The Cognition Problem of Surroundings for the Agent Based on Direction Measurement

Authors: yingjing  shi
rui  li
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-21 10:46:58

Abstract: This paper investigates the cognition problem of surroundings for the agent only with the knowledge of angle measurement. In this paper, in virtue of the local motion information of the agent, the problem determining the agent’s position and attitude in surroundings with only angle measurement is attributed to solve linear algebraic equations. Based on the obtained position and attitude information, the localization problem of the object in surroundings is handled by solving overdetermined linear algebraic equations. Finally, the proposed methods are verified by the experiments.

Main paper: 
 i. introduction: 
 For the past decade, cooperative control of multi-agent systems has been an active research area. As one of the fundamental cooperative control problems, the localization problem has been extensively studied. Potential applications of localization algorithms can be found in security and surveillance, and environmental monitoring and sampling. A crucial task in these applications is to find the (possibly time-varying) locations of all the agents based on sensor measurement data available to the agents.Based on the type of measurement data available, localization algorithms can be classified into three categories: positionbased schemes [1] - [5] , distance-based schemes [6] - [10] and direction-based scheme. Due to the complex sensing hardware or global positioning system, full information may not be used in some environment. In the distance-based schemes, the relative distances between certain neighboring agents are available for localization purposes. In comparison, the directionbased localization schemes use angle measurements instead of relative distances, which are often cheaper and more accessible than position measurements. Thus, the localization problem has been widely addressed.In [11] , the problem of bearing-only localization of a single target is formulated as a constrained optimization problem using synchronous measurements from multiple sensors, which is solved by an iterative algorithm. In [12] , self-localization and formation control tasks are considered when the agents This work is supported by the National Natural Science Foundation of China (Grant 61973055); and Natural Science Foundation of Sichuan Province of China (Grant 2023NSFSC0511).have limited sensing capabilities. [13] considers the problem of localization and circumnavigation of a slowly drifting target with unknown speed using the agents known position and the bearing angle of the target. In [14] , the authors further investigate the scenario where the agent moves with constant speed using bearing measurements. [15] investigates the problem of bearing measurement based distributed localization for sensor networks where a common sense of north is not be shared. [16] develops a distributed source localization scheme based on bearing angle measurements for a group of unicycle-type agents without the need of GPS and compass.In practical application, the global coordinate frame shared among all the agents are required for coordination control of the multi-agent systems. However, since data received from each agent is represented in its own local frame, there is no global coordinate frame shared among all the agents. Therefore, how to construct the transformation relationship between local coordinate frame and its global frame is needed. Most literatures assume that the coordinate transformation relationship mentioned above is known by default or the relative position information can be directly detected, and then focus on developing control algorithms, thus ignoring the systems distributed characteristic. However, in practice, how to obtain the transformation relationship is challenging, especially when the system is based on incomplete information where the sensors can only measure bearing information. In [17] , authors discussed preliminarily the calibration problem of the transformation relationship between local coordinate frames. However, localizing the object in the surroundings is not considered.In this paper, we first investigate how to determine the agent's position and attitude in surroundings with only angle measurement, which is transformed to solve linear algebraic