Open AccessArticle

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Alexey Mashtakov

^*,†

and

Yuri Sachkov

^†

Ailamazyan Program Systems Institute of Russian Academy of Sciences, Pereslavl-Zalessky 152021, Russia

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2023, 11(18), 3931; https://doi.org/10.3390/math11183931

Submission received: 22 July 2023 / Revised: 11 September 2023 / Accepted: 13 September 2023 / Published: 15 September 2023

(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)

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Abstract

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in

{SE}_{2}

and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part.

Keywords:

geometric control; model of a car; extremal trajectories; Pontryagin maximum principle; group of motions of a plane

MSC:

49K15

1. Introduction

Consider a car model that can move forward on a plane and turn with a given minimum turning radius (see Figure 1). The car has two wheels, equidistant from the axle of the wheelset. Both wheels have independent drives that can rotate so that the corresponding rolling of the wheels occurs without slipping. The configuration of the system is described by the triple

q = (x, y, θ) = R^{2} \times S^{1}

, where

(x, y) \in R^{2}

is the central point, and

θ \in S^{1}

is the orientation angle of the car. In such a way, the configuration space forms the Lie group of roto-translations

{SE}_{2} ≃ R^{2} \times S^{1} .

The car has two controls: the tangential velocity

u_{1}

and the angular velocity

u_{2}

. Consider the configuration

Id = (0, 0, 0)

, when the car is located at the origin and oriented along the positive direction of

O x

. An infinitesimal translation is generated by the vector

\partial_{x}

and rotation by the vector

\partial_{θ}

. They are possible motions controlled by

u_{1}

and

u_{2}

. The remaining direction

\partial_{y}

is forbidden since the immediate motion of the car in a direction perpendicular to its wheels is not possible. Thus, the dynamics of the car at the origin is given by

\dot{x} = u_{1}

\dot{y} = 0

, and

\dot{θ} = u_{2}

The origin Id is the unit element of the group

{SE}_{2}

(see Section 2). Any element

q \in {SE}_{2}

is generated by the left translation

L_{q}

Id. The dynamics at an arbitrary configuration q are

\dot{q} = u_{1} X_{1} (q) + u_{2} X_{2} (q),

(1)

where

X_{i}

represents left-invariant vector fields (see Section 2).

Various sets of admissible controls

U ∋ (u_{1}, u_{2})

lead to different models (see Figure 2). The time-optimal problem for

$u_{1} = 1$ , $| u_{2} | \leq κ$ , $κ > 0$ leads to Dubins car [1];
$| u_{1} | = 1$ , $| u_{2} | \leq κ$ , $κ > 0$ leads to Reeds–Shepp car [2];
$u_{1} \geq 1$ , $u_{1} + | u_{2} | \leq κ < 1$ leads to a generalized Dubins car, studied by Ardentov [3];
$u_{1}^{2} + u_{2}^{2} \leq 1$ leads to the model whose solutions are sub-Riemannian length minimals, studied by Sachkov [4];
$u_{1}^{2} + u_{2}^{2} \leq 1$ , $u_{1} \neq 0$ leads to the model studied by Berestovskii [5];
$u_{1} \geq 0$ , $u_{1}^{2} + u_{2}^{2} \leq 1$ leads to the model of a car moving forward and turning in place, proposed by Duits [6];
$u_{1} = r cos ϕ$ , $u_{2} = r sin ϕ$ , $0 \leq r \leq 1$ , $| ϕ | \leq α$ leads to the general model of a car with control in a circular sector, which is studied in this paper.

In 1957, Dubins described [1] the problem of finding the shortest path for a car in plane moving with no reverse gear from an initial configuration (position and direction) to a final configuration penalizing the curvature of its trajectory in the plane. Later in 1990, Reeds and Shepp studied [2] the same problem, but for a car that has reverse gear. Both papers are devoted to a description of the general shape of the optimal paths, without providing explicit solutions for given boundary conditions. It was shown that the optimal trajectories of Dubins and Reeds–Shepp cars are given by concatenation of arcs of a circle of minimum possible radius (circular trajectories) and segments of a straight line (straight trajectories). In a recent study [3], Ardentov refines the Dubins model by considering the bounded angular speed of two wheels, which results in maximal tangential speed when the car is moving straight. For the refined model, extremal trajectories have been found. They are also given by concatenation of circular and straight trajectories. Note that such trajectories have bounded curvature: the upper bound is given by reciprocal to the minimal turning radius.

In 2011, Sachkov [4] obtained optimal synthesis in a sub-Riemannian problem in

{SE}_{2}

. This problem can be seen as the shortest path problem for a car that can move forward/backward and turn in place. In such a model, the curvature of trajectories is unbounded, and it was shown that optimal trajectories may contain cusp points, the points where the direction of motions is switching to the opposite one. Berestovskii [5] studied segments of the trajectories that contain no cusps and can be parameterized by the length of the planar projection.

In 2018, Duits and coauthors [6] studied a modification of the sub-Riemannian problem obtained by the restriction of the reverse gear. In the Duits model, the car can move forward and rotate in place, but it cannot move backward. Motivation came from image analysis applications, where the trajectories of the car are used to detect salient lines in images. An exact expression for the extremals was obtained in [7]. The original idea of the Duits model was to reduce the number of undesirable cusp points by suppressing the backward motions. However, it turned out that the cusp points are replaced by turning points (the points of in-place rotation on

π

radians), and the projection of extremal trajectories of the Duits car to the image plane coincide with the projection of sub-Riemannian geodesics in

{SE}_{2}

. This observation shows that in order to entirely solve the “cusp problem”, the model needs further adaptation. In the present paper, we propose a model of a car that can move forward, stay in place, and turn with a given minimal turning radius. Trajectories of this system a priori cannot have cusp points in the planar projection.

The problem of transferring the car between the given initial and final configurations is known in robotics as a motion planning problem. There are different approaches to tackle this problem [8]. Introducing a cost function for the system leads to an optimal control problem. Such optimal control problems are tackled by various numerical methods, see, e.g., [9]. However, exact solutions (optimal synthesis) are known only for a few of the simplest models in robotics [4,10]. In the present paper, we obtain explicit expressions for extremals in a more complicated model with control in a circular sector.

System (1) arises in the modeling of the human visual system. An important discovery of the neurophysiology of vision was made by Hubel and Wiesel in 1959 [11], who showed that in the striate cortex of a cat, there exist groups of neurons sensitive to positions and directions (orientations). In the first stage of processing, the image is lifted by the brain to the extended space of positions and directions. In [12], a sub-Riemannian structure on the Heisenberg group was proposed for contour perception and completion. The model was refined in [13] by taking into account the global nature of the orientation angle. The roto-translation group

{SE}_{2}

endowed with a sub-Riemannian metric has been proposed to model the functional architecture of the primary visual cortex. This model is known as the Petiot–Citti–Sarti model, where sub-Riemannian geodesics (extremal trajectories) are used for the completion of occluded contours. The usage of sub-Riemannian geodesics for modeling the association field in the psychophysiology of vision was studied in [14]. It was shown that good candidates for the association field lines are given by sub-Riemannian geodesics with cuspless planar projection. Application of sub-Riemannian geodesics for modeling illusory contours in geometrical optical illusions has been studied in [15,16].

The principles of biological visual systems are actively used in computer vision. Based on these principles, effective image processing methods are created, e.g., image reconstruction [17] and detection of salient lines in images [18,19]. In [18], optimal trajectories of a System (1) are used to detect salient lines in images. The motivation for our study is to solve the problem of cusp points that appear in the salient line detection method [18], which is based on tracking via sub-Riemannian geodesics. At such points, the car reverses its direction of motion, and a detected salient line has an undesirable cusp point. In our model with control in a circular sector with an acute opening angle, the cusp points are impossible.

Our work generalizes the sub-Riemannian problem [4] by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. In addition to its importance for applications, the problem under study is of independent interest in geometric control theory [20] as a model example of an optimal control problem in which zero control lies on the boundary of the set of control parameters.

The problem can be seen as a rolling geodesic problem [21,22], where a disc is rolling on a plane. We explicitly derive the geodesics and analyze their optimality.

In the present paper, we study the time-optimal problem for System (1) with control in a circular sector. We generalize the results [7], where we studied the special case of admissible controls in a half-disc. The present paper is an extended version of the preliminary work [23], where we studied the case of the sector opening angle less than

π

and parameterized the extremal trajectories by the length of their planar projection (s–parametrization). Now we study the general case of the sector opening angle and derive the explicit formulas for the time-parameterized extremals (t–parametrization), which is natural for time-optimal problems. We also provide an analysis of their optimality.

The present paper is organized as follows. In Section 2, we give preliminary materials on the Lie group

{SE}_{2}

. In Section 3, we formulate an optimal control problem under consideration. In Section 4, we study local and global controllability (see Definitions 1–3) of the system and the existence of a solution for given arbitrary boundary conditions. In Section 5, we apply a necessary optimality condition, the Pontryagin maximum principle (PMP), and describe a phase portrait of the Hamiltonian system of PMP. In Section 6, we integrate the Hamiltonian system and obtain an explicit expression for the extremals. In Section 7, we partially study the optimality of the extremals and provide estimates for the cut time.

2. Preliminaries

The roto-translation group

{SE}_{2}

is the group of proper motions of the Euclidean plane. Any such motion

q = (x, y, θ) \in {SE}_{2}

consists of a rotation around a given point on angle

θ \in S^{1} = R / 2 π Z

and a parallel translation on a vector

(x, y) \in R^{2}

The composition of two motions results in the product of two elements

q^{'}, q \in {SE}_{2}

q^{'} \cdot q = (x cos θ^{'} - y sin θ^{'} + x^{'}, x sin θ^{'} + y cos θ^{'} + y^{'}, θ + θ^{'}) .

The identical transformation of the plane is the unit element

Id = (0, 0, 0) \in {SE}_{2}

For any

q \in {SE}_{2}

, there exists

q^{- 1} \in {SE}_{2}

such that

q^{- 1} \cdot q = q \cdot q^{- 1} = Id

given by

q^{- 1} = (- x cos θ - y sin θ, x sin θ - y cos θ, - θ) .

Note that the group operation is not commutative. The left translation is defined as

L_{q^{'}} q = q^{'} \cdot q .

The tangent space

T_{Id} {SE}_{2} = span (\partial_{x}, \partial_{y}, \partial_{θ})

with the Lie bracket operation

[\partial_{x}, \partial_{y}] = 0, [\partial_{x}, \partial_{θ}] = - \partial_{y}, [\partial_{y}, \partial_{θ}] = \partial_{x}

forms the Lie algebra

{se}_{2}

. It is isomorphic to the Lie algebra of the left-invariant vector field

span (X_{1}, X_{2}, X_{3})

, where the vector fields

X_{i}

are obtained via push-forward

\begin{matrix} X_{1} (q) = L_{q *} \partial_{x} = cos θ \partial_{x} + sin θ \partial_{y}, \\ X_{2} (q) = L_{q *} \partial_{θ} = \partial_{θ}, \\ X_{3} (q) = L_{q *} (- \partial_{y}) = sin θ \partial_{x} - cos θ \partial_{y} . \end{matrix}

3. Statement of the Problem

For a given angle

α \in [0, π]

, consider the following control system:

\{\begin{matrix} \dot{x} = u_{1} cos θ, \\ \dot{y} = u_{1} sin θ, \\ \dot{θ} = u_{2}, \end{matrix} \begin{matrix} (x, y, θ) = q \in {SE}_{2}, \\ (u_{1}, u_{2}) \in U, \\ U = {(r cos ϕ, r sin ϕ) | 0 \leq r \leq 1, | ϕ | \leq α} . \end{matrix}

(2)

For given boundary conditions

q_{0}

q_{1} \in {SE}_{2}

, we aim to find the controls

u_{1} (t)

u_{2} (t) \in L^{\infty} ([0, T], R)

such that the corresponding trajectory

q : [0, T] \to {SE}_{2}

transfers the system from the initial

q_{0}

to the final configuration

q_{1}

in minimal time:

q (0) = q_{0}, q (T) = q_{1}, T = \int_{0}^{T} d t \to min .

(3)

Remark 1.

The problem is invariant under the left action of

{SE}_{2}

since the vector fields

X_{1}

and

X_{2}

are left-invariant. Due to this property without loss of generality, we set

q (0) = Id

4. Existence of the Solution

4.1. Controllability and Existence of Optimal Controls

In this section, we study the existence of the solution in Problem (2), (3). First, we give some necessary definitions.

Denote by

A

the attainable set [20] of System (2) from Id for any non-negative time.

Definition 1.

