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The control development is based on new local potential functions, which attain the minimum value when the desired formation is achieved, and are equal to infinity when a collision occurs. Several simulation examples are included to illustrate the approach throughout the paper.
ISSN: 1859-2171 TNU Journal of Science and Technology 192(16): 73 - 86 FORMATION STABILIZATION OF MOBILE AGENTS USING LOCAL POTENTIAL FUNCTIONS Khac-Duc Do1,*, Dang-Binh Nguyen2, Van-Vi Nguyen2, Van-Hung Nguyen2 Curtin University, Austrailia; Viet Bac University, 1B street, Dongbam ward, ThaiNguyen City ABSTRACT We present a constructive method to design cooperative controllers that force a group of N mobile agents to stabilize at a desired location in terms of both shape and orientation while guaranteeing no collisions between the agents The control development is based on new local potential functions, which attain the minimum value when the desired formation is achieved, and are equal to infinity when a collision occurs Several simulation examples are included to illustrate the approach throughout the paper Keywords: Formation stabilization, mobile agents, local potential functions, ocean vehicles Received: 12/11/2018; Revised: 19/11/2018; Approved: 28/12/2018 ỔN ĐỊNH HỢP TÁC CÁC THIẾT BỊ DI ĐỘNG DÙNG CÁC HÀM THẾ NĂNG NHÂN TẠO CỤC BỘ Đỗ Khắc Đức1,*, Nguyễn Đăng Bình2, Nguyễn Văn Vị2, Nguyễn Văn Hùng2 Đại học Curtin, Úc; Trường Đại học Việt Bắc, Đường 1B, Phường Đồng Bẩm, Thành phố Thái Nguyên TÓM TẮT Trình bày phương pháp hệ thống để thiết kế điều khiển ổn định phối hợp cho nhóm N thiết bị di động vị trí định trước hình dạng hướng, đảm bảo khơng có va chạm thiết bị Các điều khiển thiết kế dựa hàm nhân tạo có giá trị cực thiểu thiết bị di động ổn định vị trí định trước đạt giá trị vơ hạn xảy va chạm Bài báo bao gồm số ví dụ minh họa Từ khóa: Ổn định nhóm, thiết bị di động, phương tiện giao thơng đường biển Ngày nhận bài: 12/11/2018; Hoàn thiện: 19/11/2018; Duyệt dăng: 28/12/2018 (*) Corresponding author: Email: 254282c@curtin.edu.au http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 73 Do Khac Duc et al TNU Journal of Science and Technology INTRODUCTION Formation control of multiple mobile agents has received a lot of attention from the control community over the last few years Applications of vehicle formation control include the coordination of multiple robots, unmanned air/ocean vehicles, satellites, aircraft and spacecraft 0-[32] For example, a group of mobile vehicles can be used to carry out tasks that are difficult or not effective for a single vehicle to perform alone In the literature, there are roughly three methods to formation control of multiple vehicles: leaderfollowing, behavioral and virtual structure Each method has its own advantages and disadvantages In the leader-following approach, some vehicles are considered as leaders, whist the rest of robots in the group act as followers 0, 0, 0, The leaders track predefined reference trajectories, and the followers track transformed versions of the states of their nearest neighbors according to given schemes An advantage of the leaderfollowing approach is that it is easy to understand and implement In addition, the formation can still be maintained even if the leader is perturbed by some disturbances However, a disadvantage is that there is no explicit feedback to the formation, that is, no explicit feedback from the followers to the leader in this case If the follower is perturbed, the formation cannot be maintained Furthermore, the leader is a single point of failure for the formation In the behavioral approach 0, 0, 0, 0, 0, 0, 0, few desired behaviors such as collision/obstacle avoidance and goal/target seeking are prescribed for each vehicle and the formation control is calculated from a weighting of the relative importance of each behavior The advantages of this approach are: it is natural to derive control strategies when vehicles have multiple competing objectives, and an explicit feedback is included through communication between neighbors The 74 192(16): 73 - 86 disadvantages are: the group behavior cannot be explicitly defined, and it is difficult to analyze the approach mathematically and guarantee the group stability In the virtual structure approach, the entire formation is treated as a single entity 0, 0, 0, When the structure moves, it traces out desired trajectories for each agent in the group to track Some similar ideas based on the perceptive reference frame, the virtual leader, and the formation reference point are given in 0, 0, respectively The advantages of the virtual structure approach are: it is fairly easy to prescribe the coordinated behavior for the group, and the formation can be maintained very well during the manoeuvres, i.e the virtual structure can evolve as a whole in a given direction with some given orientation and maintain a rigid geometric relationship among multiple vehicles However requiring the formation to act as a virtual structure limits