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1/47 Introduction Robotics dr Dragan Kosti ć WTB Dynamics and Control September - October 2009 Introduction Robotics, lecture 2 of 7 2/47 New Plan for the Lecture Rooms • Every Tuesday, 3 e+4 e hours • Week 37+40+42+43 in AUD.10 • Week 38+39 in AUD.14 • Week 41 in AUD.15 Introduction Robotics, lecture 2 of 7 3/47 Outline • Recapitulation • Parameterization of rotations • Rigid motions • Homogenous transformations • Robotic manipulators • Forward kinematics problem Introduction Robotics, lecture 2 of 7 4/47 Recapitulation Introduction Robotics, lecture 2 of 7 5/47 Representing rotations in 3D Each axis of the frame o1x1y1z1 is projected onto o0x0y0z0: 0 R1 ∈ SO(3) Introduction Robotics, lecture 2 of 7 6/47 Basic rotation matrices Introduction Robotics, lecture 2 of 7 7/47 Introduction Robotics, lecture 2 of 7 8/47 Introduction Robotics, lecture 2 of 7 9/47 Parameterization of rotations Rigid motions Homogenous transformations Introduction Robotics, lecture 2 of 7 10/47 Introduction Robotics, lecture 2 of 7 11/47 Introduction Robotics, lecture 2 of 7 12/47 Introduction Robotics, lecture 2 of 7 13/47 Introduction Robotics, lecture 2 of 7 14/47 Introduction Robotics, lecture 2 of 7 15/47 Introduction Robotics, lecture 2 of 7 16/47 Introduction Robotics, lecture 2 of 7 17/47 Introduction Robotics, lecture 2 of 7 18/47 Basic homogenous transformations Introduction Robotics, lecture 2 of 7 19/47 Robotic Manipulators Introduction Robotics, lecture 2 of 7 20/47 Robot manipulators • Kinematic chain is series of links and joints. SCARA geometry • Types of joints: – rotary (revolute, θ) – prismatic (translational, d). Introduction Robotics, lecture 2 of 7 schematic representations of robot joints 21/47 Anthropomorphic geometry (RRR) • Resembles geometry of human arm. • Features at least Upper arm 3 rotary joints. • Also known as Staubli articulated (ex Unimation) kinematic PUMA robot scheme. http://www.ifr.org/pictureGallery/robType.htm Introduction Robotics, lecture 2 of 7 22/47 Spherical geometry (RRP) • Robot axes form a polar coordinate system. Stanford Arm - 1969 Introduction Robotics, lecture 2 of 7 23/47 SCARA geometry (RRP) • Robot having 2 parallel rotary joints to provide compliance in x-y direction (suitable for assembly tasks); sufficiently rigid in z-direction. • SCARA stands for Selective Compliant Assembly Robot Arm or Selective Compliant Articulated Robot Arm . Introduction Robotics, lecture 2 of 7 24/47 Cylindrical geometry (RPP) • Robot axes form a cylindrical coordinate system. Introduction Robotics, lecture 2 of 7 25/47 Cartesian geometry (PPP) • Robot having 3 prismatic joints, whose axes are coincident with a Cartesian coordinate frame. Introduction Robotics, lecture 2 of 7 26/47 Parallel geometry • Robot whose arms have concurrent prismatic or revolute joints. • Some subset of robot links form a closed kinematic chain. • There are at least two open kinematic chains connecting the base to the end-effector. Introduction Robotics, lecture 2 of 7 27/47 Workspace per robot geometry (1/2) Introduction Robotics, lecture 2 of 7 28/47 Workspace per robot geometry (2/2) Introduction Robotics, lecture 2 of 7 29/47 Forward Kinematics Introduction Robotics, lecture 2 of 7 30/47 Scope • Formulate problem of Forward Kinematics (FK) in robotics. • Introduce Denavits-Hartenberg (DH) formalism to assign coordinate frames to robot joints. • Solve FK problem using matrix manipulation of homogenous transformations. Introduction Robotics, lecture 2 of 7 31/47 Forward kinematics problem • Determine position and orientation of the end-effector as function of displacements in robot joints. Introduction Robotics, lecture 2 of 7 32/47 Illustration: planar manipulator with 2 joints • Position of end-effector: • Orientation of end-effector: Introduction Robotics, lecture 2 of 7 33/47 Conventions (1/2) 1. there are n joints and hence n + 1 links; joints 1, 2, …, n; links 0, 1, …, n, 2. joint i connects link i − 1 to link i, 3. actuation of joint i causes link i to move, 4. link 0 (the base) is fixed and does not move, 5. each joint has a single degree-of-freedom (dof): Introduction Robotics, lecture 2 of 7 34/47 Conventions (2/2) 6. frame oixiyizi is attached to link i; regardless of motion of the robot, coordinates of each point on link i are constant when expressed in frame oixiyizi, 7. when joint i is actuated, link i and its attached frame oixiyizi experience resulting motion. Introduction Robotics, lecture 2 of 7 35/47 DH convention for homogenous transformations (1/2) • An arbitrary homogeneous transformation is based on 6 independent variables: 3 for rotation + 3 for translation. • DH convention reduces 6 to 4, by specific choice of the coordinate frames. • In DH convention, each homogeneous