System (2) is called globally controllable if

A = {SE}_{2}

In other words, a control system is globally controllable if any two points of its configuration space can be connected by an admissible trajectory.

Let

t \geq 0

. Denote by

A_{\leq t}

the attainable set of System (2) from

Id \in {SE}_{2}

for time

\leq t

. Denote by

int A_{\leq t}

the interior of the set

A_{\leq t}

Definition 2.

System (2) is called small-time locally controllable at Id if for all

t > 0

, there holds the inclusion

int A_{\leq t} ∋ Id .

Definition 3.

System (2) is called locally controllable for time

t > 0

at Id if

int A_{\leq t} ∋ Id .

Next, we provide an analysis of global and small-time local controllability of System (2) and study the existence of optimal control. We show that there are four different possible cases depending on the domain of angle

α

. The result is gathered in Theorem 1.

Theorem 1.

Consider Problem (2), (3) with

q_{0} = Id

. The following statements hold:

1.: For $α = 0$ , the system is not globally controllable. The attainable set is $A = {(x, 0, 0) | x \geq 0}$ . For any $q_{1} \in A$ , there exists a unique optimal trajectory;
2.: For $α \in (0, \frac{π}{2}]$ , the system is globally controllable, but not small-time locally controllable. For any $q_{1} \in {SE}_{2}$ , there exists an optimal trajectory;
3.: For $α \in (\frac{π}{2}, π)$ , the system is globally controllable and small-time locally controllable. An optimal trajectory does not exist for some boundary conditions;
4.: For $α = π$ , the system is globally and small-time locally controllable. For any $q_{1} \in {SE}_{2}$ , there exists an optimal trajectory.

Proof.

First, we show that the system is not globally controllable if

α = 0

. Indeed, when

α = 0

, System (2) is reduced to

\dot{q} = u_{1} X_{1} (q)

u_{1} \in [0, 1]

q (0) = Id

, and the attainable set

A_{\leq t}

is a segment of the trajectory

e^{τ X_{1}} (Id) = (τ, 0, 0)

0 \leq τ \leq t

. Consequently, the attainable set for any non-negative time is a ray

A = {(x, 0, 0) | x \geq 0}

. The optimal control is given by

u_{1} \equiv 1

, and there exists a unique optimal trajectory, a segment of a straight line, which transfers the system from Id to any element

q_{1} \in A

Next, we provide three different proofs of global controllability for

α \neq 0

. The global controllability of System (2) for

α \neq 0

follows from the global controllability of Dubins car (see [1]). Indeed, the sets of admissible trajectories of the two systems are the same, since the convex cones of the sets of admissible controls in these problems coincide with one another. Thus, the controllability of the Dubins car is equivalent to the controllability of our problem. The second proof follows from the general statement in [24]. According to [24], a left-invariant system is globally controllable on

{SE}_{2}

if and only if the system is full-rank (bracket generating), i.e., the Lie algebra at every point forms the full tangent space. We have

Lie (cos α X_{1} + sin α X_{2}, cos α X_{1} - sin α X_{2}) = span (X_{1}, X_{2}, X_{3})

, thus System (2) is globally controllable. The third proof of controllability is given in [23]. It relies on the Lie saturation method (see [20]), which is standard in geometric control theory.

Now we study the case

α \in (0, \frac{π}{2}]

. For any

u \in U

, we have

u_{1} \geq 0

. Thus,

x (t) = \int_{0}^{t} u_{1} (τ) cos θ (τ) d τ \geq 0 for small t > 0 .

Consequently, System (2) is not small-time locally controllable for

α \in (0, \frac{π}{2}]

. Further, in Theorem 2, we give a precise time estimate for local controllability in this case. The existence of an optimal trajectory that transfers the system for

α \in (0, \frac{π}{2}]

from Id to any

q_{1} \in {SE}_{2}

is guaranteed by the Filippov theorem [20]. Indeed, all the conditions of the Filippov theorem hold due to the compactness and convexity of U and the global controllability of the system.

Next, we analyze the case

α \in (\frac{π}{2}, π)

. First, we show that the system is small-time locally controllable. To this end, we consider a convex closure

V = co (U)

of the set U and the corresponding relaxed control system (2) with the set of admissible controls

V ∋ (u_{1}, u_{2})

. Denote by

B_{\leq t}

the attainable set of the relaxed system by time

\leq t

. Since

0 \in int V

and the relaxed system is full-rank, it is small-time locally controllable, i.e., for all

t > 0

, we have

int B_{\leq t} ∋ Id

. By Thm. 8.2 of [20], the attainable sets of the original and the relaxed systems are related by

B_{\leq t} \subset \bar{A_{\leq t}}

, where

\bar{S}

is the closure of a set S. Thus,

int \bar{A_{\leq t}} ∋ Id

. Since the original system is full-rank, we have

int A_{\leq t} ∋ Id

, and hence we prove that the system is small-time locally controllable.

In the case

α \in (\frac{π}{2}, π)

, the set U is not convex. Now we show that there exists a boundary condition

q_{1}

for which an optimal trajectory does not exist. Consider

q_{1} = (cos α, 0, 0)

. We show that it satisfies the following conditions:

(a): $inf {t_{1} > 0 | there exists a trajectory q (\cdot), s . t . q (0) = Id, q (t_{1}) = q_{1}} \leq 1$ ;
(b): A trajectory $q (\cdot)$ , s.t. $q (0) = Id, q (1) = q_{1}$ does not exist.

It is obvious that Conditions (a) and (b) imply the nonexistence of optimal trajectory connecting Id and

q_{1}

. Let us prove Item (a). Let

n \in N

. Consider the following control:

\begin{matrix} t \in [\frac{i}{n}, \frac{2 i + 1}{2 n}) \Rightarrow u_{1} (t) = cos α, u_{2} (t) = sin α, \end{matrix}

(4)

\begin{matrix} t \in [\frac{2 i + 1}{2 n}, \frac{i + 1}{n}] \Rightarrow u_{1} (t) = cos α, u_{2} (t) = - sin α . \end{matrix}

(5)

For

u_{1} = cos α

u_{2} = \pm sin α

, the corresponding trajectory satisfies the following ODE:

\dot{x} = cos α cos θ, \dot{y} = cos α sin θ, \dot{θ} = \pm sin α, (x, y, θ) (0) = (x_{0}, y_{0}, θ_{0}),

and is given by

\begin{matrix} x (t) = x_{0} \pm cot α (sin (θ_{0} \pm t sin α) - sin θ_{0}), \\ y (t) = y_{0} \mp (cos (θ_{0} \pm t sin α) - cos θ_{0}), \\ θ (t) = θ_{0} \pm t sin α . \end{matrix}

Thus, the trajectory

(x, y, θ) (t)

with Controls (4) and (5) departing from Id has an end point

\begin{matrix} x (1) = x^{n} = 2 n cot α sin (\frac{sin α}{2 n}) \to cos α, n \to + \infty, \\ y (1) = y^{n} = 2 n (1 - cos (\frac{sin α}{2 n})) \to 0, n \to + \infty, \\ θ (t) = 0 . \end{matrix}

Therefore, point

q^{n} = (x^{n}, y^{n}, 0)

is attainable from Id by time

t_{1} = 1

, and we have

q^{n} \to q_{1}

when

n \to + \infty

. However, the system is small-time locally controllable at any point, thus for any

ε > 0

, an attainable set in reverse time from point

q_{1}

for time

\leq ε

contains a neighborhood of point

q_{1}

. Therefore, for any

ε > 0

, point

q_{1}

is attainable from

q_{0}

by time

1 + ε

, and Item (a) is proved.

Now we prove Item (b). Assume that there exists a trajectory that transfers the system from Id to

q_{1}

for time

t_{1} = 1

. The corresponding control has the form

u_{1} (t) = r (t) cos ϕ (t), u_{2} (t) = r (t) sin ϕ (t), r (t) \in [0, 1], ϕ (t) \in [- α, α] .

Then,

\begin{matrix} x (1) = cos α = \int_{0}^{1} r (t) cos ϕ (t) cos θ (t) d t, \end{matrix}

(6)

\begin{matrix} θ (1) = 0 = \int_{0}^{1} r (t) sin ϕ (t) d t . \end{matrix}

(7)

Assume that the function

cos θ (t)

changes its sign on the segment

t \in [0, 1]

. Then, for some

t \in [0, 1]

, we have

θ (t) \in (\frac{π}{2}, π] \cup [- π, - \frac{π}{2})

. However,

| \dot{θ} | = | u_{2} | \leq 1

; therefore,

t > \frac{π}{2} > 1

, and we obtain a contradiction. Thus,

cos θ (t) > 0

for

t \in [0, 1]

. Then, we have

r (t) cos θ (t) cos ϕ (t) \geq cos α

, and from (6), we conclude that

| ϕ (t) | \equiv α

r (t) \equiv 1

θ (t) \equiv 0

, which contradicts (7), and Item (b) is proved.

The case

α = π

coincides with the well-known sub-Riemannian problem in

{SE}_{2}

studied in [4]. In this case, the system is globally and small-time locally controllable. For any

q_{1} \in {SE}_{2}

, there exists an optimal trajectory (sub-Riemannian length minimizer) from Id to

q_{1}

, which is explicitly computed in [4]. Note that for some boundary conditions

q_{1} \in {SE}_{2}

, there exist two distinct length minimizers. Such points are located on the so-called Maxwell set. In the general case, when

q_{1}

does not belong to the Maxwell set, there exists a unique optimal trajectory from Id to

q_{1}

. □

Remark 2.

In Theorem 1, we showed that

α = 0

and

α \in (\frac{π}{2}, π)

, Problem (2), (3) is ill-posed, and the case

α = π

gives rise to a well-known sub-Riemannian problem in

{SE}_{2}

where the optimal synthesis is known. In the remainder of this paper, we consider the case

α \in (0, \frac{π}{2}]

, which is an open, well-posed problem.

Remark 3.

For

α \in (\frac{π}{2}, π)

, a natural modification of the ill-posed time minimization problem in a circular sector is given by replacing the set of admissible controls U by its convex closure.

4.2. Local Controllability

In this subsection, we study the local controllability of System (2) for

α \in (0, \frac{π}{2}]

. In Theorem 1, we showed that the system is not small-time locally controllable. In this subsection, we find an instant of time

T > 0

at which the local controllability is attained.

We denote

By $A_{t}$ , the attainable set of System (2) from Id for a time $t > 0$ ;
By $A_{\leq t}$ , the attainable set of System (2) from Id for time not greater than t;
By $A_{\leq t}^{q}$ , the attainable set of System (2) from $q \in {SE}_{2}$ for time not greater than t.

Theorem 2.

Let

α \in (0, \frac{π}{2}]

, and let

T = \frac{2 π}{sin α}

. Then, for any

ε > 0

, System (2) is locally controllable at Id for time not greater than

T + ε

, i.e.,

Id \in int A_{\leq T + ε} .

The proof of this theorem follows from the next Lemmas 1–3.

Lemma 1.

For any

t > 0

, the point

(- t, 0, 0) \in A_{T + t}

Lemma 2.

For any

t > 0

and any

ε > 0

, the point

(t, 0, 0) \in int A_{\leq t + ε}

Lemma 3.

For any

t > 0

and any

ε > 0

, the point

Id \in int A_{\leq t + ε}^{(- t, 0, 0)}

Now we prove Theorem 2 on the basis of Lemmas 1–3.

Proof.

Fix any

ε > 0

, and denote

q_{1} = (- ε / 2, 0, 0) \in {SE}_{2}

. It follows from Lemma 1 that

q_{1} \in A_{T + ε / 2}

. Further, it follows from Lemma 3 that

Id \in A_{\leq ε / 2}^{q_{1}}

. However

A_{\leq ε / 2}^{q_{1}} \subset A_{\leq T + ε / 2 + ε / 2} = A_{\leq T + ε},

thus

Id \in int A_{\leq T + ε}

. □

Now we prove Lemma 1.

Proof.

Fix any

t > 0

. Consider the following admissible control:

u (s) = \{\begin{matrix} (cos α, sin α), & s \in [0, T / 2] \cup [T / 2 + t, T + t], \\ (1, 0), & s \in (T / 2, T / 2 + t) . \end{matrix}

Then, immediate computation shows that the corresponding trajectory

q (s)

passes through the points

q (T / 2) = (0, 2 cot α, π), q (T / 2 + t) = (- t, 2 cot α, π), q (T + t) = (- t, 0, 0) .

□

Further, we prove Lemma 2.

Proof.

Denote the vector field

X^{φ} = cos φ X_{1} + sin φ X_{2}

| φ | \leq α

. Consider the mapping

F (s, φ_{1}, φ_{2}) = exp (s X^{φ_{2}}) \circ exp (s X^{φ_{1}}) (Id) \in A_{2 s}, s > 0, | φ_{i} | \leq α .

