Page 305 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

280 MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

• v 1 is a single point, O;
• v 2 is a unit line segment, Oa;
• v 3 is a unit square, Oabc;
• V 1 is a cylinder whose (link) cross section is s 1 and whose length is 2l 1max ;
• V 2 is a slab of length 2l 2max formed by all possible motions of the front and
rear ends of link l 2 within the joint limits of l 1 and l 2 ;
• V 3 is a “cubicle” of height 2l 3max formed by all possible motions of the
front and rear ends of link l 3 within the joint limits of l 1 , l 2 ,and l 3 .

The total volume V W of W-space is hence V W = V 1 ∪ V 2 ∪ V 3 . Out of this,
the set {l}={l ∈ [0,l max ]},where l max = (l 1max ,l 2max ,l 3max ), represents points
reachable by the arm end effector; {l} is a closed set.
An obstacle in W-space, called W-obstacle, presents a set of points, none of
which can be reached by any point of the robot body. This may include some
areas of W-space which are actually free of obstacles but still not reachable
by the arm because of interference with obstacles. Such areas are called the
shadows of the corresponding obstacles. A W-obstacle is thus the sum of volumes
of the corresponding physical obstacle and the shadows it produces. The word
“interference” refers here only to the cases where the arm can apply a force to
the obstacle at the point of contact. For example, if link l 1 in Figure 6.2 happens
to be sliding along an obstacle (which is not so in this example), it cannot apply
any force onto the obstacle, the contact would not preclude the link from the
intended motion, and so it would not constitute an interference. W-obstacles that
correspond to the three physical obstacles—O 1 , O 2 ,and O 3 —of Figure 6.2 are
shown in Figure 6.4.

C-Space, C-Point, and C-Obstacle. The vector of joint variables l =
(l 1 ,l 2 ,l 3 ) forms the robot configuration space (C-space or C). In C-space,
the arm is presented as a single point, called the C-point.The C-space of our
Cartesian arm presents a parallelepiped, or generalized cubicle, and the mapping
4
W → C is unique. For the example of Figure 6.2, the corresponding C-space is
shown in Figure 6.5. For brevity, we will refer to the sides of the C-space cubicle
as its floor (in Figure 6.2 this is the side Oabc), its ceiling (side edgf), and its
walls, the remaining four sides. C-obstacle is the mapping of a W-obstacle into
C. In the algorithm, the planning decisions will be based solely on the fact of con-
tact between the links and obstacles and will never require explicit computation
of positions or geometry of W-obstacles or C-obstacles.
M-Line, M-Plane, and V-Plane. As before, a desired path, called the main line
(M-line), is introduced as a simple curve connecting points S and T (start and
target) in W-space. The M-line presents the path that the arm end effector would
4 In general, the mapping W → C is not unique. In some types of kinematics, such as arm manipulators
with revolute joints, a point in W may correspond to one, two, or even an infinite number of points
in C [107].