On this page we demonstrate an automatic control of the parallel parking maneuver that you tried manually at the beginning of this unit. To come up with an algorithm to prescribe the appropriate driving and steering commands to make the car park as desired, we use the non-reciprocal shape change discussed in the previous section.

As previously, we model the car as a single wheel (a unicycle). A unicycle is easier to maneuver than a front-wheel drive car (we are not concerned with balance here!), since we can essentially turn on a dime, unlike in a front-wheel drive car. Note, however, that we have made the problem nontrivial by making the parking spot very tight given the size of the car. That is, one cannot simply drive all the way in head on and turn in place without a crashing in to the curb or other cars.

The first step in our algorithm is simply to drive the car so that we are aligned as close as possible to the parking spot. The problem then remains to move the car sideways and in to the parking spot. To make the car move sideways, we perform over and over again the non-reciprocal shape change sequence of motions not unlike the sequence defined by the transformation matrix N. The choice of k and α are determined by the clearance between the car and the parking spot. Since in each repetition, the car moves sideways by the amount kα, the number of repetitions J is then determined such that J(kα) is equal to the total distance sideways that we need to move the car.

The fact that the car takes so many repetitions to get into the parking spot is a result of our making the parking spot so tight.

After you view the sample automatically controlled parking maneuver, try going back and using the shape change concept to perform the manual park maneuver.

You need the Flash 6 plug-in to play with the car. Internet Explorer automatically instralls the plug in, or you may install it following the link: http://www.macromedia.com/shockwave/download/index.cgi? P1_Prod_Version=ShockwaveFlash&P2_Platform=Win32

If the Flash application isn't working on your computer you can use the old version of car parking applet below:

Our algorithm uses a "smoothed out" version of the sequence of maneuvers defined by the transformation matrix N. That is, we don't actually stop driving while steering nor stop steering while driving. However, the fundamentals of the non-reciprocal shape change defined by N are unchanged. In the plots at the bottom of the applet, we show the steering signal and the driving signal as functions of time during a single repetition. Here, u_r represents steering speed and u_f represents driving speed.

Your browser is not compatible this applet.

In this applet the car will park itself! Press the "Sample" button and the car will go to an initial position and orientation. Then press the same button to start the simulation. Press the same button again to stop the simulation at any time.