advertisement

Rose-Hulman Institute of Technology Department of Mechanical Engineering ME 536 Comput Intell in Control Engr Assignment I: Course Value: 100 points Deadline: March 18, 2016 a) Matlab / Simulink Review: The 'bicycle model' of a car backing up with a trailer in tow is illustrated below. For convenience, we define the ground reference frame as positive x to the right and positive y 'up'. The body fixed axis is then defined as x pointing in the direction of rear axis travel, and y pointing to the driver's right. The car 'heading angle' is defined counterclockwise from the ground x axis, and denoted by ψ. The steering angle is defined counter clockwise from the body negative x axis. The dimensions given below are based on a 1998 GMC Savana towing a 30 ft trailer. For simplicity, we assume that the wheels are located at the corners of the vehicle body. The rear axle is assumed to have a constant velocity, resulting in the equations for velocity of point B. x& B = v cosψ y& B = v sinψ The y component of velocity of point A at the center of the ‘front’ axle is given by: y& A = v sin (ψ + β ) cos β The x component of velocity at point A is then computed from the constraint that the length of the vehicle must remain constant. x& A = tanψ ( y& B − y& A ) + x& B The car heading angle equation is: ψ& = − v tan β L The trailer axle speed is given by Vt = V cos(ψ − θ ) The trailer heading angle is governed by V sin (θ − ψ ) Lt θ& = Construct a Matlab/Simulink model that accepts the velocity and steering angle as inputs, and gives the position of points A, B and C (in the ground reference frame) as outputs. Take L = 3.0293m, Lt = 9.172m. Use the plotting code carplot_trailer.m available on the course web page to plot the path of the van and trailer for a backup speed of 1 m/s and a steering angle defined by up and down pulses of 45 degrees (see plot). (I have the plotting routine set up to plot the vehicles as rectangles.) Note the velocities of points B and C can be rotated from the body frame to the ground frame using finite rotations: v Bx cos(ψ ) − sin (ψ ) v Bx = v By I sin (ψ ) cos(ψ ) v By B and vCx cos(θ ) = Vt sin (θ ) vCy I b) Unconstrained optimization from An Engineer's Guide to Matlab, Magrab et al. Prentice Hall, 2005. Three carts, interconnected by springs and initially at an unstressed equilibrium state, are subjected to the loads P1, P2, and P3 as shown below. The displacements of the carts from their original equilibrium position (xi = 0: for all i) are sought by minimizing the potential energy of the system (PE): PE = 1 T T X KX − X P 2 where k1 + k 3 + k 4 K= −k 3 −k 4 The input data are k 1 = 4500 N/m k 2 = 1650 N/m k 3 = 1100 N/m − k4 −k 5 k4 + k 5 + k6 −k 3 k 2 + k3 + k 5 −k 5 T P = [P1 P2 P3 ] X = [x1 x2 x3 ] k 4 = 2250 N/m k 5 = 550 N/m k 6 = 9300 N/m T P1 = 1100 N P2 = 1800 N P3 = 3300 N Find the equilibrium position of the carts using Newton method with cubic line search with the initial guess of ( x1 , x 2 , x 3 ) = (0, 0, 0). [Answer: [ x1 , x 2 , x 3 ] = [0.322, 0.714, 0.365]] x2 x1 x3 P1 P3 k4 k6 k2 P2 k1 k3 1 k5 2 Figure 13.22 Spring-mass system of part b 3