# Triangulation in automotive ultrasonic park-assist systems

The time-cycled distance algorithm described in the previous section is inherently inaccurate. This is because the correct distance between the vehicle and the object is not the distance measured by the transducers from themselves to the object. Rather, the correct distance between the vehicle and the object is the closest distance from the bumper to the object, which is along the perpendicular line from the bumper to the object.

* Figure 2* geometrically illustrates this inherent inaccuracy. In this figure,

*T1*and

*T2*at a distance

*x*from each other, are the two ultrasonic transducers in the car’s rear bumper.

The ultrasonic transducer

*T1*measures distance

*l1*while the transducer

*T2*measures distance

*l2*, while the correct distance between the vehicle and object is

*d*.

**Figure 2: Geometry of the ultrasonic transducers and the object**A simple example is used to illustrate this error. Consider an example in which two ultrasonic transducers are installed in the car’s rear bumper. Assume that the distance between the transducers is two feet, and the object is one-and-a-half feet away from the vehicle, rather the perpendicular distance between the rear bumper and the object is one-and-a-half feet.

In this example, if the object is located exactly perpendicular to

*T1*, or

*x1*in

*is zero, then transducer*

**Figure 2***T1*measures the correct distance. However, if the object is located exactly perpendicular to

*T2*, or

*x1*in

*is two feet away, then the relative error of the distance measured by*

**Figure 2***T1*is large.

**shows this relative error of the distance measured by transducer**

*Figure 3**T1*as a function of the object’s position as indicated by

*x1*. This figure shows that the relative error can be as large as 67 percent!

Note that the relative error in the distance measured by

*T2*has a similar profile. However, this profile is a mirror image of the curve in

**. That is, the relative error of the distance measured by**

*Figure 3**T2*is large when the object is located exactly perpendicular to

*T1*, while the distance measured by

*T2*is correct – if the object is located exactly perpendicular to

*T2*.

*Figure 3: Relative error in the measurement of distance*If the object is located exactly perpendicular to one of the transducers, then the system is able to inform the driver (or the parking control system) the exact distance. However, if the object is located such that the perpendicular line from the vehicle to the object is between the two transducers, errors in the measured distances by both transducers are large.