Kinematics of Robot Manipulator .


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Prologue to Apply autonomy. Kinematics of Robot Controller. Jizhong Xiao Branch of Electrical Building City School of New York jxiao@ccny.cuny.edu. Diagram. Survey Robot Controllers Robot Arrangement Robot Determination Number of Tomahawks, DOF Accuracy, Repeatability Kinematics
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Prologue to ROBOTICS Kinematics of Robot Manipulator Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu

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Outline Review Robot Manipulators Robot Configuration Robot Specification Number of Axes, DOF Precision, Repeatability Kinematics Preliminary World casing, joint edge, end-effector outline Rotation Matrix, composite pivot framework Homogeneous Matrix Direct kinematics Denavit-Hartenberg Representation Examples Inverse kinematics

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Review What is a robot? By general understanding a robot is: A programmable machine that emulates the activities or appearance of a canny creature–usually a human. To qualify as a robot, a machine must have the capacity to: 1) Sensing and recognition: get data from its surroundings 2) Carry out various assignments: Locomotion or control, accomplish something physical–such as move or control objects 3) Re-programmable: can do distinctive things 4) Function independently or potentially associate with people Why utilize robots? Perform 4A assignments in 4D conditions 4A: Automation, Augmentation, Assistance, Autonomous 4D: Dangerous, Dirty, Dull, Difficult

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Manipulators Robot arms, mechanical robot Rigid bodies (joins) associated by Joints: revolute or kaleidoscopic Drive: electric or water driven End-effector (device) mounted on a rib or plate secured to the wrist joint of robot

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Manipulators Robot Configuration: Cartesian: PPP Cylindrical: RPP Spherical: RRP Hand arrange: n: ordinary vector; s : sliding vector; a : approach vector, typical to the instrument mounting plate SCARA: RRP (Selective Compliance Assembly Robot Arm) Articulated: RRR

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Manipulators Motion Control Methods Point to point control a succession of discrete focuses spot welding, pick-and-place, stacking & emptying Continuous way control take after an endorsed way, controlled-way movement Spray painting, Arc welding, Gluing

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Manipulators Robot Specifications Number of Axes Major tomahawks, (1-3) => Position the wrist Minor tomahawks, (4-6) => Orient the device Redundant, (7-n) => stretching around obstructions, maintaining a strategic distance from undesirable setup Degree of Freedom (DOF) Workspace Payload (stack limit) Precision v.s. Repeatability Which one is more imperative?

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What is Kinematics Forward kinematics Given joint factors End-effector position and introduction, - Formula?

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What is Kinematics Inverse kinematics End effector position and introduction Joint factors - Formula?

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Example 1

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Preliminary Robot Reference Frames World edge Joint edge Tool outline T P W R

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O, O\' Preliminary Coordinate Transformation Reference facilitate outline OXYZ Body-appended outline O\'uvw Point spoke to in OXYZ: Point spoke to in O\'uvw: Two casings correspond ==>

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Mutually opposite Unit vectors Preliminary Properties: Dot Product Let and be subjective vectors in and be the edge from to , then Properties of orthonormal arrange outline

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Preliminary Coordinate Transformation Rotation just How to relate the organize in these two edges?

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Preliminary Basic Rotation , and speak to the projections of onto OX, OY, OZ tomahawks, individually Since

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Preliminary Basic Rotation Matrix Rotation about x-pivot with

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Preliminary Is it True? Pivot about x hub with

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Basic Rotation Matrices Rotation about x-hub with Rotation about y-hub with Rotation about z-hub with

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Preliminary Basic Rotation Matrix Obtain the organize of from the facilitate of Dot items are commutative! <== 3X3 personality network

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Example 2 An indicate is joined a turning casing, the casing pivots 60 degree about the OZ hub of the reference outline. Discover the directions of the guide relative toward the reference outline after the pivot.

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Example 3 A point is the facilitate w.r.t. the reference organize framework, locate the relating point w.r.t. the turned OU-V-W arrange framework on the off chance that it has been pivoted 60 degree about OZ hub.

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Composite Rotation Matrix A succession of limited turns framework increases don\'t drive rules: if pivoting coordinate O-U-V-W is turning about chief hub of OXYZ casing, then Pre-duplicate the past (resultant) revolution network with a proper essential turn grid if pivoting coordinate OUVW is turning about its own important tomahawks, then post-duplicate the past (resultant) turn lattice with a suitable fundamental revolution framework

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Example 4 Find the revolution network for the accompanying operations: Pre-increase if pivot about the OXYZ tomahawks Post-duplicate if pivot about the OUVW tomahawks

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Coordinate Transformations position vector of P in { B } is changed to position vector of P in { A } portrayal of { B } as observed from an onlooker in { A } Rotation of { B } regarding { A } Translation of the source of { B } concerning root of { A }

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Coordinate Transformations Two Special Cases 1. Interpretation just Axes of { B } and { A } are parallel 2. Turn just Origins of { B } and { A } are correspondent

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Homogeneous Representation Coordinate change from { B } to { A } Homogeneous change lattice Rotation framework Position vector Scaling

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Homogeneous Transformation Special cases 1. Interpretation 2. Revolution

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O, O\' O, O\' h Example 5 Translation along Z-hub with h:

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Example 6 Rotation about the X-hub by

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Homogeneous Transformation Composite Homogeneous Transformation Matrix Rules: Transformation (pivot/interpretation) w.r.t (X,Y,Z) (OLD FRAME), utilizing pre-duplication Transformation (turn/interpretation) w.r.t (U,V,W) (NEW FRAME), utilizing post-increase

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Example 7 Find the homogeneous change framework (T) for the accompanying operations:

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Homogeneous Representation An edge in space (Geometric Interpretation) (z\') (y\') (X\') Principal hub n w.r.t. the reference facilitate framework

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Homogeneous Transformation Translation

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? Homogeneous Transformation Composite Homogeneous Transformation Matrix Transformation network for nearby facilitate outlines Chain result of progressive organize change lattices

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Example 8 For the figure demonstrated as follows, locate the 4x4 homogeneous change grids and for i=1, 2, 3, 4, 5 Can you discover the appropriate response by perception in light of the geometric understanding of homogeneous change framework?

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Orientation Representation Rotation grid portrayal needs 9 components to totally depict the introduction of a turning unbending body. Any simple way? Euler Angles Representation

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Euler Angle I Euler Angle II Roll-Pitch-Yaw Sequence about OZ hub about OZ axis about OX hub of about OU pivot about OV hub about OY hub Rotations about OW hub about OW hub about OZ hub Orientation Representation Euler Angles Representation ( , ) Many diverse sorts Description of Euler edge portrayals

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Euler Angle I, Animated w " = z w\'"= w " f v\'" v "  v " y u \'"  u " =u" x

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Orientation Representation Euler Angle I

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Euler Angle I Resultant eulerian revolution network:

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Euler Angle II, Animated w " = z w " w"\' =  v "\'  v " =v "  y u"\' u" Note the inverse (clockwise) feeling of the third turn, f . u " x

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Orientation Representation Matrix with Euler Angle II Quiz: How to get this lattice ?

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Orientation Representation Description of Roll Pitch Yaw Z Y X Quiz: How to get pivot framework ?

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Thank you! Homework 1 is posted on the web. Due: Sept. 16 , 200 8 , before class Next class: kinematics II

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