On the Length of Dubins Path with Any Initial and Terminal Configurations

Dubins has proved in 1957 that the minimum length path between an initial and a terminal configuration can be found among the six paths {LSL, RSR, LSR, RSL, RLR, LRL}. Skel and Lumelsky have studied the length of Dubins path with the initial configuration (0, 0; α) and the terminal configuration (d, 0; β) and the minimal turning radius ρ=1 in 2001. We extended the Skel and Lumelsky’s results to the case that the initial and terminal configuration is 0 0 ( , ; ) x y α , 1 1 ( , ; ) x y β , respectively (where 0 0 1 1 , , , x y x y ∈ R ), and the minimal turning radius is 0 ρ > .


Introduction
The problem of finding the shortest smooth path between two configurations in the plane appear in various applications, such as when joining pieces of railways [1] or planning two and three-dimensional pipe networks. In robotics, this problem plays a central role in most of the work on nonholonomic motion planning [2][3][4] . In unmanned aerial vehicle (UAV), the kinematics of the UAV can be approximated by the Dubins vehicle, too [5] . The Dubins distance in the path planning for UAV is needed to be computed [5][6][7] . Let the initial and terminal configurations is ( , ) x y is the position, α and β is the orientation angle (heading). The task is to find the shortest smooth path from 0 0 ( , ; ) x y α to 1 1 ( , ; ) x y β such that the path curvature is limited by 1 / ρ , where ρ is the minimal turning radius. The problem of finding the shortest smooth path between two configurations in the plane was firstly considered by Dubins [8] . The classical 1957 result by Dubins gives a sufficient set of paths (Dubins path)  , where L and R are arc of the minimal allowed radius ρ turning left or turning right, respectively, S is a line segment.
Shkel and Lumelsky have considered the length of the six Dubins path in 2001 [9] with the initial configuration (0, 0; ) α , the terminal configuration ( , 0; ) d β and the minimal turning radius 1 ρ = . Also they studied the logical classification scheme which allows one to extract the shortest path from the Dubins set directly, without explicitly calculating the candidate paths for 'long path case' and 'short path case'.
In this paper, we studied the length of Dubins path with initial configuration 0 0 ( , ; ) x y α and terminal configuration 1 1 ( , ; ) x y β , and the minimal turning radius 0 ρ > . Although we can convert this situation into [9] by coordinate transformation. It is still worthy of study. For example, in order to find out the optimal tour of DTSP (Dubins Traveling Salesman Problem, which has been attracted a lot of attention [5,[10][11][12][13][14][15][16] for it is a useful abstractions for the study of problems related to motion planning and task assignment for uninhabited vehicles. ), we need to do n different coordinate transformations and this will waste computing time.

The Length of Dubins Path with =
The length of the Dubins path is calculated as l t p q = + + . So, the value of , , t p q is the key problem. To get the value of , , t p q , we will now consider elements of Dubins set one-by-one and derive the operator equations for the length of each path.
x y and 1 1 ( , ) x y in the plane. Let ρ be the minimal radius. Firstly, let 1 ρ = and 1 ρ ≠ will be discussed later in Theorem 1. 1.
. By applying the corresponding operators (1), we have the following equations: The solutions of this system with respect to the segments , t p and q is found as The length of LSL as a function of the boundary conditions can be now written as LSL l t p q = + + , where , , t p q are calculated by (2). 2.
By applying the corresponding operators (1), we have the following equations: The solutions of this system with respect to the segments , t p and q is found as The length of RSR as a function of the boundary conditions can be now written as RSR l t p q = + + , where , , t p q are calculated by (3). 3.
By applying the corresponding operators (1), we have the following equations: The solutions of this system with respect to the segments , t p and q is found as The length of LSR as a function of the boundary conditions can be now written as LSR l t p q = + + , where , , t p q are calculated by (4). 4. . By applying the corresponding operators (1), we have the following equations: The solutions of this system with respect to the segments , t p and q is found as , ( )(mod 2 ), The length of RSL as a function of the boundary conditions can be now written as RSL l t p q = + + , where , , t p q are calculated by (5). 5.
The solutions of this system with respect to the segments , t p and q is found as the minimum of the following two cases. Let sin sin The length of RLR as a function of the boundary conditions can be now written as where 1 1 1 , , t p q are calculated by (6) and 2 2 2 , , t p q are calculated by (7).
It is worth to noting that both of the two cases are needed to be considered (but [9] only considered the case 1  Fig. 1(b).   ,  sin  sin  0,  cos  cos  arctan  ,  sin  sin  0. sin sin Case2. arccos , The length of LRL as a function of the boundary conditions can be now written as where 1 1 1 , , t p q are calculated by (8) and 2 2 2 , , t p q are calculated by (9).

The Length of Dubins Path with
That is .
. Now (11) can be rewritten as x y β and any minimal turning radius 0 ρ > . Next, we consider its application in DTSP. For the instance considered in this section, the vertices are generated randomly, independently and uniformly in a 5 by 5 square (see Fig.2). The vertices are labeled as 1,2, ,10 ⋯ and their positions and headings are shown in Tab. 1.  The Dubins tour for the 10 vertex instance with different turning radius 0.1, 0.5,1 ρ = are presented in Fig.3. For this instance, the length of ETSP is 12.6239. From Fig.3, we can see that the Dubins length are all larger than the length of ETSP and it is increasing rapidly as the minimal turning radius ρ goes up.

Summary
In this paper, the formula to calculate , , t p q for the length of Dubins path is provided which is applicable to any initial point 0 0 ( , ; ) x y α , any final point 1 1 ( , ; ) x y β and any minimal turning radius 0 ρ > . We have proved the formula theoretically and experimentally with its application in DTSP. Numeral experiment has shown that the formulas are correct and efficient.