Dimensional Synthesis of Bennett Linkages
|
|
- Tyrone Ferdinand Burns
- 5 years ago
- Views:
Transcription
1 Dimensional Synthesis of Bennett Linkages DETC ASME Design Engineering Technical Conferences September 000 Baltimore, Maryland, USA Alba Perez J. M. McCarthy Robotics and Automation Laboratory Department of Mechanical and Aerospace Engineering University of California Irvine, California 9697 Introduction The Bennett linkage The workspace of the RR chain The cylindroid The design equations Bennett linkage coordinates Solving the design equations Examples Conclusions 1
2 Literature review Synthesis Bennett, G.T., (1903) A New Mechanism. Engineering, vol. 76, pp Veldkamp, G.R., (1967) Canonical Systems and Instantaneous Invariants in Spatial Kinematics. Journal of Mechanisms, vol. 3, pp Suh, C.H., (1969) On the Duality in the Existence of R-R Links for Three Positions. J. Eng. Ind. Trans. ASME, vol. 91B, pp Tsai, L.W., and Roth, B.,(1973) A Note on the Design of Revolute-Revolute Cranks. Mechanisms and Machine Theory, vol. 8, pp Perez, A., McCarthy, J.M., (000), Dimensional Synthesis of Spatial RR Robots. 7th. International Symposium on Advances in Robot Kinematics, Piran-Portoroz, Slovenia. Analysis Suh, C.H., and Radcliffe, C.W., (1978), Kinematics and Mechanisms Design. John Wiley. Yu, H.C., (1981) The Bennett Linkage, its Associated Tetrahedron and the Hyperboloid of its Axes. Mechanism and Machine Theory, vol. 16, pp Huang, C., (1996), The Cylindroid Associated with Finite Motions of the Bennett Mechanism. Proc. of the ASME Design Engineering Technical Conferences, Irvine, CA. Baker, J.E., (1998), On the Motion Geometry of the Bennett Linkage. Proc. 8th Internat. Conf. on Engineering Computer Graphics and Descriptive Geometry, Austin, Texas, USA, vol., pp Linear Systems of Screws Hunt, K.H.,(1978), Kinematic Geometry of Mechanisms. Clarendon Press. Parkin, I.A., (1997), Finding the Principal Axes of Screw Systems. Proc. of the ASME Design Engineering Technical Conferences, Sacramento, CA.
3 The Bennett Linkage - Spatial closed 4R chain W α H a γ g G U Ground link Formed by connecting the end links of two spatial RR chains {G, W} and {H, U} to form a coupler link. Mobility: For a general 4R closed spatial chain: M =6(n 1) m p k c k = = k=1 However, the Bennett linkage can move with one degree of freedom. Special geometry: Link lenght and twist angle (a, α) and (g, γ) must be the same for opposite sides. Ratio of the sine of the twist angle over the link length: sin α a = sin γ g The joints of the Bennett linkage form the vertices of a tetrahedron 3
4 The Design Theory - RR Chain The Workspace of the Synthesis Theory- Find the kinematic chain that reaches exactly a number of specified positions {M 3 } {M } Moving Axis {M 1 } {F} Fixed Axis End Effector The specified positions must lie on the workspace of the chain. The kinematics equation for the RR chain defines its workspace 4
5 The Kinematics Equation for the RR Chain {M 3 } {M } {F} [G] a θ α φ {M 1 } [H] The kinematics equation in matrix form:the set of displacements [D(θ, φ)] of the RR chain. [D] =[G][Z(θ, 0)][X(α, a)][z(φ, 0)][H] If we choose a reference configuration [D 1 ], we can write the workspace of the relative displacements [D 1i ]=[D i ][D 1 ] 1. [D 1i ]=[T(θ i,g)][t (φ i, W)] where [T ( θ, G)]=[G][Z(θ, 0)][Z(θ 0, 0)] 1 [G] 1, [T ( φ, W)]=([G][[Z(θ 0, 0)][X(ρ, r)])[z(φ, 0)][Z(φ 0, 0)] 1 ([G][[Z(θ 0, 0)][X(ρ, r)]) 1 5
6 The Kinematics Equation for the RR Chain Dual quaternion formulation We can also formulate the workspace using dual quaternions to express the relative displacements. The dual quaternion form of the workspace is given by: ˆD 1i = Ĝ( θ)ŵ ( φ) cos( ˆψ 1i θ φ θ φ ) = cos cos sin sin G W, sin( ˆψ 1i )S 1i = sin θ φ φ cos G + sin θ θ φ cos W + sin sin G W where S 1i is the screw axis of the relative displacement and ˆψ 1i =(φ 1i,d 1i ) is the associated rotation about and slide along this axis for each displacement in the workspace. Every pair of values θ and φ defines a screw axis S 1i that represents a relative displacement from position 1 to position i. 6
