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T
T
F
W
F
W
FW
FW
Figure 2.3. Two different configurations that place FT (the gripper) in the same
position with the same orientation relative FW .
represented by a point in C. From here on it is assumed that a goal configuration
of A is given by the location and orientation of FT relative FW , ignoring the actual
configuration of the rigid bodies that make up A. Why this assumption is valid will
be described later.
For every point qc ∈ C there exists a unique mapping qc → qw ∈ R6 that
gives the location and orientation of FT relative to FW . This mapping is called the
forward-kinematics (FK) of A and there exists a unique solution for all points in C
for any physical agent. The inverse mapping qw → qc ∈ C is called the inversekinematics (IK) and is in general not trivial. In general, unique solutions to the IK
mapping do not exist and the solutions may give rise to singularities. From here on
it is assumed that the FK of A is known and the IK is not.
Consider the manipulator in figure 2.3 and note that the gripper is positioned
in the same way in both the left and right figures. If the manipulator’s task was
to pick up an object located at the origin of the coordinate-system denoted by FT
both configurations would be acceptable. This is often the case in path planning, a
frame FT is attached to some specific point on the agent and the task of the path
planner is to put FT at a specific location and in a specific orientation relative to
the a fixed frame FW in the world. Figure 2.3 shows the relation between FT and
FW . Throughout this thesis a goal configuration in W , denoted qw
goal , is a vector
specifying the position and orientation of FT relative to FW , thus dim(qw
goal ) ≤ 6.
w
For an agent consisting of a single rigid body a configuration q in W completely
specifies the closed and bounded region occupied by the agent in the workspace W .
Let Wf ree ⊂ W denote the subset of W in which the agent may place a part of
its body, i.e. the part of W not occupied by obstacles. The task of the path planner
is to find a feasible and unobstructed path in C such that agent moves from a given
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