Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field

Abstract

We consider the optimal synthesis of the Zermelo–Markov–Dubins problem, that is, the problem of steering a vehicle with the kinematics of the Isaacs–Dubins car in minimum time in the presence of a drift field. By using standard optimal control tools, we characterize the family of control sequences that are sufficient for complete controllability and necessary for optimality for the special case of a constant field. Furthermore, we present a semianalytic scheme for the characterization of an optimal synthesis of the minimum-time problem. Finally, we establish a direct correspondence between the optimal syntheses of the Markov–Dubins and the Zermelo–Markov–Dubins problems by means of a discontinuous mapping.

Appendix A

1.1 A.1 $\mathsf{b}^{+}_{\alpha}\mathsf{s}_{\beta}\mathsf {b}^{+}_{\gamma } [\mathsf{b}^{-}_{\alpha}\mathsf{s}_{\beta}\mathsf {b}^{-}_{\gamma } ]$ Paths

The coordinates x _f , y _f of a state in ℜ_ZMD(b ⁺ sb ⁺;θ _f )[ℜ_ZMD(b ⁻ sb ⁻;θ _f )], can be expressed as functions of the parameters α and β as follows:

(27)

(28)

where T _f =α+β+γ, and $\gamma/\rho= (\theta_{\mathsf{f}}- \alpha/\rho) \operatorname{mod}2\pi [ \gamma/\rho= (2\pi-\theta_{\mathsf{f}}-\alpha/\rho) \operatorname{mod}2\pi ]$.

Conversely, given a state (x _f ,y _f ,θ _f )∈ℜ_ZMD(b ⁺ sb ⁺;θ _f )[ℜ_ZMD(b ⁻ sb ⁻;θ _f )], we can determine (α,β)∈[0,2πρ]×[0,∞[. In particular, after some algebraic manipulation, it follows that β satisfies the following quadratic equation, which is decoupled from α,

(29)

where $A(x_{\mathsf{f}},\theta_{\mathsf{f}}) = x_{\mathsf{f}}- [+] \rho\sin\theta_{\mathsf{f}}- w_{x} \rho \hat{\theta_{\mathsf{f}}}$, $B(y_{\mathsf{f}},\theta_{\mathsf{f}}) = [-]y_{\mathsf{f}}+ \rho (\cos\theta_{\mathsf{f}}- 1) -[+] w_{y} \rho\hat{\theta_{\mathsf{f}}}$, and

$$ \hat{\theta}_{\mathsf{f}}= \begin{cases} \theta_{\mathsf{f}}[2\pi- \theta_{\mathsf{f}}], & \mathrm{if}\ \alpha\leq\rho\theta_{\mathsf{f}}[\alpha\leq(2\pi -\theta_{\mathsf{f}})\rho],\\ 2\pi+ \theta_{\mathsf{f}}[4\pi-\theta_{\mathsf{f}}], & \mathrm {if}\ \alpha > \rho\theta_{\mathsf{f}}[\alpha> (2\pi-\theta_{\mathsf{f}})\rho]. \end{cases} $$

(30)

Note that for each (x _f ,y _f ,θ _f )∈ℜ_ZMD(b ⁺ sb ⁺;θ _f )[ℜ_ZMD(b ⁻ sb ⁻;θ _f )], there exist at most two solutions of (29). If β is one solution of (29), then α is determined with back substitution in Eqs. (27)–(28). In particular, after some algebraic manipulation, it follows that $\alpha= \hat{\alpha}\rho$, where $\hat{\alpha}\in[0,2\pi]$ satisfies

(31)

when β≠0, whereas α=ρθ _f  [ρ(2π−θ _f )], otherwise. In this way, for a given (x _f ,y _f ,θ _f )∈P(θ _f ), we find pairs (α,β) and the corresponding final time T _f (b ⁺ sb ⁺)[T _f (b ⁻ sb ⁻)]=α+β+γ(α), and subsequently, we associate the state (x _f ,y _f ,θ _f )∈P(θ _f ) with the pair (α ^∗,β ^∗) that gives the minimum time T _f (b ⁺ sb ⁺)[T _f (b ⁻ sb ⁻)].

1.2 A.2 $\mathsf{b}^{+}_{\alpha}\mathsf{s}_{\beta}\mathsf {b}^{-}_{\gamma } [\mathsf{b}^{-}_{\alpha}\mathsf{s}_{\beta}\mathsf {b}^{+}_{\gamma} ]$ Paths

If (x _f ,y _f ,θ _f )∈ℜ_ZMD(b ⁺ sb ⁻;θ _f )[ℜ_ZMD(b ⁻ sb ⁺;θ _f )], then

(32)

(33)

where $T_{\mathsf{f}}= \alpha+ \beta+ \gamma,~\gamma/\rho= (\alpha /\rho- \theta_{\mathsf{f}}) \operatorname{mod}2\pi~[\gamma/\rho= (\alpha/\rho+ \theta_{\mathsf{f}}) \operatorname{mod}2\pi ]$.

