Page 275 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

250 MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS

length, α i is the link twist, θ i is the angle between links L i and L i−1 ,and l i is
the distance between links L i and L i−1 , i = 1,... ,d [8].
Assume that a revolute joint has no joint limit, and a translational (prismatic)
joint has its lower and upper limits. Let θ i denote a revolute joint variable, let
l i be a translational joint variable, and let j i be a (revolute or translational)
joint variable when the exact type is not important; i = 1,. ..,d. Assume for
1
1
simplicity, and without loss of generality, that l i ∈ I and θ i ∈ S (a 1-circle).
1
The configuration space (CS) is defined as C = C 1 ×· · · × C n ,where C i = I if
1
the ith joint is translational and C i = S if it is revolute. In all combinations of
cases, C = I d t × S d r for all d-DOF robots, where d t and d r are respectively the
∼
numbers of independent translational and revolute joints, d t + d r = d.
Lemma 5.8.1. CS is compact and is of finite volume (area).

Proof: The compactness is obvious since, by definition, CS is the cross product
of a finite number of unit intervals (length 1) and circles (length 2π). The volume
d r
of CS is (2π) . Q.E.D.
1
For example, for a robot with two revolute joints, C = S × S 1 with area
∼
2
2π · 2π = 4π ; for a robot with two revolute joints and one prismatic joint,
1
2
1
1
∼
C = S × S × I with volume 2π · 2π · 1 = 4π .
We define the 3D robot workspace (denoted by WS or W) as follows (its 2D
3
2
counterpart can be defined accordingly by replacing with ):
Definition 5.8.1. A robot link L i , i = 1,...,d, is defined as the interior of a
3
connected and compact subset of homeomorphic to an open ball; for any
3
3
point x ∈ L i ,let x(j) ∈ be the point that x would occupy in when the joint
! 3
vector of the robot is j ∈ C.Let L i (j) = x(j). Then, L i (j) ⊂ is a set
x∈L i
of points the ith link occupies when the robot’s joint vector is j ∈ C. Similarly,
! n 3
L(j) = L i (j) ⊂ is a set of points the whole robot occupies when its joint
i
vector is j ∈ C. The workspace is defined as
"
W = L(j)
j∈C
where L(j) is the closure of L(j).
We assume that L i has a finite volume; thus, W is bounded.
The robot workspace may contain obstacles; each obstacle is a rigid body of
an arbitrary shape. In the 2D case, an obstacle is of finite area and its boundary
presents a simple closed curve. In the 3D case, an obstacle has a finite volume, its
surface has a finite area, and it presents one or more orientable 2D manifolds. The
assumption that WS has a finite volume (area) implies that the number of obstacles
present in WS must be finite. Being rigid bodies, obstacles cannot intersect. We
define 3D obstacles as follows (2D obstacles are defined accordingly):