An accelerometer records acceleration for a device that is determined to be equivalent to the function below: a(t) = -1.5 sin(2t)| Given the following initial conditions, x(t = 0) = 0, and v(t = 0) = 1[m/s], determine the position of the device when t = 3.5s.

Context The needle of a sewing machine makes a repetitive up and down motion which is described by the equation y(t)=Acos(wt) To choose the settings of the device that will ensure that the needle successfully pierces the textile, you are asked to model the speed in ? of the needle as a function of time. Constraints and information The parameter A represents the range of motion (in m);The parameter w represents the angular frequency. It tells us about the number of back and forth movements that the needle makes per second;The parameter w is given in rad/s, so your calculator absolutely must be in radians mode to do the calculations in this problem;Modelization Establish the model for the y velocity of the needle as a function of time and the parameters A and w. Then test your model by calculating the speed in y of the needle at the moment and with the given parameters: Moment: 4.1 s Amplitude: 0.012 m Angular frequency: 17.9 rad/s

5. Determine the validity of a second-order approximation of the system below: 450 / [(s +9)*(s² + 3s +70)]. If valid, give the approximation.

1 Overview In this project, you'll use several robot datasets for estimation and you will implement a Kalman Filter (KF) for state estimation. Consider the vehicle depicted in Fig. 1 with m = 0.5, k = 3.5, b = 2. The position s and velocity v represent the state x of the system and the input ƒ (t) is the force often denoted with u. F b m Figure 1: Robot cart 0 s(t) f(t) 1. Write the system in state space form and consider additive Gaussian white noise to the model n N (0, Q). Discretize the system state space form using Euler approach (30 points).