3.1. Generation of trajectories of mobile robots through parametric curves

Predicting the movement of a robot is important as it implies the computation of a proper path that meets the kinematic and dynamic properties of the robot. Simply moving a mobile robot from an initial position (xi,yi,θi) to a final position (xg,yg,θg) safely implicates many research fields, which are involved in the generation of efficient path planning algorithms.

Many researchers consider parametric curves very useful in the construction of trajectories of wheeled robots, due to their advantageous properties able to improve trajectories produced by path planning techniques.

obstacles. This algorithm is based on the use of seventh-order Bézier curves that connect the vertices of the tree. In this way, the generated paths meet the main kinematic constraint of the vehicle: the smoothness of the acceleration is guaranteed for the entire path by controlling the values of the curvature of the endpoints of each Bézier curve composing the tree. The proposed algorithm provides a rapid convergence to the final result. In addition, the number of vertices of the tree is reduced because the method allows the connections between the vertices of the tree with an unlimited range. The properties of seventh-order Bézier curves are also used to avoid static obstacles in the environment. This method was simulated with a small UAV. Since then, B-splines and Beziers curves have been used to generate search trees by a

Path Planning Based on Parametric Curves http://dx.doi.org/10.5772/intechopen.72574 133

Recent efforts are being made to merge Bézier curves with numerical optimization, [4, 5]. In these works, a teleoperation is carried out where the operator indicates some points. The proposed algorithm calculates the path to continuity of curvature C1 and C2. In [6], something similar is proposed: nodes/points are initially generated between the start and the goal (collision-free) and then are joined by cubic Bézier curves with curvature constraints. Finally, cubic Bézier curves are used in [8] to solve the problem of roundabouts for automated vehicles:

In 1989, B-spline curves were incorporated in the design of robot trajectories. In [13], segments were added with the aim of generating the entire path near the desired one. This new trajectory did not go through the exact points. Later, in 1994, the work in [14] used B-splines for path planning but adding a temporal variable. In this case, the speed of the robot was controlled by the same B-spline. The same year, in [15], a fuzzy controller is designed to emulate spline curves for generating smooth motion trajectories. In 1999, the work [16] also used a B-spline curve to calculate the trajectory of a mobile robot by generating many points from a spline for the robot to follow them in the form of succession. Additionally, in [17], kinematic constraints were introduced in the path planning using B-spline curves to find the optimal temporal

Lately, in 2007, the works [18–20] developed a method to solve the path planning problem using cubic splines to avoid the obstacles. This method iteratively refined the path to be followed in order to obtain in real time a collision-free feasible path in unstructured environments. In [20], the path planning implementation based on B-spline is detailed. The use of splines allows to restrict the polynomials since the first derivative of P1,…, Pn-1 is continuous across the entire boundary. In addition, some constraints can be introduced on the first and last points to force a particular value of the derivative. These characteristics of the splines offer many advantageous properties to plan a suitable path. If a value of the derivative is imposed, a path can be generated starting from a specific position and having a direction imposed by the value of the derivative. Therefore, they can be generated and initialized from the current position and direction of the vehicle. The first derivative is proportional to the direction of the vehicle, then a non-continuous derivative could be obtained and, as a consequence, a nonfeasible path for that type of vehicle. As the second derivative is proportional to the direction

large number of researchers, see [7].

entry, departure, and crossing.

trajectory in a static environment.

3.3. Trajectories of mobile robots defined by B-spline

## 3.2. Trajectories of mobile robots defined by Bézier curves

The first relevant publications in robotics using Bézier curves are published in 1997 and 1998 [29, 30]. These works combine path planning and reactive control for a non-holonomic mobile robot introducing the concept of "Bubble Band" (bubble path). With an appropriate metric, bubbles are connected with Bézier curves, generating a path. These bubbles are the maximum free space reachable in any direction without risk of collision. This is due to the property of the convex hull and implies that if the control points are within the bubble, then the path approximation remains within the bubble. The planner, using a model of the environment, generates an initial path connecting the start and goal positions that may not be adequate. Next, the proposed algorithm generates a sequence of bubbles connecting both ends and replacing the original path, the bubble band. This band is exposed to the forces in the environment, and as a consequence, the band is modified.

