported AKT/GSK3β-mediated c-myc T58 phosphorylation in control of its turnover [41, 42, 43] is relevant to our own studies: (i) NQO2 is involved in AKT/GSK3β-mediated cyclin D1 T286 phosphorylation and degradation and (ii) NQO2 knockdown CWR22Rv1-sh25 cells show a 37% decrease in chymotrypsin-like proteasome activity compared to control CWR22Rv1-sh08 cells [44]. These results suggest a hitherto-never-considered aspect of control of c-myc stability by NQO2. Next, we will discuss a proposed study to test the potential role NQO2 plays as the mediator of control of c-myc stability via AKT/GSK3β-c-myc T-58 phosphorylation, and by regulation of activity and functioning of the proteasome.
\nOur previous studies showing that resveratrol exerts its effects via its target protein NQO2 provide the impetus for testing that down regulation of c-myc by resveratrol requires the participation of NQO2. Based on our data that NQO2 affects cyclin D1 turnover, we expect that NQO2 will increase c-myc degradation, that is, ↓t1/2 instead of conferring protection on c-myc stability by competing against proteasome as has been reported for C/EBPα whose degradation by 20S proteasome is attenuated by NQO2 [35]. Since c-myc degradation by proteasomes is known to involve a multitude of mechanisms, there is a possibility resveratrol or NQO2 may directly affect c-myc degradation by exerting control on ubiquitin ligases like Fbw7 and Skp2 or the deubiquitinating enzyme, USP28. As to whether NQO2 interacts with AKT to affect AKT-GSK3β-mediated c-myc degradation, our expectation is that the accumulated c-myc protein in MG132-treated cells will show a higher T58 phosphorylation as compared to nontreated control cells. As a corollary, addition of GSK3β inhibitor LiCl to treated cells should significantly reduce T58 phosphorylated c-myc protein, in parallel with an increase in c-myc protein accumulation. siRNA-knockdown of GSK-3α or -3β in NQO2 expression cells compared to NQO2 knockdown cells will further confirm the role of GSK3β in mediating c-myc degradation. Results of these studies will provide support for the as yet untested hypothesis regarding the indirect role of NQO2 in controlling AKT → GSK3β → c-myc T58 phosphorylation → c-myc degradation by proteasome, and the direct role of resveratrol acting as a metabolic switch to shut off c-myc-mediated metabolic reprogramming in cancer cells.
\nEpidemiological studies have shown that moderate intake of red wine is correlated with a reduced incidence of dementia and neurodegenerative disease [57]. Moreover, resveratrol, a tri-hydroxyl stilbene found in abundance in red wine, red grapes, peanuts and a number of other consumed foods in the United States, has been reported to confer protection against oxidative stress in PC-12 cells [58, 59]. It has been determined that the preeminent presence of senile plaques, composed mainly of amyloid-β (Aβ) deposits, is a pathological brain feature in individuals diagnosed with Alzheimer’s Disease (AD). The Aβ peptides are derived from cleavage of the amyloid-β precursor protein (APP) and have been shown to destabilize neurons and lead to cell death through the induction of oxidative stress, mediated by the generation of reactive oxygen species (ROS) and elevation in intracellular hydrogen peroxide [60, 61, 62]. Compelling evidence supports that Aβ peptides serve as the “primary instigator” of AD [63]. Davies and coworkers[64] used tissue culture studies combined with biochemical assays and siRNA-silencing approaches to show that resveratrol lowered the levels of secreted and intracellular Aβ-peptides in a concentration and time-dependent manner. Further, this effect occurred not by targeting the Aβ-producing enzymes β and γ-secretases or by affecting the stability of APP or the turnover of its C-terminal fragments; but instead appeared to involve the promotion of intracellular degradation of Aβ via a proteasome subunit β5-dependent mechanism. It is worth noting that resveratrol reportedly also promotes the proteasome-mediated degradation of important regulatory proteins, including cyclin D1 [65], the estrogen receptor-α [66], and the hypoxia-inducible factor-1α [67].
\nThe notion that a folded three-dimensional structure is required for the biological function of a protein is dispelled by the discovery of intrinsically disordered proteins (IDPs) in the 1990s [68]. IDPs can be viewed as proteins that have minimal structures or are devoid of an overall defined fold, either entirely or in parts and are more likely to exist in dynamic, mosaic states under physiological conditions. The absence of structural orderliness also confers plasticity and “fussiness” to IDPs for diverse protein-protein interactions; however, the very same structural flexibility may also render them difficult and challenging as druggable targets using traditional structure-function drug design approaches.
\nEukaryotic transcription factors perform important biological functions in control of gene expression. They play an essential role in identifying the target sequences on DNA located in the vicinity as well as far removed from the transcription start site through direct protein:nucleic acid interaction, and also are required for binding to a large array of co-transcription regulatory proteins via protein:protein interaction. As such, eukaryotic transcription factors have been shown to exhibit a high degree of intrinsic disorderliness; based on bioinformatic model prediction analysis it is estimated that more than 49% of human transcription factors contain intrinsic disorderliness [69]. Studies have also revealed that IDPs, because of their intrinsic destabilized nature circumventing the requirement for unfolding protein substrates for proteolysis by the 26S proteasome, are more likely to be degraded via an ubiquitin-independent mechanism using the 20S proteasomes [70].