(8)

Let

t > 0

p = (\frac{t}{2}, 0, 0) \in {(0, + \infty)}_{s} \times {[- α, α]}_{φ_{1}} \times {[- α, α]}_{φ_{2}}

, and let

q = F (p) = exp (t X_{1}) (Id) = (t, 0, 0) \in {SE}_{2}

. Then

\begin{matrix} \frac{\partial F}{\partial s} (p) = X_{1} (q), \\ \frac{\partial F}{\partial φ_{1}} (p) = \frac{t}{2} X_{2} (q), \\ \frac{\partial F}{\partial φ_{2}} (p) = (X_{2} - \frac{t}{2} X_{3}) (q), \end{matrix}

thus

\frac{\partial F}{\partial (s, φ_{1}, φ_{2})} (p) = - \frac{t^{2}}{4} \neq 0 .

(9)

Fix any

ε \in (0, t / 2)

. Consider the mapping

F : D \to {SE}_{2}, D = {(\frac{t - ε}{2}, \frac{t + ε}{2})}_{s} \times {[- α, α]}_{φ_{1}} \times {[- α, α]}_{φ_{2}}

given by (8). Notice that

p \in int D

. In view of (9), by the inverse function theorem, F is a local diffeomorphism from a neighborhood of p onto a neighborhood of

F (p) = q \in {SE}_{2}

. Thus,

q \in int F (D)

. However

F (D) \subset A_{\leq t + ε}

, so

q = (t, 0, 0) \in int A_{\leq t + ε}

. □

Finally, we prove Lemma 3.

Proof.

Fix any

t > 0

and any

ε \in (0, t / 2)

. By Lemma 2, the point

q_{2} : = (t, 0, 0) \in int A_{\leq t + ε}

. System (2) is invariant with regard to left translations

L_{q} : {SE}_{2} \to {SE}_{2}

q \in {SE}_{2}

, thus

L_{q_{2}}^{- 1} (q_{2}) = Id \in int A_{\leq t + ε}^{q_{3}}, q_{3} = L_{q_{2}}^{- 1} (Id) = (- t, 0, 0) \in {SE}_{2} .

□

Remark 4.

There is numerical evidence that the lower bound

T + ε

ε > 0

, of the time at which the local controllability is attained (see Theorem 2) is exact, i.e.,

Id \notin int A_{\leq T}

5. Pontryagin Maximum Principle

5.1. Hamiltonian System and Maximality Condition

A necessary optimality condition is given by the Pontryagin maximum principle (PMP) [20,25]. In this section, we apply the PMP to Problem (2), (3).

Let

p \in T_{q}^{*} {SE}_{2}

. Define the Pontryagin function

H_{u} (p, q) = 〈 p, u_{1} X_{1} (q) + u_{2} X_{2} (q) 〉 = u_{1} (p_{1} cos θ + p_{2} sin θ) + u_{2} p_{3},

where

(p_{1}, p_{2}, p_{3})

are canonical coordinates in the cotangent space

T_{q}^{*} {SE}_{2}

corresponding to the coordinates

(x, y, θ)

{SE}_{2}

The PMP states the following. Let

u (t)

q (t)

t \in [0, T]

be an optimal control and the corresponding optimal trajectory. Then, there exists a Lipshitzian curve

p (t)

for which

p_{1}^{2} (t) + p_{2}^{2} (t) + p_{3}^{2} (t) \neq 0

for

t \in [0, T]

(the nontriviality condition), and the following conditions hold for almost every

t \in [0, T]

1.: The Hamiltonian system

$\dot{p} (t) = - \frac{\partial H_{u}}{\partial q} (p (t), q (t)), \dot{q} (t) = \frac{\partial H_{u}}{\partial p} (p (t), q (t));$
2.: The maximality condition

$H_{u (t)} (p (t), q (t)) = max_{u \in U} H_{u} (p (t), q (t)) = H (p (t), q (t)) \geq 0 .$

The maximized Pontryagin function

H (p, q)

is called the Hamiltonian. It is the first integral of the Hamiltonian system. The case

H = 0

is called abnormal, and the case

H > 0

is called normal.

Natural coordinates for left-invariant systems [20] are given linearly on fibers of cotangent bundle Hamiltonians

h_{i}

associated with the basis left-invariant vector fields:

h_{i} (p, q) = 〈 p, X_{i} (q) 〉

p \in T_{q}^{*} {SE}_{2}

. In canonical coordinates, they read as

h_{1} = p_{1} cos θ + p_{2} sin θ, h_{2} = p_{3}, h_{3} = p_{1} sin θ - p_{2} cos θ .

The Pontryagin function takes the form

H_{u} = u_{1} h_{1} + u_{2} h_{2} .

(10)

The Hamiltonian system is given by

\{\begin{matrix} \dot{x} = u_{1} cos θ, \\ \dot{y} = u_{1} sin θ, \\ \dot{θ} = u_{2}, \end{matrix} \{\begin{matrix} {\dot{h}}_{1} = - u_{2} h_{3}, \\ {\dot{h}}_{2} = u_{1} h_{3}, \\ {\dot{h}}_{3} = u_{2} h_{1} . \end{matrix}

(11)

The subsystem for state variables x, y, and

θ

is called the horizontal part, and the subsystem for adjoint variables

h_{1}

h_{2}

, and

h_{3}

is called the vertical part of the Hamiltonian system.

Now we analyze the maximum condition

H = max_{u \in U} H_{u} = max_{u \in U} (h_{1} u_{1} + h_{2} u_{2}) .

Note that the Pontryagin function

H_{u}

can be seen as a scalar product of two vectors

(h_{1}, h_{2}) \in R^{2}

and

(u_{1}, u_{2}) \in U \subset R^{2}

. From this point of view, the maximum condition has a clear geometric interpretation (see Figure 3).

Introduce polar coordinates in the planes

O h_{1} h_{2}

and

O u_{1} u_{2}

\begin{matrix} h_{1} = ρ cos ψ, h_{2} = ρ sin ψ, & ψ \in (- π, π], ρ \geq 0; \\ u_{1} = r cos ϕ, u_{2} = r sin ϕ, & ϕ \in (0, α] \subset (0, \frac{π}{2}), r \in [0, 1] . \end{matrix}

For

| ψ | \in (\frac{π}{2} + α, π]

, we have

H = 0

and

u_{1} = u_{2} = 0

. In this case, we have trivial abnormal extremals.

For

ψ = \pm (\frac{π}{2} + α)

, we have

H = 0

u_{1} = r cos α

, and

u_{2} = \pm r sin α

. In this case, we have nontrivial abnormal extremals. We study them in detail in Section 5.2.

For

ρ = 0

, we have

H = 0

for any

(u_{1}, u_{2}) \in U

. In this case, we have abnormal extremals. Since the Hamiltonian

H = 0

is constant along the extremals, the point

ρ = 0

can lie on a trivial or a nontrivial abnormal extremal that we listed in the two cases above. Further, we consider

ρ > 0

For

\pm ψ \in (α, \frac{π}{2} + α)

, we have

u_{1} = cos α

u_{2} = \pm sin α .

In this case, we have

H = ρ cos (α \mp ψ) > 0

and the normal extremals corresponding to the motion of a car along a circle of minimal possible radius, as we will show in Section 5.3.

For

| ψ | \leq α

, we have

u_{1} = cos ψ, u_{2} = sin ψ

. In this case, we have

H = ρ > 0

that and the normal extremals corresponding to the motion of a car along sub-Riemannian geodesics, as we will show in Section 5.3.

5.2. Abnormal Case $H = 0$

Trivial abnormal extremal control

(u_{1}, u_{2}) \equiv 0

corresponds to the fixed point of the Hamiltonian system.

Nontrivial abnormal extremal controls are given by

u_{1} (t) = r (t) cos α, u_{2} (t) = sign (ψ (t)) r (t) sin α, where 0 \leq r (t) \leq 1 .

Proposition 1.

If an admissible control

u (t) = (u_{1}, u_{2}) (t) \in U

t \in [0, T]

is not arc length parameterized, i.e., there exists a positive measure set

σ \subset [0, T]

such that

u_{1}^{2} (t) + u_{2}^{2} (t) < 1

for

t \in σ

, then

u (t)

is not optimal.

Proof.

Let

r (t) = \sqrt{u_{1}^{2} (t) + u_{2}^{2} (t)}

for a given trajectory. Let us show that if

0 \leq r (t) < 1

on a set

σ \subset [0, T]

of measure

μ (σ) > 0

, then the trajectory is not optimal. First, note that if

r (t) = 0

for all

t \in σ

, then this part of the trajectory is a fixed point. Staying in place is not optimal in a time minimization problem. Now, if

r (t) \in (0, 1)

for

t \in σ

, then one can reparametrize the trajectory with

d s = r (t) d t

. In this case, the support of the trajectory stays the same, new controls are also admissible, but the new motion time S becomes strictly less than the original one:

S = \int_{0}^{S} d s = \int_{0}^{T} r (t) d t = \int_{σ} r (t) d t + \int_{[0, T] ∖ σ} r (t) d t < μ (σ) + μ ([0, T] ∖ σ) = T .

□

Further, we consider arclength parameterized abnormal extremals.

Theorem 3.

Abnormal arclength parameterized extremal controls are piecewise constant

u_{1} (t) = cos α, u_{2} (t) = s_{2} (t) sin α, where s_{2} (t) \in {- 1, 1},

with switching times (discontinuity points of

s_{2} (t)

) differing by

\frac{π}{sin α}

. The corresponding trajectories are motions of a car along circular arcs of the minimal possible radius by angle π (except the first and the last arcs, which may have angles not greater than π).

Proof.

Abnormal extremals satisfy the statements of the PMP: the maximum condition, the Hamiltonian system (11), and nontriviality condition

h_{1}^{2} (t) + h_{2}^{2} (t) + h_{3}^{2} (t) \neq 0

for all

t \in [0, T]

From the maximum condition we have

h_{1} (t) = - R (t) sin α, h_{2} (t) = s_{2} (t) R (t) cos α, R (t) \geq 0 .

(12)

Here,

s_{2} (t) = sign h_{2} (t)

The vertical part of the Hamiltonian system (11) is given by

\{\begin{matrix} {\dot{h}}_{1} (t) = - s_{2} (t) sin α h_{3} (t), \\ {\dot{h}}_{2} (t) = cos α h_{3} (t), \\ {\dot{h}}_{3} (t) = s_{2} (t) sin α h_{1} (t) . \end{matrix}

(13)

This system preserves the Hamiltonian

H = cos α h_{1} + sin α | h_{2} |

and the Casimir

E = \frac{h_{1}^{2} + h_{3}^{2}}{2}

(see Figure 4). Since

H = 0

for abnormal extremals, the nontriviality condition implies

E > 0

Denote

a = \sqrt{2 E} > 0

. Let

h_{1} = a cos φ

h_{3} = a sin φ

, and

c = h_{2}

, where

φ \in [\frac{π}{2}, \frac{3 π}{2}]

as we have

h_{1} \leq 0

(see (12)). System (13) reads

\{\begin{matrix} \dot{φ} (t) = sin α s_{2} (t), \\ \dot{c} (t) = a cos α sin φ (t) . \end{matrix}

c (t_{1}) = 0

, for some

t_{1} \in [0, T]

, then

H = 0

implies

h_{1} (t_{1}) = 0

and

φ (t_{1}) = π \pm \frac{π}{2}

. Thus,

sin φ (t_{1}) = \pm 1

and from the second equation of the system, we have

\dot{c} (t) \neq 0

for

[t_{1} - ε, t_{1} + ε]

for sufficiently small

ε > 0

. Thus, the function

c (t)

changes its sign in a neighborhood of

t_{1}

. We see that

t_{1}

is an isolated point where

c (t)

changes its sign.

c (t) \neq 0

, then

s_{2} (t) = sign c (t)

(see (12)). The extremal control

u_{2} (t) = sign c (t) sin α

is piecewise constant with interval between switchings

Δ t = t_{2} - t_{1} = \frac{π}{sin α}

, as it follows from the first equation of the system with boundary values

φ (t_{1}) = π \pm \frac{π}{2}

φ (t_{2}) = π \mp \frac{π}{2}

The corresponding trajectory of the horizontal part of (11) is easy to find by direct integration. For time intervals between switchings, the point

(x (t), y (t))

moves along the circular arc with increments of angle

Δ θ = \pm sin α Δ t = \pm π

. □

5.3. Normal Case $H > 0$

In this section, we provide a qualitative analysis of the dynamics in the normal case

H > 0

. Note that the normal case appears iff

| ψ | < \frac{π}{2} + α

. In the normal case, we set the Hamiltonian value

H = 1

without loss of generality (see [20]).

