Page 335 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

310 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

∗
Let j ∈ O J . By Definition 6.3.5, there exists a point x ∈ L such that y =
∗
x(j ) ∈ O.Since O is an open set (Definition 6.3.3), there must exist an > 0
such that the neighborhood U(y, ) ⊂ O. On the other hand, since x(j) is a con-
6
tinuous function from J to W, there exists a δ> 0 such that for all j ∈ U(j ,δ),
∗
we have x(j) ∈ U(y, ) ⊂ O; thus, U(j ,δ) ⊂ O J ,and O J is an open set.
∗

The free J-space is J f = J − O J . Theorem 6.3.1 gives rise to this corollary:
Corollary 6.3.1. J f is a closed set.
Being a closed set, J f = J f . Thus, a collision-free path can pass through
∂J f .
When the arm kinematics contains a revolute joint, due to the 2π repe-
tition in the joint position it may happen that L(j) = L(j ) for j = j .For

an RR arm, for example, given two robot arm configurations, L s and L t ,in
= (2k 1 π + θ 1s , 2k 2 π + θ 2s ) ∈ J,k 1 ,k 2 =
) = L s 0,0 = L s for j s k 1 ,k 2
W, L(j s k 1 ,k 2
= (2k 3 π + θ 1t , 2k 4 π
) = L t 0,0 = L t for j t k 3 ,k 4
0, ±1, ±2,... . Similarly, L(j t k 3 ,k 4
+ θ 2t ) ∈ J,k 3 ,k 4 = 0, ±1, ±2,... . This relationship reflects the simple fact that
in W every link comes to exactly the same position with the periodicity 2π.In
physical space this is the same position, but in J-space these are different points. 7
The result is the tiling of space by tiles of size 2π. Figure 6.14 illustrates this
situation in the plane. We can therefore state the motion planning task in J-space
as follows:
Given two robot arm configurations in W, L s and L t ,let sets S s ={j ∈ J : L(j) =
L s } and S t ={j ∈ J : L(j) = L t } contain all the J-space points that correspond to
L s and L t respectively. The task of motion planning is to generate a path p J ⊂ J f
between j s and j t for any j s ∈ S s and any j t ∈ S t or, otherwise, conclude that no
path exists if such is the case.

The motion planning problem has thus been reduced to one of moving a
point in J-space. Consider the two-dimensional RR arm shown in Figure 6.14a;
shown also is an obstacle O 1 in the robot workspace, along with the arm start-
ing and target positions, S and T . Because of obstacle O 1 , no motion from
position S to position T is possible in the “usual” sector of angles [0, 2π]. In J-
space (Figure 6.14b), this motion would correspond to the straight line between
points s 0,0 and t 0,0 in the square [0, 2π]; obstacle O 1 appears as multiple vertical
columns with the periodicity (0, 2π).
However, if no path can be found between a specific pair of positions j s and j t
in J-space, it does not mean that no paths between S and T exist. There may be
(Figure 6.14b).
paths between other pairs, such as between positions j s 0,0 and j t 1,0
On the other hand, finding a collision-free path by considering all pairs of j s and
6 If x ∈ L is the arm endpoint, then x(j) is the forward kinematics and is thus continuous.
7 In fact, in those real-world arm manipulators that allow unlimited movement of their revolute joints,
going over the 2π angle may sometimes be essential for collision avoidance.