equations. Based on the obtained position and attitude information, the localization problem of the object in surroundings is attributed to solve overdetermined linear algebraic equations. Finally, we develop experiments to verify the availability of the proposed methods.The rest of this paper is organized as follows. In Section II, objectives of this paper are introduced formally. In Section III, main results of this paper are presented. We first solve how to determine the relationship between the agent and the surroundings. Then the localization problem of the object in surroundings is solved. Experiments results are shown in Section IV. Conclusions are included in Section V.The theoretical analysis of the paper relies on technique of the coordinate transformation and the linear equation.Denote Σ s is the surrounding frame and Σ a is the local frame of agent ν. From the knowledge of coordinate transformation, we know there exist rotation matrix R (R is orthogonal) and translation vector t, s.t.,y p = Rx p + t.Here, x p and y p are the coordinate of point P , respectively, in Σ s and Σ a .Suppose there are n groups of points {µ i , ν i }, where µ i and ν i are, respectively, the points in Σ s and Σ a . Here, µ i and µ j (i ̸ = j) can be the same or not, and ν i and ν j (i ̸ = j) are as well, but µ i and νi (i = 1, 2, • • • , n) are different. Collect the data x µi , y νi and l i (i = 1, 2, • • • , n). Here, x µi is the coordinate of point µ i in Σ s , y νi is the coordinate of point ν i in Σ a , l i ∈ R 3 denotes the unit vector (in Σ a ) pointing from the point ν i to the point µ i , that is, l i = Rx µi + t -y νi ∥Rx µi + t -y νi ∥ .Then we can build the nonhomogeneous linear equation as below,EQUATIONwhereQ ∈ R 3×3 , s ∈ R 3 and λ i ∈ R are unknown variables.Proposition 1. [17] R and t are always a solution of equation (1) . Another way to think about it is if equation ( 1) has unique solution Q 0 and s 0 , then Q 0 = R and s 0 = t.Proposition 2. [17] In order to solve R and t, we only need collect data to guarantee that homogeneous linear equationEQUATIONhas only zero solution.Remark 1. Considering R T R = I is not a linear relationship, we choose to ignore the orthogonality of R and set up general linear equations, since the linear equations are easy to build and to solve.Proposition 3. [17] Ignore the orthogonality of R, if equation ( 2) has only zero solution, then x µ1 , x µ2 , • • • , x µn must be non-coplanar.The objective of this paper includes two parts: (i) Determine the position and attitude relationship between the agent and the surroundings; (ii) Localize the object in the surroundings. Therefore, we propose the following two problems. Problem 1. In 3D space, the goal is to figure out the coordinate transformation relationship between the surrounding frame Σ s and the local frame Σ a , that is, to figure out R and t.Problem 2. In 3D space, the goal is to figure out the relative relation among the points in surrounding frame, that is, figure out the coordinate of the point in Σ s .In order to satisfy Proposition 3, the following assumptions are proposed.Assumption 1. x µ1 , x µ2 , x µ3 , x µ4 are known in Σ s , and they are non-coplanar.Remark 2. From the logic of the paper, Assumption 1 is unnecessary. First, the conclusion that "not coplanar is the necessary condition for the existence of only zero solution for (2)" is proposed when the orthogonality of R is ignored. Second, even the above conclusion holds with the orthogonality of R being considered, we didn't mention that (2) has only zero solution is the necessary condition for solving R, t. Assumption 2. For t = t 1 and t = t 2 (t 1 < t 2 ), ν knows the rotation matrix and the translation vector from t 1 to t 2 (in Σ a ).Represent the local frame of agent ν as Σ T a , when t = T . This subsection will determine the relationship between Σ s and Σ T a . Theorem 1. Suppose that Assumption 1 and Assumption 2 hold, and ν can measure the directions of four points in Assumption 1. Then Problem 1 is almost always solvable.Proof. Collect data at t = 0 and t = T , where T > 0. Suppose that the directions of points measured at t = 0 by ν in Assumption 1 are, respectively, l 0 1 , l 0 2 , l 0 3 , l 0 4 (in Σ T a ), and the directions of that at t = T are, respectively,l T 1 , l T 2 , l T 3 , l T 4 (in Σ T a too). Denote [ x µ1 x µ2 x µ3 x µ4 1 1 1 1 ] as X, then from equation (2) we have [ Q s ] X = [ λ 0 1 l 0 1 λ 0 2 l 0 2 λ 0 3 l 0 3 λ 0 4 l 0 4 ] = [ λ T 1 l T 1 λ T 2 l T 2 λ T 3 l T 3 λ T 4 l T 4 ] .Obviously, for the agent who is freely moving in 3D space,l 0 i = l T i (i = 1, • • • , 4) is an Zero-Probability event, which means that λ 0 i = λ T i = 0 (i = 1, • • • , 4) is almost always true. Consequently, [ Q s ] X = 0is almost always true. As can be seen from the premise of the theorem, X is nonsingular, thus Q = 0 and s = 0 also are almost always true. From Proposition 2, Problem 1 almost always solvable.Remark 3. If premise is added to Theorem 1, which is "the agent does not move toward the observed points, that is,l 0 i ̸ = l T i (i = 1, • • • , 4)", then Problem 1 must be solvable.In the following, an algorithm is given to solve Problem 1.Algorithm 1. Represent the rotation matrix and the translation vector in Assumption 2 as Γ and τ respectively, that is, y T = Γy 0 + τ . Then, R and t in Problem 1 can be solved by following steps:1) Calculate [ l 0 1 l 0 2 l 0 3 l 0 4 ] = Γ [ l 1 (0) l 2 (0) l 3 (0) l 4 (0) ] ,where l i (0) (i = 1, • • • , 4) is the direction recorded at t = 0. 2) Calculate y 0 s = Γy s + τ , where y s is installation position of camera on the agent, that is, the coordinate of the camera position in Σ T a .{ Qx µi + s -λ 0 i l 0 i = y 0 s , Qx µi + s -λ T i l T i = y s . i = 1, • • • , 4Then, the solution of Q and s are, respectively, the rotation matrix and the translation vector between Σ s and Σ T a . Remark 4. Obviously, the rotation matrix and the translation vector between Σ s and Σ 0 a can be obtained by R, Γ, t and τ (are, respectively, Γ -1 R and Γ -1 (t -τ )).When the rotation and the translation form Σ a to Σ s are known, that is, R and t have been figured out, then we have Theorem 2 and Theorem 3 as follows.Theorem 2. Suppose that Assumption 2 holds, µ is a point in Σ s , and ν is able to measure the direction of µ. Thus Problem 2 is almost always solvable.Proof. Consider that ν collects data at t = 0 and t = T . Denote the rotation matrix and the translation vector from Σ 0 a to Σ T a as Γ and τ . The direction of µ measured by ν are, respectively, denoted as l 0 and l T (in Σ T a ). Let x denote the coordinate of µ (in Σ s ). Then we can build the nonhomogeneous linear equation as below,EQUATION)Here, y 0 s = Γy s + τ and l 0 = Γl(0). Nonhomogeneous linear equations ( 3) have unique solution if and only if homogeneous linear equations{ Rx = λ 0 l 0 , Rx = λ T l Thave only zero solution. Obviously, when l 0 and l T are noncollinear, above equations have only zero solution. Since l 0 and l T are non-collinear almost always true (as long as µ is not in the direction of ν motion), the conclusion of the theorem follows readily.In Theorem 2 the target is localized in virtue of the movement of the agent. Due to the relativity of the motion, the following result can be obtained.Theorem 3. Suppose that the position change of µ in Σ s from t = 0 to t = T is known, and ν can measure the direction of µ. Then Problem 2 is almost always solvable for stationary ν.Proof. Represent the coordinate of µ at t = 0 and t = T as x 0 and x T respectively, and let x Tx 0 = σ (in Σ s ). The directions of µ measured by ν at t = 0 and t = T are, respectively, denoted as l 0 and l T (in Σ a ). Then we can build the nonhomogeneous linear equation as below,EQUATIONwhere x T = x 0 + σ. This nonhomogeneous linear equations have unique solution if and only if homogeneous linear equations{ Rx 0 = λ 0 l 0 , Rx 0 = λ T l Thave only zero solution. When l 0 and l T are non-collinear, above equations have only zero solution. Since l 0 and l T are non-collinear almost always true (as long as µ doesnot move in the direction of l 0 ), the conclusion of the theorem follows readily.Another way to think about Theorem 3 is the following corollary.Corollary 1. Suppose that the relative position relationship between µ 1 and µ 2 in Σ s is known, and ν can measure the directions of µ 1 and µ 2 . Then Problem 2 is almost always solvable for stationary ν.In this section, three experiments are conducted to illustrate the feasibility of the proposed method. A wheeled robot platform developed in our lab which carries a 3D lidar is used as agent ν. It is noteworthy that although the whole map containing both distance and bearing information of the surroundings of ν can be obtained by the 3D lidar, only the relative bearing information is utilized to mimic the To