JOTA manuscript No. will be inserted by the editor Computing the Minimum-Time Interception of a Moving

2025-05-05 0 0 507.62KB 20 页 10玖币

侵权投诉

JOTA manuscript No.

(will be inserted by the editor)

Computing the Minimum-Time Interception of a Moving

Target

Maksim Buzikov

Received: date / Accepted: date

Abstract In this study, we propose an algorithmic framework for solving a

class of optimal control problems. This class is associated with the minimum-

time interception of moving target problems, where a plant with a given state

equation must approach a moving target whose trajectory is known a priori.

Our framework employs an analytical description of the distance from an arbi-

trary point to the reachable set of the plant. The proposed algorithm is always

convergent and cannot be improved without losing the guarantee of its conver-

gence to the correct solution for arbitrary Lipschitz continuous trajectories of

the moving target. In practice, it is diﬃcult to obtain an analytical description

of the distance to the reachable set for the sophisticated state equation of the

plant. Nevertheless, it was shown that the distance can be obtained for some

widely used models, such as the Dubins car, in an explicit form. Finally, we

illustrate the generality and eﬀectiveness of the proposed framework for simple

motions and the Dubins model.

Keywords Optimal control ·Moving target ·Reachable set ·Lipschitz

functions ·Markov-Dubins path

Mathematics Subject Classiﬁcation (2000) 49M15 ·49J15 ·65H05

1 Introduction

The problem of intercepting a moving target (PIMT) arises in both civilian

and military problem areas. Most modern guidance laws for real problems are

derived using linear quadratic optimal control theory to obtain explicit ana-

lytical feedback solutions [36]. This approach requires linear approximation of

Maksim Buzikov

V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences

Moscow, Russia

me.buzikov@physics.msu.ru

arXiv:2210.03439v2 [math.OC] 20 Jun 2023

2 Maksim Buzikov

the problem. Another way to solve a PIMT is to simplify the state equation

such that the corresponding optimal control problem can have a closed-form

solution. The solution to the relaxed problem produces a reference path for

a real plant. This requires the use of path-following control strategies [42].

Relaxation of the problem requires the selection of a state equation that si-

multaneously satisﬁes several requirements. First, in practice, the reference

path provided by the simpliﬁed model should be close to a feasible path. Sec-

ondly, path computation should be a time-rapid and memory-eﬃcient process

for onboard computers. The main motivation of this study was to investigate

a compromise method for PIMT.

In the literature, numerous models have been studied analytically. For ex-

ample, isotropic rocket [24,27,2,3,4,49,5], Dubins’ model [31,20,40,43,10,39,

25,17] (including asymmetric cases [6,37]), Dubins airplane [15,32,48], and

Reeds-Shepp model [41,46,9,44]. Such models, in comparison with linearly

approximated models, can consider the maneuverability of a real plant more

accurately. At the same time, these models retain the possibility of analytically

solving the problem of computing optimal trajectories.

In this study, we considered a class of problems involving the minimum

time interception of a target that moves along a known and predetermined

trajectory. We assume that the state equation of the plant is so simple that we

can analytically describe the distance from an arbitrary point to the reachable

set of the plant. The only restriction imposed on the trajectory of a moving

target is that it is a Lipschitz continuous function of the time. From a practical

perspective, this implies that target coordinates vary at a limited rate. Spe-

ciﬁc cases of such a problem have been reported in several studies. In [5], the

problem of intercepting a moving target using an isotropic rocket is explored.

In a series of studies [16,28,50,34,51], the same problems were investigated

for the interception of a uniformly moving target using a Dubins car. A more

general problem of intercepting by a Dubins car for an arbitrary continuous

target trajectory was considered in [51,11]. The lateral interception of a mov-

ing target by a Dubins car was explored in [47,7,23,35,52] for a uniformly

moving target, in [30,29] for a target moving on a circle or a racetrack path,

and in [33,12] for an arbitrary continuous trajectory of the target. The main

practical contribution of this study is the always convergent algorithm for cal-

culating the minimum time interception of a moving target by a Dubins car.

In comparison with [50,51], the proposed algorithm works not only with rec-

tilinear uniform movement of the target. It also calculates the minimum time

required for arbitrary Lipschitz continuous trajectories (including rectilinear

uniform trajectories).

The analytical description of a reachable set in a simple motion model is

trivial. In more complex cases, such as Dubins’ model, a series of analytical

results is obtained. For example, an analytical description of a planar reachable

set of a Dubins car can be found in [18,8,21,13,11]. Papers [39,37,38,12] have

been devoted to the study of a three-dimensional reachable set of a Dubins

car. In this study, we assume that the distance from an arbitrary point to

the reachable set of the plant is known or can be calculated eﬀectively. If the

Computing the Minimum-Time Interception of a Moving Target 3

distance to the target position is calculated, the minimum time interception

is such that the distance is not greater than the desired value of the capture

radius. The main idea of designing an always convergent algorithm to obtain

the optimal interception time was borrowed from [14,45,1,22]. To specify this

idea, we additionally used the properties of distance to the reachable set. The

novelty of this study in comparison with these works lies in the use of additional

information regarding the description of the reachable set to increase the step

of the ﬁxed-point iteration algorithm.

