# AAS Project 1- Dead Reckoning Solved

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This short project is composed by problems which are useful as a training session, for preparing you for projects during subsequent weeks in AAS.

Problem 1

Given the following approximate model of a pendulum,

 ( t ) = − A  s i n ( ( t ) ) − B   ( t ) + C  u ( t ) A=110rad, B=2.21, C=1.1 rad

s2 s s2 volt
where (t)is the angular position (expressed in radians) of the pendulum, and u(t)is the voltage (expressed in volts)

controlling the pendulum’s electric motor.
a) Obtain a valid state space representation for this system, in continuous time.
b) Obtain an approximate discrete time model (using Euler’s approximation), for a sample time dt=1ms. c) Implement a program (in plain Matlab language), for simulating the model proposed in (b).

Test your program simulating the following cases:
c.1) The pendulum is released, at time =0, having the following initial conditions: angular velocity =0 and angle = 110 degrees. The voltage of the electric motor is assumed to be constantly 0 volts (no torque being applied by the motor).
c.2) Similar to (c.1) but having the electric motor controlled with a constant voltage = 3 volts.

In both cases, perform the simulation for an interval of time from 0 to t=7 seconds. Plot the results (position and angular velocity) in a figure.

d) Using the model implemented in item c, implement a simulation in Simulink.

Problem 2

Given the following simplified 3DoF kinematic model (of a car-like wheeled platform),

x(t)=v(t)cos((t)) y(t)=v(t)sin((t))

 (t ) = tan ( (t )) v (t ) L

1. a)  Obtain an approximate discrete-time version of the model, assuming small discrete steps, e.g. of dt=0.01 seconds

(10ms). Consider the case of a vehicle that has L=2.5m.

2. b)  Implement a program for simulating the system in (a). Run it under different steering actions (sequences of

steering angles  (k ) ) and assume constant speed, v (k ) = 3.5 m/s, k . c.1) See what happen if you keep the steering angle set at a constant value.

c.2) Try to generate a path having an 8-shape (define a proper sequence of control actions to achieve it).

Advanced Autonomous Systems–Project 0 (training). Version 2021.1 . 1

c.4) Apply a small modification on the model (e.g. a small change in parameter L) and see how the result is affected, for a long-term simulation (for cases c.1 and c.2). Plot, jointly, both models’ trajectories using different colors, to appreciate the different responses.

Note: The main purpose of this task is to give the students some initial training, before the actual projects are released. This task is intended to be solved during week 1.

Advanced Autonomous Systems–Project 0 (training). Version 2021.1 . 2