Kinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction

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1 Katedra Robotyki i Mechatroniki Akademia Górniczo-Hutnicza w Krakowie Kinematics and Dynamics of Mechatronic Systems Wojciech Lisowski 1 An Introduction KADOMS KRIM, WIMIR, AGH Kraków 1

2 The course contents: - Mechanisms of the mechatronic systems synthesis and analysis (J. Cieślik) - Kinematics and dynamics of positioning mechatronic systems - the manipulator example (W. Lisowski) - Kinematics and dynamics of the mobile positioning systems - the mobile robot example (T. Buratowski) Classes: 1. Determination of position and orientation 2. Determination of joint variables for an assumed pose 3. Planning of end-effector s trajectories 4. Formulation of the dynamic model 5. Inverse dynamics 6. Dynamic model with the driving system dynamics KADOMS KRIM, WIMIR, AGH Kraków 2

3 Literature: J.J. CRAIG, INTRODUCTION TO ROBOTICS MECHANICS AND CONTROL, ADDISON-WESLEY, 1984 K.S. FU, R.C. GONZALES, C.S.G. LEE, ROBOTICS: CONTROL, SENSING, VISION, AND INTELLIGENCE, McGRAW-HILL, 1987 M.W. SPONG, M. VIDYASAGAR, ROBOT DYNAMICS AND CONTROL, JOHN WILLEY & SONS, INC., 1989 L-W TSAI, ROBOT ANALYSIS, JOHN WILLEY & SONS, INC., 1999 KADOMS KRIM, WIMIR, AGH Kraków 3

4 Problems: Object of modelling Model Aim of modelling Examples KADOMS KRIM, WIMIR, AGH Kraków 4

5 The object Industrial Robot a complex mechatronic positioning system Robot: a motorized and computer-controlled mechanical device (often resembling an "arm") that can be programmed to do automatically a variety of manufacturing tasks. KADOMS KRIM, WIMIR, AGH Kraków 5

6 Manipulating (industrial) robot manipulators End-effector Wrist Arm (Nachi) KADOMS KRIM, WIMIR, AGH Kraków 6

7 Model A Model is a theoretical description of a considered object (or a process). The properties and characteristics of the model should be convergent to the object properties and characteristics according to some criterion. The description should be simplified so that one could analyse the model with use of available methods. However the simplification level should enable to obtain some new knowledge about the modelled object basing on investigation of its model behaviour. KADOMS KRIM, WIMIR, AGH Kraków 7

8 What is modelling for? Robot manipulator modelling is performed for the following purposes: DESIGN - determine stiffness and strength of materials, - select driving systems, - select end-effector (gripper, assembly, technological or inspection tool), - characterise joint motions, workspace, velocities and accelerations, - formulate manufacturing technology, - estimate cost SIMULATION when the use of real robot is: impossible, costly, dangerous, technically complex, determine the robot manipulator behaviour by numerical calculations. KADOMS KRIM, WIMIR, AGH Kraków 8

9 - CONTROL SYNTHESIS - of end-effector free motion in robot workspace (manipulation) - of constrained motion of end-effector during its interaction with other systems (e.g. during assembly) Assumptions - manipulator possesses only rotary or prismatic joints - manipulator has open loop kinematic chain - manipulator links are straight line ones - manipulator is non-redundant (DOM=DOF) - stiff links and joints are considered - friction and clearance effects are neglected. KADOMS KRIM, WIMIR, AGH Kraków 9

10 Diagram of a Modelled system DISTURBANCES DISTURBANCES INPUT MANIPULATOR + OUTPUT INPUT DISTURBANCES MANIPULATOR OUTPUT - forces/torques - environmental influences - controller - free motion in robot WS - voltage/current - neglected phenomena - power supply - the end-effector action on pressure difference/ - measurement errors - power amplifier objects present in robot WS flow rate - actuator - motion transmission system - end-effector An object model is used to determine: - properties of the output for the command input (the direct task) - properties of the input for the command output (the inverse task) KADOMS KRIM, WIMIR, AGH Kraków 10

11 Types of mathematic models of manipulators Manipulator model types depending on the modelling aim: GEOMETRICAL MODEL a set of algebraic equations describing position and orientation of end-effector in robot workspace KINEMATIC MODEL a set of algebraic equations expressing velocity and acceleration of robot links and end-effector during motion (causes of motion forces - are neglected) STATIC MODEL a set of algebraic equations describing static force balance of a manipulator DYNAMIC MODEL - a set of differential equations that defines relationship between forces acting on manipulator, its mass distribution and its kinematic parameters that describe its motion. Mathematical model is an equation or a set of equations in general form f(x)=0 KADOMS KRIM, WIMIR, AGH Kraków 11

12 Example of geometrical model of a manipulator of SCARA (RRPR) kinematic structure z 0 y 0 x 0 z 1 y 1 x 1 y2 z 2 x 2 a 1 =0.5 m a 2 =0.35 m y3 y4 x 3 z 4 x 4 z 3 Link No. θ d a α Motion range 1 θ 1 v 0 a o 120 o 2 θ 2 v 0 a 2 π 0 o 150 o 3 0 d 3 v m 0.3 m 4 θ 4 v o 180 o KADOMS KRIM, WIMIR, AGH Kraków 12

13 0 T 4 C C + S S C S + S C 0 a C + a C S C C S C C S S 0 a S a S = d where: S 1 =sin(θ 1 ), C 1 =cos(θ 1 ), S 4 =sin(θ 4 ), C 4 =cos(θ 4 ), S 12 =sin(θ 1 +θ 2 ), C 12 =cos(θ 1 +θ 2 ), KADOMS KRIM, WIMIR, AGH Kraków 13

14 Example of dynamic model of SCARA (RRPR) manipulator ( ) ( ) ( C ) τ1 = C && 2 θ C && 2 θ2 01. && θ S && 2θθ S2θ& 2 2 τ = && θ && θ 01. && θ S & θ f = 24. d&& 24. g 3 3 τ = 01. && θ 01. && θ && θ where: θ 1, θ 2, d 3, θ 4 are joint variables τ 1, τ 2, f 3, τ 4 are generalized forces S 2 =sin(θ 2 ), C 2 =cos(θ 2 ), g gravitational acceleration & θ = i θ d i dt && θ = i d 2 i 2 dt θ D ( q )&&+ q C( q,q& ) + G( q) = τ 2 KADOMS KRIM, WIMIR, AGH Kraków 14

15 Design, manufacturing and application of industrial robots requires formulation of the appropriate models of manipulators. Remarks: phenomena not taken into account in the formulated model are considered to be disturbances of the model INPUT and/or the model OUTPUT, assessment of the modelling assumptions is possible only by pursuing a verification experiment in a general case a structure and most of all values of parameters of the formulated model are known only with limited precision, determination of these values requires carrying out an identification experiment KADOMS KRIM, WIMIR, AGH Kraków 15

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