verify Theorem 1, Theorem 2 and Theorem 3, Experiment 1, Experiment 2 and Experiment 3 are respectively conducted.Assignments: In this experiment, the surrounding frame Σ s and the local frame Σ a are defined, respectively, by a cuboidal cabinet (one vertex and its three edges are as origin and three axes of Σ s ) and the robot's body frame. Four vertices of the cabinet, which are non-coplanar, are used as µ i (i = 1, • • • , 4) in surroundings. Therefor, for the data in Theorem 1, what we know ahead are x µi , y s , Γ and τ , and what we collect are l i (0) and l T i . We list these data as TABLE I , and show the map mapping by the 3D lidar in Fig. 2 . The measured value is not so close to the truth value, since for our 3D lidar, it is hard to measure the positions of the expected points exactly.Assignments: This experiment is also in line with Experiment 1. µ in Theorem 3 is represented by µ 1 in Theorem 1, and it moves 36 units along the x-axis of Σ s . Therefore, for the data in Theorem It is concluded that the measured value is close to the truth value.This study is based on the assumption that the agent can only access the angle information of the surroundings and can not obtain the distance information. First, when the agent is with the sensing ability to know their own motion, we propose a method to solve the position and attitude of the agent in surroundings via four non-coplanar points. Second, with the obtained position and attitude of the agent, we propose a method to localize the objects in the surroundings via the motion of the agent. Finally, we extend the developed methods and establish a way to localize two objects with known relative position by agent. The proposed methods are achieved by solving the linear algebraic equations and are easy to iteratively correct, thus it suitable for the situation where rapid attitude orientation is needed. 
 
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Reviews: review: title: title: The paper is written clearly, exceptionally excellent.
main_review: main_review: This paper excels in terms of quality, clarity, originality, and significance, but I would still like to offer some suggestions.
1. Please introduce the possible future work.
2. Please provide a brief discussion on the feasibility and rationality of the assumptions made in the paper.
title: title: summarize the innovation and individual originality work
main_review: main_review: In the manuscript titled "The Cognition Problem of Surroundings for the Agent Based on Direction Measurement"presents a method for determining an agent's position and orientation using only angle measurements and local motion data, which is solved through linear algebraic equations. It also addresses object localization in the environment, with methods validated by experiments.This study contains some interesting findings and is valuable for the understanding of how an agent can determine its position, orientation, and object localization using only angle measurements and known motion, with methods that are effective for rapid adjustments and iterative corrections..However, lack of the innovations and individual originality work is the major flaw of the study .it is suggested to summarize the innovation and individual originality work point by point, in other words, add the content of this section.
title: title: Accept
main_review: main_review: This paper explores the cognitive challenge of an agent's surroundings when it possesses only angle measurement capabilities. By leveraging the agent's local motion information, the authors effectively frame the problem of determining the agent's position and orientation within its environment solely through angle measurements as a series of linear algebraic equations. Subsequently, utilizing the derived position and orientation data, the localization of objects in the environment is addressed by solving overdetermined linear algebraic equations. While the overall approach is well-conceived and the logic is sound, there are a few areas that could benefit from further refinement:
1. The introduction could provide a more thorough discussion of the limitations of existing methods in addressing similar problems, thus emphasizing the novelty and necessity of the current research.
2.The conclusion should more explicitly highlight the key contributions of the paper and offer insights into potential avenues for future research.