**Parking Maneuver** **Using** **TWO CIRCLE METHOD**

The algorithm decided upon is based on the concept that a parallel parking

Procedure is comprised of two arcs of equal length by identical circles, *C*1
and *C*2. *C*1 is R distance in only the positive y-direction from the
goal parking position. *C*2 is the circle relating to the initial position
whose center is located a distance R in the negative y direction. In both the
initial and desired positions, the line connecting the center of the robot to
its respective circle center is perpendicular to the x-axis. The initial
position begins at the center of *C*2 minus a distance R in the y
direction. These circles are equal in size, with radius R, the turning radius of
our model car.

The center of *C*1 is assumed from here on to be the origin. The robots
coordinates are then based off of this origin, plus some padding. Using angle A,
the angle that is part of the triangle connecting *C*1, *C*2 and the Y-axis, trigonometry is used to obtain the distance the center of *C*2,
(x1, *y*1) is from the origin. Calculating *C*2 (x1, *y*1) uses
the following equations:

Sin (*A*) = *y*1/2R

*A*=sin-1 (*y*1/2R)

Cos (*A*) = *x*1/2R

*x1*=2Rcos (*A*)

For each possible configuration of the two circles, the turning point is
going to create a symmetric divide forcing the two arc lengths along each circle
to have the same length. The arc length is determined using angle A through the
following equation: *Arc Len *= R[(π/2)-A]. It always performs movements in
the backward direction because that was all that was required within the scope
of this project.

Calculation Behind Parking

Arc Len = R[(π/2)-A]

A=sin-1 (y1/2R)

x1=2Rcos (A)

Y1=?

R=PQ + Spot width/Car width + In-between distance + Half Car width

Now,

Y1=R + Half Car width + PQ

=R + Half Car width + R – (Car width + Half Car width + In-between distance)

= 2R – Car width – distance between car and obstacle.