transformation has the form: Introduction Robotics, lecture 2 of 7 36/47 DH convention for homogenous transformations (2/2) • Position and orientation of coordinate frame i with respect to frame i-1 is specified by homogenous transformation matrix: q xn z 0 0 q i+1 zn qi ‘0’ n y0 αi ‘ ’ yn zi ai x0 xi di z i-1 q where i xi-1 Introduction Robotics, lecture 2 of 7 37/47 Mathematical rational (1/3) • There exists unique homogenous transformation matrix A1 that expresses coordinates from frame 1 relative to frame 0. Introduction Robotics, lecture 2 of 7 38/47 Mathematical rational (2/3) x1 ⋅ x0 y1 ⋅ x0 z1 ⋅ x0 0 = ⋅ ⋅ ⋅ ⊥ ⇒ ⋅ = R1 x1 y0 y1 x0 z1 x0 ; x1 z0 x1 z0 0 x1 ⋅ z0 y1 ⋅ z0 z1 ⋅ z0 0 • Each column and row of R1 has unique length, implying 2 2 2 2 (x1 ⋅ x0 ) + (x1 ⋅ y0 ) =1 and (y1 ⋅ z0 ) + (z1 ⋅ z0 ) =1 and in turn there exist unique θ and α such that (x1 ⋅ x0 , x1 ⋅ y0 ) = (cos θ, sin θ ) (y1 ⋅ z0 , z1 ⋅ z0 ) = (sin α, cos α) 0 0 • Other elements of R1 ensure R1 ∈SO(3) . Introduction Robotics, lecture 2 of 7 39/47 Mathematical rational (3/3) • Displacement between the origins of frames 1 and 0 is linear combination of vectors z0 and x1: 0 0 0 0 o1 = o0 + dz 0 + ax 1 = 0 0 cos θ = 0 + d 0 + asin θ = 0 1 0 a cos θ = asin θ . d Introduction Robotics, lecture 2 of 7 40/47 Physical meaning of DH parameters • Link length ai is distance from zi-1 to zi measured along xi. q xn z 0 • Link twist α is angle between z 0 q i i-1 i+1 zn qi and z measured in plane normal ‘0’ n i y0 αi ‘ ’ yn zi ai to x (right -hand rule). x0 i xi • Link offset di is distance from origin di z i-1 q of frame i-1 to the intersection x i i xi-1 with zi-1, measured along zi-1. • Joint angle θi is angle from xi-1 to xi measured in plane normal to zi-1 (right-hand rule). Introduction Robotics, lecture 2 of 7 41/47 DH convention to assign coordinate frames 1. Assign zi to be the axis of actuation for joint i+1 (unless otherwise stated zn coincides with zn-1). 2. Choose x0 and y0 so that the base frame is right-handed. 3. Iterative procedure for choosing oixiyizi depending on oi-1xi-1yi-1zi-1 (i=1, 2, …, n-1): a) zi−1 and zi are not coplanar; there is an unique shortest line segment from zi−1 to zi, perpendicular to both; this line segment defines xi and the point where the line intersects zi is the origin oi; choose yi to form a right-handed frame, b) zi−1 is parallel to zi; there are infinitely many common normals; choose xi as the normal passes through oi−1 ; choose oi as the point at which this normal intersects zi; choose yi to form a right-handed frame, c) zi−1 intersects zi; axis xi is chosen normal to the plane formed by zi and zi−1 ; it’s positive direction is arbitrary; the most natural choice of oi is the intersection of zi and zi−1 , however, any point along the zi suffices; choose yi to form a right-handed frame. Introduction Robotics, lecture 2 of 7 42/47 Forward kinematics (1/2) • Homogenous transformation matrix relating the frame oixiyizi to oi-1xi-1yi-1zi-1: Ai specifies position and orientation of oixiyizi w.r.t. oi-1xi-1yi-1zi-1. i • Homogenous transformation matrix Tj expresses position and orientation of ojxjyjzj with respect to oixiyizi: i K T j = Ai+1Ai+2 Aj−1Aj Introduction Robotics, lecture 2 of 7 43/47 Forward kinematics (2/2) • Forward kinematics of a serial manipulator with n joints can be 0 represented by homogenous transformation matrix Hn which defines position and orientation of the end-effector’s (tip) frame onxnynzn relative to the base coordinate frame o0x0y0z0: 0 0 K H n (q) = Tn (q) = A1(q1) ⋅ ⋅ An (qn ), 0 0 0 Rn (q) xn (q) H n (q) = 03×1 1 03×1 = [0 0 0]; L T q = [q1 qn ] Introduction Robotics, lecture 2 of 7 44/47 Case-study: RRR robot manipulator d3 x3 x2 y3 z3 y 2 -q 3 d a3 y 2 1 z2 x1 elbow α1 q 2 a2 z1 z0 shoulder d1 α1 - twist angle y 0 ai - link lenghts q1 waist di - link offsets Introduction Robotics, lecture 2 of 7 x0 qi - displacements 45/47 DH parameters of RRR robot manipulator Introduction Robotics, lecture 2 of 7 46/47 Forward kinematics of RRR robot manipulator (1/2) • Coordinate frame o3x3y3z3 is related with the base frame o0x0y0z0 via homogenous transformation matrix: 0 T3(q) = A1(q)A 2(q)A 3(q) = R0(q) x0(q) = 3 3 01×3 1 where T 0 T q = [q1 q2 q3] x3(q) = [x y z] 01×3 = [0 0 0] Introduction Robotics, lecture 2 of 7 47/47 Forward kinematics of RRR robot manipulator (2/2) • Position of end-effector: x = cos q1[a3cos( q2 + q3) + a2cos q2 ]+ (d2 + d3)sin q1 , y = sin q1[a3cos( q2 + q3) + a2cos q2 ]−(d2 + d3)cos q1 , z = a3sin( q2 + q3) + a2sin q2 + d1 • Orientation of end-effector: cos q1cos( q2 + q3) − cos q1sin( q2 + q3) sin q1 0 R3 = sin q1cos( q2 + q3) − sin q1sin( q2 + q3) − cos q1 sin( q2 + q3) cos( q2 + q3) 0 Introduction Robotics, lecture 2 of 7.
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