Page 349 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

324 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

where P c and P m are respectively the conventional and minimal projections as in
Definition 6.3.7 and Definition 6.3.8.

In other words, V ab contains both a and b andisparallel tothe l 3 axis. The
degenerate case where ab is parallel to the l 3 axis is simple to handle and is not
considered.

Definition 6.3.15. Let L s and L t be the given start and target configurations of

the arm, and let S ={j ∈ J|L(j) = L s }⊂ J and T ={j ∈ J|L(j) = L t }⊂ J,
respectively, be the sets of points corresponding to L s and L t .Let f : J → C be
the projection as in Definition 6.3.12. Then the vertical surface V ⊂ C is defined as

V ={f(j) ∈ C|j ∈ V st for all s ∈ S and t ∈ T}
For the RRP arm, which is the most general case among XXP arms, V consists
of four components V i , i = 1, 2, 3, 4. Each V i represents a class of vertical planes
in J and can be determined by the first two coordinates of a pair of points drawn
s
t
s
s
t
t
respectively from S and T.If j s = (θ ,θ ,l ) and j t = (θ ,θ ,l ) are the robot’s
1
2
1
2
3
3
start and target configurations, the four components of the vertical surface V can
be represented as follows:
s
t
s
t
V 1 : (θ ,θ )(θ ,θ )
1 2 1 2
s
t
s
t
t
s
V 2 : (θ ,θ )(θ ,θ − 2π × sign(θ − θ )) (6.7)
1 2 1 2 2 2
s
s
t
s
t
t
V 3 : (θ ,θ )) (θ − 2π × sign(θ − θ ), θ )
1 2 1 1 1 2
s
t
t
s
s
s
t
t
V 4 : (θ ,θ )(θ − 2π × sign(θ − θ ), θ − 2π × sign(θ − θ ))
1 2 1 1 1 2 2 2
where sign() takes the values +1or −1 depending on its argument. Each of the
components of V -surface determines a generic path, as follows:

g i = V i ∩ B, i = 1, 2, 3, 4
Since B is homeomorphic to a torus, any three of the four generic paths can be
3
!
used to form a connectivity graph. Without loss of generality, let g = g i
i=1

and denote g = B f ∩ g.A connectivity graph can be defined as G = g ∪ ∂B f .
If there exists a path in C f , then at least one such path can be found in the
connectivity graph G.
Now we give a physical interpretation of the connectivity graph G; G consists
of the following curves:
• ∂C p —the boundary curve of the floor, identified by the fact that the third
link of the robot reaches its lower joint limit (l 3 = 0) and, simultaneously,
one or both of the other two links reach their joint limits.