the class of potential applications such as when the formation shape is time-varying or needs to be frequently reconfigured, this approach may not be the optimal choice The virtual structure and leader-following approaches require that the full state of the leader or virtual structure be communicated to each member of the formation In contrast, behavior-based approach is decentralized and may be implemented with significantly less communication Formation feedback has been recently introduced in the literature 0, 0, 0, In 0, a coordination architecture for spacecraft formation control is introduced to incorporate the leader-following, behavioral, and virtual structure approaches to the multi-agent coordination problem This architecture can be extended to include formation feedback In 0, formation feedback is used for the coordinated control problem for multiple robots In 0, a Lyapunov formation function is used to define a formation error for a class of robots (double integrator dynamics) so that a constrained motion control problem of http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology multiple systems is converted into a stabilization problem for one single system The error feedback is incorporated to the virtual leader through parameterized trajectories The formation control problem for the three general approaches described above would be for each agent to move to a desired point in the formation while avoiding collisions Such a desired point may be time varying or stationary, and can be defined, for instance, relative to a leader or virtual structure The objective can be achieved through the use of centralized control, see for example 0, by using a single controller that generates collision free trajectories in the workspace Although this guarantees a complete solution, centralized schemes require high computational power (on the part of the central command and control centre) and are not robust due to the heavy dependence on a single controller On the other hand, decentralized schemes, see for example 0, require less computational effort, and is relatively more scalable to team size This approach usually involves a combination of agent based local potential fields 0, Error! Reference source not found The main problem with the decentralized approach is that it is unable or extremely difficult to predict and control the critical points, i.e the closed loop system has multiple equilibrium points It is rather difficult to design a controller such that all the equilibrium points except for the desired equilibrium ones (in the formation that the agents are to track) are unstable/saddle points Recently, following the approach presented in 0, a method based on a different navigation function provided a centralized formation stabilization control design strategy, which can potentially be extended for complete decentralization, is proposed in However, the navigation function approaches a finite value when a collision occurs, and the formation is http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 192(16): 73 - 86 stabilized to any point in workspace instead of being “tied” to a fixed coordinate frame This motivates our work presented in this paper, which derives control laws for the agents to track their desired locations within formations, and such that only the critical points at the desired locations in the formation are stable In this paper, a constructive method is proposed to design cooperative controllers to solve the problem of stabilizing a group of N mobile agents at a (pre-specified) desired location in terms of both shape and orientation while avoiding collisions between themselves The control development is based on new local potential functions guaranteeing global and complete convergence except for the set of measure zero These local potential functions are chosen such that when the controls are designed to decrease these functions, all the agents approach their desired locations and no collisions can occur Behavior of the closed loop system near equilibrium points is investigated via linearization of the inter-agent dynamics around those points We also show that the proposed control scheme is easy to extend to design bounded controllers The rest of the paper is organized as follows In the next section, we present a simple example in two-dimensional space to illustrate the approach Section presents the control design and stability analysis for formation stabilization Section concludes our paper PLANAR FORMATION STABILIZATION OF TWO AGENTS To illustrate our proposed approach to solve the problem of formation stabilization and formation tracking of N mobile agents, we begin with an examination of a group of two mobile agents whose dynamics are given by qi ui (1) 75 Do Khac Duc et al where qi [ xi yi ]T TNU Journal of Science and Technology and ui [uix uiy ]T , i 1,2 are the states and control inputs of agents and 2, respectively The control objective is to design the controls ui such that they force the agents to move from initial positions qi (t0 ) , 