Day 02 Introduction to manipulator kinematics 1 12152021

Day 02 Introduction to manipulator kinematics 1 12/15/2021

Robotic Manipulators �a robotic manipulator is a kinematic chain � i. e. an assembly of pairs of rigid bodies that can move respect to one another via a mechanical constraint � the rigid bodies are called links � the mechanical constraints are called joints 2 Symbolic Representation of Manipulators 12/15/2021

A 150 Robotic Arm link 2 3 link 3 Symbolic Representation of Manipulators 12/15/2021

Joints � most manipulator joints are one of two types revolute (or rotary) 1. like a hinge allows relative rotation about a fixed axis between two links � � � prismatic (or linear) 2. like a piston allows relative translation along a fixed axis between two links � � � axis of translation is the z axis by convention our convention: joint i connects link i – 1 to link i � � 4 axis of rotation is the z axis by convention when joint i is actuated, link i moves Symbolic Representation of Manipulators 12/15/2021

Joint Variables revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable qi : joint variable for joint i revolute � � 1. � prismatic 2. � 5 qi = qi : angle of rotation of link i relative to link i – 1 qi = di : displacement of link i relative to link i – 1 Symbolic Representation of Manipulators 12/15/2021

Revolute Joint Variable revolute � � qi = qi : angle of rotation of link i relative to link i – 1 6 qi Symbolic Representation of Manipulators 12/15/2021

Prismatic Joint Variable prismatic � � qi = di : displacement of link i relative to link i – 1 link i di 7 Symbolic Representation of Manipulators 12/15/2021

Common Manipulator Arrangments � most industrial manipulators have six or fewer joints � the first three joints are the arm � the remaining joints are the wrist � it is common to describe such manipulators using the joints of the arm � R: revolute joint � P: prismatic joint 8 Common Manipulator Arrangements 12/15/2021

Articulated Manipulator � RRR (first three joints are all revolute) � joint axes : waist � z 1 : shoulder (perpendicular to z 0) � z 2 : elbow (parallel to z 1) � z 0 z 1 z 2 q 3 shoulder forearm elbow q 1 waist 9 Common Manipulator Arrangements 12/15/2021

Spherical Manipulator � RRP � Stanford � arm http: //infolab. stanford. edu/pub/voy/museum/pictures/display/robots/IMG_2404 Arm. Front. Peeking. Out. JP G z 0 z 1 d 3 q 2 shoulder z 2 q 1 waist 10 Common Manipulator Arrangements 12/15/2021

SCARA Manipulator � RRP � Selective � Compliant Articulated Robot for Assembly http: //www. robots. epson. com/products/g-series. htm z 1 z 0 z 2 q 2 d 3 q 1 11 Common Manipulator Arrangements 12/15/2021

Forward Kinematics � given the joint variables and dimensions of the links what is the position and orientation of the end effector? a 2 q 2 a 1 q 1 12 Forward Kinematics 12/15/2021

Forward Kinematics � choose � we the base coordinate frame of the robot want (x, y) to be expressed in this frame (x, y) ? a 2 q 2 y 0 a 1 q 1 x 0 13 Forward Kinematics 12/15/2021

Forward Kinematics � notice that link 1 moves in a circle centered on the base frame origin (x, y) ? a 2 q 2 y 0 a 1 q 1 ( a 1 cos q 1 , a 1 sin q 1 ) x 0 14 Forward Kinematics 12/15/2021

Forward Kinematics � choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame (x, y) ? y 1 a 2 q 2 y 0 q 1 a 1 q 1 x 1 ( a 1 cos q 1 , a 1 sin q 1 ) x 0 15 Forward Kinematics 12/15/2021

Forward Kinematics � notice that link 2 moves in a circle centered on frame 1 (x, y) ? y 1 ( a 2 cos (q 1 + q 2), a 2 sin (q 1 + q 2) ) q 2 y 0 q 1 a 1 q 1 x 0 16 a 2 Forward Kinematics x 1 ( a 1 cos q 1 , a 1 sin q 1 ) 12/15/2021

Forward Kinematics � because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base(aframe 1 cos q 1 + a 2 cos (q 1 + q 2), a 1 sin q 1 + a 2 sin (q 1 + q 2) ) y 1 ( a 2 cos (q 1 + q 2), a 2 sin (q 1 + q 2) ) q 2 y 0 q 1 a 1 q 1 x 0 17 a 2 Forward Kinematics x 1 ( a 1 cos q 1 , a 1 sin q 1 ) 12/15/2021

Forward Kinematics � we also want the orientation of frame 2 with respect to the base frame and y 2 expressed in terms of x 0 and y 0 � x 2 y 2 x 2 a 2 q 2 y 0 q 1 a 1 q 1 x 0 18 Forward Kinematics 12/15/2021

Forward Kinematics � without proof I claim: x 2 = (cos (q 1 + q 2), sin (q 1 + q 2) ) y 2 x 2 y 2 = (-sin (q 1 + q 2), cos (q 1 + q 2) ) a 2 q 2 y 0 q 1 a 1 q 1 x 0 19 Forward Kinematics 12/15/2021

Inverse Kinematics � given the position (and possibly the orientation) of the end y 2 effector, and the dimensions of the links, what are the joint variables? x 2 (x, y) a 2 q 2 ? y 0 a 1 q 1 ? x 0 20 Inverse Kinematics 12/15/2021

Inverse Kinematics � harder than forward kinematics because there is often more than one possible solution (x, y) a 2 y 0 a 1 x 0 21 Inverse Kinematics 12/15/2021

Inverse Kinematics law of cosines (x, y) b y 0 a 2 q 2 ? a 1 x 0 22 Inverse Kinematics 12/15/2021

Inverse Kinematics and we have the trigonometric identity therefore, We could take the inverse cosine, but this gives only one of the two solutions. 23 Inverse Kinematics 12/15/2021

Inverse Kinematics Instead, use the two trigonometric identities: to obtain which yields both solutions for q 2. In many programming languages you would use th four quadrant inverse tangent function atan 2 c 2 = (x*x + y*y – a 1*a 1 – a 2*a 2) / (2*a 1*a 2); s 2 = sqrt(1 – c 2*c 2); theta 21 = atan 2(s 2, c 2); theta 22 = atan 2(-s 2, c 2); 24 Inverse Kinematics 12/15/2021

Inverse Kinematics � Exercise 25 for the student: show that Inverse Kinematics 12/15/2021