Trajectory planning of endeffector with intermediate point Planowanie trajektorii ruchu chwytaka z punktem pośrednim*


 Bartholomew Willis Hart
 2 years ago
 Views:
Transcription
1 Article citation info: Graboś A, Boryga M. Trajectory planning of endeffector with inrmedia point. Eksploatacja i Niezawodnosc Mainnance and Reliability 2013; 15 (2): Andrzej Graboś Marek Boryga Trajectory planning of endeffector with inrmedia point Planowanie trajektorii ruchu chwytaka z punkm pośrednim* The article presents the Polynomial Cross Method (PCM) for trajectory planning of an endeffector with an inrmedia point. The PCM is applicable for designing robot endeffector motion, whose path is composed of two rectilinear segments. Acceleration profile on both segments was described by the 7 th degree polynomial. The study depicts an algorithm for the method and the research results presend as the runs of resultant velocity, acceleration and linear jerk of the stationary coordina sysm. Keywords: trajectory planning, polynomial acceleration profile, jerk. W pracy zaprezentowano metodę PCM (Polynomial Cross Method) do planowania trajektorii ruchu chwytaka z punkm pośrednim. PCM ma zastosowanie do planowania ruchu chwytaka, którego tor składa się z dwóch odcinków prostoliniowych. Profil przyspieszenia na obu odcinkach opisany został wielomianem siódmego stopnia. W pracy przedstawiono algorytm metody oraz wyniki w postaci przebiegów prędkości, przyspieszenia i udaru liniowego. łowa kluczowe: planowanie trajektorii, chwytak, wielomianowy profil przyspieszenia, udar. 1. Introduction Trajectory planning proves to be the first and critical phase in the operation of robotic workstations (such as supporting of machines, painting, welding, sealing, gluing, cutting, assembly, palletization and depalletization). This problem has been an active field of research and consequently vast lirature addresses the issue. The authors have applied various chniques for trajectory generation. ome of them considered the minimization of adverse jerk that causes the practical limitation of trajectory mapping errors. The works of Visioli [10] and Dyllong and Visioli [3] highlight the unfavourable jerk effects at the initial and final point of the path for the cubic and thirdorder trigonometric splines. Inrestingly, in some cases, jerk reduction was achieved by the fourthorder trigonometric spline introduction. One of the criria for optimization of motion path design given by Choi et al. [2], was to keep the jerk within the specified limits. The obtained jerk profiles in the kinematic pairs are discontinuous and sp shaped. At the initial and final point of the trajectory, the jerk is different from zero. Red [7] using the curves applied the constant (but different from zero) jerk values at the transition period between the constant phases of acceleration and deceleration. The analysis of the link acceleration profiles for Puma 560 manipulator presend by Rubio et al. [8] indicas that negative jerk effect in the kinematic pairs occurs at the initial and final point of the trajectory. That agrees with the observations made by aramago and Ceccarelli [9] in their study on jerk runs in the kinematic joints. According to Huang et al. [5], the jerk profiles in the kinematic pairs at the both start and end points motion are close to zero. The method proposed by Olabi et al. [6], generas smooth jerk limid patrn constrained by the laws of tool motion and taking into account the joints kinematics constraints. Very inresting research results on the jerk runs in kinematic pairs were repord by Gasparetto and Zanotto [4]. They obtained not only continuous jerk for the applied fifthorderbsplines, but importantly, its values at the start and end path point were equal zero. The higher degree polynomials to describe acceleration profile were applied by Boryga and Graboś [1]. The authors analyzed the runs of velocity, acceleration and jerk for polynomials of the 5 th, 7 th and 9 th degree. On the basis of the simulation sts performed, they achieved the lowest values of the linear and angular jerks for the 7 th degree polynomials. The authors proposed the Polynomial Cross Method (PCM) algorithm, which allows the design of trajectory comprising two rectilinear segments in the robot workspace. There were formulad the following assumptions concerning the manipulator endeffector motion: acceleration profile on both rectilinear segments depicd with 7 th degree polynomial, acceleration profile at the initial and end path points is tangent to the time axis that eliminas adverse jerk effect, change of runup phase into brake one occurs at the inrmedia point, linear acceleration value for any coordina does not exceed the preset maximum value a max, endeffector motion proceeds so that resultant velocity does not change at the inrmedia point (where rectilinear segments connect). As a consequence of the presumed constant resultant velocity value at the inrmedia point, resultant acceleration is equal to zero. It is noworthy that at the inrmedia point, a direction