7 The Workspace of the Bennett Linkage α W G γ θ H φ γ φ α U Restriction to a Bennett linkage: The angles θ and φ are not independent. There exist the input/coupler angular relation: tan φ γ+α = sin sin γ α tan θ = K tan θ The workspace of the Bennett linkage: The set of screw axes obtained applying the input/coupler relation to the workspace of the RR chain, generates a cylindroid tan( ˆψ 1i )S 1i = G + K W 1 + K tan θ G W1 cot θ K tan θ G. W1 7
8
9 The cylindroid Simply-Ruled surface that has a nodal line cutting all generators at right angles. z(x + y )+(P X P Y )xy =0 It appears as generated by the real linear combination of two screws. The cylindroid has a set of principal axes located in the midpoint of the nodal line. K Y S 13 z 0 σ S 1 X The principal axes can be located from any pair of generators. tan σ = (P b P a ) cot δ + d (P b P a )+dcot δ z 0 = 1 (d (P b P a ) cos δ sin δ ) 8
10 Bennett linkage coordinates P 1 b W 1 K Q X c G Y c κ C 1 B a Yu, 1981: The Bennett linkage can be determined using a tetrahedron defined by four parameters (a, b, c, κ). The principal axes are located in the middle of the tetrahedron. The joint axes G and W 1 are given by the cross product of the edges. This ensures that the chosen points B, P 1 are on the common normal. ( ) G = K g (Q B) (P 1 B)+ɛK g B (Q B) (P 1 B) W 1 = K w (B P 1 ) (C 1 P 1 )+ɛk w P 1 ( (B P 1 ) (C 1 P 1 ) ) We obtain: and G = K g { bc sin κ bc cos κ 4ab cos κ sin κ W 1 = K w { ac sin κ ac cos κ 4ab cos κ sin κ } { } b cos κ (4a sin κ + c ) + ɛkg b sin κ (4a cos κ + c ) abc(cos κ sin κ ) } + ɛkw { a cos κ (4b sin κ + c ) a sin κ (4b cos κ + c ) abc(cos κ sin κ ) } Using the principal axes and the tetrahedron formulation, we can write the coordinates of the joint axes of the Bennett linkage with only four parameters. 9
11 The design equations for an RR dyad The constant dual angle constraint: ˆα =(α, a),the angle and distance between the fixed and moving axes,must remain constant during the movement. G [ ˆT 1i I]W 1 =0,i=,3, Usign the equivalent screw triangle formulation and separating real and dual part, 1. The direction equations. The distance equations tan ψ 1i = G (S 1i W 1 ) (S 1i G) (S 1i W 1 ), i =,3. (B P 1 ) S 1i t 1i =0, i =,3. The normal constraints: The normal line to G andw, P i B,remains the same. G ([T 1i ]P 1 B) =0, W 1 (P 1 [T 1i ] 1 B)=0,i=1,,3. Total equations: (n 1)+n Total parameters: 10 Number of positions needed for a finite number of solutions: n =3. The standard algebraic formulation of the synthesis problem consists on solving ten equations in ten parameters. 10
12 Solving the design equations in the principal axes frame The six common normal constraints are automatically satisfied. We solve system of four equations in four parameters a, b, c, κ. tan ψ 1 = G (S 1 W 1 ) (S 1 G) (S 1 W 1 ) tan ψ 13 = G (S 13 W 1 ) (S 13 G) (S 13 W 1 ) (1) () (B P 1 ) S 1 t 1 (B P 1 ) S 13 t 13 =0 (3) =0 (4) Solution for a and b :the distance equations (3) and (4) are linear in a, b. t 1 +(a b) cos δ 1 cos κ +(a+b) sin δ 1 sin κ =0 t 13 +(a b) cos δ cos κ +(a+b) sin δ sin κ =0 Defining the constraints: K s = t 1 cos δ t 13 cos δ 1 sin(δ 1 δ ) We obtain: K d = t 13 sin δ 1 t 1 sin δ sin(δ 1 δ ) a = K s sin κ + K d cos κ b = K s sin κ K d cos κ. 11
13 Solution for c : Substitute the values of a and b in the direction equations (1) and () and make the algebraic substitution y = tan κ. tan ψ 1 ( K s K d ψ y )+c tan 1 K d ( K s K d y )+ (y (cos δ 1 1) + cos δ 1 +1) cy K d (cos δ 1 + Ks sin δ 1 K d ) c K d (y (cos δ 1 1) + cos δ 1 +1) =0, tan ψ 13 ( K s K d ψ y )+c tan 13 K d ( K s K d y )+ (y (cos δ 1) + cos δ +1) cy K d (cos δ + Ks sin δ K d ) c K d (y (cos δ 1) + cos δ +1) =0, The numerator and denominator share the roots associated with c =0, which are not a solution of the spatial problem. Eliminate them from the numerator forcing the linear system to have more solutions than the trivial. [ tan ψ 1 tan ψ 13 tan ψ 1 K d tan ψ 13 K d (y c(cos δ 1 1) + cos δ 1 +1) y K d (cos δ 1 + Ks sin δ 1 K d ) (y c(cos δ 1) + cos δ +1) y K d (cos δ + Ks sin δ K d ) ] { (K Making the determinant of the matrix equal to zero we obtain a linear equation in c. Define the constants ( ) K 1 = t 1/ 1 tan ψ 1 sin δ 1 sin δ We obtain the expression for c: K 13 = t 13/ tan ψ 13 ( ) 1 sin δ 1 sin δ s /K d y ) c } =0. c =(K 13 K 1 ) sin κ 1
