Introduction to ROBOTICS Kinematics of Robot Manipulator PROF.UJWAL HARODE UJWALHARODE 1 Outline Review Robot Manipulators -Robot Configuration -Robot Specification • Number of Axes, DOF • Precision, Repeatability Kinematics -Preliminary • World frame, joint frame, end-effector frame • Rotation Matrix, composite rotation matrix • Homogeneous Matrix -Direct kinematics • Denavit-Hartenberg Representation • Examples -Inverse kinematics UJWALHARODE 2 Review What is a robot? By general agreement a robot is: A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human. To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings 2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects 3) Re-programmable: can do different things 4) Function autonomously and/or interact with human beings Why use robots? –Perform 4A tasks in 4D environments 4A: Automation, Augmentation, Assistance, Autonomous 4D: Dangerous, Dirty, Dull, Difficult UJWALHARODE 3 Manipulators • Robot arms, industrial robot • Rigid bodies (links) connected by joints • Joints: revolute or prismatic • Drive: electric or hydraulic • End-effector (tool) mounted on a flange or plate secured to the wrist joint of robot UJWALHARODE 4 Manipulators Robot Configuration Cartesian: PPP Cylindrical: RPP Spherical: RRP Hand coordinate: n: normal vector; s: sliding vector; SCARA: RRP Articulated: RRR (Selective Compliance Assembly Robot Arm) a: approach vector, normal to the tool mounting plate 5 Manipulators • Motion Control Methods • Point to point control • a sequence of discrete points • spot welding, pick-and-place, loading & unloading • Continuous path control • follow a prescribed path, controlledpath motion • Spray painting, Arc welding, Gluing UJWALHARODE 6 Manipulators • Robot Specifications • Number of Axes • Major axes, (1-3) => Position the wrist • Minor axes, (4-6) => Orient the tool • Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration • • • • Degree of Freedom (DOF) Workspace Payload (load capacity) Precision v.s. Repeatability UJWALHARODE 7 What is Kinematics Forward kinematics z Given joint variables q (q1 , q2 , q3 , q4 , q5 , q6 ,qn ) y Y ( x, y, z, O, A, T ) End-effector position and orientation, Formula? UJWALHARODE x 8 What is Kinematics Inverse kinematics End effector position and orientation ( x, y, z, O, A, T ) q (q1 , q2 , q3 , q4 , q5 , q6 ,qn ) Joint variables -Formula? UJWALHARODE 9 Example 1 Forward kinemat ics x0 l cos y0 y1 x1 y0 l sin l Inversekinemat ics x0 cos ( x0 / l ) 1 UJWALHARODE 10 Preliminary • Robot Reference Frames 1. World frame 2. Joint frame 3. Tool frame z T z y x y W T x R UJWALHARODE 11 Coordinate Transformation z Reference coordinate frame –XYZ Body-attached frame-ijk Point represented in XYZ T P [ p , p , p ] xyz x y z Pxyz p x i x p y jy p z k z P y w v Point represented in uvw: Puvw pu i u pv jv pw k w UJWALHARODE O, O’ Two frames coincide x u 12 Properties: Dot Product Let and be arbitrary vectors in be the angle from to , then and x y x y cos Properties of orthonormal co-ordinate frame Unit vectors are mutually perpendicular i j 0 i k 0 k j 0 | i | 1 | j | 1 | k | 1 UJWALHARODE 13 Coordinate Transformation Reference coordinate frame OXYZ Body-attached frame O’uvw z P Pxyz RPuvw w y v u x O, O’ How to relate the coordinate in these two frames? UJWALHARODE 14 • Basic Rotation – px , p y, and p z represent the projections of P onto OX, OY, OZ axes, respectively – Since P pu i u pv jv pwk w px i x P i x i u pu i x jv pv i x k w pw py jy P jy i u pu jy jv pv jy k w pw pz k z P k z i u pu k z jv pv k z k w pw UJWALHARODE 15 Basic Rotation Matrix px i x i u p j i y y u p z k z i u i x jv j y jv k z jv i x k w pu j y k w pv k z k w pw Rotation about x-axis with 0 1 0 Rot( x, ) 0 C S 0 S C UJWALHARODE z w P v u x y 16 Rotation about y-axis with : C Rot( y, ) 0 S Rotation about z-axis with : C Rot( z , ) S 0 UJWALHARODE S 1 0 0 C 0 S C 0 0 0 1 17 Example 2 • A point auvw (4,3,2) is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation. a xyz Rot( z ,60)auvw 0.5 0.866 0 4 0.598 0.866 0.5 0 3 4.964 0 0 1 2 2 UJWALHARODE 18 Composite Rotation Matrix • A sequence of finite rotations – matrix multiplications do not commute – rules: • if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix • if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix UJWALHARODE 19 • Find the rotation matrix for the following operations: Rotation about OY axis Rotation about OW axis Rotation about OU axis Answer... R Rot( y, ) I 3 Rot( w, ) Rot(u, ) C 0 0 1 - S 0 CC S SC S C S 0 1 0 0 0 S C 0 0 C S C 0 0 1 0 S C SS CSC CSS SC CC CS SSC CS CC SSS Pre-multiply if rotate about the OXYZ axes Post-multiply if rotate about the OUVW axes UJWALHARODE 20 Homogeneous Transformation • Special cases 1. Translation I 33 A TB 013 2. Rotation r 1 A o' A RB A TB 013 031 1 Homogeneous transformation matrix R33 P31 F 1 0 UJWALHARODE 21 Homogeneous Transformation • Composite Homogeneous Transformation Matrix • Rules: – Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplication – Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication UJWALHARODE 22 Orientation Representation • Description of Roll Pitch Yaw Z ROLL PITCH X Y YAW UJWALHARODE 23