of the resultant velocity vector gets changed due to the preset path of the endeffector. ubstantial advantage of the presend algorithm proves to be a fact that coefficients of the polynomials depicting the acceleration profile on any coordina are established solely on coordina increment and preset maximum acceleration. In general, the jerk elimination at the initial and final trajectory point influences the accuracy of trajectory mapping. That appears to be very helpful as far as chnological processes such as pick and place, painting, assembly, welding, sealing, gluing, palletization and depalletization are concerned. ayout of the paper comprises the following sections: ection 2 depicting a trajectory planning chnique with the 7 th degree polynomial application utilizing the root of an equation multiplicity; ection 3 presenting an algorithm, which was divided into initial computations, computations for a longer and shorr rectilinear segment and final computations; ection 4 demonstras the example of the proposed algorithm practi (*) Tekst artykułu w polskiej wersji językowej dostępny w elektronicznym wydaniu kwartalnika na stronie 182 Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2, 2013
2 cal employment, while ection 5 summarizes the simulation results. The conclusions are presend in the last section of the paper. 2. Trajectory planning with polynomial use The planning of robot endeffector trajectory can be accomplished by using higherdegree polynomials, that facilita acceleration profile development. The study of Boryga and Graboś [1] showed that among the polynomials of 5 th, 7 th and 9 th degree describing the acceleration profile, the lowest values of linear and angular jerks were repord for the 7 th degree polynomial. Therefore in this paper, the acceleration profile on coordina x i is described with the 7 th degree polynomial in the form of xi( t) = ai ( t) ( t 0.5 ) ( t ) (1) where: a i coefficient of polynomial on coordina x i, i =1, 2, 3 coordina number, t e time of motion end. Acceleration profile described with the 7 th degree polynomial is presend in Fig xi () t xbi a = ± i t t + t t + t t (4) If the endeffector motion is concordant with the versor of the axis i, the plus sign should appear in the equation and the minus one if it is discordant. 3. Planning trajectory with inrmedia point 3.1. Polynomial Cross Method (PCM) PCM is employed to genera endeffector trajectory whose path is composed of two connecd rectilinear segments BM and ME (Fig. 2). Implementation of polynomial acceleration profile of the robot endeffector defined with the equation (1) for the preset segments BM and ME could cause that the endeffector velocity at the inrmedia point was equal to zero. Thus, the problem of trajectory motion planning would be simplified to the motion with a stop at the inrmedia point M. For that matr, the ancillary points E and B are introduced. The E point arises from the axial symmetry of point B reflecd across the inrmedia point M, whereas the point B through the axial symmetry of the point E reflecd across the inrmedia point M. Description of acceleration defined by the equation (1) includes the segments BE and B E (rmed total segments in the algorithm). On both total segments, change of the runup phase into brake one proceeds at the inrmedia point M. Maximum acceleration of robot endeffector was limid to a max value. It was assumed that at the transition from Fig. 1. Acceleration profile described by a 7 th  degree polynomial Acceleration profile is depicd by a continuous function for each coordina of the Carsian coordina sysm x i. Change of the runup phase into brake phase proceeds at t=0.5t e and the acceleration profile for t=0, t=0.5t e and t=t e is tangent to the time axis. Thus, jerk effect is eliminad at these points. The polynomials describing the profiles of velocity and displacement dermined through analytical ingration of the dependence (1) go as follows: xi() t a = i t tt e + tt e tt e + tt e tt e (2) xi() t a = i t tt e + tt e tt e + tt e tt e (3) The obtained value x i (t) is a distance tracked by the robot endeffector on the coordina i. In order to establish the endeffector coordina at any moment of time, the following points should be taken into account initial coordina of endeffector on coordina i denod as x bi, and direction of endeffector motion concordant or discordant with the axis versor orientation. The endeffector coordina on the coordina i is defined by the equation Fig. 2. Planned trajectory BME and ME and B M ancillary segments the BM segment to the ME one (at the inrmedia point M), the resultant velocity does not change, while resultant acceleration is equal to zero. Change of the direction and orientation of the velocity vector at the point M is imposed by the predermined trajectory of endeffector motion. At the inrmedia point M, there occured a rotation of the resultant velocity vector from the BM direction towards the ME direction. The problem can be solved through the introduction of the arc connecting the rectilinear segments