14 Solution for κ : Substitute the expressions for a, b, c in one of the direction equations, (1) or (). We obtain a cubic polynomial in y. The coefficients are: P : C 3 y 6 + C y 4 + C 1 y + C 0 =0 C 3 = K d, C = K s K d +4(K 1 K 13 )(K 13 sin δ 1 K 1 sin δ ), C 1 =K s K d 4(K 1 K 13 )(K 13 cos δ 1 K 1 cos δ ), C 0 = K s. Solve the cubic polynomial for z = y. This polynomial has one and only one real positive root z 0 : P (0) = K s P ( ) = K d P( 1) = 4(K 1 K 13 ) The square root of the positive root gives the two solutions for κ. tan κ = ± z 0 13
15 The Solutions The two sets of solutions (a, b, c, +κ) and ( b, a, c, κ) correspond to both dyads of the Bennett mechanism: Solution 1 Solution G( a, b, c, κ) = H( b, a, c, κ) W( a, b, c, κ) = U( b, a, c, κ) H( a, b, c, κ) = G( b, a, c, κ) U( a, b, c, κ) = W( b, a, c, κ) The synthesis procedure yields the two RR dyads that form a Bennett linkage. 14
16 Example Tsai and Roth positions (Tsai and Roth, 1973) The specified positions: x y z θ φ ψ M M M The joint axes in the initial frame: Axis G W 1 H U 1 Line coordinates (0.36, 0.45, 0.81), (0.6, 1.05, 0.70) (0.60, 0.36, 0.7), (0.87, 0.83, 1.14) (0.60, 0.36, 0.7), (0.87, 0.83, 1.14) (0.36, 0.45, 0.81), (0.6, 1.05, 0.70) 15
17 16
18 17
19 18
20 19
21 Conclusions Using the geometry of the RR chain to formulate the problem leads to a simple convenient set of equations. The design procedure for three positions for an RR chain yields a Bennett linkage. A Mathematica notebook with the complete synthesis procedure can be downloaded from: mccarthy/pages/resprojects.html The synthesis routine is to be used in a robot design environment for continuous tasks. 0
Position and orientation of rigid bodies
Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal Reference Frames RF A A p AB B RF B rigid body position: A p AB
More informationDYNAMICS OF PARALLEL MANIPULATOR
DYNAMICS OF PARALLEL MANIPULATOR PARALLEL MANIPULATORS 6-degree of Freedom Flight Simulator BACKGROUND Platform-type parallel mechanisms 6-DOF MANIPULATORS INTRODUCTION Under alternative robotic mechanical
More informationSingle Exponential Motion and Its Kinematic Generators
PDFaid.Com #1 Pdf Solutions Single Exponential Motion and Its Kinematic Generators Guanfeng Liu, Yuanqin Wu, and Xin Chen Abstract Both constant velocity (CV) joints and zero-torsion parallel kinematic
More informationRELATION BETWEEN THIRD ORDER FINITE SCREW SYSTEMS AND LINE CONGRUENCE
INTERNATIONAL JOURNAL OF GEOETRY Vol. 3 (014), No., 5-34 RELATION BETWEEN THIRD ORDER FINITE SCREW SYSTES AND LINE CONGRUENCE ŞER IFE NUR YALÇIN, AL I ÇALIŞKAN Abstract. In this paper, by using Dimentberg
More informationRIGID BODY MOTION (Section 16.1)
RIGID BODY MOTION (Section 16.1) There are cases where an object cannot be treated as a particle. In these cases the size or shape of the body must be considered. Rotation of the body about its center
More informationTECHNICAL RESEARCH REPORT
TECHNICAL RESEARCH REPORT A Parallel Manipulator with Only Translational Degrees of Freedom by Lung-Wen Tsai, Richard E. Stamper T.R. 97-72 ISR INSTITUTE FOR SYSTEMS RESEARCH Sponsored by the National
More informationFactorization of Rational Curves in the Study Quadric and Revolute Linkages
Factorization of Rational Curves in the Study Quadric and Revolute Linkages Gabor Hegedüs, Josef Schicho, and Hans-Peter Schröcker May 10, 2012 arxiv:1202.0139v4 [math.ra] 9 May 2012 Given a generic rational
More informationGEOMETRIC DESIGN OF CYLINDRIC PRS SERIAL CHAINS
Proceedings of DETC 03 2003 ASME Design Engineering Technical Conferences September 2 6, 2003, Chicago, Illinois, USA DETC2003/DAC-48816 GEOMETRIC DESIGN OF CYLINDRIC PRS SERIAL CHAINS Hai-Jun Su suh@eng.uci.edu,
More informationOn Martin Disteli s Main Achievements in Spatial Gearing: Disteli s Diagram
On Martin Disteli s Main Achievements in Spatial Gearing: Disteli s Diagram Jorge Angeles, McGill University, Montreal Giorgio Figliolini, University of Cassino Hellmuth Stachel stachel@dmg.tuwien.ac.at
More informationRobotics I. Figure 1: Initial placement of a rigid thin rod of length L in an absolute reference frame.