In 2001, the concept of "bubble band" is used in [31]. In this case, dynamic obstacles are introduced in the environment. Simultaneously, in [32], also Bézier curves are used for local path planning. An initial path is computed using the generalized Voronoi graph (GVG) theory, which is mildly deformed maximizing the evaluation of a function. Candidates obtained as smooth paths are expressed with Bézier curves.

In 2003, a touchscreen was introduced in [33] to control a mobile robot, avoiding obstacles in real time. In this work, two algorithms are developed: the first one extracts a succession of important points, and the second one generates a path using cubic Bézier curves.

In 2007, the work in [34] introduces Bézier curves in cooperative collision avoidance for several mobile non-holonomic robots and is based on the previous contributions [35, 36]. Two tasks are developed: first, path planning based on Bézier curves for each individual robot in order to obtain its final position and, second, computation of an optimal path that minimizes a "penalty" function that accounts for the sum of the maximum times subject to the distances between the robots.

In 2008, [37] presents a preliminary framework that generates space trajectories for multiple unmanned aerial vehicles (UAVs) using 3D Bézier curves. The algorithm solves a constrained optimization problem in order to generate the trajectories. In this case, the optimization function penalizes an excessive length, as the shortest path is required, and the restrictions are the distances between the multiple UAVs. The system is non-linear, and numerical methods are applied to solve it.

It is worth mentioning the work of Choi et al. [38–46] related to the computation of trajectories of mobile robots designed from Bézier curves. In many of the publications, a constrained optimization problem is raised, where the function to be optimized is the curvature of a Bézier curve.

Finally, [47] presents a methodology based on the variation of the RRT that generates suitable trajectories for autonomous vehicles with holonomic constraints in environments with obstacles. This algorithm is based on the use of seventh-order Bézier curves that connect the vertices of the tree. In this way, the generated paths meet the main kinematic constraint of the vehicle: the smoothness of the acceleration is guaranteed for the entire path by controlling the values of the curvature of the endpoints of each Bézier curve composing the tree. The proposed algorithm provides a rapid convergence to the final result. In addition, the number of vertices of the tree is reduced because the method allows the connections between the vertices of the tree with an unlimited range. The properties of seventh-order Bézier curves are also used to avoid static obstacles in the environment. This method was simulated with a small UAV. Since then, B-splines and Beziers curves have been used to generate search trees by a large number of researchers, see [7].

Recent efforts are being made to merge Bézier curves with numerical optimization, [4, 5]. In these works, a teleoperation is carried out where the operator indicates some points. The proposed algorithm calculates the path to continuity of curvature C1 and C2. In [6], something similar is proposed: nodes/points are initially generated between the start and the goal (collision-free) and then are joined by cubic Bézier curves with curvature constraints. Finally, cubic Bézier curves are used in [8] to solve the problem of roundabouts for automated vehicles: entry, departure, and crossing.

### 3.3. Trajectories of mobile robots defined by B-spline

Many researchers consider parametric curves very useful in the construction of trajectories of wheeled robots, due to their advantageous properties able to improve trajectories produced by

The first relevant publications in robotics using Bézier curves are published in 1997 and 1998 [29, 30]. These works combine path planning and reactive control for a non-holonomic mobile robot introducing the concept of "Bubble Band" (bubble path). With an appropriate metric, bubbles are connected with Bézier curves, generating a path. These bubbles are the maximum free space reachable in any direction without risk of collision. This is due to the property of the convex hull and implies that if the control points are within the bubble, then the path approximation remains within the bubble. The planner, using a model of the environment, generates an initial path connecting the start and goal positions that may not be adequate. Next, the proposed algorithm generates a sequence of bubbles connecting both ends and replacing the original path, the bubble band. This band is exposed to the forces in the environment, and as a

In 2001, the concept of "bubble band" is used in [31]. In this case, dynamic obstacles are introduced in the environment. Simultaneously, in [32], also Bézier curves are used for local path planning. An initial path is computed using the generalized Voronoi graph (GVG) theory, which is mildly deformed maximizing the evaluation of a function. Candidates obtained as

In 2003, a touchscreen was introduced in [33] to control a mobile robot, avoiding obstacles in real time. In this work, two algorithms are developed: the first one extracts a succession of

In 2007, the work in [34] introduces Bézier curves in cooperative collision avoidance for several mobile non-holonomic robots and is based on the previous contributions [35, 36]. Two tasks are developed: first, path planning based on Bézier curves for each individual robot in order to obtain its final position and, second, computation of an optimal path that minimizes a "penalty" function that accounts for the sum of the maximum times subject to the distances between the robots.