The vertical part has the first integrals: the Hamiltonian

1 = H = \{\begin{matrix} h_{1} cos α + | h_{2} | sin α, & for α < | ψ | < \frac{π}{2} + α, \\ \sqrt{h_{1}^{2} + h_{2}^{2}}, & for | ψ | \leq α, \end{matrix}

(14)

and the Casimir

E = \frac{h_{1}^{2}}{2} + \frac{h_{3}^{2}}{2} .

(15)

In Figure 5, we show variants of the mutual arrangement of the level surface of the Hamiltonian

H = 1

, which consists of two half-planes glued with a segment of the cylinder, and the level surface of the Casimir

E \geq 0

, which is a cylinder.

For the description of the phase portrait of the vertical part, we use techniques of convex trigonometry [26]. The polar set to U is

\begin{matrix} U^{o} = \{(h_{1}, h_{2}) \in R^{2 *}| u_{1} h_{1} + u_{2} h_{2} \leq 1, (u_{1}, u_{2}) \in U\} = \\ \{(h_{1}, h_{2}) = (ρ cos ψ, ρ sin ψ) |\begin{matrix} for | ψ | \leq α : \sqrt{h_{1}^{2} + h_{2}^{2}} \leq 1, \\ for α < ψ < α + \frac{π}{2} : h_{1} cos α + h_{2} sin α \leq 1, \\ for - α - \frac{π}{2} < ψ < - α : h_{1} cos α - h_{2} sin α \leq 1 \end{matrix}\} . \end{matrix}

The corresponding functions of convex trigonometry are

\begin{matrix} {cos}_{U^{o}} ϕ^{o} = \{\begin{matrix} cos ϕ^{o}, & for | ϕ^{o} | \leq α, \\ cos α - (ϕ^{o} - α) sin α, & for | ϕ^{o} | > α, \end{matrix} \\ {sin}_{U^{o}} ϕ^{o} = \{\begin{matrix} sin ϕ^{o}, & for | ϕ^{o} | \leq α, \\ \pm (sin α + (ϕ^{o} - α) cos α), & for | ϕ^{o} | > α, \end{matrix} \end{matrix}

where

\pm = sign (ϕ^{o})

Along the extremal trajectories, we have

\begin{matrix} u_{1} = cos ϕ, u_{2} = sin ϕ, h_{1} = {cos}_{U^{o}} ϕ^{o}, h_{2} = {sin}_{U^{o}} ϕ^{o} . \end{matrix}

Denote

K (ϕ^{o}) = \frac{1}{2} {cos}_{U^{o}}^{2} ϕ^{o}

. The Casimir E can be seen as a total energy integral (sum of potential and kinetic energy)

E = \frac{h_{1}^{2}}{2} + \frac{h_{3}^{2}}{2} = \frac{h_{3}^{2}}{2} + K (ϕ^{o})

of conservative system with one degree of freedom [27]

\dot{ϕ^{o}} = h_{3}, {\dot{h}}_{3} = - K^{'} (ϕ^{o}) .

(16)

The phase portrait of this system is depicted in Figure 6.

Analyzing the phase portrait of (16), we conclude:

E = 0 \Rightarrow (ϕ^{o}, h_{3}) \equiv (\pm (α + cot α), 0)

is stable equilibrium;

E \in (0, \frac{1}{2}) \cup (\frac{1}{2}, + \infty) \Rightarrow

the trajectory

(ϕ^{o}, h_{3}) (t)

is periodic;

E = \frac{1}{2} \Rightarrow

either

(ϕ^{o}, h_{3}) \equiv (0, 0)

is unstable equilibrium or

(ϕ^{o}, h_{3}) (t)

is a separatrix of the saddle point.

6. Explicit Expression for Normal Extremals

6.1. Stratification of the Hamiltonian System Adjoint Variables Domain

In this section, we derive explicit expressions for the normal extremals. To this end, we stratify the domain of adjoint variables

h_{1}

h_{2}

, and

h_{3}

into five subdomains and introduce the coordinates

(ξ, k)

that rectify the phase portrait of the vertical part in each subdomain. Next, we integrate the horizontal part and derive explicit expressions for the extremal trajectories. This method was applied for the construction of optimal synthesis in several optimal control problems [28].

In Figure 7, we stratify the domain of the vertical part with respect to the value of the Casimir E, the value of

ϕ^{o}

, and the sign

s_{i} = sign h_{i}

i = 2, 3

. From the analysis in the previous section, we know that there are two generic types of motion:

| ϕ^{0} | < α

and

| ϕ^{0} | > α

, which are equivalent to

h_{1} > cos α

and

h_{1} < cos α

, respectively. We denote by the letter S the domain with the first type of motion and by O the domain with the second type of motion. Such notions are motivated by S — sub-Riemannian, and O — circle, as we will show next that the corresponding extremal trajectory (solution to the horizontal part) is given by arcs of sub-Riemannian geodesics in

{SE}_{2}

[4] and arcs of circular extremals (which correspond to motion of a car along circles of minimal possible radius).

Denote

E = \sqrt{2 E}

. There are five qualitative types of normal extremal trajectories:

1.: Arcs of noninflectional sub-Riemannian geodesics in ${SE}_{2}$ , joined by arcs of the circular extremals, when $cos α < E < 1$ (the subdomain $S_{1}^{\pm} \cup O_{1}^{\pm}$ ; $\pm = sign h_{2}$ );
2.: Arcs of inflectional sub-Riemannian geodesics in ${SE}_{2}$ joined by arcs of the circular extremals, when $E > 1$ (the subdomain $S_{2}^{\pm} \cup O_{2}^{\pm}$ ; here, $\pm = sign h_{3}$ in the S-domain, and $\pm = sign h_{2}$ in the O-domain);
3.: Arcs of the separatrix sub-Riemannian geodesics in ${SE}_{2}$ joined by an arc of the circular extremal, when $E = 1$ and $h_{1} < 1$ (the subdomain $S_{3}^{\pm \pm} \cup O_{3}^{\pm}$ ; here, $(\pm, \pm) = (sign h_{3}, sign h_{2})$ in the S-domain, and $\pm = sign h_{2}$ in the O-domain);
4.: The circular extremals, when $E \leq cos α$ (the subdomains $O_{4}^{\pm}$ ; here, $\pm = sign h_{2}$ correspond to the motion of the car clockwise or counterclockwise);
5.: The straight extremal (the ray), when $E = 1$ and $h_{1} = 1$ (the subdomain $S_{5}$ ).

Next, we derive explicit formulas for the extremals in each case. In order to avoid difficulties in the interpretation of formulas of convex trigonometry, which appeared very useful for qualitative analysis of the Hamiltonian system in the previous section, we return to the classical coordinates

(h_{1}, h_{2}, h_{3}) \in C \subset R^{3}

, where the vertical part (16) takes the form

\{\begin{matrix} {\dot{h}}_{1} = - h_{2} h_{3}, \\ {\dot{h}}_{2} = h_{1} h_{3}, \\ {\dot{h}}_{3} = h_{2} h_{1}, \end{matrix} for h_{1} > cos α; \{\begin{matrix} {\dot{h}}_{1} = - s_{2} sin α h_{3}, \\ {\dot{h}}_{2} = cos α h_{3}, \\ {\dot{h}}_{3} = s_{2} sin α h_{1}, \end{matrix} for h_{1} < cos α;

(17)

where

s_{2} = sign h_{2}

. The extremal controls are given by

(u_{1}, u_{2}) = (h_{1}, h_{2}), for h_{1} > cos α; (u_{1}, u_{2}) = (cos α, s_{2} sin α), for h_{1} < cos α;

(18)

and the horizontal part is given by

\{\begin{matrix} \dot{x} = u_{1} cos θ, \\ \dot{y} = u_{1} sin θ, \\ \dot{θ} = u_{2} . \end{matrix}

(19)

Note that for

h_{1} > cos α

, System (17)–(19) coincides with the system for sub-Riemannian geodesics in

{SE}_{2}

[4]. Thus, the solutions in the S-domains are given by arcs of sub-Riemannian geodesics. Denote by

σ : R \times C \to {SE}_{2} : (t, h^{0}) \mapsto σ (t, h^{0})

the flow from Id along the sub-Riemannian geodesic with the initial covector

h^{0} \in C

. The exact formula for the operator

σ

in the different domains is presented in the subsections below.

For

h_{1} < cos α

, System (17)–(19) is easily integrated, and the solutions are given by arcs of circles of radius

cot α

, which is the minimal possible turning radius for the car model. Denote by

ω : R \times C \to {SE}_{2} : (t, h^{0}) \mapsto ω (t, h^{0})

the corresponding flow from Id:

ω (t, h^{0}) = (cot α sin (t sin α), s_{2} cot α (1 - cos (t sin α)), s_{2} t sin α) .

(20)

The case

h_{1} = cos α

is joined to the case

h_{1} > cos α

when

s_{2} s_{3} < 0

, and joined to the case

h_{1} < cos α

, when

s_{2} s_{3} > 0

(see Figure 7). Here,

s_{2} = sign h_{2}

, and

s_{3} = sign h_{3}

To obtain a general solution for the horizontal part we separate the timeline by the intervals, in which the S or the O mode holds. We denote by

t_{0 i}

i \in N

, the instances of time when the dynamics switches (see Figure 8). Denote by

h^{0 i} \in C

the corresponding value of the adjoint covector, and by

q_{0 i} = q (t_{0 i}) \in {SE}_{2}

the corresponding point of the extremal trajectory. Due to left-invariance (see Remark 1), the arc of the trajectory starting from

q_{0 i}

is given by

\begin{matrix} q (t) = L_{q_{0 i}} σ (t - t_{0 i}, h^{0 i}) = q_{0 i} \cdot σ (t - t_{0 i}, h^{0 i}), & in S-domain, \\ q (t) = L_{q_{0 i}} ω (t - t_{0 i}, h^{0 i}) = q_{0 i} \cdot ω (t - t_{0 i}, h^{0 i}), & in O-domain . \end{matrix}

(21)

6.2. The Domain $cos α < E < 1$

Here, we consider the case when the initial covector

h^{0}

belongs to the subdomain

S_{1}^{\pm} \cup O_{1}^{\pm}

, where

\pm = sign h_{2} = : s_{2}

. Notice that in this case, the trajectory of the vertical part is periodic, and

s_{2}

is constant (see Figure 7). Denote the period by

T_{o} + T_{s}

, where

T_{o}

is the full time of motion in the O-domain, and

T_{s}

is the full time of motion in the S-domain.

Denote

k : = E

. Introduce the coordinates

(ξ, k) \in [- \frac{T_{s}}{2}, \frac{T_{s}}{2} + T_{o}] \times (cos α, 1)

(see Figure 9). Here,

ξ

is periodic with the period

T_{s} + T_{o}

For

h^{0} = (h_{10}, h_{20}, h_{30})

S_{1}^{\pm}

, where

\pm = s_{2} = sign h_{20}

\{\begin{matrix} h_{1} = k sn (ξ_{c} + ξ), \\ h_{2} = s_{2} dn (ξ_{c} + ξ), \\ h_{3} = - s_{2} k sn (ξ_{c} + ξ), \end{matrix}

(22)

where

ξ_{c} = K

ξ \in [- \frac{T_{s}}{2}, \frac{T_{s}}{2}]

. The time

T_{s}

of the full motion is computed as

T_{s} = 2 (F (arg (- \sqrt{k^{2} - {cos}^{2} α} + i cos α)) - K), where arg \in (\frac{π}{2}, π),

(23)

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = F (arg (- s_{2} h_{30} + i h_{10})) - ξ_{c}, where arg \in (0, π) .

Here, sn, cn, and dn are Jacobi elliptic functions; F is the elliptic integral of the first kind; and

K = F (\frac{π}{2})

is the complete elliptic integral of the first kind.

For

h^{0} = (h_{10}, h_{20}, h_{30})

O_{1}^{\pm}

, where

\pm = s_{2} = sign h_{20}

\{\begin{matrix} h_{1} = k cos (sin α (ξ_{c} + ξ)), \\ h_{2} = \frac{s_{2}}{sin α} (1 - k cos α cos (sin α (ξ_{c} + ξ))), \\ h_{3} = s_{2} k sin (sin α (ξ_{c} + ξ)), \end{matrix}

(24)

where

ξ_{c} = \frac{φ}{sin α} - \frac{T_{s}}{2}

ξ \in [\frac{T_{s}}{2}, \frac{T_{s}}{2} + T_{o}]

. The time

T_{o}

of the full motion is computed as

T_{o} = \frac{2}{sin α} (π - φ), where φ = arg (cos α + i \sqrt{k^{2} - {cos}^{2} α}) \in (0, \frac{π}{2}),

(25)

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = arg (h_{10} + i s_{2} h_{30}) / sin α - ξ_{c}, where arg \in (0, \frac{π}{2}) \cup (\frac{3 π}{2}, 2 π) .

