ROBOTICS Kinematics TEMPUS IV Project 158644 JPCR Development

ROBOTICS Kinematics TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS

• Task of kinematics is to describe the location of systems in space, as well as the situation changes as a function of time. Kinematics may therefore be regarded as a geometry of spatial relations and movements. Position Orientation Kinematics Speed Acceleration Forces Dynamics Torques TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 2

Basic problem of robot kinematics Motors coordinates Motor movement Transformation 1 Joint coordinates Internal coordinates Joint movement Transformation 2 Cartesian coordinates External coordinates End-Effector

Transformation 1 Joint coordinates φ[Grad] l [mm] Motors-Coordinates n [1/min] l [mm] Translation 1 : 1 Rotation 1 : 1 “Direct Drive” Drive (Without gear ratio ) Simple relations 1 Revolution = x mm 1/x Transmission Translation a² = b² + c² - 2 bc cos(a) Cosinus Rate lever + Spindle Translation No mathematical difficulties TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 4

Transformation 2 Forward Kinematics Joint-Coordinates Transformation Cartesian. Coordinates Inverse Kinematics TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 5

Description of the mechanism Mechanical arrangement Image Replacement Position part Orientation part (Wrist) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 6

Kinematic chain Open chain Serial mechanism Closed chain Parallel mechanism -Good dexterity -compact design -high flexibility -high accuracy -large working space -high stiffness -good accessibility -high payload TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 7

Constraints of the kinematics chain • The positioning accuracy decreases with the number of axles. • Each additional axle means additional costs - (Engine, transmission, brakes, measuring system, storage, power amplifiers, position control loop). • The carrying capacity is reduced with the increasing number of axles. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 8

Arrangement Dominant kinematics chains Sphärisch SCARA Gelenkarm Parallel Working space Substitute image Kartesisch Zylindrisch TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 9

Redundant mechanisms • High flexibility of movement - Several combinations of the mechanism for the same position • All the robot with more than 6 axles are redundant Avoid the singularity Hyper redundanted mechanism ► yet in practice TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 10

Representation of the orientation Matrix R Elements of the matrix R are not independent In total there are nine elements All are orthogonal and have uniform length Because these six terms are only three redundant with the matrix R Representation of the orientation with three independent variables Euler-Angle (Φ, θ, ψ) • Fi, Theta, Psi Yaw, pitch, roll angle Yaw – ψ, Pitch – θ, Roll – Φ Representation of the orientation with four independent variables Quaternions q 0, q 1, q 2, q 3 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 11

Euler-Angles R(Φ, θ, ψ) = Rz(Φ) Rx’(θ) Rz”(ψ) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 12

Yaw, Pitch, Roll R(ψ, θ, Φ) = Rz(ψ) Rx’(θ) Ry”(Φ) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 13

Literature / references: 1) Jennifer Kay, Rowan: Introduction to Homogeneous Transformations & Robot Kinematics, University Computer Science Department, January 2005 2) J. M. Selig: Introductory Robotics, Prentice Hall, 1992. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 14