5 Chapter 2 Robot Kinematics: Position Analysis DENAVIT-HARTENBERG REPRESENTATIONSymbol Terminologies :Chapter 2 Robot Kinematics: Position Analysis⊙ : 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.
9 Joints are numbered to represent hierarchy Ui-1 is parent of Ui
10 Parameter ai-1 is outgoing limb length of joint Ui-1
11 Joint angle, qi, is rotation of xi-1 about zi-1 relative to xi
12 Link twist, ai-1, is the rotation of ith z-axis about xi-1-axis relative to z-axis of i-1th frame
13 Link offset, di-1, specifies the distance along the zi-1-axis (rotated by ai-1) of the ith frame from the i-1th x-axis to the ith x-axis
14 DENAVIT-HARTENBERG REPRESENTATION PROCEDURES Start point:Assign joint number n to the first shown joint.Assign a local reference frame for each and every joint before orafter these joints.Y-axis is not used in D-H representation.
15 DENAVIT-HARTENBERG REPRESENTATION Procedures for assigning a local reference frame to each joint:٭ All joints are represented by a z-axis.(right-hand rule for rotational joint, linear movement for prismatic joint)The common normal is one line mutually perpendicular to any two skew lines.Parallel z-axes joints make a infinite number of common normal.Intersecting z-axes of two successive joints make no common normal between them(Length is 0.).
16 Chapter 2 Robot Kinematics: Position Analysis DENAVIT-HARTENBERG REPRESENTATIONThe necessary motions to transform from one reference frame to the next.Chapter 2 Robot Kinematics: Position Analysis(I) Rotate about the zn-axis an able of n+1. (Coplanar)(II) Translate along zn-axis a distance of dn+1 to make xn and xn+1colinear.(III) Translate along the xn-axis a distance of an+1 to bring the originsof xn+1 together.(IV) Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axiswith zn+1-axis.
17 Denavit - Hartenberg Parameters – a general explanation
18 Denavit-Hartenberg Notation Only and d are joint variablesZ(i - 1)Y(i -1)Y iZ iX ia ia(i - 1 )d iX(i -1) i( i - 1)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)
19 The a(i-1) Parameter 1) a(i-1) You can align the two axis just using the 4 parametersZ(i - 1)Y(i -1)Y iZ iX ia ia(i - 1 )d iX(i -1) i( 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.The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines.
20 The alpha a(i-1) Parameter 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 iY iX ia 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)
21 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 iY iX ia 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)
22 The (i-1) Parameter 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 Zi axis.Positive rotation follows the right hand rule.The (i-1) ParameterZ(i - 1)X(i -1)Y(i -1)( i - 1)a(i - 1 )Z iY iX ia id i i
23 3) d(i-1) The d(i-1) Parameter The i Parameter Technical Definition: The displacement along the Zi axis needed to align the a(i-1) common perpendicular to the ai common perpendicular.In other words, displacement along the Zi to align the X(i-1) and Xi axes.4) iAmount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi axis.The d(i-1) ParameterThe i ParameterZ(i - 1)X(i -1)Y(i -1)( i - 1)a(i - 1 )Z iY iX ia id i iThe same table as last slide
24 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 )Z iY iX ia id i iPut 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)
25 Example: Calculating the final DH matrix with the DH Parameter Table
26 Denavit-Hartenberg Link Parameter Table The DH Parameter TableExample with three Revolute JointsZ0X0Y0Z1X2Y1Z2X1Y2d2a0a1Denavit-Hartenberg Link Parameter TableNotice 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.We calculate with respect to previous
27 Denavit-Hartenberg Link Parameter Table Example with three Revolute JointsZ0X0Y0Z1X2Y1Z2X1Y2d2a0a1Denavit-Hartenberg Link Parameter TableNotice 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
28 d2 a0 a1 Note: T is the D-H matrix with (i-1) = 0 and i = 1. Z0X0Y0Z1X2Y1Z2X1Y2d2a0a1The same table as last slideNote: T is the D-H matrix with (i-1) = 0 and i = 1.World coordinatestool coordinatesThese matrices T are calculated in next slide
29 This is just a rotation around the Z0 axis The same table as last slideThis is just a rotation around the Z0 axisThis is a translation by a1 and then d2 followed by a rotation around the X2 and Z2 axisThis is a translation by a0 followed by a rotation around the Z1 axis
32 Forward Kinematics Problem The Situation:You have a robotic arm that starts out aligned with the xo-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:1. Geometric ApproachThis 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.2. Algebraic ApproachInvolves coordinate transformations.
33 Example Problem with H matrices: You have a three link arm that starts out aligned in the x-axis.Each link has lengths l1, l2, l3, respectively.You tell the first one to move by 1 , and so on as the diagram suggests.Find the Homogeneous matrix to get the position of the yellow dot in the X0Y0 frame.Y3 3l2l3Y2X3 2X2H = Rz( 1 ) * Tx1(l1) * Rz( 2 ) * Tx2(l2) * Rz( 3 )Rotating by 1 will put you in the X1Y1 frame.Translate in the along the X1 axis by l1.Rotating by 2 will put you in the X2Y2 frame.and so on until you are in the X3Y3 frame.The position of the yellow dot relative to the X3Y3 frame is(l3, 0).Multiplying H by that position vector will give you thecoordinates of the yellow point relative the X0Y0 frame.Y0l1X1 1Y1X0
34 Slight variation on the last solution: Make the yellow dot the origin of a new coordinate X4Y4 frameY3Y4323Y2X32X2addedX4H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) * Tx3(l3)This takes you from the X0Y0 frame to the X4Y4 frame.The position of the yellow dot relative to the X4Y4 frame is (0,0).Y01X11Y1X0
38 THE INVERSE KINEMATIC SOLUTION OF ROBOT We calculate all angles from px, py, a1, a2, ni, oi, etc
39 INVERSE KINEMATIC PROGRAM: a predictable path on a straight line 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 breakthe line into many small sections.٭ All unnecessary computations should be eliminated.Fig Small sections of movement for straight-line motions
41 DEGENERACY AND DEXTERITY Degeneracy : The robot looses a degree of freedomand 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-axesof two similar joints becomes collinear.Dexterity : The volume of points where one canposition the robot as desired, but notorientate it.Fig An example of a robot in a degenerate position.
42 THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATION Defect of D-H presentation : D-H cannot represent any motion aboutthe y-axis, because all motions are about the x- and z-axis.TABLE 2.3 THE PARAMETERS TABLE FOR THESTANFORD ARM#da11-9022d1903445566Fig The frames of the Stanford Arm.