**2.1. Classification of motion planning problems**

There still does not exist unified MP algorithm that can robustly solve MP problems in any scenarios such as time optimality, path optimality, moving target, non-holonomic motion, etc. However, with the active recent development of MP, a variety of MP algorithm families are invented to deal with the mentioned scenarios. We will provide detail MP algorithm family classifications based on problem type and therefore demonstrate the location of search-based paradigm in MP.

**Figure 1** describes the family tree of MP algorithms based on problem-type classification.

As can be seen, MP with non-holonomic (velocity and kinodynamic) constraints, which is handled by sampling-based paradigm, is still an open research area due to the hardness of transforming high DOF robot and surroundings into configuration space. This configuration space problem has been proved to be NP hard, and computing configuration space operation has Search-Based Planning and Replanning in Robotics and Autonomous Systems http://dx.doi.org/10.5772/intechopen.71663 65

In general, the problem statement of MP can be generalised as follows: Given the initial defined world space and the robot's configuration space, the MP algorithm must generate a series of consecutive collision-free configurations of the robot that connects start configuration and goal configuration. This series configuration must satisfy any inherent motion or

To cope with a wide range of environment characteristics, MP can be divided into two categories: gross MP and fine MP [7]. The gross MP concerns with the scenarios when world space is much wider than obstacles' size and positional error of the robot, whereas the fine MP solves

This manuscript presents the development of gross MP algorithm family, in particular searchbased planning and replanning paradigm. The foundation concepts of MP, configuration space representations, and the position of mentioned paradigm in MP big picture is presented in Section 2. Section 3 describes historical basis of search-based algorithm family. Section 4 demonstrates the properties and pitfall of D\* Lite, which is one of the most crucial algorithms to plan path in dynamic environment. After that, the variants of D\* Lite, which improve D\* Lite's optimality and performance, are presented. To confirm the improvements, we provide experimental results of recent path planning algorithms and their comparisons in terms of performance and optimality in Section 5. Section 6 will discuss about the future development

This section will provide an overview of the basic elements that every MP problem must involve. These elements are configuration space of robot and obstacles, environment representation, MP method and search method. The mentioned factors must be analysed consecu-

There still does not exist unified MP algorithm that can robustly solve MP problems in any scenarios such as time optimality, path optimality, moving target, non-holonomic motion, etc. However, with the active recent development of MP, a variety of MP algorithm families are invented to deal with the mentioned scenarios. We will provide detail MP algorithm family classifications based on problem type and therefore demonstrate the location of search-based

**Figure 1** describes the family tree of MP algorithms based on problem-type classification.

As can be seen, MP with non-holonomic (velocity and kinodynamic) constraints, which is handled by sampling-based paradigm, is still an open research area due to the hardness of transforming high DOF robot and surroundings into configuration space. This configuration space problem has been proved to be NP hard, and computing configuration space operation has

tively in order to apply suitable MP algorithm family for each scenario.

the planning problems in narrow space that requires high accuracy.

non-motion constraints of the robot.

64 Advanced Path Planning for Mobile Entities

of MP and provide conclusion.

paradigm in MP.

**2. Motion planning concepts**

**2.1. Classification of motion planning problems**

**Figure 1.** Classification of MP algorithm families based on problem type; the deepest leaves of algorithm tree are representatives for their families.

exponential lower bound [7]. Until recently, the mainstream of non-holonomic MP research is developed based on random rapidly exploring random tree planner (RRT). For example, heuristics property of A\* [8] has been applied to RRT for faster trajectory convergence [9]. Fast Marching Square method was developed for non-holonomic car-like robot based on RRT that produces smoother trajectory than RRT [10].

