Dimensional Synthesis of Bennett Linkages

Size: px
Start display at page:

Download "Dimensional Synthesis of Bennett Linkages"

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

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 information

DYNAMICS OF PARALLEL MANIPULATOR

DYNAMICS 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 information

Single Exponential Motion and Its Kinematic Generators

Single 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 information

RELATION BETWEEN THIRD ORDER FINITE SCREW SYSTEMS AND LINE CONGRUENCE

RELATION 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 information

RIGID BODY MOTION (Section 16.1)

RIGID 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 information

TECHNICAL RESEARCH REPORT

TECHNICAL 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 information

Factorization of Rational Curves in the Study Quadric and Revolute Linkages

Factorization 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 information

GEOMETRIC DESIGN OF CYLINDRIC PRS SERIAL CHAINS

GEOMETRIC 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 information

On Martin Disteli s Main Achievements in Spatial Gearing: Disteli s Diagram

On 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 information

Robotics I. Figure 1: Initial placement of a rigid thin rod of length L in an absolute reference frame.

Robotics 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 information

Kinematic Analysis of the 6R Manipulator of General Geometry

Kinematic 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 information

Kinematic Analysis of a Pentapod Robot

Kinematic 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 information

WEEK 1 Dynamics of Machinery

WEEK 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 information

Three Types of Parallel 6R Linkages

Three 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 information

Screw Theory and its Applications in Robotics

Screw 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 information

Interpolated Rigid-Body Motions and Robotics

Interpolated 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 information

Robotics & 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 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 information

Differential Kinematics

Differential 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 information

EDEXCEL 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 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 information

Error Analysis of Four Bar Planar Mechanism Considering Clearance Between Crank & Coupler

Error 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 information

MECH 576 Geometry in Mechanics November 30, 2009 Kinematics of Clavel s Delta Robot

MECH 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 information

Ch. 5: Jacobian. 5.1 Introduction

Ch. 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 information

Framework Comparison Between a Multifingered Hand and a Parallel Manipulator

Framework 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 information

ME Machine Design I. EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, September 30th, 2009

ME 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 information

PLANAR RIGID BODY MOTION: TRANSLATION &

PLANAR 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 information

ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

ROBOTICS: 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 information

Exponential and Cayley maps for Dual Quaternions

Exponential 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 information

2. Preliminaries. x 2 + y 2 + z 2 = a 2 ( 1 )

2. 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 information

Theory of Vibrations in Stewart Platforms

Theory 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 information

Robotics I Kinematics, Dynamics and Control of Robotic Manipulators. Velocity Kinematics

Robotics 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 information

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.

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. 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 information

UNIT 2 KINEMATICS OF LINKAGE MECHANISMS

UNIT 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 information

Chapter 3 Velocity Analysis

Chapter 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 information

arxiv: v2 [cs.ro] 29 Jun 2015

arxiv: 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 information

Mathematical review trigonometry vectors Motion in one dimension

Mathematical 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 information

ON JACK PHILLIP'S SPATIAL INVOLUTE GEARING

ON 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 information

Main theorem on Schönflies-singular planar Stewart Gough platforms

Main 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 information

The Numerical Solution of Polynomial Systems Arising in Engineering and Science

The 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 information

arxiv: v1 [cs.ro] 19 Sep 2013

arxiv: 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 information

Homogeneous Transformations

Homogeneous 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 information

Outline. Outline. Vector Spaces, Twists and Wrenches. Definition. ummer Screws 2009

Outline. 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 information

Robotics & 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 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 information

2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.