Robotics I September, 7 Exercise Consider the rigid body in Fig., a thin rod of length L. The rod will be rotated by an angle α around the z axis, then by an angle β around the resulting x axis, and finally
More informationKinematic Analysis of the 6R Manipulator of General Geometry
Kinematic Analysis of the 6R Manipulator of General Geometry Madhusudan Raghavan Powertrain Systems Research Lab General Motors R&D Center Warren, MI 48090-9055 and Bernard Roth, Professor Design Division
More informationKinematic Analysis of a Pentapod Robot
Journal for Geometry and Graphics Volume 10 (2006), No. 2, 173 182. Kinematic Analysis of a Pentapod Robot Gert F. Bär and Gunter Weiß Dresden University of Technology Institute for Geometry, D-01062 Dresden,
More informationWEEK 1 Dynamics of Machinery
WEEK 1 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J. Uicker, G.R.Pennock ve J.E. Shigley, 2003 Makine Dinamiği, Prof. Dr. Eres SÖYLEMEZ, 2013 Uygulamalı Makine Dinamiği, Jeremy
More informationThree Types of Parallel 6R Linkages
PDFaid.Com #1 Pdf Solutions Three Types of Parallel 6R Linkages Zijia Li and Josef Schicho Abstract In this paper, we consider a special kind of overconstrained 6R closed linkages which we call parallel
More informationScrew Theory and its Applications in Robotics
Screw Theory and its Applications in Robotics Marco Carricato Group of Robotics, Automation and Biomechanics University of Bologna Italy IFAC 2017 World Congress, Toulouse, France Table of Contents 1.
More informationInterpolated Rigid-Body Motions and Robotics
Interpolated Rigid-Body Motions and Robotics J.M. Selig London South Bank University and Yuanqing Wu Shanghai Jiaotong University. IROS Beijing 2006 p.1/22 Introduction Interpolation of rigid motions important
More informationRobotics & Automation. Lecture 17. Manipulability Ellipsoid, Singularities of Serial Arm. John T. Wen. October 14, 2008
Robotics & Automation Lecture 17 Manipulability Ellipsoid, Singularities of Serial Arm John T. Wen October 14, 2008 Jacobian Singularity rank(j) = dimension of manipulability ellipsoid = # of independent
More informationDifferential Kinematics
Differential Kinematics Relations between motion (velocity) in joint space and motion (linear/angular velocity) in task space (e.g., Cartesian space) Instantaneous velocity mappings can be obtained through
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA ADVANCED MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 18 NQF LEVEL 3
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA ADVANCED MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 18 NQF LEVEL 3 OUTCOME 3 BE ABLE TO DETERMINE RELATIVE AND RESULTANT VELOCITY IN ENGINEERING SYSTEMS Resultant
More informationError Analysis of Four Bar Planar Mechanism Considering Clearance Between Crank & Coupler
Error nalysis of Four Bar Planar Mechanism Considering Clearance Between Crank & Coupler.M.VIDY 1, P.M. PDOLE 2 1 Department of Mechanical & Production Engineering, M.I.E.T.,GONDI, India. 2 Department
More informationMECH 576 Geometry in Mechanics November 30, 2009 Kinematics of Clavel s Delta Robot
MECH 576 Geometry in Mechanics November 3, 29 Kinematics of Clavel s Delta Robot The DELTA Robot DELTA, a three dimensional translational manipulator, appears below in Fig.. Figure : Symmetrical (Conventional)
More informationCh. 5: Jacobian. 5.1 Introduction
5.1 Introduction relationship between the end effector velocity and the joint rates differentiate the kinematic relationships to obtain the velocity relationship Jacobian matrix closely related to the
More informationFramework Comparison Between a Multifingered Hand and a Parallel Manipulator
Framework Comparison Between a Multifingered Hand and a Parallel Manipulator Júlia Borràs and Aaron M. Dollar Abstract In this paper we apply the kineto-static mathematical models commonly used for robotic
More informationME Machine Design I. EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, September 30th, 2009
ME - Machine Design I Fall Semester 009 Name Lab. Div. EXAM. OPEN BOOK AND CLOSED NOTES. Wednesday, September 0th, 009 Please use the blank paper provided for your solutions. Write on one side of the paper
More informationPLANAR RIGID BODY MOTION: TRANSLATION &
