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Manipulators consist of nearly rigid links, which are interconnected by joints. Each of the joints allows a relative motion between neighboring links. Robots are usually equipped with internal position sensor (e.g. joint encoders) in order to measure the relative position of two neighboring links.
The various robot configurations are based on either prismatic or revolute joints whose position is described by a translation or rotation respectively, measured by a feedback element such as a potentiometer or optical encoder. The servo control systems drive each joint to eliminate the error signal formed by subtracting the measured from the commanded position.
The state of the robot is defined in joint space by the set of values representing each joint position. This may be different from actuator space if gear ratios vary or some drives are coupled. It is more convenient for the user to specify positions in natural coordinate systems such as world coordinates fixed with respect to the ground, or tool coordinates datumed to the gripper flange. In a Cartesian system, 3 translations specify the gripper position and also 3 rotations are needed to describe the gripper orientation, making 6 degrees of freedom in all. More may be needed to specify the state of the gripper itself. Fewer degree of freedom (d.o.f.) are possible when some motions are excluded.
In most cases, robot configurations, poses and motions are described in the robot's base frame, which is meant to be fixed at the location where the robot is attached to the floor. The base frame is usually defined in cartesian space. It is a virtual frame, because it is not related to any mechanical components of the robot. Its origin is inside the robot and therefore it is not accessible for measurement purpose.
The robot's tool frame is fixed to the handling device. Its origin is in the tip of the tool with 3 mutually orthogonal unit vectors representing the tool orientation. Position and orientation of the tool frame is always referred to the base frame.
In robotics, the kinematics descriptions of manipulators and their assigned tasks are used to set up the fundamental equations for joint control.
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Figure 1.1: Planar two-link mechanism |
The fundamental problems of motion description can be introduced with
reference to the planar two-link-mechanism introduced in figure 1.1. The
position of the two links is described via two joint angles
.
It is assumed that the tip of the second (outer) link is expected to move on a
straight path from position 
to
position 
with
a constant velocity
.
The first problem for any given pair of joint angles
is
to find the corresponding 
position
in cartesian space. This is usually called the forward kinematics problem.
Its solution provides a transformation from joint space to cartesian space.
The inverse kinematics problem calculates all possible sets of joint
angles
for
a given cartesian position of the tip.
Its solution is a transformation from cartesian space to joint space.
In the case of the planar mechanism in figure 1, it becomes obvious that two
different joint positions
and
result
in the same cartesian position.
So, two solutions are possible for the inverse kinematics problem, the problem
becomes indefinite.
