Page 45 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

20 MOTION PLANNING—INTRODUCTION

Cartesian coordinates are (x, y, θ 3 ), and its configuration space coordinates are
(θ 1 ,θ 2 ,θ 3 ).
Typically, the user is interested in defining the robot’s paths in terms of Carte-
sian coordinates. Robot’s motors are controlled, however, in terms of joint values
(that is, configuration space coordinates). Hence a standard task in robot motion
planning and control is translation from one reference system to the other. This
process gives rise to two problems: (a) direct kinematics—given joint values, find
the corresponding Cartesian coordinates—and (b) inverse kinematics—given the
robot’s Cartesian coordinates, find the corresponding joint values. As we will see
in the next chapter, calculation of inverse kinematics is usually significantly more
difficult than the calculation of direct kinematics.

1.2.5 Motion Control
The robot’s path is a curve that the robot’s end effector (or possibly its some
other part) moves along in the robot workspace. To be physically realizable, each
point of the path must be associated with the joint values that fully describe the
robot position and orientation in the respective configuration. The term trajec-
tory is used sometimes to designate a path geometry plus timing, velocity, and
acceleration information along the path. 3
As used in robotics, the term motion control or motion control system refers to
the lower-level control functions, such as algorithmic and electronic and mechan-
ical means that direct individual motors, as opposed to motion planning,which
signifies the upper-level control—that is, control that requires some intelligence.
This is not to say that motion control is a simple matter—robot controllers are
often quite sophisticated. The control means are used to realize a given path or
trajectory with required fidelity. While control means are beyond the scope of
this book, for completeness we will review them briefly in the next chapter.
Depending on the number of DOF available for motion planning, we distin-
guish between three types of systems:

• Holonomic Systems. These have enough DOF for an arbitrary motion. The
minimum number of those is equal to the dimensionality of the correspond-
ing C-space: For example, 6 is the minimum number of DOF a 3D arm
manipulator needs to realize an arbitrary motion in space without obstacles.
• Nonholonomic Systems. These are systems with constraints on their motion.
For example, a car is a nonholonomic system: with its 2-DOF con-
trol—forward motion and steering—it cannot execute a lateral motion; this
creates a well-known difficulty in parallel car parking. Note that a car’s
C-space is 3D, with its axes being two position variables (x, y) plus the
orientation angle.
• Redundant Systems. Those with the number of DOF well above the mini-
mum necessary for holonomic motion. Humans, animals, and some complex
robots present redundant systems.

3 In some books, and also here, terms “trajectory” and “path” are used interchangeably.