Systems Modelling and Analysis - Assignment 1 Due: Friday 25/08/2022 by 5:00:00 pm. To be submitted individually on Canvas and Gradescope. Part 1: Dartboard Positioning System The following link shows an interesting video of a dartboard positioning system: youtube.com/watch?v=MHTizZ_XcUM. Over the next few weeks, these assignments will guide you through the modelling and analysis of this system, the first steps to the construction of this impressive application. Figure 1 shows a simple schematic of this positioning system, which you will be using as the basis of your model. The positioning system consists of a DC motor, two pulleys with fixed axes of rotation connected by a belt and a dartboard. The equilibrium position of the system is such that the dartboard is positioned halfway between both pulleys (y = 0). Figure 2 defines the various physical parameters of the model. Assume that the motor shaft has no inertia and that the dartboard can be treated like a point mass moving in the ty direction only (movement in the x direction is controlled by a separate mechanism independent of y). + DC Motor L www ww www 2k 4/2 2k E 5 20 +y 13 19 3 17 Figure 1: Schematic of the dartboard positioning system 02 parameter Description r L m J J2 k b bp Ra Rb C La k ke y x 01 82 ia Radius of each pulley Distance between pulleys Mass of dartboard Mass moment of inertia of the powered pulley Mass moment of inertia of the unpowered pulley Spring constant of elastic band between pulleys Rotational damping coefficient of each pulley Damping coefficient on the dartboard for modelling air resistance Armature resistance of the motor Internal resistance of the battery Filter capacitor Armature inductance of the motor Motor torque constant Motor voltage (back EMF) constant Horizontal position of dartboard Vertical position of dartboard Angle of powered pulley i.e. motor angle Angle of unpowered pulley Armature current Motor input voltage V Ve Back EMF voltage Figure 2: Parameters of the positioning system model