The direct computation shows that (17) is rectified in the coordinates

(ξ, k)

ξ (t) = ξ_{t} = ξ_{0} + t, k = const .

Remark 5.

There is a symmetry between the solutions for

s_{2} = 1

and

s_{2} = - 1

: the substitution

(h_{1}, h_{2}, h_{3}) \to (h_{1}, - h_{2}, - h_{3})

translates a trajectory in

S_{1}^{\pm} \cup O_{1}^{\pm}

to the trajectory in

S_{1}^{\mp} \cup O_{1}^{\mp}

Remark 6.

The values

T_{s}

and

T_{o}

are computed as the minimal positive time instances when

\begin{matrix} (h_{1} (0), h_{3} (0)) = (cos α, - \sqrt{k^{2} - h_{1}^{2} (0)}), & (h_{1} (T_{s}), h_{3} (T_{s})) = (cos α, \sqrt{k^{2} - h_{1}^{2} (0)}), \\ (h_{1} (0), h_{3} (0)) = (cos α, \sqrt{k^{2} - h_{1}^{2} (0)}), & (h_{1} (T_{o}), h_{3} (T_{o})) = (cos α, - \sqrt{k^{2} - h_{1}^{2} (0)}) . \end{matrix}

Now we integrate the horizontal part and find the exact expression for the extremal trajectories. Recall that in the O-domain, the extremal trajectories

q (t) = ω (t, h^{0})

are given by arcs of the circles (20), and in the S-domain, the extremal trajectories

q (t) = σ (t, h^{0})

are given by arcs of sub-Riemannian geodesics. Exact formulas for the sub-Riemannian geodesics are found in [4]. In our notation, they read as:

\begin{matrix} σ (t, h^{0}) = (x (t), y (t), θ (t)), \\ x (t) = \frac{1}{k} (s β (t - E (am (ξ_{c} + ξ_{t})) + E (β)) - c β (dn (ξ_{c} + ξ_{t}) - \sqrt{1 - k^{2} s β^{2}})), \\ y (t) = \frac{s_{2}}{k} (c β (t - E (am (ξ_{c} + ξ_{t})) + E (β)) + s β (dn (ξ_{c} + ξ_{t}) - \sqrt{1 - k^{2} s β^{2}})), \\ θ (t) = s_{2} (- β + am (ξ_{c} + ξ_{t})), \end{matrix}

(26)

where

β = am (ξ_{0} + ξ_{c}), s β = sin β = sn (ξ_{0} + ξ_{c}), c β = cos β = cn (ξ_{0} + ξ_{c}) .

Here, am, sn, cn, and dn are Jacobi elliptic functions, and E is the elliptic integral of the second kind.

To obtain a general solution for the horizontal part, we separate the timeline by the intervals

[0, t] = [0, t_{01}) \cup [t_{01}, t_{02}) \cup [t_{02}, t_{03}) \cup \dots \cup [t_{0 (2 m - 1)}, t_{0 (2 m + 1)}) \cup [t_{0 (2 m + 1)}, t_{0 (2 m + 1)} + \tilde{t}],

in which the S or the O mode holds (see Figure 10). Here, we have

\tilde{t} = t - t_{01} - m (T_{o} + T_{s}), m = [\frac{t - t_{01}}{T_{o} + T_{s}}], t_{01} = min \{t > 0 | h_{1} (t) = cos α\},

(27)

where the square brackets denote the integer part.

Denote

h_{o}^{\pm} = (cos α, \pm sin α, \pm \sqrt{k^{2} - {cos}^{2} α}), h_{s}^{\pm} = (cos α, \pm sin α, \mp \sqrt{k^{2} - {cos}^{2} α})

(28)

the initial covector for the full arc of O and S segment of the extremals (see Figure 7), and

q_{ω}^{\pm} = ω (T_{o}, h_{o}^{\pm}), q_{σ}^{\pm} = σ (T_{s}, h_{s}^{\pm})

(29)

the end point of the corresponding extremal trajectory.

Now we obtain the resulting formula for the extremal trajectories by usage of (21).

Theorem 4.

For

h^{0} \in S_{1}^{\pm}

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} σ (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot {(q_{σ}^{\pm} \cdot q_{ω}^{\pm})}^{m} \cdot ω (\tilde{t}, h_{o}^{\pm}), & for 0 \leq \tilde{t} < T_{o}, \\ q_{01} \cdot {(q_{σ}^{\pm} \cdot q_{ω}^{\pm})}^{m} \cdot q_{ω}^{\pm} \cdot σ (\tilde{t} - T_{o}, h_{s}^{\pm}), & for T_{o} \leq \tilde{t} < T_{o} + T_{s}, \end{matrix}

where

q_{01} = σ (t_{01}, h^{0})

For

h^{0} \in O_{1}^{\pm}

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} ω (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot {(q_{σ}^{\pm} \cdot q_{ω}^{\pm})}^{m} \cdot σ (\tilde{t}, h_{s}^{\pm}), & for 0 \leq \tilde{t} < T_{s}, \\ q_{01} \cdot {(q_{σ}^{\pm} \cdot q_{ω}^{\pm})}^{m} \cdot q_{σ}^{\pm} \cdot ω (\tilde{t} - T_{s}, h_{o}^{\pm}), & for T_{s} \leq \tilde{t} < T_{o} + T_{s}, \end{matrix}

where

q_{01} = ω (t_{01}, h^{0})

For both cases, the operators σ and ω are defined by (26) and (20); the values

T_{s}

and

T_{o}

are defined by (23) and (25); the values

\tilde{t}

, m, and

t_{01}

are defined by (27); and the initial covectors

h_{o}^{\pm}

and

h_{s}^{\pm}

and the corresponding end points

q_{σ}^{\pm}

and

q_{ω}^{\pm}

are defined by (28) and (29).

In Figure 11, we show an example of the extremal trajectories in the domain

S_{1}^{\pm} \cup O_{1}^{\pm}

Remark 7.

The power of an element

q \in {SE}_{2}

is defined as

q^{m} = \{\begin{matrix} Id, & for m = 0, \\ \overset{m times}{\overset{︷}{q \cdot \dots \cdot q}}, & for m \geq 1 . \end{matrix}

6.3. The Domain $E > 1$

Here, we consider the case when the initial covector

h^{0}

belongs to the subdomain

S_{2}^{\pm} \cup O_{2}^{\pm}

, where

\pm = sign h_{3} = : s_{3}

in the S-domain, and

\pm = sign h_{2} = : s_{2}

in the O-domain. Notice that in this case, the trajectory of the vertical part is periodic (see Figure 7). Denote the period by

2 (T_{o} + T_{s})

, where

T_{o}

is the full time of motion in the O-domain, and

T_{s}

is the full time of motion in the S-domain.

Denote

k : = \frac{1}{E}

. Introduce the coordinates

(ξ, k) \in [- \frac{T_{s}}{2}, \frac{3 T_{s}}{2} + 2 T_{o}] \times (0, 1)

(see Figure 12). Here,

ξ

is periodic with the period

2 (T_{s} + T_{o})

For

h^{0} = (h_{10}, h_{20}, h_{30})

S_{2}^{\pm}

, where

\pm = s_{3} = sign h_{30}

\{\begin{matrix} h_{1} = s_{3} sn (\frac{ξ_{c} + ξ}{k}), \\ h_{2} = - cn (\frac{ξ_{c} + ξ}{k}), \\ h_{3} = \frac{s_{3}}{k} dn (\frac{ξ_{c} + ξ}{k}), \end{matrix}

(30)

where the time

T_{s}

of the full motion is computed as

T_{s} = 2 k (F (α + \frac{π}{2}) - K) .

(31)

For

h^{0} \in S_{2}^{+}

, we have

ξ_{c} = k K, ξ \in [- \frac{T_{s}}{2}, \frac{T_{s}}{2}],

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = k F (arg (- h_{20} + i h_{10})) - ξ_{c}, where arg \in (0, π) .

For

h^{0} \in S_{2}^{-}

, we have

ξ_{c} = k (F (\frac{π}{2} - 2 K)) - (T_{o} + \frac{T_{s}}{2}), ξ \in [\frac{T_{s}}{2} + T_{o}, \frac{3 T_{s}}{2} + T_{o}],

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = k F (arg (- h_{20} - i h_{10})) - ξ_{c}, where arg \in (- π, 0) .

For

h^{0} = (h_{10}, h_{20}, h_{30})

O_{2}^{\pm}

, where

\pm = s_{2} = sign h_{20}

\{\begin{matrix} h_{1} = \frac{1}{k} cos (sin α (ξ_{c} + ξ)), \\ h_{2} = \frac{s_{2}}{sin α} (1 - \frac{cos α}{k} cos (sin α (ξ_{c} + ξ))), \\ h_{3} = \frac{s_{2}}{k} sin (sin α (ξ_{c} + ξ)), \end{matrix}

(32)

where the time

T_{o}

of the full motion is computed as

T_{o} = \frac{2}{sin α} (π - φ), where φ = arg (k cos α + i \sqrt{1 - k^{2} {cos}^{2} α}) \in (0, \frac{π}{2}) .

(33)

For

h^{0} \in O_{2}^{+}

, we have

ξ_{c} = \frac{φ}{sin α} - \frac{T_{s}}{2}, ξ \in [\frac{T_{s}}{2}, \frac{T_{s}}{2} + T_{o}],

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = arg (h_{10} + i h_{30}) / sin α - ξ_{c}, where arg \in (- \frac{π}{2}, \frac{π}{2}) .

For

h^{0} \in O_{2}^{-}

, we have

ξ_{c} = \frac{φ}{sin α} - (\frac{T_{s}}{2} + T_{o}), ξ \in [\frac{3 T_{s}}{2} + T_{o}, \frac{3 T_{s}}{2} + 2 T_{o}],

and the initial value

ξ_{0}

is expressed via

h^{0}

as follows:

ξ_{0} = arg (h_{10} - i h_{30}) / sin α - ξ_{c}, where arg \in (0, \frac{π}{2}) \cup (\frac{3 π}{2}, 2 π) .

The direct computation shows that (17) is rectified in the coordinates

(ξ, k)

ξ (t) = ξ_{t} = ξ_{0} + t, k = const .

Now we integrate the horizontal part and find the exact expression for the extremal trajectories. Recall that in the O-domain, the extremal trajectories

q (t) = ω (t, h^{0})

are given by arcs of the circles (20), and in the S-domain, the extremal trajectories

q (t) = σ (t, h^{0})

are given by arcs of sub-Riemannian geodesics. Exact formulas for the sub-Riemannian geodesics are found in [4]. In our notation, they read as:

\begin{matrix} σ (t, h^{0}) = (x (t), y (t), θ (t)), \\ x (t) = s_{3} k (s β (E (β) - E (am (\frac{ξ_{c} + ξ_{t}}{k})) + \frac{t}{k}) + \sqrt{1 - k^{2} s β^{2}} (c β - cn (\frac{ξ_{c} + ξ_{t}}{k}))), \\ y (t) = s_{3} (k^{2} s β (c β - cn (\frac{ξ_{c} + ξ_{t}}{k})) - \sqrt{1 - k^{2} s β^{2}} (E (β) - E (am (\frac{ξ_{c} + ξ_{t}}{k})) + \frac{t}{k})), \\ cos θ (t) = k^{2} s β sn (\frac{ξ_{c} + ξ_{t}}{k}) + \sqrt{1 - k^{2} s β^{2}} dn (\frac{ξ_{c} + ξ_{t}}{k}), \\ sin θ (t) = k (s β dn (\frac{ξ_{c} + ξ_{t}}{k}) - \sqrt{1 - k^{2} s β^{2}} sn (\frac{ξ_{c} + ξ_{t}}{k})), \end{matrix}

(34)

where

β = am (\frac{ξ_{0} + ξ_{c}}{k}), s β = sin β = sn (\frac{ξ_{0} + ξ_{c}}{k}), c β = cos β = cn (\frac{ξ_{0} + ξ_{c}}{k}) .

Here, am, sn, cn, and dn are Jacobi elliptic functions, and E is the elliptic integral of the second kind.

