AAS - Advanced Autonomous Systems - Project 0 - Solved

Problem 1
 

Given the following approximate model of a pendulum,

 (t) = −Asin((t))− B(t) +C u t ( )

                                                                                     rad                   1                     rad 

                                                               A =110 s2 ,    B = 2.2 s ,  C =1.1 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)) v t( )

                                                                                     (t) = tan((t))      L
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.

 

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( )= 3.5m/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).                                                      .            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.