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Links and Joints

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Joints: Links End Effector Robot Basis 2 DOF’s

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Denavit – Hartenberg details and examples

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Chapter 2 Robot Kinematics: Position Analysis ⊙ : A rotation about the z-axis. d ⊙ d : The distance on the z-axis. a ⊙ a : The length of each common normal (Joint offset). ⊙ : The angle between two successive z-axes (Joint twist) Only and d are joint variables. DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies :

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Z-axis aligned with joint Joints U Links S

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X-axis aligned with outgoing limb

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Y-axis is orthogonal

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Joints are numbered to represent hierarchy U i-1 is parent of U i

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Parameter a i-1 is outgoing limb length of joint U i-1

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Joint angle, i, is rotation of x i-1 about z i- 1 relative to x i

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Link twist, i-1, is the rotation of i th z-axis about x i-1 -axis relative to z-axis of i-1 th frame

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Link offset, d i-1, specifies the distance along the z i-1 -axis (rotated by i-1 ) of the i th frame from the i-1 th x-axis to the i th x-axis

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Start point: Assign joint number n to the first shown joint. Assign a local reference frame for each and every joint before or after these joints. Y-axis is not used in D-H representation. DENAVIT-HARTENBERG REPRESENTATION PROCEDURES

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1.٭ All joints are represented by a z-axis. (right-hand rule for rotational joint, linear movement for prismatic joint) 2.The common normal is one line mutually perpendicular to any two skew lines. 3. Parallel z-axes joints make a infinite number of common normal. 4.Intersecting z-axes of two successive joints make no common normal between them(Length is 0.). DENAVIT-HARTENBERG REPRESENTATION Procedures for assigning a local reference frame to each joint:

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Chapter 2 Robot Kinematics: Position Analysis (I) Rotate about the z n -axis an able of n+1. (Coplanar) (II) Translate along z n -axis a distance of d n+1 to make x n and x n+1 colinear. (III) Translate along the x n -axis a distance of a n+1 to bring the origins of x n+1 together. (IV) Rotate z n -axis about x n+1 axis an angle of n+1 to align z n -axis with z n+1 -axis. DENAVIT-HARTENBERG REPRESENTATION from one reference frame The necessary motions to transform from one reference frame to the next.

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Denavit - Hartenberg Parameters – a general explanation

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Denavit-Hartenberg Notation Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d i i IDEA: Each joint is assigned a coordinate frame. Using the Denavit-Hartenberg notation, you need 4 parameters to describe how a frame (i) relates to a previous frame ( i -1 ). THE PARAMETERS/VARIABLES: , a, d, ⊙ : A rotation about the z-axis. ⊙ d : The distance on the z-axis. ⊙ a : The length of each common normal (Joint offset). ⊙ : The angle between two successive z-axes (Joint twist) Only and d are joint variables

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The a (i-1) Parameter Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i align You can align the two axis just using the 4 parameters 1) a (i-1) 1) a (i-1) Technical Definition: a (i-1) is the length of the perpendicular between the joint axes. The joint axes are the axes around which revolution takes place which are the Z (i-1) and Z (i) axes. These two axes can be viewed as lines in space. perpendicular to both The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines.

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⊙ : A rotation about the z-axis. ⊙ d : The distance on the z-axis. ⊙ a : The length of each common normal (Joint offset). ⊙ : The angle between two successive z-axes (Joint twist) a (i-1) cont... Visual Approach - “A way to visualize the link parameter a (i-1) is to imagine an expanding cylinder whose axis is the Z (i-1) axis - when the cylinder just touches the joint axis i the radius of the cylinder is equal to a (i-1). ” (Manipulator Kinematics) Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i The alpha a (i-1) Parameter

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It’s Usually on the Diagram Approach - If the diagram already specifies the various coordinate frames, then the common perpendicular is usually the X (i-1) axis. So a (i-1) is just the displacement along the X (i-1) to move from the (i-1) frame to the i frame. If the link is prismatic, then a (i-1) is a variable, not a parameter. Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i ⊙ : A rotation about the z-axis. ⊙ d : The distance on the z-axis. ⊙ a : The length of each common normal (Joint offset). ⊙ : The angle between two successive z-axes (Joint twist)

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2) (i-1) Technical Definition: Amount of rotation around the common perpendicular so that the joint axes are parallel. i.e. How much you have to rotate around the X (i-1) axis so that the Z (i-1) is pointing in the same direction as the Z i axis. Positive rotation follows the right hand rule. Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i The Parameter The (i-1) Parameter

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3) d (i-1) Technical Definition: The displacement along the Z i axis needed to align the a (i-1) common perpendicular to the a i common perpendicular. In other words, displacement along the Z i to align the X (i-1) and X i axes. 4) i Amount of rotation around the Z i axis needed to align the X (i-1) axis with the X i axis. Z (i - 1) X (i -1) Y (i -1) ( i - 1) a (i - 1 ) Z i Y i X i a i d id i i The Parameter The d (i-1) Parameter The Parameter The i Parameter The same table as last slide

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The Denavit-Hartenberg Matrix Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame. Z (i - 1) X (i - 1) Y (i - 1) ( i - 1) a (i - 1 ) ZiZi Y i XiXi aiai didi i Put the transformation here ⊙ : A rotation about the z-axis. ⊙ d : The distance on the z-axis. ⊙ a : The length of each common normal (Joint offset). ⊙ : The angle between two successive z-axes (Joint twist)

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Example: Calculating the final DH matrix with the DH Parameter Table