Unlike sampling-based paradigm, search-based paradigm, which represents for path planning algorithms, has a long history of evolution, from basic graph searching to dynamic motion planner with constraints. In this paradigm, robot is treated as point or scalar robot that is able to move in any direction at any time interval. Hence, the configuration obstacle space has the same dimension with the environment, and the generated trajectory is just a path in operating environment. Search-based paradigm is divided into time-invariant and timevariant environment categories. A\* is the representative for time-invariant algorithm family; its cost function is incorporated with heuristic property for faster optimal path planning. When dealing with time-variant problem, although we can ensure the optimality and correctness of path solution, we cannot just rerun A\* from the point that the robot detects changes in environment due to high latency. To efficiently path replanning in dynamic environment, incremental property is combined with heuristic property to develop D\* Lite algorithm; this algorithm is the basis for future development of search-based replanning. Many variants of D\* Lite for different MP problems are presented in **Table 1**.

The development of search-based algorithm family is described detail in Section 3 and Section 4.


hardness of representing orientation dimension and other parameters such as motion constraints on computer. Therefore, a C-space is needed, which incorporates all independent parameters that completely define the position of all points on the robot and specifies global constraints of the robot as Cartesian space. **Figure 2** [17] shows a mapping between an effector of 2DOF robot arm and a set of possible two angle parameters that constitutes C-space

Search-Based Planning and Replanning in Robotics and Autonomous Systems

http://dx.doi.org/10.5772/intechopen.71663

67

After computing the C-space, all MP problems are basically reduced to finding a series of configuration that connects start configuration and goal configuration. In other word, the problem is reduced to finding a path for a point robot from start to goal. The number of parameters that defines robot position is the dimension of C-space. The method to compute

For simplicity, to follow the scope of this chapter, we will treat C-space of point robot the same as world space; the reason is that search-based paradigm deals with holonomic MP

After transforming world space to C-space, we still cannot apply search-based algorithms to C-space. The problem is that search-based algorithms like A\* or D\* Lite work on graphlike structures; hence, applying search-based algorithms on continuous C-space is intractable. However, other MP algorithm families such as sampling based can apply directly to C-space. Unfortunately, the path optimality and performance of sampling-based algorithms are cur-

**Figure 2.** Configuration space of 2DOF robot arm that represents a set of collision-free angles in white and specific object

problem in which the size of robot is neglected compared to operating environment.

of the effector.

C-space is mentioned in [7].

*2.3.2. Continuous to discrete approximation*

collided in colours (a) Workspace, (b) C-space.

rently worse than state-of-the-art search-based algorithms.

**Table 1.** Different families of D\* Lite variants.

#### **2.2. Problem statement formulation**

The general MP problem can be formulated as the following six terms:


General MP is viewed as a search for path (a series of configuration) in state space W that connects start configuration *xinit* to goal configuration region *Xgoal*. The robot is incorporated with a set of global constraints (small discrete headings, velocity, balancing, etc.). We denote *Wfree* as a set of configuration that satisfies global constraints, and the generated path must be in *Wfree*. The incremental rules can be considered as discrete-time response system, and together with input space, it defines possible robot state transitions. Metric can affect heavily to the algorithm's optimality and performance; it indicates the distance between pair of points in topological space. One can construct MP algorithm to deal with specific constraints in certain environment by following these basic terms.

#### **2.3. Environment representation**

This section will describe the transformation of world space to state space. This is the first step to formulate a MP algorithm; it creates an operation environment for MP algorithm and a way to represent physical world information as data structure in computer.

#### *2.3.1. Configuration space (C-space) transformation*

The world space (physical space) is where the robot and obstacles exist; it is a map of the practical world. However, we cannot apply directly MP algorithm to this space due to the hardness of representing orientation dimension and other parameters such as motion constraints on computer. Therefore, a C-space is needed, which incorporates all independent parameters that completely define the position of all points on the robot and specifies global constraints of the robot as Cartesian space. **Figure 2** [17] shows a mapping between an effector of 2DOF robot arm and a set of possible two angle parameters that constitutes C-space of the effector.

After computing the C-space, all MP problems are basically reduced to finding a series of configuration that connects start configuration and goal configuration. In other word, the problem is reduced to finding a path for a point robot from start to goal. The number of parameters that defines robot position is the dimension of C-space. The method to compute C-space is mentioned in [7].

For simplicity, to follow the scope of this chapter, we will treat C-space of point robot the same as world space; the reason is that search-based paradigm deals with holonomic MP problem in which the size of robot is neglected compared to operating environment.