PLANAR RIGID BODY MOTION: TRANSLATION & Today s Objectives : ROTATION Students will be able to: 1. Analyze the kinematics of a rigid body undergoing planar translation or rotation about a fixed axis. In-Class
More informationROBOTICS: ADVANCED CONCEPTS & ANALYSIS
ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 4 KINEMATICS OF PARALLEL ROBOTS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian Institute of Science
More informationExponential and Cayley maps for Dual Quaternions
Exponential and Cayley maps for Dual Quaternions J.M. Selig Abstract. In this work various maps between the space of twists and the space of finite screws are studied. Dual quaternions can be used to represent
More information2. Preliminaries. x 2 + y 2 + z 2 = a 2 ( 1 )
x 2 + y 2 + z 2 = a 2 ( 1 ) V. Kumar 2. Preliminaries 2.1 Homogeneous coordinates When writing algebraic equations for such geometric objects as planes and circles, we are used to writing equations that
More informationTheory of Vibrations in Stewart Platforms
Theory of Vibrations in Stewart Platforms J.M. Selig and X. Ding School of Computing, Info. Sys. & Maths. South Bank University London SE1 0AA, U.K. (seligjm@sbu.ac.uk) Abstract This article develops a
More informationRobotics I Kinematics, Dynamics and Control of Robotic Manipulators. Velocity Kinematics
Robotics I Kinematics, Dynamics and Control of Robotic Manipulators Velocity Kinematics Dr. Christopher Kitts Director Robotic Systems Laboratory Santa Clara University Velocity Kinematics So far, we ve
More informationQuiz No. 1: Tuesday Jan. 31. Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.
Quiz No. 1: Tuesday Jan. 31 Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.20 Chapter 3 Vectors and Two-Dimensional Kinematics Properties of
More informationUNIT 2 KINEMATICS OF LINKAGE MECHANISMS
UNIT 2 KINEMATICS OF LINKAGE MECHANISMS ABSOLUTE AND RELATIVE VELOCITY An absolute velocity is the velocity of a point measured from a fixed point (normally the ground or anything rigidly attached to the
More informationChapter 3 Velocity Analysis
Chapter 3 Velocity nalysis The position of point with respect to 0, Fig 1 may be defined mathematically in either polar or Cartesian form. Two scalar quantities, the length R and the angle θ with respect
More informationarxiv: v2 [cs.ro] 29 Jun 2015
7R Darboux Linkages by Factorization of Motion Polynomials Z. Li RICAM Austrian Academy of Sciences Linz, Austria J. Schicho RISC Johannes Kepler University Linz, Austria June 30, 2015 H.-P. Schröcker
More informationMathematical review trigonometry vectors Motion in one dimension
Mathematical review trigonometry vectors Motion in one dimension Used to describe the position of a point in space Coordinate system (frame) consists of a fixed reference point called the origin specific
More informationON JACK PHILLIP'S SPATIAL INVOLUTE GEARING
The th International Conference on Geometry and Graphics, -5 August, 004, Guangzhou, China ON JACK PHILLIP'S SPATIAL INVOLUTE GEARING Hellmuth STACHEL Vienna University of Technology, Vienna, AUSTRIA ABSTRACT
More informationMain theorem on Schönflies-singular planar Stewart Gough platforms
Main theorem on Schönflies-singular planar Stewart Gough platforms Georg Nawratil Institute of Discrete Mathematics and Geometry Differential Geometry and Geometric Structures 12th International Symposium
More informationThe Numerical Solution of Polynomial Systems Arising in Engineering and Science
The Numerical Solution of Polynomial Systems Arising in Engineering and Science Jan Verschelde e-mail: jan@math.uic.edu web: www.math.uic.edu/ jan Graduate Student Seminar 5 September 2003 1 Acknowledgements
More informationarxiv: v1 [cs.ro] 19 Sep 2013
Four-Pose Synthesis of Angle-Symmetric 6R Linkages Gábor Hegedüs Applied Mathematical Institute, Antal Bejczy Center for Intelligent Robotics, Obuda University hegedus.gabor@nik.uni-obuda.hu Josef Schicho
More informationHomogeneous Transformations
Purpose: Homogeneous Transformations The purpose of this chapter is to introduce you to the Homogeneous Transformation. This simple 4 x 4 transformation is used in the geometry engines of CAD systems and
More informationOutline. Outline. Vector Spaces, Twists and Wrenches. Definition. ummer Screws 2009
