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