and cosines of the angles. In this section, we describe
some of the techniques that can be used for this task.
Certain simple arm geometries allow a closed form
solution. For such arms, one of the equations will have the form
aS+bC=d
where S and C are the sine and cosine, respectively, of some angle a. If the constants a, b, and d are known, then there are two possible solutions when a2 + b2 d2
e = atan2[d,1(a2+b2-d2)] atan2(b,a)
where atan2(v,w) returns the angle arctan(v/w) adjusted to the proper quadrant according to the sign of the real numbers v and w.
A special case occurs when a = 0 or b = 0. The
equation can then be solved for S or C separately. The
other variable can be obtained from the Pythagorean identity
S2 + C2. .
S +C= 1 (4. 17)
with a sign ambiguity. Again, this leads to two possible values for the angle 8,
e = atan2(S, 1(1 S2)) if S is computed or
8 = atan2( /(l C2), C) if C is the known
variable.
A value of 0 can be directly and uniquely obtained when two linear equations in the sine and cosines of one angle
..