or an alrnative stop in the inrmedia point. Coefficients of polynomials depicting acceleration profile on the total segments are dermined separaly for each coordina x i. The motion time is calculad using only the path increments and preset maximum acceleration a max. Velocity value at the inrmedia point is established performing the substition of t=0.5t e into the dependence describing velocity profile (2). The resultant velocity vector at the inrmedia point displaces from one segment to the other and projects on the axes of the stationary coordina sysm. That facilitas the dermination of coefficients of a polynomial depicting aceleration profile on the other total segment. As the motion time on both total segments may vary, it is necessary to perform an Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2,
3 appropria translation in the time of acceleration, velocity, displacement and jerk profile PCM algorithm Initial computations p 1. Assumption of coordinas of the initial, inrmedia and fi b b b m m m nal points are denod by B( x1; x2; x 3), M( x1 ; x2 ; x 3 ) and e e e E( x1; x2; x 3). The points should belong to the workspace. p 2. Dermination of the coordinas of ancillary points b b b B( x1 ; x2 ; x 3 ) and e e e E( x1 ; x2 ; x3 ) are made on the grounds of dependence b m e xi = 2xi xi for i = 1, 2, 3 (5) e m b xi = 2xi xi for i = 1, 2, 3 (6) The B and E ancillary points need not belong to the workspace. The ancillary distances B M and ME are used only to construct an appropria form of the acceleration profile. p 3. Dermination of path increments on each coordina of the total segments BE e b xi xi xi = for i = 1, 2, 3 (7) BE e b xi xi xi = for i = 1, 2, 3 (8) p 4. cheduling of the coordina increments starting from the highest, with denotation by subscript in the brackets, in the schedule sequence BE BE BE x{1} x{2} x{3} (9) BE BE BE x{1} x{2} x{3} (10) p 5. Dermination of maximum coordina increment out of BE and B E segments and denoting it as x{1}, that is, BE B E x{1} = max{ x{1}, x{1} } (11) In the x{1} denotation, a superscript describes the longer total BE B E BE segment. If x{1} = x{1} increments are equal then x{1} = x{1}. p 6. Assumption of endeffector maximum acceleration a max on the coordina of the maximum path increment, x{1}. Thus, the accelerations on the other coordinas will not exceed the preset acceleration a max that results from lower or equal path increments on these coordinas Computations longer total segment () p 1. Dermination of polynomial a {1} coefficient and the end time of motion t e on coordina x {1} requires solution of the equation sysm where: 1 9 a{1} ( ) = x{1} c a{ 1} ( ct 2 e ) ( ct 2 e 0.5 ) ( ct 2 e t e ) =amax Having solved the above equation sysm, the below was obtained: 5 c1 a a max {1} = 9 ( cc 3 13) amax ( x{1} ) x{1} cc 1 3 amax x{1} amax (12) (13) = (14) c 1 = 10080, c2 =  21, c3 =  c2(2c21) ( c21) p 2. Dermination of polynomial coefficients for the other coordinas [1] c1 a{} i x 9 {} i ( ) = for i = 2, 3 (15) p 3. Dermination of components and endeffector resultant velocity at M point 8 ( t ) x e {} im= a{} i 6144 for i = 1, 2, 3 (16) 8 3 ( t ) 2 x e M = ( a{} i ) 6144 i= 1 (17) Computations shorr total segment () p 1. Dermination of direction cosines between the speed vector in the M point and the axes of the stationary coordina sysm x cos( α ) i e xi b i = if 3 ( xi e xi b 2 ) i= 1 x cos( α ) i e xi b i = 3 ( xi e xi b 2 ) i= 1 B x{1} > x{1} (18) B if x{1} x{1} (19) 184 Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2, 2013
4 where: α 1 angle between the speed vector at M point and axis x i of the stationary coordina sysm. peed vector orientation comes from a motion direction on a segment. p 2. Dermination of speed components in M point point x im = x M cos( α i) for i = 1, 2, 3 (20) According to the assumption, the resultant speed value in the M x M = x M = x M, does not change. Table 1. The point coordinas for the planned trajectory and ancillary points Point Point coordinas [m] denotation x 1 x 2 x 3 B M E B E p 3. Dermination of motion time for shorr total segment 105 x t i e = 64 x im for i = 1 (21) Formula (21) results from a sysm of equations formed from the dependences (15) and (16). The same motion time is obtained when appropria coordina increments and appropria velocity components at point M are substitud simultaneously. p 4. Dermination of polynomial coefficients on each coordina ai Final computations 6144 x 8 im ( ) = for i =1, 2, 3 (22) p 1. Dermination of motion start time on the B E segment t t e b =± (23) 2 The time is established so as to obtain the same velocity at exactly the same moment in the M point for both move segments. If B x{1} > x{1}, dependence (23) should acquire the minus sign, the opposi case should acquire the plus sign. p 2. Time displacement of the polynomial depicting the acceleration profile on the coordinas of B E distance by t b value B xi( t) = ai ( t tb) ( t 0.5 tb) ( t tb) if x{1} > x{1} (24) B xi( t) = ai ( t tb) ( t 0.5 tb) ( t tb) if x{1} x{1} (25) Analogical time displacement