3.4. Trajectories of mobile robots defined by NURBS

This type of parametric curve is used in the reconstruction of trajectories with the aim of generating smooth paths that approximate the real movement of the robot. In [22], the advantages and disadvantages of the NURBS curves are highlighted, providing a detailed study of their properties. In the field of robotics, the work [23] highlights advantageous properties of NURBS for path planning in both 2D and 3D.

In other works, such as [24–26], NURBS curves approximate or describe the path described by a robot arm PUMA 560. Programming by Demonstration is used to program the behavior of the robot, a good solution to automatically transfer the human knowledge to a robot. However, the NURBS trajectory does not guarantee the obstacle avoidance.

More recently, in [9–12] a predefined NURBS curve is used to improve its properties adjusting the weights.

#### 3.5. Trajectories of mobile robots defined by RBC

In [27], an off-line methodology is presented to approximate a Clothoid (Fresnel integrals) to an RBC. Subsequently, [28] presents a method to obtain trajectories in real time with Clothoids. To do this, two steps are involved: the off-line definition of approximations of Clothoids with RBCs and the generation of online paths by scaling, rotating and moving the previous off-line curves. One of the advantages of this method is the off-line calculation since it considerably reduces the computational time. Throughout the process, the weight coefficients and control points remain invariant. In this work, it is guaranteed that an RBC has the same behavior as a Clothoid using a low order for the curve.

## 3.6. Current trends in the use of parametric curves in robotics

of the angle, some discontinuities could force the vehicle to stop at each control point to adjust

B-splines curves allow an easy construction of smooth paths through control points. In order to avoid obstacles, control points are introduced near them, and methods are developed to move

Earlier methods also worked with splines to generate smooth paths also avoiding the surrounding obstacles [20, 21]. Nevertheless, these previous methods had a high computational cost when evaluating the overall path. In [18], the computational time and the viability of one of these algorithms are analyzed, since it is executed with an iterative method. Monte Carlo simulations indicate a high degree of success for complex environments. The running time is also measured and increases with the complexity of the environment. Finally, in [19], experimental results are provided. The main disadvantage of this algorithm is that the obstacle-free path is computed by means of an iterative method. Thus, the computational time will always

A large number of researchers have also used parametric curves, and particularly B-splines

This type of parametric curve is used in the reconstruction of trajectories with the aim of generating smooth paths that approximate the real movement of the robot. In [22], the advantages and disadvantages of the NURBS curves are highlighted, providing a detailed study of their properties. In the field of robotics, the work [23] highlights advantageous properties of

In other works, such as [24–26], NURBS curves approximate or describe the path described by a robot arm PUMA 560. Programming by Demonstration is used to program the behavior of the robot, a good solution to automatically transfer the human knowledge to a robot. How-

More recently, in [9–12] a predefined NURBS curve is used to improve its properties adjusting

In [27], an off-line methodology is presented to approximate a Clothoid (Fresnel integrals) to an RBC. Subsequently, [28] presents a method to obtain trajectories in real time with Clothoids. To do this, two steps are involved: the off-line definition of approximations of Clothoids with RBCs and the generation of online paths by scaling, rotating and moving the previous off-line curves. One of the advantages of this method is the off-line calculation since it considerably reduces the computational time. Throughout the process, the weight coefficients and control points remain invariant. In this work, it is guaranteed that an RBC has the same behavior as a

ever, the NURBS trajectory does not guarantee the obstacle avoidance.

these control points away from the obstacles and move them to the free space.

increase with respect to other non-iterative methods.

3.4. Trajectories of mobile robots defined by NURBS

and Béziers, to generate search trees as in [7].

NURBS for path planning in both 2D and 3D.

3.5. Trajectories of mobile robots defined by RBC

Clothoid using a low order for the curve.

its direction.

134 Advanced Path Planning for Mobile Entities

the weights.

This comprehensive study of the use of the parametric curves evidences its importance in the design of trajectories of a mobile robot. They are not only used for interpolating points in the global map but also being integrated into global planners and in numerical optimizations. Although non-rational curves have a lower approximation capacity, researchers prefer them for their simplicity and easy manipulation. Among them, we must highlight the Bézier curves, which are the most used.

However, when the parametric curve is used as an approximation, the use of rational curves is significantly greater, as in the approximation to the clothoid and the circle. Recently, predefined rational curves are being used, where only the weights are modified. This can transform rational curves into manageable curves in comparison to non-rational curves.

Along the lines of merging the use of parametric curves with other types of algorithms in an intelligent navigation system, it is not only important to define the path of the robot, but also to avoid obstacles in the environment. Consequently, the initial trajectory must be modified in real time so that the mobile robot avoids the possible dynamic obstacles that may appear. In this sense, the Bézier trajectory deformation (BTD) algorithm, described in the next section, introduces the possibility of deforming a Bézier curve through a vector field, which can be used in mobile robotics. The temporal parameter is introduced in the Bézier curve to transform it into a path and a vector field is needed to modify the initial path.