\nThe intrinsically unstructured protein (IUP), disorderly theme is also found in transcription factor c-myc; indeed, c-myc only attains an ordered structure after binding to its disordered partner MAX protein (myc-associated factor). Bioinformatics and experimental approaches have estimated that c-myc contains more than 45% of residues which have high probability for disordered structure formation [71]. The possession of intrinsically disordered regions allows c-myc to be degraded independently of ubiquitin, which may account for its observed short half-life at the mRNA [72, 73, 74] and protein levels [75, 76, 77], and is dynamically aligned with its switch on/off master transcriptional role independent of collaborative interaction with the pool of cellular ubiquitin that drives 26S proteasome-mediated protein degradation.
\nIt should be noted that the function of the 20S proteasome can also be modulated by interaction with NAD(P)H:quinone oxidoreductase 1 and 2 (NQO1 and NQO2). Previous studies have reported that NQO1 physically binds the 20S proteasome in an NADH-dependent manner [78, 79, 80] and to protect IDPs from degradation [81]. A double negative feedback mechanism exists between NQO1 and the 20S proteasome [78]. On the one hand, NQO1 acts to attenuate the proteolytic activity of the proteasome; on the other hand, the proteasome degrades the NQO1 FAD-free apo form which manifests as a partially unfolded structure and a substrate for the 20S proteasome [81]. Studies have shown that NQO2 confers protection against proteolytic degradation by the 20S proteasome [35, 82] albeit by a mechanism independent of NQO1 [83].
\nResveratrol has been shown to inhibit the uptake/transport of glucose or glutamine, and decrease the expression of c-myc in cancer cells. Rewiring of metabolite:gene circuitry is a key event in tumorigenesis that has been known for decades [7, 15, 16, 17], however, with an incompletely understood underlying mechanism. In this chapter, we discuss the aversion of c-myc-mediated reprogrammed cancer cell metabolism by targeting the expression and stability of c-myc using a chemical/genetic disruptive approach focusing on resveratrol and its high affinity target NQO2 we identified [36]. Additionally, the hypothesis we propose broadens the classical function of NQO2 in quinone detoxification to AKT/c-myc-mediated metabolic reprogramming observable in a clinical setting. Taken together, resveratrol/NQO2 in a c-myc controlling role to block metabolic addiction represents a novel diet-based chemoprevention approach in concept, and is transformative in implications warranting further investigation. The results will lay foundation for discovery of drugs able to disrupt AKT/c-myc-mediated reorganized metabolism using NQO2 inhibitors.
\nIn summary, we advance the thesis to avert c-myc-mediated metabolic reprogramming in cancer cells by targeting the control of c-myc and the uptake and metabolism of glucose and glutamine and their downstream effectors using resveratrol. We propose to focus on control of phosphorylation of c-myc T58 by GSK3β, shown to be critical for proteasome mediated c-myc degradation, by the resveratrol target protein NQO2 which we have previously shown to act as a modulator of AKT/GSK3β proteasome mediated degradation of cyclin D1 [44].
\nThe dual role resveratrol plays in disrupting c-myc-mediated metabolic reprogramming in cancer cells—a direct role targeting suppression of c-myc expression and an indirect role involving NQO2-AKT-GSK 3β mediated increase in c-myc T58-phosphorylation for increased degradation by proteasome—is illustrated in Figure 1.
\nResveratrol and NQO2 exert dual control of c-myc-mediated glucose/glutamine adaptation in cancer cells by transcription/translation suppression of c-myc expression and by control of proteasome-dependent c-myc stability according to the sequence: NQO2 binds AKT, reducing AKT kinase and increasing GSK 3β activity, resulting in increase in c-myc T58 phosphorylation and facilitating an increase in c-myc degradation by proteasome.
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\n\nOur reputation – Everything we publish goes through a two-stage peer review process. We’re proud to count Nobel laureates among our esteemed authors. We meet European Commission standards for funding, and the research we’ve published has been funded by the Bill and Melinda Gates Foundation and the Wellcome Trust, among others. IntechOpen is a member of all relevant trade associations (including the STM Association and the Association of Learned and Professional Society Publishers) and has a selection of books indexed in Web of Science's Book Citation Index.
\n\nOur expertise – We’ve published more than 4,500 books by more than 118,000 authors and editors.
\n\nOur reach – Our books have more than 130 million downloads and more than 184,650 Web of Science citations. We increase citations via indexing in all the major databases, including the Book Citation Index at Web of Science and Google Scholar.
\n\nOur services – The support we offer our authors and editors is second to none. Each book in our program receives the following:
\n\nOur end-to-end publishing service frees our authors and editors to focus on what matters: research. We empower them to shape their fields and connect with the global scientific community.
\n\n"In developing countries until now, advancement in science has been very limited, because insufficient economic resources are dedicated to science and education. These limitations are more marked when the scientists are women. In order to develop science in the poorest countries and decrease the gender gap that exists in scientific fields, Open Access networks like IntechOpen are essential. Free access to scientific research could contribute to ameliorating difficult life conditions and breaking down barriers." Marquidia Pacheco, National Institute for Nuclear Research (ININ), Mexico
\n\nInterested? Contact Ana Pantar (book.idea@intechopen.com) for more information.
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