Vector Spaces, Twists and Wrenches Dimiter Zlatanov DIMEC University of Genoa Genoa, Italy Z-2 Definition A vector space (or linear space) V over the field A set V u,v,w,... of vectors with 2 operations:
More informationRobotics & Automation. Lecture 06. Serial Kinematic Chain, Forward Kinematics. John T. Wen. September 11, 2008
Robotics & Automation Lecture 06 Serial Kinematic Chain, Forward Kinematics John T. Wen September 11, 2008 So Far... We have covered rigid body rotational kinematics: representations of SO(3), change of
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationDepartment of Physics, Korea University Page 1 of 8
Name: Department: Student ID #: Notice +2 ( 1) points per correct (incorrect) answer No penalty for an unanswered question Fill the blank ( ) with ( ) if the statement is correct (incorrect) : corrections
More informationDepartment of Physics, Korea University
Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an
More informationRobotics I. June 6, 2017
Robotics I June 6, 217 Exercise 1 Consider the planar PRPR manipulator in Fig. 1. The joint variables defined therein are those used by the manufacturer and do not correspond necessarily to a Denavit-Hartenberg
More informationRobotics, Geometry and Control - Rigid body motion and geometry
Robotics, Geometry and Control - Rigid body motion and geometry Ravi Banavar 1 1 Systems and Control Engineering IIT Bombay HYCON-EECI Graduate School - Spring 2008 The material for these slides is largely
More informationA Robust Forward-Displacement Analysis of Spherical Parallel Robots
A Robust Forward-Displacement Analysis of Spherical Parallel Robots Shaoping Bai, Michael R. Hansen and Jorge Angeles, Department of Mechanical Engineering Aalborg University, Denmark e-mail: {shb;mrh}@me.aau.dk
More informationA STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME
Bull. Korean Math. Soc. 49 (), No. 3, pp. 635 645 http://dx.doi.org/.434/bkms..49.3.635 A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME N ihat Ayyildiz and Tunahan Turhan
More informationDETC MECHANISM BRANCHES, TURNING CURVES, AND CRITICAL POINTS
Proceedings of IDETC/CIE 01 ASME 01 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference August 1-15, 01, Chicago, Illinois, USA DETC01-7077 MECHANISM
More informationLecture 10. Rigid Body Transformation & C-Space Obstacles. CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University
CS 460/560 Introduction to Computational Robotics Fall 017, Rutgers University Lecture 10 Rigid Body Transformation & C-Space Obstacles Instructor: Jingjin Yu Outline Rigid body, links, and joints Task
More information2- Scalars and Vectors
2- Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement,
More informationGeneralized Forces. Hamilton Principle. Lagrange s Equations
Chapter 5 Virtual Work and Lagrangian Dynamics Overview: Virtual work can be used to derive the dynamic and static equations without considering the constraint forces as was done in the Newtonian Mechanics,
More information3. ANALYTICAL KINEMATICS
In planar mechanisms, kinematic analysis can be performed either analytically or graphically In this course we first discuss analytical kinematic analysis nalytical kinematics is based on projecting the
More informationAnalytical and Numerical Methods Used in Studying the Spatial kinematics of Multi-body Systems. Bernard Roth Stanford University
Analytical and Numerical Methods Used in Studying the Spatial kinematics of Multi-body Systems Bernard Roth Stanford University Closed Loop Bound on number of solutions=2 6 =64 Open chain constraint
More informationLecture Note 4: General Rigid Body Motion
ECE5463: Introduction to Robotics Lecture Note 4: General Rigid Body Motion Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture
More informationKinematic Isotropy of the H4 Class of Parallel Manipulators
Kinematic Isotropy of the H4 Class of Parallel Manipulators Benoit Rousseau 1, Luc Baron 1 Département de génie mécanique, École Polytechnique de Montréal, benoit.rousseau@polymtl.ca Département de génie
More informationThe Principle of Virtual Power Slide companion notes
The Principle of Virtual Power Slide companion notes Slide 2 In Modules 2 and 3 we have seen concepts of Statics and Kinematics in a separate way. In this module we shall see how the static and the kinematic
More informationPART 1: USING SCIENTIFIC CALCULATORS (50 PTS.)