should be done for polynomials, describing the level of velocity, displacement, and jerk. p 3. Dermination of motion time on the segments along the BME path + t t e e = (26) 2 4. Numerical example The point coordinas for the planned endeffector trajectory B, M, E and ancillary points E and B are presend in Table 1. The path BE BE BE increments on each coordina are: x1 = 0, x2 = x3 = 0.5 m, BE BE x1 = x3 = 0.5 m, be the first segment to study. BE x2 = 0. ince BE B E x{1} = x{1}, BE will The maximum acceleration set is a max = 2 m/s 2 on coordina x 2. The polynomial coefficients depicting acceleration level on each coordina as well as motion time go as following: 9 a{1} = a{2} = m/ s, a {3} = 0, = s. As for the BE distance, path increments are recorded for the coordinas x 2 and x 3 so consequently, the established coefficients a {1} and a {2} refer to these coordinas. Resultant speed in M point is x M = m/ s. The direction cosines of a speed vector in M point for the B E segment go as follows cos α 1 = 2 / 2, cosα 2 = 0, cos α 3 = 2 / 2. The velocity components in M point on the B E segment are x 1 M = x 3 M = m/ s, x 2 M = 0, whereas 1, 2 and 3 indices refer to the axis of the stationary coordina sysm. Polynomial coefficients describing acceleration profile on each coordina of the B E segment are 9 a1 = a3 = m/ s, a 2 = 0. Move time recorded on the B E segment was = s. Time of motion along the BME path is = s, while t = imulation sts results b According to the simulation sts performed, the following courses of kinematic characristics of endeffector were recorded (Fig. 3 5). In each figure presend, a continuous line indicas the runs of kinematic characristics of motion for the designed trajectory BME, while a dashed line the courses for ancillary segments (ME and B M). A planned motion path and ancillary distances are displayed in the stationary coordina sysm x 1 x 2 x 3, whereas kinematic characristics of motion at two planes perpendicular to the plane dermined by the points of the generad BME trajectory. Fig. 3 displays the runs of endeffector speeds on the BE and B E segments. The maximum velocity x M = x M = m/ s is obtained at the point M. The speed value at transition from the BM segment to ME did not change, while a direction of the resultant velocity vec Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2,
5 Fig. 3. Resultant velocity course along planned BME path and ancillary segments Fig. 5. Resultant jerk course along planned BME path and ancillary segments Fig. 4. Resultant acceleration course along planned BME trajectory and ancillary segments tor changes from BM path to ME. The runs of endeffector resultant linear acceleration on the distances BE and B E are presend in Fig. 4. In the points B, M, E acceleration is equal to zero. The obtained absolu maximum acceleration at each coordina does not surpass the set value a max = 2 m/s 2 and the maximum resultant value reaches 2.83 m/ s 2. A jerk value at points B and E is equal zero (Fig. 5). The maximum jerk value is 19.8 m/s Conclusions On the basis of the simulation sts of the manipulator endeffector motion according to the PCM, the following conclusions were formulad: a) The profiles of resultant velocity, acceleration and jerk obtained by the PCM application are continuous on the BM and ME segments. At the inrmedia point M, resultant velocity value does not change, consisntly with the underlying assumption. The velocity vector direction changes according to the motion direction (from BM towards ME). b) Generation of trajectory according to the PCM may be utilized in some chnological processes (pick and place, painting, assembly, welding, sealing, gluing, palletization and depalletization), where it is critical to elimina jerk effect in the initial and final point of the trajectory. c) If deformability of kinematic chain occurs, jerk elimination will result in vibration limitation that guaranes lower tracking errors. Our further research will focus on the effect of trajectory of the robot manipulator endeffector (planned using PCM) on kinematics and manipulator dynamics. References 1. Boryga M, Graboś A. Planning of manipulator motion trajectory with higherdegree polynomials use. Mechanism and Machine Theory 2009; 44: Choi YK, Park JH, Kim H, Kim JH. Optimal trajectory planning and sliding mode control for robots using evolution stragy. Robotica 2000; 18: Dyllong E, Visioli A. Planning and realtime modifications of a trajectory using spline chniques. Robotica 2003; 1: Gasparetto A, Zanotto V. Optimal trajectory planning for industrial robots. Advances in Engineering oftware 2010; 41: Huang P, Xu Y, iang B. Global minimumjerk trajectory planning of space manipulator. Inrnational Journal of Control, Automation, and ysms 2006; 4 no. 4: Olabi A, Béarée R, Gibaru O, Damak M. Feedra planning for machining with industrial sixaxis robots. Control Engineering Practice 2010; 18(5): Red E. A dynamic optimal trajectory generator for Carsian Path following. Robotica 2000; 18: Rubio FJ, Valero FJ, uñer J, Mata V. imultaneous algorithm to solve the trajectory planning problem. Mechanism and Machine Theory 2009; 44: Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2, 2013