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Example with three Revolute Joints Z0Z0 X0X0 Y0Y0 Z1Z1 X2X2 Y1Y1 Z2Z2 X1X1 Y2Y2 d2d2 a0a0 a1a1 Denavit-Hartenberg Link Parameter Table Notice that the table has two uses: 1) To describe the robot with its variables and parameters. 2) To describe some state of the robot by having a numerical values for the variables. The DH Parameter Table We calculate with respect to previous

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Example with three Revolute Joints Z0Z0 X0X0 Y0Y0 Z1Z1 X2X2 Y1Y1 Z2Z2 X1X1 Y2Y2 d2d2 a0a0 a1a1 Denavit-Hartenberg Link Parameter Table Notice that the table has two uses: 1) To describe the robot with its variables and parameters. 2) To describe some state of the robot by having a numerical values for the variables. The same table as last slide

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Z0Z0 X0X0 Y0Y0 Z1Z1 X2X2 Y1Y1 Z2Z2 X1X1 Y2Y2 d2d2 a0a0 a1a1 Note: T is the D-H matrix with (i-1) = 0 and i = 1. These matrices T are calculated in next slide The same table as last slide World coordinates tool coordinates

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This is just a rotation around the Z 0 axis This is a translation by a 0 followed by a rotation around the Z 1 axis This is a translation by a 1 and then d 2 followed by a rotation around the X 2 and Z 2 axis The same table as last slide

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World coordinates tool coordinates Conclusions

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ForwardKinematics

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The Situation: You have a robotic arm that starts out aligned with the x o -axis. You tell the first link to move by 1 and the second link to move by 2. The Quest: What is the position of the end of the robotic arm? Solution:. Geometric Approach 1. Geometric Approach For robots with more links and whose arm extends into 3 dimensions the geometry gets much more tedious This might be the easiest solution for the simple situation. However, notice that the angles are measured relative to the direction of the previous link. (The first link is the exception. The angle is measured relative to it’s initial position.) For robots with more links and whose arm extends into 3 dimensions the geometry gets much more tedious. Algebraic Approach 2. Algebraic Approach coordinate Involves coordinate transformations. Forward Kinematics Problem

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X2X2 X3X3 Y2Y2 Y3Y3 1 1 2 2 3 3 l1 l2l3 Example Problem with H matrices: 1.You have a three link arm that starts out aligned in the x-axis. 2.Each link has lengths l 1, l 2, l 3, respectively. 3.You tell the first one to move by 1, and so on as the diagram suggests. 4.Find the Homogeneous matrix to get the position of the yellow dot in the X 0 Y 0 frame. H = R z ( 1 ) * T x1 (l 1 ) * R z ( 2 ) * T x2 (l 2 ) * R z ( 3 ) 1.Rotating by 1 will put you in the X 1 Y 1 frame. 2.Translate in the along the X 1 axis by l 1. 3.Rotating by 2 will put you in the X 2 Y 2 frame. 4. and so on until you are in the X 3 Y 3 frame. The position of the yellow dot relative to the X 3 Y 3 frame is (l 3, 0). Multiplying H by that position vector will give you the coordinates of the yellow point relative the X 0 Y 0 frame. X1X1 Y1Y1 X0X0 Y0Y0

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Slight variation on the last solution Slight variation on the last solution : Make the yellow dot the origin of a new coordinate X 4 Y 4 frame X2X2 X3X3 Y2Y2 Y3Y3 11 22 3 X1X1 Y1Y1 X0X0 Y0Y0 X4X4 Y4Y4 H = R z ( 1 ) * T x1 (l 1 ) * R z ( 2 ) * T x2 (l 2 ) * R z ( 3 ) * T x3 (l 3 ) This takes you from the X 0 Y 0 frame to the X 4 Y 4 frame. The position of the yellow dot relative to the X 4 Y 4 frame is (0,0). added

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THE INVERSE KINEMATIC SOLUTION OF A ROBOT

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each joint Determine the value of each joint to place the arm at a desired position and orientation. THE INVERSE KINEMATIC SOLUTION OF ROBOT Multiply both sides by A1 -1 RHS

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THE INVERSE KINEMATIC SOLUTION OF ROBOT A1 -1

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THE INVERSE KINEMATIC SOLUTION OF ROBOT We calculate all angles from px, py, a1, a2, ni, oi, etc

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A robot has a predictable path on a straight line, Or an unpredictable path on a straight line. ٭ A predictable path is necessary to recalculate joint variables. (Between 50 to 200 times a second) ٭ To make the robot follow a straight line, it is necessary to break the line into many small sections. ٭ All unnecessary computations should be eliminated. Fig Small sections of movement for straight- line motions INVERSE KINEMATIC PROGRAM: INVERSE KINEMATIC PROGRAM: a predictable path on a straight line

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PROBLEMS with DH

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Degeneracy : The robot looses a degree of freedom and thus cannot perform as desired. ٭ When the robot ’ s joints reach their physical limits, and as a result, cannot move any further. ٭ In the middle point of its workspace if the z-axes of two similar joints becomes collinear. Fig An example of a robot in a degenerate position. Dexterity : The volume of points where one can position the robot as desired, but not orientate it. DEGENERACY AND DEXTERITY

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cannot represent any motion about Defect of D-H presentation : D-H cannot represent any motion about the y-axis, the y-axis, because all motions are about the x- and z-axis. Fig The frames of the Stanford Arm. # da 1 1 22 d1d d1d 4 5 66 000 TABLE 2.3 THE PARAMETERS TABLE FOR THE STANFORD ARM THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATION

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