More details and examples on robot arms and kinematicsDenavit-Hartenberg Notation

INTRODUCTIONForward Kinematics: to determine where the robots hand is? (If all joint variables are known)

Inverse Kinematics: to calculate what each joint variable is? (If we desire that the hand be located at a particular point)

Direct Kinematics

Direct Kinematics with no matricesDirect Kinematics:HERE!

Direct KinematicsPosition of tip in (x,y) coordinates

Direct Kinematics Algorithm1) Draw sketch2) Number links. Base=0, Last link = n3) Identify and number robot joints4) Draw axis Zi for joint i5) Determine joint length ai-1 between Zi-1 and Zi6) Draw axis Xi-17) Determine joint twist i-1 measured around Xi-18) Determine the joint offset di9) Determine joint angle i around Zi10+11) Write link transformation and concatenateOften sufficient for 2D

Kinematic Problems for ManipulationReliably position the tip - go from one position to another position

Dont hit anything, avoid obstacles

Make smooth motions at reasonable speeds and at reasonable accelerations

Adjust to changing conditions - i.e. when something is picked up respond to the change in weight

ROBOTS AS MECHANISMs

Robot Kinematics: ROBOTS AS MECHANISMFig. 2.1 A one-degree-of-freedom closed-loop four-bar mechanismMultiple type robot have multiple DOF. (3 Dimensional, open loop, chain mechanisms)Fig. 2.2 (a) Closed-loop versus (b) open-loop mechanism

Chapter 2Robot Kinematics: Position AnalysisFig. 2.3 Representation of a point in space A point P in space : 3 coordinates relative to a reference frameRepresentation of a Point in Space

Chapter 2Robot Kinematics: Position AnalysisFig. 2.4 Representation of a vector in space A Vector P in space : 3 coordinates of its tail and of its headRepresentation of a Vector in Space

Chapter 2Robot Kinematics: Position AnalysisFig. 2.5 Representation of a frame at the origin of the reference frame Each Unit Vector is mutually perpendicular. : normal, orientation, approach vectorRepresentation of a Frame at the Origin of a Fixed-Reference Frame

Chapter 2Robot Kinematics: Position AnalysisFig. 2.6 Representation of a frame in a frame Each Unit Vector is mutually perpendicular. : normal, orientation, approach vectorRepresentation of a Frame in a Fixed Reference Frame The same as last slide

Chapter 2Robot Kinematics: Position AnalysisFig. 2.8 Representation of an object in space An object can be represented in space by attaching a frame to it and representing the frame in space. Representation of a Rigid Body

Chapter 2Robot Kinematics: Position AnalysisA transformation matrices must be in square form.

It is much easier to calculate the inverse of square matrices. To multiply two matrices, their dimensions must match.HOMOGENEOUS TRANSFORMATION MATRICES

Representation of Transformations of rigid objects in 3D space

Chapter 2Robot Kinematics: Position AnalysisFig. 2.9 Representation of an pure translation in space A transformation is defined as making a movement in space. A pure translation. A pure rotation about an axis. A combination of translation or rotations.Representation of a Pure Translation identitySame value a

Chapter 2Robot Kinematics: Position AnalysisFig. 2.10 Coordinates of a point in a rotating frame before and after rotation around axis x. Assumption : The frame is at the origin of the reference frame and parallel to it.Fig. 2.11 Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis. Representation of a Pure Rotation about an Axis Projections as seen from x axisx,y,z n, o, a

Fig. 2.13 Effects of three successive transformations A number of successive translations and rotations.Representation of Combined Transformations Order is importantx,y,z n, o, anioi aiT1T2T3

Fig. 2.14 Changing the order of transformations will change the final result Order of Transformations is important x,y,z n, o, atranslation

Chapter 2Robot Kinematics: Position AnalysisFig. 2.15 Transformations relative to the current frames. Example 2.8Transformations Relative to the Rotating Frametranslationrotation

MATRICES FOR FORWARD AND INVERSE KINEMATICS OF ROBOTSFor positionFor orientation

Chapter 2Robot Kinematics: Position AnalysisFig. 2.17 The hand frame of the robot relative to the reference frame. Forward Kinematics Analysis: Calculating the position and orientation of the hand of the robot. If all robot joint variables are known, one can calculate where the robot is at any instant. . FORWARD AND INVERSE KINEMATICS OF ROBOTS

Chapter 2Robot Kinematics: Position AnalysisForward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates. Forward and Inverse Kinematics Equations for Position

Chapter 2Robot Kinematics: Position AnalysisIBM 7565 robot All actuator is linear. A gantry robot is a Cartesian robot. Fig. 2.18 Cartesian Coordinates.Forward and Inverse Kinematics Equations for Position (a) Cartesian (Gantry, Rectangular) Coordinates

Chapter 2Robot Kinematics: Position Analysis2 Linear translations and 1 rotation translation of r along the x-axis rotation of about the z-axis translation of l along the z-axis Fig. 2.19 Cylindrical Coordinates.Forward and Inverse Kinematics Equations for Position:Cylindrical Coordinatescosinesine