To obtain a general solution for the horizontal part, we separate the timeline by the intervals

[0, t] = [0, t_{01}) \cup [t_{01}, t_{02}) \cup [t_{02}, t_{03}) \cup \dots \cup [t_{0 (2 m - 1)}, t_{0 (2 m + 1)}) \cup [t_{0 (2 m + 1)}, t_{0 (2 m + 1)} + \tilde{t}],

in which the S or the O mode holds (see Figure 10). Here, we have

\tilde{t} = t - t_{01} - [\frac{m}{2}] (2 T_{o} + 2 T_{s}), m = [\frac{t - t_{01}}{T_{o} + T_{s}}], t_{01} = min \{t > 0 | h_{1} (t) = cos α\},

(35)

where the square brackets denote the integer part.

Denote

h_{o}^{\pm} = (cos α, \pm sin α, \pm \sqrt{\frac{1}{k^{2}} - {cos}^{2} α}), h_{s}^{\pm} = (cos α, \pm sin α, \mp \sqrt{\frac{1}{k^{2}} - {cos}^{2} α})

(36)

the initial covector for the full arc of O and S segment of the extremals (see Figure 7), and denote

q_{ω}^{\pm} = ω (T_{o}, h_{o}^{\pm}), q_{σ}^{\pm} = σ (T_{s}, h_{s}^{\pm})

(37)

the end point of the corresponding extremal trajectory.

Now we obtain the resulting formula for the extremal trajectories by usage of (21).

Theorem 5.

For

h^{0} \in S_{2}^{\pm}

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} σ (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot q_{m} \cdot ω (\tilde{t}, h_{o}^{\pm}), & for 0 \leq \tilde{t} < T_{o}, \\ q_{01} \cdot q_{m} \cdot q_{ω}^{\pm} \cdot σ (\tilde{t} - T_{o}, h_{s}^{\mp}), & for T_{o} \leq \tilde{t} < T_{o} + T_{s}, \\ q_{01} \cdot q_{m} \cdot q_{ω}^{\pm} \cdot q_{σ}^{\mp} \cdot ω (\tilde{t} - T_{o} - T_{s}, h_{o}^{\mp}), & for T_{o} + T_{s} \leq \tilde{t} < 2 T_{o} + T_{s}, \\ q_{01} \cdot q_{m} \cdot q_{ω}^{\pm} \cdot q_{σ}^{\mp} \cdot q_{ω}^{\mp} \cdot σ (\tilde{t} - 2 T_{o} - T_{s}, h_{s}^{\pm}), & for 2 T_{o} + T_{s} \leq \tilde{t} < 2 T_{o} + 2 T_{s}, \end{matrix}

where

q_{01} = σ (t_{01}, h^{0})

, and

q_{m} = {(q_{ω}^{\pm} \cdot q_{σ}^{\mp} \cdot q_{ω}^{\mp} \cdot q_{σ}^{\pm})}^{[\frac{m}{2}]}

For

h^{0} \in O_{2}^{\pm}

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} ω (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot q_{m} \cdot σ (\tilde{t}, h_{s}^{\mp}), & for 0 \leq \tilde{t} < T_{s}, \\ q_{01} \cdot q_{m} \cdot q_{σ}^{\mp} \cdot ω (\tilde{t} - T_{s}, h_{o}^{\mp}), & for T_{s} \leq \tilde{t} < T_{s} + T_{o}, \\ q_{01} \cdot q_{m} \cdot q_{σ}^{\mp} \cdot q_{ω}^{\mp} \cdot σ (\tilde{t} - T_{s} - T_{o}, h_{s}^{\pm}), & for T_{s} + T_{o} \leq \tilde{t} < 2 T_{s} + T_{o}, \\ q_{01} \cdot q_{m} \cdot q_{σ}^{\mp} \cdot q_{ω}^{\mp} \cdot q_{σ}^{\pm} \cdot ω (\tilde{t} - 2 T_{s} - T_{o}, h_{o}^{\pm}), & for 2 T_{s} + T_{o} \leq \tilde{t} < 2 T_{s} + 2 T_{o}, \end{matrix}

where

q_{01} = ω (t_{01}, h^{0})

, and

q_{m} = {(q_{σ}^{\mp} \cdot q_{ω}^{\mp} \cdot q_{σ}^{\pm} \cdot q_{ω}^{\pm})}^{[\frac{m}{2}]} .

For both cases, the operators σ and ω are defined by (34) and (20); the values

T_{s}

and

T_{o}

are defined by (31) and (33); the values

\tilde{t}

, m, and

t_{01}

are defined by (35); and the initial covectors

h_{o}^{\pm}

and

h_{s}^{\pm}

and the corresponding end points

q_{σ}^{\pm}

and

q_{ω}^{\pm}

are defined by (36) and (37).

In Figure 13, we show an example of the extremal trajectories in the domain

S_{2}^{\pm} \cup O_{2}^{\pm}

6.4. The Domain $E = 1$ and $h_{1} < 1$

In this subsection, we consider the case when the initial covector

h^{0} = (h_{10}, h_{20}, h_{30})

belongs to the subset

S_{3}^{s_{3} s_{2}} \cup O_{3}^{s_{2}}

, where

(s_{2}, s_{3}) = (sign h_{2}, sign h_{3})

. Notice that in this case, the trajectory of the vertical part is a separatrix of the saddle point (see Figure 7). It consists of not greater than three arcs of the types S, O, and S.

Introduce the coordinate

ξ \in (- \infty, \infty)

by (39), (40), and (39) (see Figure 14) in the domains

\{\begin{matrix} ξ \in (- \infty, - \frac{T_{o}}{2}), & for s_{2} s_{3} > 0, h_{10} > cos α; \\ ξ \in [- \frac{T_{o}}{2}, \frac{T_{o}}{2}], & for h_{10} \leq cos α; \\ ξ \in (\frac{T_{o}}{2}, \infty), & for s_{2} s_{3} < 0, h_{10} > cos α, \end{matrix}

where

T_{o} = \frac{2}{sin α} (π - α), T_{s} = - arctanh (cos α) .

(38)

For

h^{0} \in S_{3}^{s_{2} s_{3}}

, we have:

\{\begin{matrix} h_{1} = - s_{2} s_{3} tanh (ξ_{c} + ξ), \\ h_{2} = s_{2} sech (ξ_{c} + ξ), \\ h_{3} = s_{3} sech (ξ_{c} + ξ), \end{matrix}

(39)

where

ξ_{c} = s_{2} s_{3} (\frac{T_{o}}{2} + T_{s}) .

For

h^{0} \in O_{3}^{s_{2}}

\{\begin{matrix} h_{1} = cos (sin α (ξ_{c} + ξ)), \\ h_{2} = \frac{s_{2}}{sin α} (1 - cos α cos (sin α (ξ_{c} + ξ))), \\ h_{3} = s_{2} sin (sin α (ξ_{c} + ξ)), \end{matrix}

(40)

where

ξ_{c} = \frac{π}{sin α} .

The direct computation shows that (17) is rectified:

ξ (t) = ξ_{t} = ξ_{0} + t,

where

ξ_{0} = \{\begin{matrix} - s_{2} s_{3} arctanh (h_{10}) - (\frac{T_{o}}{2} + T_{s}), & for h_{10} > cos α; \\ arg (h_{10} + i s_{2} s_{3} \sqrt{1 - h_{10}^{2}}) / sin α - ξ_{c}, & for h_{10} \leq cos α, \end{matrix}

where

arg \in (0, 2 π) .

Now we integrate the horizontal part and find the exact expression for the extremal trajectories. Recall that in the O-domain, the extremal trajectories

q (t) = ω (t, h^{0})

are given by arcs of the circles (20), and in the S-domain, the extremal trajectories

q (t) = σ (t, h^{0})

are given by arcs of sub-Riemannian geodesics. Exact formulas for the sub-Riemannian geodesics are found in [4]. In our notation, they read as:

\begin{matrix} σ (t, h^{0}) = (x (t), y (t), θ (t)), \\ x (t) = - s_{2} s_{3} (sech (\bar{ξ_{0}}) \tilde{x} (t) + tanh (\bar{ξ_{0}}) \tilde{y} (t)), \\ y (t) = s_{3} (tanh (\bar{ξ_{0}}) \tilde{x} (t) - sech (\bar{ξ_{0}}) \tilde{y} (t)), \\ cos θ (t) = sech (\bar{ξ_{0}}) sech (\bar{ξ_{0}} + t) + tanh (\bar{ξ_{0}}) tanh (\bar{ξ_{0}} + t), \\ sin θ (t) = sech (\bar{ξ_{0}}) sech (\bar{ξ_{0}} + t) - sech (\bar{ξ_{0}} + t) tanh (\bar{ξ_{0}}), \end{matrix}

(41)

where

\begin{matrix} \tilde{x} (t) = sech (\bar{ξ_{0}}) - sech (\bar{ξ_{0}} + t), \\ \tilde{y} (t) = t + tanh (\bar{ξ_{0}}) - tanh (\bar{ξ_{0}} + t), \\ \bar{ξ_{0}} = ξ_{0} + s_{2} s_{3} (\frac{T_{o}}{2} + T_{s}) . \end{matrix}

Denote the first switching time by

t_{01} = min \{t > 0 | h_{1} (t) = cos α\} .

(42)

The initial covector for the full O segment of the extremals (see Figure 7), denoted by

h_{o}^{\pm} = (cos α, \pm sin α, \pm sin α)

(43)

and the end point of the corresponding trajectory denoted by

q_{ω}^{\pm} = ω (T_{o}, h_{o}^{\pm}) .

(44)

Denote

h_{s}^{\pm} = (cos α, \pm sin α, \mp sin α)

(45)

the covector, when switching from the O to the S segment appears.

Now we obtain the resulting formula for the extremal trajectories by usage of (21).

Theorem 6.

For

h^{0} \in S_{3}^{s_{3} s_{2}}

s_{3} s_{2} > 0

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} σ (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot ω (t - t_{01}, h_{o}^{s_{2}}), & for t_{01} \leq t < t_{01} + T_{o}, \\ q_{01} \cdot q_{ω}^{s_{2}} \cdot σ (t - T_{o} - t_{01}, h_{s}^{s_{2}}), & for t \geq t_{01} + T_{o}, \end{matrix}

where

q_{01} = σ (t_{01}, h^{0})

For

h^{0} \in O_{3}^{s_{2}}

, the corresponding extremal trajectory is given by

q (t) = \{\begin{matrix} ω (t, h^{0}), & for t \in [0, t_{01}), \\ q_{01} \cdot σ (t - t_{01}, h_{s}^{s_{2}}), & for t \geq t_{01}, \end{matrix}

where

q_{01} = ω (t_{01}, h^{0})

For

h^{0} \in S_{3}^{s_{3} s_{2}}

s_{3} s_{2} < 0

, the corresponding extremal trajectory is given by

\begin{matrix} q (t) = σ (t, h^{0}), & for t \geq 0 . \end{matrix}

For all cases, the operators σ and ω are defined by (41) and (20), the values

T_{s}

and

T_{o}

are defined by (38), the value

t_{01}

is defined by (42), the initial covectors

h_{o}^{\pm}

and

h_{s}^{\pm}

are defined by (43) and (45), and the corresponding end point

q_{ω}^{\pm}

is defined by (44).

In Figure 15, we show an example of the extremal trajectories in the set

S_{3} \cup O_{3}

6.5. The Domain $E \leq cos α$

When

h^{0} \in O_{4}^{\pm} = O_{4}^{s_{2}}

, the Hamiltonian system (17)–(19) is easily integrated:

\{\begin{matrix} h_{1} (t) & = h_{10} cos (sin α t) - h_{30} s_{2} sin (sin α t), \\ h_{2} (t) & = s_{2} cot α (h_{10} + h_{20} s_{2} tan α - h_{10} cos (sin α t) + s_{2} h_{30} sin (sin α t)), \\ h_{3} (t) & = h_{30} cos (sin α t) + h_{10} s_{2} sin (sin α t), \end{matrix}

(46)

(x (t), y (t), θ (t)) = (cot α sin (t sin α), s_{2} cot α (1 - cos (t sin α)), s_{2} t) .

The extremal trajectories are given by circles (20) (see Figure 16).

Theorem 7.

For

h^{0} \in O_{4}^{s_{2}}

, the corresponding extremal trajectory is given by the circle

x (t) = cot α sin (t sin α), y (t) = s_{2} cot α (1 - cos (t sin α)), θ (t) = s_{2} t sin α, t \geq 0 .