2014 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 information

Department of Physics, Korea University Page 1 of 8

Department 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 information

Department of Physics, Korea University

Department 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 information

Robotics I. June 6, 2017

Robotics 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 information

Robotics, Geometry and Control - Rigid body motion and geometry

Robotics, 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 information

A Robust Forward-Displacement Analysis of Spherical Parallel Robots

A 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 information

A STUDY ON A RULED SURFACE WITH LIGHTLIKE RULING FOR A NULL CURVE WITH CARTAN FRAME

A 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 information

DETC MECHANISM BRANCHES, TURNING CURVES, AND CRITICAL POINTS

DETC 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 information

Lecture 10. Rigid Body Transformation & C-Space Obstacles. CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University

Lecture 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 information

2- Scalars and Vectors

2- 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 information

Generalized Forces. Hamilton Principle. Lagrange s Equations

Generalized 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 information

3. ANALYTICAL KINEMATICS

3. 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 information

Analytical 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 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 information

Lecture Note 4: General Rigid Body Motion

Lecture 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 information

Kinematic Isotropy of the H4 Class of Parallel Manipulators

Kinematic 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 information

The Principle of Virtual Power Slide companion notes

The 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 information

PART 1: USING SCIENTIFIC CALCULATORS (50 PTS.)

PART 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 information

Analytic Trigonometry. Copyright Cengage Learning. All rights reserved.

Analytic 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 information

7. FORCE ANALYSIS. Fundamentals F C

7. 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 information

Summer Assignment MAT 414: Calculus

Summer 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 information

MECH 314 Dynamics of Mechanisms February 17, 2011 Offset Slider Crank Analysis and Some Other Slider Systems

MECH 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 information

On Spatial Involute Gearing

On 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 information

Author's version. Optimal Design of N-UU Parallel Mechanisms. 1 Introduction. Yuanqing Wu 1 and Marco Carricato 2

Author'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 information

arxiv: v2 [cs.ro] 29 Jun 2015

arxiv: 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 information

RELATIVE MOTION ANALYSIS: VELOCITY (Section 16.5)

RELATIVE 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 information

Robotics I. Test November 29, 2013

Robotics 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 information

Course Notes Math 275 Boise State University. Shari Ultman

Course 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 information

Triangular Plate Displacement Elements

Triangular Plate Displacement Elements Triangular Plate Displacement Elements Chapter : TRIANGULAR PLATE DISPLACEMENT ELEMENTS TABLE OF CONTENTS Page. Introduction...................... Triangular Element Properties................ Triangle

More information

Lecture Note 7: Velocity Kinematics and Jacobian

Lecture 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 information

MEAM 520. More Velocity Kinematics

MEAM 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 information

MR. YATES. Vocabulary. Quadratic Cubic Monomial Binomial Trinomial Term Leading Term Leading Coefficient

MR. 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 information

Design Flow-Chart of Slider-Crank Mechanisms and Applications

Design 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 information

Closed-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 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 information

Mathematical 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: 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 information

Lecture Note 7: Velocity Kinematics and Jacobian

Lecture 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 information

Representation Theory

Representation 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 information

CLASS X FORMULAE MATHS

CLASS 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 information

CHAPTER 5: Analytic Trigonometry

CHAPTER 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 information

Kinematic analysis of Active Ankle using computational algebraic geometry

Kinematic 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 information

Position and orientation of rigid bodies

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 z A p AB B RF B z B x B y A rigid body

More information

Lecture 7: Kinematics: Velocity Kinematics - the Jacobian

Lecture 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 information

Kinematics of a UR5. Rasmus Skovgaard Andersen Aalborg University

Kinematics 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 information

Lecture «Robot Dynamics»: Kinematics 2

Lecture «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 information

Figure 1. A planar mechanism. 1

Figure 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 information

Chapter 2 Describing Motion: Kinematics in One Dimension

Chapter 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 information

9. Stress Transformation

9. 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 information

ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

ROBOTICS: 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 information

Static Unbalance. Both bearing reactions occur in the same plane and in the. After balancing no bearing reaction remains theoretically.

Static 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 information

Chapter 3 Kinematics in Two Dimensions; Vectors

Chapter 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 information

MIDTERM 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

MIDTERM 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 information

Calculus I Exam 1 Review Fall 2016

Calculus 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 information

Chapter 6. Differentially Flat Systems

Chapter 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 information

1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to

1. 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 information

MCE/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 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 information

On Spatial Involute Gearing

On 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 information

Singular Foci of Planar Linkages

Singular 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 information

Cambridge International Examinations CambridgeOrdinaryLevel

Cambridge 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 information

Advanced Robotic Manipulation

Advanced 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