Given a state (x _f ,y _f ,θ _f )∈ℜ_ZMD(b ⁺ sb ⁻;θ _f )[ℜ_ZMD(b ⁻ sb ⁺;θ _f )], it can be shown that α satisfies the following transcendental equation (decoupled from β)

(34)

where

and where

$$ \hat{\theta}_{\mathsf{f}}= \begin{cases} \theta_{\mathsf{f}}, & \mathrm{if}\ \alpha\geq \rho\theta_{\mathsf{f}} [ \alpha\leq\rho(2\pi- \theta_{\mathsf{f}}) ],\\ \theta_{\mathsf{f}}-2\pi, & \mathrm{if}\ \alpha< \rho\theta_{\mathsf{f}} [ \alpha> \rho(2\pi- \theta_{\mathsf{f}}) ]. \end{cases} $$

(35)

Furthermore, it can be shown that β satisfies the following equation:

(36)

1.3 A.3 $\mathsf{b}^{+}_{\alpha}\mathsf{b}^{-}_{\beta}\mathsf {b}^{+}_{\gamma} [ \mathsf{b}^{-}_{\alpha}\mathsf{b}^{+}_{\beta }\mathsf{b}^{-}_{\gamma} ]$ and $\mathsf{b}^{+}_{\alpha}\tilde{\mathsf{b}}^{-}_{\beta}\mathsf {b}^{+}_{\gamma} [ \mathsf{b}^{-}_{\alpha}\tilde{\mathsf{b}}^{+}_{\beta}\mathsf {b}^{-}_{\gamma} ]$ Paths

The coordinates of a state (x _f ,y _f ,θ _f ) in ℜ_ZMD(b ⁺ b ⁻ b ⁺;θ _f )[ℜ_ZMD(b ⁻ b ⁺ b ⁻;θ _f )] or $\mathfrak{R}_{\mathrm{ZMD}}(\mathsf{b}^{+}\tilde{\mathsf{b}}{}^{-}\mathsf {b}^{+};\theta_{\mathsf{f}}) [ \mathfrak{R}_{\mathrm{ZMD}}(\mathsf{b}^{-}\tilde{\mathsf{b}}{}^{+}\mathsf {b}^{-};\theta_{\mathsf{f}}) ]$ are given by

(37)

(38)

where $T_{\mathsf{f}}= \alpha+ \beta+ \gamma,~\gamma/\rho= (\theta_{\mathsf{f}}-\alpha/\rho+ \beta/\rho) \operatorname{mod}2\pi [ \gamma/\rho= (-\theta_{\mathsf{f}}-\alpha/\rho+ \beta/\rho) \operatorname{mod}2\pi ]$.

Conversely, given (x _f ,y _f ,θ _f ) in ℜ_ZMD(b ⁺ b ⁻ b ⁺;θ _f )[ℜ_ZMD(b ⁻ b ⁺ b ⁻;θ _f )] or $\mathfrak{R}_{\mathrm{ZMD}}(\mathsf{b}^{+}\tilde{\mathsf {b}}^{-}\mathsf{b}^{+};\theta_{\mathsf{f}}) [ \mathfrak{R}_{\mathrm{ZMD}}(\mathsf{b}^{-}\tilde{\mathsf{b}}{}^{+}\mathsf {b}^{-};\theta_{\mathsf{f}}) ]$, it follows after some algebra that β satisfies the following transcendental equation, which is decoupled from α,

(39)

where

and

$$ \hat{\theta_{\mathsf{f}}} = \begin{cases}\theta_{\mathsf{f}}, & \mathrm{if}\ 0\leq[-]\theta_{\mathsf{f}}-\frac{\alpha}{\rho} +\frac{\beta}{\rho} <2\pi, \\[4pt] -2\pi[-4\pi] + \theta_{\mathsf{f}}, & \mathrm{if}\ 2\pi[-4\pi ]\leq [-]\theta_{\mathsf{f}}- \frac{\alpha}{\rho} + \frac{\beta}{\rho} < 4\pi[-2\pi], \\[4pt] [-]2\pi+ \theta_{\mathsf{f}}, & \mathrm{if}\ -2\pi\leq[-]\theta_{\mathsf{f}}- \frac{\alpha}{\rho} + \frac{\beta}{\rho} <0. \end{cases} $$

Given β∈[0,2πρ], it follows after some algebraic manipulation that α satisfies

(40)

where M(β;x _f ,θ _f )=A(x _f ,θ _f )−2βw _x , N(β;y _f ,θ _f )=B(y _f ,θ _f )+[−]2βw _y .