6.6. The Domain $E = 1$ and $h_{1} = 1$

When

h^{0} \in S_{5}

, i.e.,

h^{0} = (1, 0, 0)

, the Hamiltonian system (17)–(19) is reduced to

\dot{x} = 1, \dot{y} = \dot{θ} = 0, {\dot{h}}_{1} = {\dot{h}}_{2} = {\dot{h}}_{3} = 0,

which is easily integrated

x (t) = t, y (t) = 0, θ (t) = 0, h_{1} (t) = 1, h_{2} (t) = 0, h_{3} (t) = 0 .

Theorem 8.

For

h^{0} \in S_{5}

, i.e.,

h^{0} = (1, 0, 0)

, the corresponding extremal trajectory is the ray

x (t) = t, y (t) = θ (t) = 0, t \geq 0 .

(47)

The extremal trajectory for

h^{0} \in S_{5}

is depicted in Figure 17.

7. Optimality of Extremal Trajectories

7.1. General Upper Bound of Cut Time

The limitation of the Pontryagin maximum principle is that it is only a necessary but not a sufficient condition of optimality. It provides a Hamiltonian system for the extremals, which are first-order candidates for being optimal among all admissible trajectories of a control system. An extremal loses its optimality at a cut point [20].

In this section, we provide an upper bound of the cut time for a time-optimal problem

\begin{matrix} \dot{q} = u_{1} X_{1} + u_{2} X_{2}, q = (x, y, θ) \in {SE}_{2}, u = (u_{1}, u_{2}) \in U \subset R^{2}, \end{matrix}

(48)

\begin{matrix} X_{1} = cos θ \frac{\partial}{\partial x} + sin θ \frac{\partial}{\partial y}, X_{2} = \frac{\partial}{\partial θ}, \end{matrix}

(49)

\begin{matrix} q (0) = q_{0}, q (t_{1}) = q_{1}, \end{matrix}

(50)

\begin{matrix} t_{1} \to min . \end{matrix}

(51)

Recall that the cut time for a trajectory

q (\cdot)

of an optimal control problem is defined as

t_{cut} (q (\cdot)) = sup \{T > 0 ∣ {q|}_{[0, T]} is optimal\} .

A trajectory

q (t) = (x (t), y (t), θ (t))

t \in R

, of the control system (48) is called quasiperiodic with a period

T > 0

if the following two conditions hold:

(1): $θ (t) = θ (t + T)$ for all $t \in R$ ;
(2): There exists a linear function $l : R^{2} \to R$ , $l \neq 0$ such that

$l (x (t + T), y (t + T)) = l (x (t), y (t)) for all t \in R .$

(52)

Proposition 2.

Let the time-optimal Problem (48)–(51) satisfies the hypotheses:

(1): $u_{1}^{2} + u_{2}^{2} \leq 1$ and $u_{1} > 0$ for any $u = (u_{1}, u_{2}) \in U$ ;
(2): $(1, 0) \in U$ .

Let a trajectory

q (t) = (x (t), y (t), θ (t))

t \in R

, of the control system (48) be quasiperiodic with a period

T > 0

. Assume moreover that the curve

γ (t) = (x (t), y (t))

t \in R

, is not a straight line.

Then, the cut time of the trajectory

q (\cdot)

for the time-optimal problem (48)–(51) admits an upper bound

t_{cut} (q (\cdot)) \leq 2 T .

(53)

Proof.

Consider the function l from Condition (52). The function

f (t) = l (x (t), y (t))

is T-periodic and continuous, thus it attains a maximum

M = f (τ)

τ \in [0, T]

. The curve

γ (t)

is tangent to the straight line

l^{- 1} (M)

for each value of the parameter

t = τ + n T

n \in Z

. Moreover,

γ

is distinct from the line

l^{- 1} (M)

, thus the control

u (\cdot)

corresponding to the trajectory

q (\cdot)

satisfies the inequality

u (t) ≢ 1

. On the other hand, the vector

{(\dot{x}, \dot{y})|}_{τ} = {u_{1} (cos θ, sin θ)|}_{τ}

is collinear with the line

l^{- 1} (M)

, as well as the vector

{(cos θ, sin θ)|}_{τ}

Consider now the trajectory

\hat{q} (\cdot)

of the control system (48) with the initial condition

\hat{q} (0) = q (τ)

and the control

u \equiv (1, 0)

. There exists

{\hat{t}}_{1} > 0

such that

\hat{q} ({\hat{t}}_{1}) = q (τ + T)

. Since

u (t) ≢ 1

, then

{\hat{t}}_{1} < T

. Indeed,

(\hat{x}, \hat{y}) (t)

t \in [0, {\hat{t}}_{1}]

, is a straight line segment connecting the points

(x, y) (τ)

and

(x, y) (τ + T)

, while

(x, y) (t)

t \in [τ, τ + T]

, is a curvilinear arc connecting the same points (see Figure 18). Thus, the trajectory

{q|}_{[τ, τ + T]}

is not optimal for Problem (48)–(51).

The arc

{q|}_{[0, 2 T]}

is not optimal as well since it contains a non-optimal arc

{q|}_{[τ, τ + T]}

. The required bound (53) follows. □

Remark 8.

It is obvious that Proposition 2 is applicable to the time-optimal Problem (2), (3) on

{SE}_{2}

with control in a circular sector.

7.2. Optimality of Extremals for $cos α < E < 1$

In this subsection, we analyze optimality of the extremal trajectories for

h^{0} \in S_{1}^{\pm} \cup O_{1}^{\pm}

, described in Section 6.2.

Proposition 3.

Let

q (t)

t \in R^{+}

, be an extremal trajectory in the time-optimal Problem (2), (3). Assume that the corresponding initial covector

h^{0} \in S_{1}^{\pm} \cup O_{1}^{\pm}

, and let

T_{o}

and

T_{s}

be the traveling time along full O and full S arc, given by (25) and (23).

The trajectory

q (t)

t \in [0, t_{1}]

, is not optimal if

t_{1} > 2 (T_{o} + T_{s})

. In particular,

t_{cut} (q (\cdot)) \leq 2 (T_{o} + T_{s}) .

Proof.

First, we fix the sign

h^{0} \in S_{1}^{+} \cup O_{1}^{+}

. The proof for

h^{0} \in S_{1}^{-} \cup O_{1}^{-}

follows by Remark 5.

To prove the proposition, we rely on the general upper bound of cut time in Proposition 2, where obviously the Hypotheses (1) and (2) hold. Let us show that the extremal trajectory

q (t)

is quasiperiodic with the period

T = T_{o} + T_{s}

. To this end, without loss of generality, we fix the initial covector

h^{0} = h_{o}^{+} = (cos α, sin α, \sqrt{k^{2} - {cos}^{2} α})

corresponding to the beginning of the circular arc of the extremal

q (t)

. The result for another choice of initial covector follows by left-invariance (21). Denote by

h_{s}^{+} = (cos α, sin α, - \sqrt{k^{2} - {cos}^{2} α})

the covector at the end of the circular arc (see Figure 7). Periodicity by

θ

follows by direct computation from Theorem 4:

ω^{θ} (T_{o}, h_{o}^{+}) + σ^{θ} (T_{s}, h_{s}^{+}) = 2 π,

where by

ω^{θ} (t, h_{o}^{+})

and

σ^{θ} (t, h_{s}^{+})

, we denote the

θ

-component of the operators

ω (t, h_{o}^{+})

and

σ (t, h_{s}^{+})

, which are the flows from Id along the O and S arcs, respectively. Thus,

θ (T) - θ (0) = 2 π \Rightarrow θ (T) = θ (0) (mod 2 π), \sin ce θ \in S^{1} = R / 2 π Z .

Now, note that the extremal control

(u_{1}, u_{2}) (t)

is periodic with the period

T = T_{o} + T_{s}

, and thus the components

(x, y) (t)

of the corresponding extremal trajectory satisfy

(x, y) (2 T) - (x, y) (T) = (x, y) (T) - (x, y) (0) .

Thus, by Proposition 2, the extremal trajectory

q (t)

is quasiperiodic and admits the upper bound for the cut time

t_{cut} (q (\cdot)) \leq T = 2 (T_{o} + T_{s})

. □

7.3. Optimality of Extremals for $E > 1$

In this subsection, we analyze optimality of the extremal trajectories for

h^{0} \in S_{2}^{\pm} \cup O_{2}^{\pm}

, described in Section 6.3.

Proposition 4.

Let

q (t)

t \in R^{+}

, be an extremal trajectory in the time-optimal Problem (2), (3). Assume that the corresponding initial covector

h^{0} \in S_{2}^{\pm} \cup O_{2}^{\pm}

, and let

T_{o}

and

T_{s}

be the traveling time along full O and full S arc, given by (33) and (31).

The trajectory

q (t)

t \in [0, t_{1}]

, is not optimal if

t_{1} > 4 (T_{o} + T_{s})

. In particular,

t_{cut} (q (\cdot)) \leq 4 (T_{o} + T_{s}) .

Proof.

The proof is similar to the proof of Proposition 4. The only difference appears in the periodicity by

θ

. In the case

h^{0} \in S_{2}^{\pm} \cup O_{2}^{\pm}

, we have

θ (T) - θ (0) = 0

T = 2 (T_{o} + T_{s})

, which follows from Theorem 5. □

7.4. Optimality of Separatrix Extremals ( $E = 1$ , $h_{10} \neq 1$ )

In this subsection, we analyze optimality of extremal trajectories for

h^{0} \in S_{3} \cup O_{3}

, described in Section 6.4.

Proposition 5.

Let

q (t)

t \in R^{+}

, be an extremal trajectory in Problem (2), (3). Assume that the initial covector

h^{0} \in S_{3} \cup O_{3}

. The trajectory

q (t)

is optimal for

t \in [0, t_{01}]

, i.e, it is optimal before the first switching. In particular, for

h^{0} \in S_{3}^{\pm \mp}

, the trajectory

q (t)

is optimal up to infinity.

Proof.

The extremal trajectory

q (t)

is given by Theorem 6. For

h^{0} \in O_{2}^{\pm}

, the corresponding trajectory

q (t)

t \in [0, t_{01}]

is given by an arc of circular extremal, which is optimal as we prove in Proposition 6. For

h^{0} \in S_{3}

, the corresponding extremal trajectory before the first switching is given by an arc of separatrix sub-Riemannian geodesic in

{SE}_{2}

, which is optimal, as proved in [4]. In particular, for

h^{0} \in S_{3}^{\pm \mp}

, the trajectory

q (t)

is optimal up to infinity since it has no switchings. □

Remark 9.

An upper bound of the cut time for the separatrix extremals of general type, which have the switching points, remains an open problem.

7.5. Optimality of Circular Trajectories, $E \leq$ `cos` $α$

In this subsection, we analyze optimality of circular extremals described in Section 6.5.

Proposition 6.

Let

q (t)

t \in R

, be an extremal trajectory in the time-optimal Problem (2), (3). Assume that

q (\cdot)

is of circular type and let

T_{o} = \frac{2 π}{sin α}

be its period.

A trajectory

q (t)

t \in [0, t_{1}]

, is optimal iff

t_{1} \in [0, T_{o})

. In particular,

t_{cut} (q (\cdot)) = T_{o} .

Moreover, for any

t_{1} \in [0, T_{o})

, the trajectory

q (t)

t \in [0, t_{1}]

, is a unique optimal trajectory connecting

q_{0}

q (t_{1})

Proof.

First of all, a full circular arc

q (t)

t \in [0, T_{o}]

, is not optimal since it connects the same points

q (0) = q (T_{o})

in a positive time

T_{o}

Now we prove that for any

t_{1} \in (0, T_{o})

, the trajectory

q (t)

t \in [0, t_{1}]

, is a unique optimal trajectory connecting

q_{0}

q (t_{1})

. By contradiction, suppose that there is another trajectory

\tilde{q} (t) = (\tilde{θ}, \tilde{x}, \tilde{y}) (t)

t \in [0, {\tilde{t}}_{1}]

, with a control

({\tilde{u}}_{1}, {\tilde{u}}_{2}) (t)

such that

{\tilde{t}}_{1} \leq t_{1}

and

\tilde{q} (0) = q_{0}

\tilde{q} ({\tilde{t}}_{1}) = q (t_{1})

. We have

\dot{\tilde{θ}} (t) = {\tilde{u}}_{2} (t) \leq sin α

, moreover, this inequality becomes strict on a nonempty interval since the trajectory

\tilde{q} (\cdot)

is not circular. Thus,

t_{1} sin α = θ (t_{1}) = \tilde{θ} ({\tilde{t}}_{1}) = \int_{0}^{{\tilde{t}}_{1}} {\tilde{u}}_{2} (t) d t < {\tilde{t}}_{1} sin α \leq t_{1} sin α,

whence

t_{1} sin α < t_{1} sin α

, a contradiction. □

7.6. Optimality of the Straight Trajectory ( $E = 1$ , $h_{10} = 1$ )

In this subsection, we analyze the optimality of the straight extremal (47) (see Section 6.6).