Chapter 2Robot Kinematics: Position Analysis2 Linear translations and 1 rotation translation of r along the z-axis rotation of about the y-axis rotation of along the z-axis Fig. 2.20 Spherical Coordinates.Forward and Inverse Kinematics Equations for Position (c) Spherical Coordinates

Chapter 2Robot Kinematics: Position Analysis3 rotations -> Denavit-Hartenberg representation Fig. 2.21 Articulated Coordinates.Forward and Inverse Kinematics Equations for Position (d) Articulated Coordinates

Chapter 2Robot Kinematics: Position Analysis

Roll, Pitch, Yaw (RPY) angles Euler angles Articulated joints

Forward and Inverse Kinematics Equations for Orientation

Chapter 2Robot Kinematics: Position AnalysisFig. 2.22 RPY rotations about the current axes.Forward and Inverse Kinematics Equations for Orientation (a) Roll, Pitch, Yaw(RPY) Angles

Chapter 2Robot Kinematics: Position AnalysisFig. 2.24 Euler rotations about the current axes.Forward and Inverse Kinematics Equations for Orientation (b) Euler Angles

Chapter 2Robot Kinematics: Position Analysis Assumption : Robot is made of a Cartesian and an RPY set of joints. Assumption : Robot is made of a Spherical Coordinate and an Euler angle.Another Combination can be possibleDenavit-Hartenberg RepresentationForward and Inverse Kinematics Equations for OrientationRoll, Pitch, Yaw(RPY) Angles

Forward and Inverse Transformations for robot arms

Fig. 2.16 The Universe, robot, hand, part, and end effecter frames. Steps of calculation of an Inverse matrix: Calculate the determinant of the matrix. Transpose the matrix. Replace each element of the transposed matrix by its own minor (adjoint matrix). Divide the converted matrix by the determinant.INVERSE OF TRANSFORMATION MATRICES

Identity Transformations

We often need to calculate INVERSE MATRICES

It is good to reduce the number of such operations

We need to do these calculations fast

How to find an Inverse Matrix B of matrix A?

Inverse Homogeneous Transformation

Homogeneous CoordinatesHomogeneous coordinates: embed 3D vectors into 4D by adding a 1More generally, the transformation matrix T has the form:

a11 a12 a13 b1a21 a22 a23 b2a31 a32 a33 b3c1 c2 c3 sfIt is presented in more detail on the WWW!

For various types of robots we have different maneuvering spaces

For various types of robots we calculate different forward and inverse transformations

For various types of robots we solve different forward and inverse kinematic problems

Forward and Inverse Kinematics: Single Link Example

Forward and Inverse Kinematics: Single Link Exampleeasy

Denavit Hartenberg idea

Denavit-Hartenberg Representation : @ Simple way of modeling robot links and joints for any robot configuration, regardless of its sequence or complexity.@ Transformations in any coordinates is possible. @ Any possible combinations of joints and links and all-revolute articulated robots can be represented. DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT

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

Links are in 3D, any shape associated with Zi always

Only rotationOnly translationOnly offsetOnly offsetOnly rotationAxis alignment

DENAVIT-HARTENBERG REPRESENTATION for each link

4 link parameters

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

Example with three Revolute JointsDenavit-Hartenberg Link Parameter TableThe DH Parameter TableApply firstApply last

i

((i-1)

a(i-1)

di

(i

0

0

0

0

(0

1

0

a0

0

(1

2

-90

a1

d2

(2

Denavit-Hartenberg Representation of Joint-Link-Joint Transformation

Notation for Denavit-Hartenberg Representation of Joint-Link-Joint TransformationAlpha applied first

Four Transformations from one Joint to the NextOrder of multiplication of matrices is inverse of order of applying themHere we show order of matricesJoint-Link-Joint

Denavit-Hartenberg Representation of Joint-Link-Joint TransformationAlpha is applied firstHow to create a single matrix A n

EXAMPLE: Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Type 1 LinkFinal matrix from previous slidesubstitutesubstituteNumeric or symbolic matrices

The Denavit-Hartenberg Matrix for another link typeSimilarity to Homegeneous: 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.

Put the transformation here for every link

In DENAVIT-HARTENBERG REPRESENTATION we must be able to find parameters for each linkSo we must know link types

Links between revolute joints

ln=0Type 3 LinkJoint n+1Joint ndn=0Link nxn-1xn

ln=0dn=0Type 4 LinkOrigins coinciden-1Joint n+1Joint nPart of dn-1Link nxn-1yn-1xnn

Links between prismatic joints

Forward and Inverse Transformations on Matrices

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

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.). DENAVIT-HARTENBERG REPRESENTATION Procedures for assigning a local reference frame to each joint:

Chapter 2Robot 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.DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies Reminder:

Chapter 2Robot 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+1 colinear.(III) Translate along the xn-axis a distance of an+1 to bring the origins of xn+1 together. (IV) Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis with zn+1-axis. DENAVIT-HARTENBERG REPRESENTATION The necessary motions to transform from one reference frame to the next.

****