Math 210--Linear Algebra: Project 3, Spring 1999

A bit of a reach...

by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, April 1999

permission granted to use and distribute free in an academic setting.

National Space Agency

the Heptagon
Tonwashing, CD 012000

19 April 1999

Linear Mathematics, Inc.
Lonlinc, SK 04685

Dear LMI:

As you know, we are here at the National Space Agency in the process of completing plans for the new international space station Omfreed. An integral part of the station as envisioned by some designers is an articulated arm, shown in figure 1, for the manipulation of loose space station parts and other items such as scientific experiments. As this is controlled remotely from within the space station using the limited visual clues provided by video monitoring cameras, it is essential that we be able to predict, based on known current locations, the position of the ``elbow'' and ``hand'' of the arm after an extension through some specified set of angles theta1, theta2 and phi (as shown in figure 1). In addition, we need to know what set of points in space the arm will be able to reach (each of the joints is sufficiently flexible as to admit motions through a range of angles 0<=theta1, theta2, phi <= pi radians. The current arm design has the two articulations shown, which are of equal length.

Table 1: series of movement commands
 angle initial move 1 move 2 move 3 move 4 move 5 move 6 move 7 theta1 0 0 +pi/5 +pi/10 -pi/15 +pi/20 0 0 theta2 0 +pi/6 +pi/6 -pi/8 +pi/16 0 -pi/25 +pi/30 phi 0 0 0 +pi/6 +pi/15 +pi/15 +pi/20 -pi/25

Our preliminary consultation with the ethereal Dr. P. Gavin LaRose (whose rates, we might add, are as astronomical as our goals) suggested that it should be possible to develop a mathematical formulation that will allow the new position of the ``elbow'' and ``hand'' to be calculated from any known positions. We expect this to allow the determination of these new positions after arbitrary changes in the angles theta1, theta2, and phi and after arbitrary numbers of those changes. We would like your company to develop such a formulation and demonstrate its application to the sample series of movements given in table 1, obtaining the desired positions after each movement.

In addition, we would like you to use this formulation to develop the set of all possible positions of the ``elbow'' and ``hand.'' We need your final, written report on this by 7 May, and to facilitate its completion have arranged for you to be able to consult with Dr. LaRose at any time between now and 5 May should you find that you have questions on your work. Please note, however, that you must have established contact with him by 28 April to take advantage of such consultation.

Yours most sincerely
John G. Lenn