The figure below shows a simplified model of an single-link robotic manipulator. N2 + Ra Ia + La Tp Wp b Tm Om V.. Km A Wm N₁ (m l Given an input voltage Va, the permanent magnet DC motor drives the motor shaft, which then drives the (pendulum) robotic link shaft through the gearbox. Important: In the schematic, the total rotating mass of the system (of the shafts, gearbox and the link) is represented by the lumped inertia J- In this system, having accounted for the mass in this way, the pendulum only acts to impose an angle-varying, nonlinear torque Tp of the form: Tp = Img sin (Op) Where is the angle of the robotic link, measured from the downward position. The motor and link shafts are initialised such that when Op=00, the pendulum is in the downward position. These angular posititons are always related via the gear ratio N₂ N₁ N = . The angular velocities of these shafts are related the same way. A linear viscous rotating friction b represents all of the losses in the mechanical system (e.g. friction due to misalignment, bearings, gearbox, etc.). The following parameters characterise the system; ■ La=0.1 [H] ■Ra=1 [9] ■ Km 0.4 [VS] ■b=0.2 [Nms] ■ Jm = 0.1[kgm2] ■N N₂/N₁-7 [-] ■m=5 [kg] ■ 1-1 [m] ⚫g=9.8 [m/s²] Your task is as follows: 1. Show that the state-space model given below, when using the following states: 2₁ = a, 22Lm 230m, describes the dynamics of the system in the figure. Take the driving voltage as the only input to the system u₁ Va and (pendulum) link position as the only output y₁ =0p 2. 3. Ra. Km 21- Km 62=> 21- ما 22 Img sin (23) b22- Jm N 23=> 22 Keep your working you will be required to submit a hard-copy. Calculate the two equilibrium points of this system, which correspond to the downward and upward position of the pendulum for u₁ = Va=0. Store these points in the vectors Zie and Z2e respectively in the code below. Keep your working - you will be required to submit a hard-copy. Linearise this system about the equilibrium point corresponding to the downward position of the pendulum Zle with input u₁ = V₁ = 0. Remeber that for a system in the form: z = f(z, v) The linear approximation can be computed using: z = f(20)+(220) + where -z is the state. Zo. Vo define the point at which the linearisation is computed (the equilibrium point in general). off O is the Jacobian matrix (matrix of gradients) evaluated at the linearisation point. The linearised model can then be written in the form: x = Ax+ Bu y = Cx + Du where x =z-Zo and uv-ve, and matrices are obtained from the Jacobian matrices. Keep your working -- you will be required to submit a hard-copy.