Proposition 7.

The straight extremal trajectory (47) in Problem (2), (3) is optimal up to infinity.

Proof.

Since

u_{1} \leq 1

, an end point

(t, 0, 0)

cannot be reached from Id by time

< t

. □

7.7. Lower Bound of Cut Time

Proposition 8.

Let

q (t)

t \in [0, t_{1}]

, be an extremal trajectory in the time-optimal problem on

{SE}_{2}

with control in a circular sector. Assume that

q (\cdot)

is of sub-Riemannian type.

q (\cdot)

is optimal for the sub-Riemannian problem on

{SE}_{2}

, then it is optimal in the time-optimal problem on

{SE}_{2}

with control in a circular sector. Thus,

t_{cut} (q (\cdot)) \geq t_{cut}^{S R} (q (\cdot)),

where

t_{cut}^{S R} (q (\cdot))

is the cut time of the trajectory

q (\cdot)

for the sub-Riemannian problem on

{SE}_{2}

(see [4]).

In particular, there exists

ε > 0

such that

t_{cut} (q (\cdot)) > ε

Proof.

The sub-Riemannian problem on

{SE}_{2}

can be stated as a time-optimal problem with a set of control parameters given by the full disc. Now the statements of this proposition are obvious since if a trajectory is optimal for a problem with a bigger set of control parameters, then it is also optimal for a problem with a smaller set of control parameters. □

Now, Propositions 6 and 8 imply the following.

Corollary 1.

Let

q (t)

t \in [0, t_{1}]

, be an extremal trajectory in the time-optimal problem on

{SE}_{2}

with control in a circular sector. Then, there exists

ε > 0

such that

t_{cut} (q (\cdot)) > ε

8. Conclusions

We considered the time minimization Problem (2), (3) in the roto-translation group

{SE}_{2}

with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. The model generalizes the sub-Riemannian length minimizers problem by adding a restriction on the velocity vector to lie in a circular sector with the opening angle

2 α \in S^{1}

We studied the local and global controllability of the system and the existence of the solution for given arbitrary boundary conditions in Theorem 1. We obtained the following.

1.: For $α = 0$ , the system is not globally controllable.
2.: For $α \in (0, \frac{π}{2}]$ , the system is globally but not small-time locally controllable.
3.: For $α \in (\frac{π}{2}, π)$ , the problem is ill-posed. The system is globally and small-time locally controllable, but an optimal trajectory does not exist for some boundary conditions.
4.: For $α = π$ , the system is globally and small-time locally controllable. This case coincides with the sub-Riemannian length minimizers problem in ${SE}_{2}$ .

Then, we considered the well-posed case

α \in (0, \frac{π}{2}]

We obtained a lower bound for local controllability time

T = \frac{2 π}{sin α}

(see Theorem 2).

We applied PMP and obtained explicit expressions for the extremals.

We showed that arclength parameterized abnormal extremals are given by joining of the half period arcs of the circular extremals (see Theorem 3).

We showed that there are five qualitative types of normal extremal trajectories.

1.: Arcs of noninflectional sub-Riemannian geodesics in ${SE}_{2}$ , joined by arcs of circular extremals. The exact expression is given by Theorem 4. An upper bound for the cut time is given by Proposition 3.
2.: Arcs of inflectional sub-Riemannian geodesics in ${SE}_{2}$ , joined by arcs of circular extremals. The exact expression is given by Theorem 5. An upper bound for the cut time is given by Proposition 4.
3.: Arcs of the separatrix sub-Riemannian geodesics in ${SE}_{2}$ joined by an arc of the circular extremal. The exact expression is given by Theorem 6. The extremals before the first switching are optimal (see Proposition 5).
4.: The circular extremals. The exact expression is given by Theorem 7. The cut time is given by Proposition 6.
5.: The straight extremal. The exact expression is given by Theorem 8. It is optimal up to infinity (see Proposition 7).

We showed that small arcs of the normal extremals are optimal (see Corollary 1).

As a further extension of the present work, we plan to use the obtained exact expressions for the extremals and the bounds for the cut time to construct the optimal synthesis; similarly, it is performed for the sub-Riemannian length minimizer problem in

{SE}_{2}

[4].

Author Contributions

Conceptualization, A.M. and Y.S.; methodology, A.M. and Y.S.; software, A.M.; validation, A.M. and Y.S.; formal analysis, A.M. and Y.S.; investigation, A.M. and Y.S.; resources, A.M. and Y.S.; data curation, A.M. and Y.S.; writing—original draft preparation, A.M. and Y.S.; writing—review and editing, A.M. and Y.S.; visualization, A.M.; supervision, Y.S.; project administration, A.M.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.

Funding

The work is supported by the Russian Science Foundation under grant 22-11-00140 (https://rscf.ru/project/22-11-00140/, accessed on 14 September 2023) and performed in Ailamazyan Program Systems Institute of Russian Academy of Sciences.

Data Availability Statement

Not applicable.

Acknowledgments

The authors thank Andrei Ardentov for valuable discussions resulting to clarification of the structure of abnormal extremals.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

PMP	Pontryagin maximum principle

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Figure 1. A model of a car that can move forward and turn within a given minimal radius. The control

u_{1}

is responsible for moving forward and

u_{2}

for the turn.

Figure 1. A model of a car that can move forward and turn within a given minimal radius. The control

u_{1}

is responsible for moving forward and

u_{2}

for the turn.

Figure 2. Set of admissible controls for various models of a car on a plane: Dubins car [1]; Reeds–Shepp car [2]; Ardentov model [3] (generalized Dubins car); Sachkov model [4] (sub-Riemannian problem); Berestovskii model [5]; Duits model [6]; our model with control in a sector.

Figure 3. The maximum condition.

Figure 4. Abnormal case. (Left) Level surfaces of the Hamiltonian

H = 0

(in green) and the Casimir

E > 0

(in red). (Center) Phase portrait on the surface

H = 0

. (Right) An abnormal extremal trajectory.

Figure 4. Abnormal case. (Left) Level surfaces of the Hamiltonian

H = 0

(in green) and the Casimir

E > 0

(in red). (Center) Phase portrait on the surface

H = 0

. (Right) An abnormal extremal trajectory.

Figure 5. Level surfaces of the Hamiltonian H (in green) and the Casimir E (in red). (Left)

E < \frac{1}{2}

. (Center)

E = \frac{1}{2}

. (Right)

E > \frac{1}{2}

Figure 5. Level surfaces of the Hamiltonian H (in green) and the Casimir E (in red). (Left)

E < \frac{1}{2}

. (Center)

E = \frac{1}{2}

. (Right)

E > \frac{1}{2}

Figure 6. The phase portrait on the level surfaces of the Hamiltonian

H = 1

Figure 6. The phase portrait on the level surfaces of the Hamiltonian

H = 1

Figure 7. Stratification of the domain of the vertical part.

Figure 8. Timeline with indicated instances of switching and the corresponding trajectory.

Figure 9. Rectified coordinates in the domain

cos α < E < 1

Figure 9. Rectified coordinates in the domain

cos α < E < 1

Figure 10. Timeline for the trajectory with indicated instances of switches.

Figure 11. Two extremal trajectories in

S_{1}^{\pm} \cup O_{1}^{\pm}

with

α = \frac{3 π}{7}

for

t \in [0, 15.4]

: for

h^{0} = (0.32, - 0.85, - 0.66)

and

h^{0} = (0.52, 0.85, - 0.46)

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 11. Two extremal trajectories in

S_{1}^{\pm} \cup O_{1}^{\pm}

with

α = \frac{3 π}{7}

for

t \in [0, 15.4]

: for

h^{0} = (0.32, - 0.85, - 0.66)

and

h^{0} = (0.52, 0.85, - 0.46)

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 12. Rectified coordinates in the domain

E > 1

Figure 12. Rectified coordinates in the domain

E > 1

Figure 13. Two extremal trajectories in

S_{2}^{\pm} \cup O_{2}^{\pm}

with

α = \frac{π}{4}

: for

h^{0} = (0.7, - 0.714, - 1.05)

and

h^{0} = (0.7, 0.714, - 0.85)

, for

t \in [0, 16]

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 13. Two extremal trajectories in

S_{2}^{\pm} \cup O_{2}^{\pm}

with

α = \frac{π}{4}

: for

h^{0} = (0.7, - 0.714, - 1.05)

and

h^{0} = (0.7, 0.714, - 0.85)

, for

t \in [0, 16]

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 14. Rectified coordinates in the domain

E = 1

h_{1} < 1

Figure 14. Rectified coordinates in the domain

E = 1

h_{1} < 1

Figure 15. Three extremal trajectories in

S_{3}^{\pm \pm} \cup O_{3}^{\pm}

with

α = \frac{π}{4}

: for

(h_{10}, s_{20}, s_{30}) = (0.95, - 1, 1)

and

T = 12

; for

(h_{10}, s_{20}, s_{30}) = (0.99999, 1, 1)

and

T = 21

; for

(h_{10}, s_{20}, s_{30}) = (0.9, 1, 1)

and

T = 15.5

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 15. Three extremal trajectories in

S_{3}^{\pm \pm} \cup O_{3}^{\pm}

with

α = \frac{π}{4}

: for

(h_{10}, s_{20}, s_{30}) = (0.95, - 1, 1)

and

T = 12

; for

(h_{10}, s_{20}, s_{30}) = (0.99999, 1, 1)

and

T = 21

; for

(h_{10}, s_{20}, s_{30}) = (0.9, 1, 1)

and

T = 15.5

. The sub-Riemannian arcs are depicted in red, and arcs of the circles are depicted in blue.

Figure 16. Extremal trajectories in

O_{4}^{\pm}

Figure 16. Extremal trajectories in

O_{4}^{\pm}

Figure 17. Extremal trajectory in

S_{5}

Figure 17. Extremal trajectory in

S_{5}

Figure 18. Nonoptimal arc of the extremal trajectory.

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Mashtakov, A.; Sachkov, Y. Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector. Mathematics 2023, 11, 3931. https://doi.org/10.3390/math11183931

AMA Style

Mashtakov A, Sachkov Y. Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector. Mathematics. 2023; 11(18):3931. https://doi.org/10.3390/math11183931

Chicago/Turabian Style

Mashtakov, Alexey, and Yuri Sachkov. 2023. "Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector" Mathematics 11, no. 18: 3931. https://doi.org/10.3390/math11183931

APA Style

Mashtakov, A., & Sachkov, Y. (2023). Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector. Mathematics, 11(18), 3931. https://doi.org/10.3390/math11183931

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Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

Abstract

1. Introduction

2. Preliminaries

3. Statement of the Problem

4. Existence of the Solution

4.1. Controllability and Existence of Optimal Controls

4.2. Local Controllability

5. Pontryagin Maximum Principle

5.1. Hamiltonian System and Maximality Condition

5.2. Abnormal Case H = 0

5.3. Normal Case H > 0

6. Explicit Expression for Normal Extremals

6.1. Stratification of the Hamiltonian System Adjoint Variables Domain

6.2. The Domain cos α < E < 1

6.3. The Domain E > 1

6.4. The Domain E = 1 and h 1 < 1

6.5. The Domain E ≤ cos α

6.6. The Domain E = 1 and h 1 = 1

7. Optimality of Extremal Trajectories

7.1. General Upper Bound of Cut Time

7.2. Optimality of Extremals for cos α < E < 1

7.3. Optimality of Extremals for E > 1

7.4. Optimality of Separatrix Extremals ( E = 1 , h 10 ≠ 1 )

7.5. Optimality of Circular Trajectories, E ≤ cos α

7.6. Optimality of the Straight Trajectory ( E = 1 , h 10 = 1 )

7.7. Lower Bound of Cut Time

8. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

References

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5.2. Abnormal Case $H = 0$

5.3. Normal Case $H > 0$

6.2. The Domain $cos α < E < 1$

6.3. The Domain $E > 1$

6.4. The Domain $E = 1$ and $h_{1} < 1$

6.5. The Domain $E \leq cos α$

6.6. The Domain $E = 1$ and $h_{1} = 1$

7.2. Optimality of Extremals for $cos α < E < 1$

7.3. Optimality of Extremals for $E > 1$

7.4. Optimality of Separatrix Extremals ( $E = 1$ , $h_{10} \neq 1$ )

7.5. Optimality of Circular Trajectories, $E \leq$ `cos` $α$

7.6. Optimality of the Straight Trajectory ( $E = 1$ , $h_{10} = 1$ )