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 SPRING 2018 KUNIYUKI 150 POINTS TOTAL: 50 FOR PART 1, AND 100 FOR PART 2 Show all work, simplify as appropriate,
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.4 Basic Trigonometric Equations Copyright Cengage Learning. All rights reserved. Objectives Basic Trigonometric Equations Solving
More information7. FORCE ANALYSIS. Fundamentals F C
ME 352 ORE NLYSIS 7. ORE NLYSIS his chapter discusses some of the methodologies used to perform force analysis on mechanisms. he chapter begins with a review of some fundamentals of force analysis using
More informationSummer Assignment MAT 414: Calculus
Summer Assignment MAT 414: Calculus Calculus - Math 414 Summer Assignment Due first day of school in September Name: 1. If f ( x) = x + 1, g( x) = 3x 5 and h( x) A. f ( a+ ) x+ 1, x 1 = then find: x+ 7,
More informationMECH 314 Dynamics of Mechanisms February 17, 2011 Offset Slider Crank Analysis and Some Other Slider Systems
MCH 314 ynamics of Mechanisms February 17, 011 ffset Slider Crank nalysis and Some ther Slider Systems 1 ffset Slider Crank Position nalysis Position and velocity analysis for the common slider-crank has
More informationOn Spatial Involute Gearing
6 th International Conference on Applied Informatics Eger, Hungary, January 27 31, 2004. On Spatial Involute Gearing Hellmuth Stachel Institute of Discrete Mathematics and Geometry, Vienna University of
More informationAuthor's version. Optimal Design of N-UU Parallel Mechanisms. 1 Introduction. Yuanqing Wu 1 and Marco Carricato 2
Optimal Design of N-UU Parallel Mechanisms Yuanqing Wu 1 and Marco Carricato 2 1 Dept. of Industrial Engineering, University of Bologna, Italy, e-mail: yuanqing.wu@unibo.it 2 Dept. of Industrial Engineering,
More informationarxiv: v2 [cs.ro] 29 Jun 2015
Factorisation of Rational Motions: A Survey with Examples and Applications Z. Li RICAM Austrian Academy of Sciences Linz, Austria T.-D. Rad Unit Geometry and CAD University of Innsbruck Innsbruck, Austria
More informationRELATIVE MOTION ANALYSIS: VELOCITY (Section 16.5)
RELATIVE MOTION ANALYSIS: VELOCITY (Section 16.5) Today s Objectives: Students will be able to: a) Describe the velocity of a rigid body in terms of translation and rotation components. b) Perform a relative-motion
More informationRobotics I. Test November 29, 2013
Exercise 1 [6 points] Robotics I Test November 9, 013 A DC motor is used to actuate a single robot link that rotates in the horizontal plane around a joint axis passing through its base. The motor is connected
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationTriangular Plate Displacement Elements
Triangular Plate Displacement Elements Chapter : TRIANGULAR PLATE DISPLACEMENT ELEMENTS TABLE OF CONTENTS Page. Introduction...................... Triangular Element Properties................ Triangle
More informationLecture Note 7: Velocity Kinematics and Jacobian
ECE5463: Introduction to Robotics Lecture Note 7: Velocity Kinematics and Jacobian Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018
More informationMEAM 520. More Velocity Kinematics
MEAM 520 More Velocity Kinematics Katherine J. Kuchenbecker, Ph.D. General Robotics, Automation, Sensing, and Perception Lab (GRASP) MEAM Department, SEAS, University of Pennsylvania Lecture 12: October
More informationMR. YATES. Vocabulary. Quadratic Cubic Monomial Binomial Trinomial Term Leading Term Leading Coefficient
ALGEBRA II WITH TRIGONOMETRY COURSE OUTLINE SPRING 2009. MR. YATES Vocabulary Unit 1: Polynomials Scientific Notation Exponent Base Polynomial Degree (of a polynomial) Constant Linear Quadratic Cubic Monomial
More informationDesign Flow-Chart of Slider-Crank Mechanisms and Applications
Design Flow-hart of Slider-rank Mechanisms and Applications Giorgio Figliolini, Pierluigi Rea, Marco onte DiMSA, University of assino G. Di iasio 43, 03043 assino, Italy E-mail: figliolini@unicas.it Keywords:
More informationClosed-Form Solution to the Position Analysis of Watt-Baranov Trusses Using the Bilateration Method
Closed-Form Solution to the Position Analysis of Watt-Baranov Trusses Using the Bilateration Method Nicolás Rojas and Federico Thomas Institut de Robòtica i Informàtica Industrial CSIC-UPC Llorens Artigas
More informationMathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: Exercise Answers
Mathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: 978-1-118-71219-1 Updated /17/15 Exercise Answers Chapter 1 1. Four right-handed systems: ( i, j, k), ( i, j,
More informationLecture Note 7: Velocity Kinematics and Jacobian
ECE5463: Introduction to Robotics Lecture Note 7: Velocity Kinematics and Jacobian Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018
More informationRepresentation Theory
Frank Porter Ph 129b February 10, 2009 Chapter 3 Representation Theory 3.1 Exercises Solutions to Problems 1. For the Poincare group L, show that any element Λ(M,z) can be written as a product of a pure
More informationCLASS X FORMULAE MATHS
