
340 Chapter 5 Controller-Based Animation
goal to the initial point of the manipulator is larger than the sum of the lengths of the
segments. Even if the goal is attainable, there may be multiple solutions. Thus, the
inverse problem is generally ill-posed.
Obtaining a closed-form representation of the joint angles in terms of G, P
0
, and
the L
i
is a hard problem. To show the complexity, consider the case of two segments.
TheequationtobeinvertedisG = P
0
+ L
0
U
0
+ L
1
U
1
,whereU
0
= (cos θ
0
, sin θ
0
)
and U
1
= (cos(θ
0
+ θ
1
), sin(θ
0
+ θ
1
)). Define (a, b)
⊥
= (−b, a). Using the double-
angle identities from trigonometry, U
1
= (cos θ
1
)U
0
+ (sin θ
1
)U
⊥
0
. The equation to
be solved ...