Page 344 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

THREE-LINK XXP ARM MANIPULATORS 319

1, 2, 3,where % is the modular operation. The quotient space C = J/F is called
the configuration space (C-space), with normal quotient space topology assigned,
see Ref. 57, A.1. Let c = Fj represent an equivalence class; then the project
f : J → C is given by f(j) = Fj.

Theorem 6.3.7. The configuration space C is compact and of finite volume (area).

By definition, J = J 1 × J 2 × J 3 . Define equivalence relations F i in J i such

that j i F i j if and only if (j i − j )%2π = 0. Define C i = J i /F i and the project

i i
1
f i : J i → C i given by f i (j) = F i j. Apparently, C i = S with length v i = 2π if
∼
1
1
∼
J i = ,and C i = I with length v i = 1if J i = I . Because f is the product of
f i ’s, f i ’s are both open and closed, and the product topology and the quotient
topology on C 1 × C 2 × C 3 are the same (see Ref. 57, Proposition A.3.1); therefore,
∼
C = C 1 × C 2 × C 3 is of finite volume v 1 · v 2 · v n .
1
1
2
For an RR arm, for example, C = S × S with area 2π · 2π = 4π ;for an
∼
1
1
1
2
∼
RRP arm, C = S × S × I with volume 2π · 2π · 1 = 4π .
−1
For c ∈ C,we define L(c) = L(j),where j ∈ f (c), to be the area the robot
arm occupies in W when its joint vector is j.
Definition 6.3.13. The configuration space obstacle (C-obstacle) is defined as

O C ={c ∈ C : L(c) ∩ O =∅}

The free C-space is C f = C − O C .
The proof for the following theorem and its corollary are analogous to those for
Theorem 6.3.1.
Theorem 6.3.8. A C-obstacle is an open set.
Corollary 6.3.3. The free C-space C f is a closed set.
The configuration space obstacle O C may have more than one component.
For convenience, we may call each component an obstacle.
Theorem 6.3.9. Let L(c s ) = L s and L(c t ) = L t . If there exists a collision-free
path (motion) between L s and L t in W, then there is a path p C ⊂ C f connecting
c s and c t , and vice versa.

If there exists a motion between L s and L t in W, then there must be a path
p J ⊂ J f between two points j s ,j t ∈ J f such that L(j s ) = L s and L ( j t ) = L t .
Then, p C = f(p J ) ⊂ C f is a path between c s = f(j s ) and c t = f(j t ). The other
half of the theorem follows directly from the definition of C f .