INTRODUCTION TO ROBOTICS CPSC 460 Lecture 2 A

INTRODUCTION TO ROBOTICS CPSC - 460 Lecture 2 A – DH Notation

TEXTBOOK Robotics, Vision and Control Fundamental Algorithms in MATLAB, Peter Corke, Springer, 2011 Author: Peter Corke Publisher: Springer ISBN: 978 -3 -642 -20144 -8

KINEMATICS Kinematics is the branch of mechanics that studies the motion of a body or a system of bodies without considering its mass or the forces acting on it. Forward (direct) Kinematics Given: The values of the joint variables. Required: The position and the orientation of the end effector. Inverse Kinematics Given : The position and the orientation of the end effector. Required : The values of the joint variables.

DESCRIBING A SERIAL MANIPULATOR A serial manipulator comprises a set of bodies, called links, in a chain and connected by joints. Each joint has one degree of freedom, either translational (prismatic joint) or rotational (a revolute joint). Motion of the joint changes the relative angle or position of its neighboring links. The joint structure of a robot can be described by a string such as “RRRRRR” for the Puma � “RRPRRR” for the Stanford arm �

KINEMATICS For a manipulator with N joints numbered from 1 to N, there are N +1 links, numbered from 0 to N Any given joint j connects links j-1 and j Link 0 is the base of the manipulator and link N carries the end-effector

WHY DH NOTATION A systematic way of describing the geometry of a serial chain of links and joints was proposed by Denavit and Hartenberg in 1955 and is known today as Denavit. Hartenberg notation or simply D-H notation. A very simple way of modeling robot links and joints that can be used for any kind of robot configuration. This technique has became the standard way of representing robots and modeling their motions.

DH TECHNIQUES 7

DH TECHNIQUES A link j can be specified by two parameters, its length aj and its twist αj Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θj is the rotation of one link with respect to the next about the joint axis.

DH TECHNIQUES 9

DH TECHNIQUES 10

DH TECHNIQUES • Link twist αi : the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the righthand rule. • Joint angle θi the angle between the Xi-1 and Xi axes about the Zi-1 axis.

DH TECHNIQUES The four parameters for each link ai: link length αi: Link twist di : Link offset θi : joint angle With the ith joint, a joint variable is qi associated where 12

DH TECHNIQUES All joints are represented by the z-axis. If the joint is revolute, the z-axis is in the direction of the axis of rotation as followed by the right hand rule. If the joint is prismatic, the z-axis for the joint is along the direction of the liner movement.

DH TABLE - EXAMPLE I THE TWO LINK PLANAR ARM • Base frame O 0 • All Z ‘s are normal to the page 14

DH TABLE - EXAMPLE 2 15