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I NTRODUCTION TO R OBOTICS CPSC - 460 Lecture 2A – DH Notation

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

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K INEMATICS 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.

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D ESCRIBING A S ERIAL M ANIPULATOR 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

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K INEMATICS 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

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W HY DH N OTATION 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.

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DH T ECHNIQUES 7

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A link j can be specified by two parameters, its length aj and its twist α j 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 θ j is the rotation of one link with respect to the next about the joint axis.

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DH T ECHNIQUES 9

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DH T ECHNIQUES Link twist α i :the angle from the Z i-1 axis to the Z i axis about the X i axis. The positive sense for α is determined from z i-1 and z i by the right-hand rule. Joint angle θ i the angle between the X i-1 and X i axes about the Z i-1 axis.

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DH T ECHNIQUES The four parameters for each link a i : link length α i : Link twist d i : Link offset θ i : joint angle With the i th joint, a joint variable is q i associated where 12

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DH T ECHNIQUES 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.

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DH T ABLE - E XAMPLE I T HE TWO LINK PLANAR ARM 14 Base frame O 0 All Z ‘s are normal to the page

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DH T ABLE - E XAMPLE 2 15

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