6 9. aramago FP, Ceccarelli M. An optimum robot path planning with payload constraints. Robotica 2002; 20: Visioli A. Trajectory planning of robot manipulators by using algebraic and trigonometric splines. Robotica 2000; 18: Andrzej Graboś, Ph.D. (Eng.) Marek Boryga, Ph.D. (Eng.) The Department of Mechanical Engineering and Automation, The Faculty of Production Engineering, University of ife ciences in ublin, Doświadczalna str. 50A, ublin, Poland s: Eks p l o a t a c j a i Ni e z a w o d n o s c Ma in t e n a n c e a n d Reliability Vo l.15, No. 2,
INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users
INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE
More informationDesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist
DesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot
More informationACTUATOR DESIGN FOR ARC WELDING ROBOT
ACTUATOR DESIGN FOR ARC WELDING ROBOT 1 Anurag Verma, 2 M. M. Gor* 1 G.H Patel College of Engineering & Technology, V.V.Nagar388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda391760,
More informationSimulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)
Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute
More informationA PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT ENDEFFECTORS
A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT ENDEFFECTORS Sébastien Briot, Ilian A. Bonev Department of Automated Manufacturing Engineering École de technologie supérieure (ÉTS), Montreal,
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMOBILE ROBOT TRACKING OF PREPLANNED PATHS. Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.
MOBILE ROBOT TRACKING OF PREPLANNED PATHS N. E. Pears Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.uk) 1 Abstract A method of mobile robot steering
More informationForce/position control of a robotic system for transcranial magnetic stimulation
Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme
More informationPath Tracking for a Miniature Robot
Path Tracking for a Miniature Robot By Martin Lundgren Excerpt from Master s thesis 003 Supervisor: Thomas Hellström Department of Computing Science Umeå University Sweden 1 Path Tracking Path tracking
More informationWe can display an object on a monitor screen in three different computermodel forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More information3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials
3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationRobot TaskLevel Programming Language and Simulation
Robot TaskLevel Programming Language and Simulation M. Samaka Abstract This paper presents the development of a software application for Offline robot task programming and simulation. Such application
More informationDev eloping a General Postprocessor for MultiAxis CNC Milling Centers
57 Dev eloping a General Postprocessor for MultiAxis CNC Milling Centers Mihir Adivarekar 1 and Frank Liou 1 1 Missouri University of Science and Technology, liou@mst.edu ABSTRACT Most of the current
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationIndustrial Robotics. Training Objective
Training Objective After watching the program and reviewing this printed material, the viewer will learn the basics of industrial robot technology and how robots are used in a variety of manufacturing
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationAnimations in Creo 3.0
Animations in Creo 3.0 ME170 Part I. Introduction & Outline Animations provide useful demonstrations and analyses of a mechanism's motion. This document will present two ways to create a motion animation
More informationPractical Work DELMIA V5 R20 Lecture 1. D. Chablat / S. Caro Damien.Chablat@irccyn.ecnantes.fr Stephane.Caro@irccyn.ecnantes.fr
Practical Work DELMIA V5 R20 Lecture 1 D. Chablat / S. Caro Damien.Chablat@irccyn.ecnantes.fr Stephane.Caro@irccyn.ecnantes.fr Native languages Definition of the language for the user interface English,
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationThis is an authordeposited version published in: http://sam.ensam.eu Handle ID:.http://hdl.handle.net/10985/7459
Science Arts & Métiers (SAM) is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. This is an authordeposited
More informationIntroduction to Computer Graphics MariePaule Cani & Estelle Duveau
Introduction to Computer Graphics MariePaule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More informationOn Motion of Robot EndEffector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot EndEffector using the Curvature
More informationOnline trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions
Online trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions Ignacio Herrera Aguilar and Daniel Sidobre (iherrera, daniel)@laas.fr LAASCNRS Université Paul Sabatier
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationIn order to describe motion you need to describe the following properties.
Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1D path speeding up and slowing down In order to describe motion you need to describe the following properties.
More informationDESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED CONTROL SYSTEM FOR A SERPENTINE ROBOTIC MANIPULATOR
Proceedings of the American Nuclear Society Ninth Topical Meeting on Robotics and Remote Systems, Seattle Washington, March 2001. DESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED
More informationAlgebra II Semester Exam Review Sheet
Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationF f v 1 = c100(10 3 ) m h da 1h 3600 s b =
14 11. The 2Mg car has a velocity of v 1 = 100km>h when the v 1 100 km/h driver sees an obstacle in front of the car. It takes 0.75 s for him to react and lock the brakes, causing the car to skid. If
More informationVirtual CRASH 3.0 Staging a Car Crash
Virtual CRASH 3.0 Staging a Car Crash Virtual CRASH Virtual CRASH 3.0 Staging a Car Crash Changes are periodically made to the information herein; these changes will be incorporated in new editions of
More informationRobot coined by Karel Capek in a 1921 sciencefiction Czech play
Robotics Robot coined by Karel Capek in a 1921 sciencefiction Czech play Definition: A robot is a reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices
More informationSelecting and Sizing Ball Screw Drives
Selecting and Sizing Ball Screw Drives Jeff G. Johnson, Product Engineer Thomson Industries, Inc. Wood Dale, IL 5406333549 www.thomsonlinear.com Thomson@thomsonlinear.com Fig 1: Ball screw drive is a
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationMachine Tool Control. Besides these TNCs, HEIDENHAIN also supplies controls for other areas of application, such as lathes.
Machine Tool Control Contouring controls for milling, drilling, boring machines and machining centers TNC contouring controls from HEIDENHAIN for milling, drilling, boring machines and machining centers
More informationCATIA V5 Tutorials. Mechanism Design & Animation. Release 18. Nader G. Zamani. University of Windsor. Jonathan M. Weaver. University of Detroit Mercy
CATIA V5 Tutorials Mechanism Design & Animation Release 18 Nader G. Zamani University of Windsor Jonathan M. Weaver University of Detroit Mercy SDC PUBLICATIONS Schroff Development Corporation www.schroff.com
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationDesign of a Universal Robot Endeffector for Straightline Pickup Motion
Session Design of a Universal Robot Endeffector for Straightline Pickup Motion Gene Y. Liao Gregory J. Koshurba Wayne State University Abstract This paper describes a capstone design project in developing
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More information5Axis TestPiece Influence of Machining Position
5Axis TestPiece Influence of Machining Position Michael Gebhardt, Wolfgang Knapp, Konrad Wegener Institute of Machine Tools and Manufacturing (IWF), Swiss Federal Institute of Technology (ETH), Zurich,
More informationChapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations
Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.
