On Motion Planning for Point-to-Point Maneuvers for a Class of Sailing Vehicles

Table 1

Classification of wearing trajectories.

Conditions	Trajectory Formulations
If 𝜃0=atan2((𝑦𝑇−𝑦0),(𝑥𝑇−𝑥0))	Straight line motion (see Lemma 1)
If 𝑦𝑇≥𝑦0
If 𝜃0∈(0,𝜋/2)
If 𝑥𝑇<𝑥0+((𝑦𝑇−𝑦0)/tan𝜃0)+(𝑟/tan(𝜃0/2))	𝐿1, 𝑅, and 𝐿2 (see Lemma 2 and Figure 10(a))
If 𝑥𝑇≥𝑥0+((𝑦𝑇−𝑦0)/tan𝜃0)+(𝑟/tan(𝜃0/2))	𝐿1, 𝑅, and 𝐿2 (see Lemma 3 and Figure 10(b))
If 𝜃0=0
If 𝑥𝑇>𝑥0−((𝑦𝑇−𝑦0)/tan(𝜋−𝜃𝑇))−(𝑟/tan((𝜋−𝜃𝑇)/2))	𝐿1, 𝑅, and 𝐿2 (see Lemma 4 and Figure 11(a))
If 𝑥𝑇≤𝑥0−((𝑦𝑇−𝑦0)/tan(𝜋−𝜃𝑇))−(𝑟/tan((𝜋−𝜃𝑇)/2))	𝐿1, 𝑅1, 𝐿2, 𝑅2, and 𝐿3 (see Lemma 5 and Figure 11(b))
If 𝜃0∈(𝜋/2,𝜋)
If 𝑥𝑇>𝑥0−((𝑦𝑇−𝑦0)/tan(𝜋−𝜃0))−(𝑟/tan((𝜋−𝜃0)/2))	𝐿1, 𝑅, and 𝐿2 (see Figure 12(a))
If 𝑥𝑇≤𝑥0−((𝑦𝑇−𝑦0)/tan(𝜋−𝜃0))−(𝑟/tan((𝜋−𝜃0)/2))	𝐿1, 𝑅, and 𝐿2 (see Figure 12(b))
If 𝜃0=±𝜋
If 𝑥𝑇<𝑥0+((𝑦𝑇−𝑦0)/tan𝜃𝑇)+(𝑟/tan(𝜃𝑇/2))	𝐿1, 𝑅, and 𝐿2 (see Figure 13(a))
If 𝑥𝑇≥𝑥0+((𝑦𝑇−𝑦0)/tan𝜃𝑇)+(𝑟/tan(𝜃𝑇/2))	𝐿1, 𝑅1, 𝐿2, 𝑅2, and 𝐿3 (see Figure 13(b))
If 𝜃0∈[−𝜋/2,0)	Path from 𝐶0 to 𝐶^′0, with 𝑦𝑇>𝑦^′0 and 𝜃^′0∈(0,𝜋/2) (see Figure 14(a))
If 𝜃0∈(−𝜋,−𝜋/2)	Path from 𝐶0 to 𝐶^′0, with 𝑦𝑇>𝑦^′0 and 𝜃^′0∈(𝜋/2,𝜋) (see Figure 14(b))
If 𝑦𝑇<𝑦0
If 𝜃0∈[−𝜋/2,0)	Path from 𝐶0 to 𝐶^′0, with 𝑦𝑇>𝑦^′0 and 𝜃^′0∈(0,𝜋/2) (see Figure 15(a))
If 𝜃0∈[−𝜋,−𝜋/2)	Path from 𝐶0 to 𝐶^′0, with 𝑦𝑇>𝑦^′0 and 𝜃^′0∈(𝜋/2,𝜋) (see Figure 15(b))
If 𝜃0∈[0,𝜋/2)	Path from 𝐶0 to 𝐶^′0, with 𝜃^′0∈(−𝜋/2,0) (see Figure 16(a))
If 𝜃0∈(𝜋/2,𝜋]	Path from 𝐶0 to 𝐶^′0, with 𝜃^′0∈(−𝜋,−𝜋/2) (see Figure 16(b))