The remainder of this paper is organized as follows. In Section 2, we for-

mally describe the minimum time interception of a moving target problem.

In Section 3, we investigate some properties of reachable sets and universal

lower estimators of interception time. In addition, we proposed an iterative

algorithm based on universal lower estimators. In Section 4, we demonstrate

the proposed algorithmic framework using two examples of plants: a simple

motion and Dubins car. Additionally, we present numerical experiments for

these two examples. Finally, Section 5 concludes the paper with a discussion

and brief description of future work.

2 Problem Formulation

Let t∈R+

0denote a time moment and x= stack(y,z) denote the state vector

function. Here, x(t)∈ X =Y ⊕Z.Xis a state space. Yis a ﬁnite-dimensional

normed space with norm ∥·∥ :Y → R+

0. We assume that y(t)∈ Y corresponds

to coordinates that are important for interception. The remaining coordinates

z(t)∈ Z are arbitrary when the target is intercepted. The state equation

(dynamic constraints) of the plant is as follows:

y=f(x,u),˙

z=g(x,u).(1)

Here, u(t)∈ U is a control input and Uis a compact restraint set of admissible

values of the control inputs. The class of all measurable admissible control

inputs is denoted as A(u∈ A). Throughout this paper, we assume that the

control input and state equation are subject to the requirements of Theorem 2

from [26, pp. 242–244]. In addition, we assumed that the velocity of the plant

is uniformly restricted. Without loss of generality, we claim ∥f(x,u)∥ ≤ 1 for

all1x∈ X and u∈ U.

Let yT∈Lipv(R+

0,Y)2describe the trajectory of the target (Fig. 1). The

constant v∈R+

0denotes the maximum speed of the target. We assumed that

the trajectory of the target is known a priori. However, the algorithm for

obtaining the solution should work with any trajectory yT∈Lipv(R+

0,Y) and

use only knowledge about the maximal speed of the target.

1We will use italic letters x,y,u... for functions and roman letters x,y,u... for points

of the corresponding spaces.

2Lipv(F,G) denotes a set of v-Lipschitz continuous functions. If y∈Lipv(F,G), then

y:F → G, and ∥y(x2)−y(x1)∥G≤v∥x2−x1∥Ffor all x1,x2∈ F.

4 Maksim Buzikov

y(·)

yT(·)

∥y(t)−yT(t)∥ ≤ ℓ

Fig. 1 Trajectory of interception (solid line) of the moving target (dotted line). The dashed

lines around the trajectory of interception deﬁne the capture set (radius ℓ). The target enters

the capture set at t. Space Yis depicted as a line for simplicity

Let x(0) = x0∈ X. Without loss of generality, we set y(0) = 0∈ Y. Unless

stated otherwise, we assume that x= stack(y,z) is an absolutely continuous

solution of the state equation with a control input u∈ A and initial conditions

x(0) = x0. We deﬁne the problem of the minimum-time interception of a

moving target by ﬁnding the minimum value of the following functional:

J[u;yT]def

= min t∈R+

0:∥y(t)−yT(t)∥ ≤ ℓ→inf

u∈A .(2)

Here, ℓ∈R+

0denotes the capture radius. If ℓ= 0, then the interception of

the target is a coincidence of the plant and target y-coordinates. Throughout

this paper, it is convenient to deﬁne min ∅= +∞for the uniformity of the

notation.

3 Interception by Reachable Sets

A reachable set at ﬁxed time tis a set of all states in the state space that

can be reached at time tfrom a given initial state by employing an admissible

control input [39]. Further, we use the projection of the reachable set on Y

(Fig. 2):

R(t)def

=







Z

[0,t]

f(x(τ),u(τ))dτ:u∈ A









.(3)

Note that R:R+

0→2Yis a multivalued mapping, and R(t)⊂ Y. If

Z=∅,R(t)⊂ Y =Xis a reachable set in the classical sense of the word.

According to the general results of optimal control theory, R(t) is a compact

set [26, Theorem 2, pp. 242–244].

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

JOTAmanuscriptNo.(willbeinsertedbytheeditor)ComputingtheMinimum-TimeInterceptionofaMovingTargetMaksimBuzikovReceived:date/Accepted:dateAbstractInthisstudy,weproposeanalgorithmicframeworkforsolvingaclassofoptimalcontrolproblems.Thisclassisassociatedwiththeminimum-timeinterceptionofmovingtargetproblem...

展开>> 收起<<

JOTA manuscript No. will be inserted by the editor Computing the Minimum-Time Interception of a Moving.pdf

共20页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

JOTA manuscript No. will be inserted by the editor Computing the Minimum-Time Interception of a Moving

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: