1. Introduction

In the last years, intelligent vehicles have increased their capacity up to the point of being able to navigate autonomously in structured environments. Implementations, such as Google [1] (with more than 700,000 hours of autonomous navigation in different scenarios), are an example of the effort made in this area. However, there is still a long way to go until we found real

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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autonomous cars on the roads, as there are both technical and legal problems involved [2, 3]. The intelligent system is composed of three different groups and subgroups: acquisition and perception, decision and actuation-control.

• Numerical optimization is generally used to minimize or maximize a numerical function that depends on different variables such as smoothness, continuity, velocity, acceleration,

Path Planning Based on Parametric Curves http://dx.doi.org/10.5772/intechopen.72574 127

In [3], the use of parametric curves is included in the category of interpolating curve planners. The most commonly used parametric curves in robotics are Béziers, B-splines, rational Bézier curves (RBCs), and non-uniform rational B-splines (NURBSs). A summary of their properties can be low computational cost, intrinsic softness, easy malleability through control points, and universal approximation. For these reasons, parametric curves are not only relevant as interpolators, but also recently they are being used in combination with many other algorithms that

The chapter is organized as follows. Section 2 provides a mathematical definition of the most used parametric curves as well as a description of their properties (Bézier, B-spline, RBC, and NURBS). Section 3 offers a state of the art of the use of parametric curves in robotics and an overview of current trends. Along the lines of the new trends in the use of these curves, Section 5 proposes a method of deformation of parametric curves aimed at modifying the trajectory in

Curves in both space and plane are a part of the geometry necessary to represent certain shapes in different areas. Curves arise in many applications, such as art, industrial design,

real time in order to avoid collisions. Section 6 presents the reader the conclusions.

2.1. Different ways of defining a curve. Advantages and disadvantages

There are different ways of defining a curve: implicit, explicit, and parametric.

The coordinates (x, y) of the points of an implicitly defined plane curve verify that:

The explicit representation of a curve clears one of the variables as a function of the other. In the plane, the coordinates (x,y) of the points in the curve explicitly defined satisfy either.

ð1Þ

ð2Þ

, then the curve must satisfy these two conditions

have effects on all the other blocks of an intelligent system [3].

2. Definitions: parametric curves

mathematics, architecture, engineering, etc.

2.1.1. Implicit and explicit expression of a curve

for some function F. If the curve is in R<sup>3</sup>

simultaneously:

jerk, curvature, etc.

Although the vast majority of the literature often depicted the problems by focusing mainly on these groups or subgroups of processes, functionality in intelligent vehicles, or in mobile robots in general, cannot be conceived as composed of separate blocks, and therefore, a sufficiently efficient system can only be achieved if all the systems work in unison.

This chapter is devoted to the use of parametric curves in the field of robotics. Parametric curves are mainly used in the decision block when the path is defined. However, they are also employed in other blocks, and some of their properties are beneficial for other processes.

Focusing on the decision-making block, the "path planning" or the design of the path to follow has been the subject of study in the last decades, where many authors divide the problem into global and local path planning. On the one hand, the global path planning generates an overall path composed of a set of points to be followed, covering large distances and considering static obstacles in the environment. On the other hand, the local path planning constructs a short path with much more precision, even in continuous form, taking into account unexpected obstacles that may appear.

In general, path planning techniques can be grouped into four large groups: graph search, sampling, interpolating and numerical optimization, see [3]:


• Numerical optimization is generally used to minimize or maximize a numerical function that depends on different variables such as smoothness, continuity, velocity, acceleration, jerk, curvature, etc.

In [3], the use of parametric curves is included in the category of interpolating curve planners. The most commonly used parametric curves in robotics are Béziers, B-splines, rational Bézier curves (RBCs), and non-uniform rational B-splines (NURBSs). A summary of their properties can be low computational cost, intrinsic softness, easy malleability through control points, and universal approximation. For these reasons, parametric curves are not only relevant as interpolators, but also recently they are being used in combination with many other algorithms that have effects on all the other blocks of an intelligent system [3].

The chapter is organized as follows. Section 2 provides a mathematical definition of the most used parametric curves as well as a description of their properties (Bézier, B-spline, RBC, and NURBS). Section 3 offers a state of the art of the use of parametric curves in robotics and an overview of current trends. Along the lines of the new trends in the use of these curves, Section 5 proposes a method of deformation of parametric curves aimed at modifying the trajectory in real time in order to avoid collisions. Section 6 presents the reader the conclusions.