Real numbers: Euclid s division lemma Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 r < b. Euclid s division algorithm: This is based on Euclid s division
More informationCHAPTER 5: Analytic Trigonometry
) (Answers for Chapter 5: Analytic Trigonometry) A.5. CHAPTER 5: Analytic Trigonometry SECTION 5.: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) csc( x) sin x Reciprocal
More informationKinematic analysis of Active Ankle using computational algebraic geometry
Kinematic analysis of Active Ankle using computational algebraic geometry Shivesh Kumar 1a, Abhilash Nayak, Bertold Bongardt 1a, Andreas Mueller 3 and Frank Kirchner 1ab 1a Robotics Innovation Center,
More informationPosition and orientation of rigid bodies
Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal Reference Frames RF A A z A p AB B RF B z B x B y A rigid body
More informationLecture 7: Kinematics: Velocity Kinematics - the Jacobian
Lecture 7: Kinematics: Velocity Kinematics - the Jacobian Manipulator Jacobian c Anton Shiriaev. 5EL158: Lecture 7 p. 1/?? Lecture 7: Kinematics: Velocity Kinematics - the Jacobian Manipulator Jacobian
More informationKinematics of a UR5. Rasmus Skovgaard Andersen Aalborg University
Kinematics of a UR5 May 3, 28 Rasmus Skovgaard Andersen Aalborg University Contents Introduction.................................... Notation.................................. 2 Forward Kinematics for
More informationLecture «Robot Dynamics»: Kinematics 2
Lecture «Robot Dynamics»: Kinematics 2 151-0851-00 V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) Marco Hutter,
More informationFigure 1. A planar mechanism. 1
ME 352 - Machine Design I Summer Semester 201 Name of Student Lab Section Number EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, July 2nd, 201 Use the blank paper provided for your solutions. Write on one
More informationChapter 2 Describing Motion: Kinematics in One Dimension
Chapter 2 Describing Motion: Kinematics in One Dimension Units of Chapter 2 Reference Frames and Displacement Average Velocity Instantaneous Velocity Acceleration Motion at Constant Acceleration Solving
More information9. Stress Transformation
9.7 ABSOLUTE MAXIMUM SHEAR STRESS A pt in a body subjected to a general 3-D state of stress will have a normal stress and shear-stress components acting on each of its faces. We can develop stress-transformation
More informationROBOTICS: ADVANCED CONCEPTS & ANALYSIS
ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 5 VELOCITY AND STATIC ANALYSIS OF MANIPULATORS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian
More informationStatic Unbalance. Both bearing reactions occur in the same plane and in the. After balancing no bearing reaction remains theoretically.
BALANCING Static Unbalance A disk-shaft system on rigid rails An unbalanced disk-shaft system mounted on bearings and rotated For the unbalanced rotating system: Dynamic Bearing Reactions + Inertia Forces
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Two Dimensions; Vectors Vectors and Scalars Addition of Vectors Graphical Methods (One and Two- Dimension) Multiplication of a Vector by a Scalar Subtraction of Vectors Graphical
More informationMIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 2018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART Show all work, simplify as appropriate, and
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationChapter 6. Differentially Flat Systems
Contents CAS, Mines-ParisTech 2008 Contents Contents 1, Linear Case Introductory Example: Linear Motor with Appended Mass General Solution (Linear Case) Contents Contents 1, Linear Case Introductory Example:
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 2: Rigid Motions and Homogeneous Transformations
MCE/EEC 647/747: Robot Dynamics and Control Lecture 2: Rigid Motions and Homogeneous Transformations Reading: SHV Chapter 2 Mechanical Engineering Hanz Richter, PhD MCE503 p.1/22 Representing Points, Vectors
More informationOn Spatial Involute Gearing
6 th International Conference on Applied Informatics Eger, Hungary, January 27 3, 2004. On Spatial Involute Gearing Hellmuth Stachel Institute of Discrete Mathematics and Geometry, Vienna University of
More informationSingular Foci of Planar Linkages
Singular Foci of Planar Linkages Charles W. Wampler General Motors Research & Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055, USA Abstract The focal points of a curve traced
More informationCambridge International Examinations CambridgeOrdinaryLevel
Cambridge International Examinations CambridgeOrdinaryLevel * 2 5 4 0 0 0 9 5 8 5 * ADDITIONAL MATHEMATICS 4037/12 Paper1 May/June 2015 2 hours CandidatesanswerontheQuestionPaper. NoAdditionalMaterialsarerequired.
More informationAdvanced Robotic Manipulation
Advanced Robotic Manipulation Handout CS37A (Spring 017) Solution Set #3 Problem 1 - Inertial properties In this problem, you will explore the inertial properties of a manipulator at its end-effector.
More information