More informationIntroduction to Robotics Analysis, Systems, Applications
Introduction to Robotics Analysis, Systems, Applications Saeed B. Niku Mechanical Engineering Department California Polytechnic State University San Luis Obispo Technische Urw/carsMt Darmstadt FACHBEREfCH
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationMS Excel as Tool for Modeling, Dynamic Simulation and Visualization ofmechanical Motion
MS Excel as Tool for Modeling, Dynamic Simulation and Visualization ofmechanical Motion MARIE HUBALOVSKA, STEPAN HUBALOVSKY, MARCELA FRYBOVA Department of Informatics Faculty of Science University of Hradec
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationDevelopment of Easy Teaching Interface for a Dual Arm Robot Manipulator
Development of Easy Teaching Interface for a Dual Arm Robot Manipulator Chanhun Park and Doohyeong Kim Department of Robotics and Mechatronics, Korea Institute of Machinery & Materials, 156, GajeongbukRo,
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
More informationSPINDLE ERROR MOVEMENTS MEASUREMENT ALGORITHM AND A NEW METHOD OF RESULTS ANALYSIS 1. INTRODUCTION
Journal of Machine Engineering, Vol. 15, No.1, 2015 machine tool accuracy, metrology, spindle error motions Krzysztof JEMIELNIAK 1* Jaroslaw CHRZANOWSKI 1 SPINDLE ERROR MOVEMENTS MEASUREMENT ALGORITHM
More informationIntelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and Motion Optimization for Maritime Robotic Research
20th International Congress on Modelling and Simulation, Adelaide, Australia, 1 6 December 2013 www.mssanz.org.au/modsim2013 Intelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and
More informationDIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems
DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems Luigi Biagiotti Luigi Biagiotti luigi.biagiotti@unimore.it Introduction
More informationNonlinear analysis and formfinding in GSA Training Course
Nonlinear analysis and formfinding in GSA Training Course Nonlinear analysis and formfinding in GSA 1 of 47 Oasys Ltd Nonlinear analysis and formfinding in GSA 2 of 47 Using the GSA GsRelax Solver
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers prerequisite material and should be skipped if you are
More informationStirling Paatz of robot integrators Barr & Paatz describes the anatomy of an industrial robot.
Ref BP128 Anatomy Of A Robot Stirling Paatz of robot integrators Barr & Paatz describes the anatomy of an industrial robot. The term robot stems from the Czech word robota, which translates roughly as
More informationLab # 3  Angular Kinematics
Purpose: Lab # 3  Angular Kinematics The objective of this lab is to understand the relationship between segment angles and joint angles. Upon completion of this lab you will: Understand and know how
More informationRIA : 2013 Market Trends Webinar Series
RIA : 2013 Market Trends Webinar Series Robotic Industries Association A market trends education Available at no cost to audience Watch live or archived webinars anytime Learn about the latest innovations
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationIntegrated sensors for robotic laser welding
Proceedings of the Third International WLTConference on Lasers in Manufacturing 2005,Munich, June 2005 Integrated sensors for robotic laser welding D. Iakovou *, R.G.K.M Aarts, J. Meijer University of
More informationAdequate Theory of Oscillator: A Prelude to Verification of Classical Mechanics Part 2
International Letters of Chemistry, Physics and Astronomy Online: 213919 ISSN: 22993843, Vol. 3, pp 11 doi:1.1852/www.scipress.com/ilcpa.3.1 212 SciPress Ltd., Switzerland Adequate Theory of Oscillator:
More informationNew Developments in Tooth Contact Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives
New Developments in Tooth Contact Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives Dr. Qi Fan and Dr. Lowell Wilcox This article is printed with permission of the copyright holder
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationPlate waves in phononic crystals slabs
Acoustics 8 Paris Plate waves in phononic crystals slabs J.J. Chen and B. Bonello CNRS and Paris VI University, INSP  14 rue de Lourmel, 7515 Paris, France chen99nju@gmail.com 41 Acoustics 8 Paris We
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationKyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.
MECHANICS: STATICS AND DYNAMICS KyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,
More informationX On record with the USOE.
Textbook Alignment to the Utah Core Algebra 2 Name of Company and Individual Conducting Alignment: Chris McHugh, McHugh Inc. A Credential Sheet has been completed on the above company/evaluator and is
More informationProjectile motion simulator. http://www.walterfendt.de/ph11e/projectile.htm
More Chapter 3 Projectile motion simulator http://www.walterfendt.de/ph11e/projectile.htm The equations of motion for constant acceleration from chapter 2 are valid separately for both motion in the x
More informationComputer Animation. Lecture 2. Basics of Character Animation
Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationRigid body dynamics using Euler s equations, RungeKutta and quaternions.
Rigid body dynamics using Euler s equations, RungeKutta and quaternions. Indrek Mandre http://www.mare.ee/indrek/ February 26, 2008 1 Motivation I became interested in the angular dynamics
More informationParametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) YanBin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationFigure 3.1.2 Cartesian coordinate robot
Introduction to Robotics, H. Harry Asada Chapter Robot Mechanisms A robot is a machine capable of physical motion for interacting with the environment. Physical interactions include manipulation, locomotion,
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationPower Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.
Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture  1 Electric Drive Today, we will start with the topic on industrial drive
More informationG U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationDRAFT. Algebra 1 EOC Item Specifications
DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as
More information