Page 43 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

18 MOTION PLANNING—INTRODUCTION

When we develop our motion planning algorithms based on tactile sensing, this
does not mean we suggest tactile sensing as a preferable sensing media, nor does
it mean that the algorithms are applicable solely to tactile sensing. As we will
see, expanding algorithms to more complex sensing is usually relatively easy, and
usually results in higher efficiency.

1.2.4 Degrees of Freedom. Coordinate Systems

It is known from mechanics that depending on space dimensionality and object
complexity, there is a minimum number of independent variables one needs to
define the object’s position and orientation in a unique way. These variables are
called the object’s degrees of freedom (DOF). The reference (coordinate) system
expressed in terms of object’s DOF is called the configuration space (C-space).
C-space is hence a special representation of the robot workspace (W-space).
From a textbook on mechanics, the minimum number of DOF that a rigid body
needs for an arbitrary motion is

In 2D, if only translation is allowed: 2
In 2D, translation plus orientation allowed: 3
In 3D, if only translation is allowed: 3
In 3D, translation plus orientation allowed: 6

For example, for a planar (2D) case with a rigid object free to translate and
rotate, the object is defined by three DOF (x, y, θ): two Cartesian coordinates
(x, y) that define the object’s position, plus its orientation angle θ.
A robot arm manipulator’s DOF also determine its ability to move around.
Specific values of all robot’s DOF signify the specific arm configuration of its
links and joints. Shown in Figure 1.2a is a revolute planar (2D) arm with two
links. Its two DOF, two rotation angles, allow an arbitrary position of its endpoint
in the robot workspace, but not an arbitrary orientation. The 3-link 3-DOF planar
arm manipulator shown in Figure 1.2b can provide an arbitrary position and an
arbitrary orientation at its endpoint.
The DOF that a robot arm possesses are usually realized via independent
control means, such as actuators (motors), located in the arm’s joints. Joints
connect together the arm’s links. Links and joints can be designed in different
configurations: The most common are the sequential linkage, which is similar to
the kinematics of a human arm, like in Figure 1.2, and the parallel linkage,where
links form a parallel structure. The latter is used in some spatial applications,
such as a universal positioner for various platforms. We will be interested in
only sequential linkages.
The most popular types of joints are revolute joints, where one link rotates
relative to the other (like in the human elbow joint), and sliding joints (also called
prismatic joints), where one link slides relative to the other. The arm shown in
Figure 1.2a has two revolute joints, of which the first joint is located in the arm’s
fixed base. The freely moving distal link or links on a typical arm manipulator
is called the end effector. The end effector can carry a tool for doing the robot’s