## Controlling a Robot

Robotic devices are often used where reasons of safety or convenience make it impractical to rely on humans. To estimate its location, a robot measures the angles to various known landmarks, and then uses this information with a built-in computer program to calculate its likely position. Since the information available to the robot is often somewhat uncertain (a common circumstance known informally as "noisy data," often due to imprecision in measurements), the robot's position can be identified only within a small region, not at an exact location.

For example, suppose a robot is moving over a flat field to find land mines left over from a war zone. Three distant trees are used as the known landmarks, forming a triangle whose base is 2520 feet long with base angles of 65 deg 30' and 52 deg 40' with the third tree. The field to be searched is approximately rectangular, situated mostly within the triangle formed by the three trees but with some area extending outside the triangle. The sensors on the robot can determine angles to the known landmarks with an accuracy of plus or minus 2 degrees, but they have no way of measuring distance. How accurately can the robot determine its position when it is near the center of the triangle formed by the trees, when it is near one of the trees, and when it is outside the triangle? In which part of the field will the robot's calculations pinpoint its position with greatest accuracy?

A fourth tree is located outside the original triangle at a position that forms angles of 38 deg 50' and 95 deg 10' with the two base trees. If this information were added to the robot's "known landmark" data file, how much more accurate would that make its estimation of position?

Comment: This is a very complex problem using routine trigonometry, but with realistic (not oversimplified) data. The narrative form of the question requires a narrative type of answer since it asks for general statements, not just specific calculations. The problem requires some amount of interpretation and judgment, and gets into common yet difficult areas (due to over-determined and possibly contradictory data) about which students may not have had any experience. A computer (even just a spreadsheet) could be used to help organize the calculations, as could a system of complex simultaneous equations. (In real situations, this problem has a tougher twist--not just to find the data, but to do it so quickly that the robot will instantly know its current position. This is a more advanced question about computer algorithms that requires linear algebra.)

Excerpted from "Mathematics for Work and Life" by Susan L. Forman and Lynn Arthur Steen (in Seventy-Five Years of Progress: Prospects for School Mathematics, NCTM, 1995).