In 2008, [37] presents a preliminary framework that generates space trajectories for multiple unmanned aerial vehicles (UAVs) using 3D Bézier curves. The algorithm solves a constrained optimization problem in order to generate the trajectories. In this case, the optimization function penalizes an excessive length, as the shortest path is required, and the restrictions are the distances between the multiple UAVs. The system is non-linear, and numerical

It is worth mentioning the work of Choi et al. [38–46] related to the computation of trajectories of mobile robots designed from Bézier curves. In many of the publications, a constrained optimization problem is raised, where the function to be optimized is the curvature of a Bézier curve. Finally, [47] presents a methodology based on the variation of the RRT that generates suitable trajectories for autonomous vehicles with holonomic constraints in environments with

important points, and the second one generates a path using cubic Bézier curves.

path planning techniques.

132 Advanced Path Planning for Mobile Entities

consequence, the band is modified.

methods are applied to solve it.

smooth paths are expressed with Bézier curves.

3.2. Trajectories of mobile robots defined by Bézier curves

In 1989, B-spline curves were incorporated in the design of robot trajectories. In [13], segments were added with the aim of generating the entire path near the desired one. This new trajectory did not go through the exact points. Later, in 1994, the work in [14] used B-splines for path planning but adding a temporal variable. In this case, the speed of the robot was controlled by the same B-spline. The same year, in [15], a fuzzy controller is designed to emulate spline curves for generating smooth motion trajectories. In 1999, the work [16] also used a B-spline curve to calculate the trajectory of a mobile robot by generating many points from a spline for the robot to follow them in the form of succession. Additionally, in [17], kinematic constraints were introduced in the path planning using B-spline curves to find the optimal temporal trajectory in a static environment.

Lately, in 2007, the works [18–20] developed a method to solve the path planning problem using cubic splines to avoid the obstacles. This method iteratively refined the path to be followed in order to obtain in real time a collision-free feasible path in unstructured environments. In [20], the path planning implementation based on B-spline is detailed. The use of splines allows to restrict the polynomials since the first derivative of P1,…, Pn-1 is continuous across the entire boundary. In addition, some constraints can be introduced on the first and last points to force a particular value of the derivative. These characteristics of the splines offer many advantageous properties to plan a suitable path. If a value of the derivative is imposed, a path can be generated starting from a specific position and having a direction imposed by the value of the derivative. Therefore, they can be generated and initialized from the current position and direction of the vehicle. The first derivative is proportional to the direction of the vehicle, then a non-continuous derivative could be obtained and, as a consequence, a nonfeasible path for that type of vehicle. As the second derivative is proportional to the direction of the angle, some discontinuities could force the vehicle to stop at each control point to adjust its direction.

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

which are the most used.

of dynamic obstacles.

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,

Path Planning Based on Parametric Curves http://dx.doi.org/10.5772/intechopen.72574 135

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 trans-

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

form rational curves into manageable curves in comparison to non-rational curves.

4. Properties of parametric curves and its applications in robotics

In mobile robotics, two main needs have arisen when dealing with path planning of a mobile robot: definition of the initial path to follow and the possibility of modifying it in the presence

In the next paragraphs, the BTD algorithm is described [48, 49], which solves the abovementioned needs. It offers the possibility of defining the trajectory of a mobile robot through a Bézier curve and then modifies it by means of the repulsive forces derived from a predictive potential field (PF) method. Reactive methods or potential field methods generate obstacle-free paths for the robot. In these methods, the movement of the robot is determined by repulsive forces associated with obstacles and attractive forces associated to the goal position of the mobile robot. In this work,

The set of discrete points provided by the posture prediction of the mobile robot is considered as initial points Si of the original Bézier curve. These points belong to a reference path in the BTD algorithm. Subsequently, the set of repulsive forces obtained by the PFP is transformed into displacements by a dynamic particle model, which generates endpoints Ti that determines the modification of the original Bézier trajectory with the BTD. A modified Bezier trajectory free of obstacles is obtained that passes through the endpoints, as displayed in Figure 2.

it into a path and a vector field is needed to modify the initial path.

the potential field projection method (PFP) has been used [50, 51].

The definition of the BTD algorithm requires two steps:

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

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 increase with respect to other non-iterative methods.

A large number of researchers have also used parametric curves, and particularly B-splines and Béziers, to generate search trees as in [7].