192(16): 73 - 86 -The goal function i is designed such that it puts penalty on the stabilization error for the agent i , and is equal to zero when the agent is at its final position A simple choice of this function is i 0.5 || qi qif ||2 (3) t0 to final positions qif [ xif yif ] while avoiding collisions between the agents It is indeed assumed that the initial and the final positions of the agent are different from those of the agent 2, i.e || q1 (t0 ) q2 (t0 ) || and || q1 f q2 f || , where || || denotes the -The related collision avoidance function i is designed such that it is equal to infinity a collision occurs, and attains the minimum value when the agents move in the desired formation A possible choice of this function is standard Euclidian norm of Control design Consider the following potential function (2) i i i where is a positive tuning constant, i and i are the goal and related collision avoidance functions, respectively They are specified below (4) T i ij , (i, j ) (1,2), i j ijf ij where ij 0.5 || qi q j ||2 , ijf 0.5 || qif q jf ||2 (5) To design the controls ui [uix uiy ]T , differentiating both sides of (2) along the solutions of (1) gives i ix uix iy uiy 1/ ijf2 1/ ij2 ( xi x j )u jx 1/ ijf2 1/ ij2 ( yi y j )u jy where 1/ ( y y ) ix xi xif 1/ ijf2 1/ ij2 ( xi x j ) iy yi yif ijf 1/ ij2 i (6) (7) j The equation (6) suggests that we choose the controls ui [uix uiy ]T as uix cix uiy ciy (8) where c is a positive constant Substituting (8) into (6) yields i c(ix2 iy2 ) 1/ ijf2 1/ ij2 ( xi x j )u jx 1/ ijf2 1/ ij2 ( yi y j )u jy (9) Indeed, substituting (8) into (1) results in the closed loop system qi ci , i 1,2 (10) where i [ix iy ]T Remark The control pairs (u1x , u2 x ) and (u1 y , u2 y ) have a special feature in the sense that the first terms (see first square brackets in ix and iy in (7)) play the role of driving the agents to their final positions while the second terms (see second square brackets in ix and iy in (7)) act as both attractive and repulsive forces to attract the agents when the distance between them is larger than the desired distance, i.e when ( x1 x2 )2 ( y1 y2 )2 ( x1 f x2 f )2 ( y1 f y2 f )2 (11) and push the agents away from each other when the distance smaller than the desired one, i.e when 76 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology 192(16): 73 - 86 ( x1 x2 )2 ( y1 y2 )2 ( x1 f x2 f )2 ( y1 f y2 f )2 (12) The second terms act as gyroscopic forces to steer the agents away from each other when they come to close to each other Stability analysis In this subsection, we show that the controls ui [uix uiy ]T given in (8) guarantees no collisions occur, the solutions of the closed loop system (10) exist, and the agents move to their desired positions asymptotically -Proof of no collisions and existence of solutions Consider the following global potential function ( i 0.5 i ) (13) i 1 The function is a proper function since substituting (2) and (4) into (13) results in 12 (14) 12 f 12 which is positive definite, radially unbounded with respect to the stabilization errors || q1 q1 f || 1 and || q2 q2 f || , and is equal to infinity when a collision between the agent and agent occurs Differentiating both sides of (14) along the solutions of the closed loop system (10) results in c Ti i (15) i 1 From (15), we have Integrating both sides of this inequality gives 12 (t ) 12 f 12 (t ) i (t ) i 1 12 (t0 ) (t ) i i 1 12 f , t t0 12 (t0 ) (16) where i (t ) 0.5 || qi (t ) qif ||2 , i (t0 ) 0.5 || qi (t0 ) qif (t0 ) ||2 , i 1,2 (17) 12 (t ) 0.5 || q1 (t ) q2 (t ) ||2 , 12 (t0 ) 0.5 || q1 (t0 ) q2 (t0 ) ||2 Since || q1 (t0 ) q2 (t0 ) || and || q1 f q2 f || , i.e 12 (t0 ) and 12 f , the right hand side of (16) is bounded As a result, the left hand side of (16) must also be bounded This means that 12 (t ) , t t0 , i.e no collisions between the agents can occur Boundedness of the left hand side of (16) also implies that of || q1 (t ) || and || q2 (t ) || , i.e the solutions of the closed loop system (10) exist Furthermore, applying Barbalat’s lemma found in [ ] to (15) gives (18) limt i (t ) 0, i 1,2 -Behavior near equilibrium points At the steady state, we have 1 and 2 These equations have two set of roots q1 q1 f , q2 q2 f and q1 q1c , q2 q2c Since the obstacle function is specified in terms of relative distance between the agents, it is easier to investigate behavior of the closed loop system near the equilibrium points by considering the inter-agent dynamics instead of dynamics of each individual agent Defining q12 q1 q2 and differentiating this equation along the solutions of the closed loop system (10) yield (19) q12 c12 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 77 Do Khac Duc et al TNU Journal of Science and Technology 192(16): 73 - 86 where 12 1 2 We can write 12 as a vector function of q12 and q12 f q1 f q2 f as because of the fact that at the equilibrium point q12 f all the forces (attractive and 12 q12 q12 f 2 1/ 122 f 1/ 122 q12 At repulsive) are equal to zero while at the critical point q12c the sum of attractive and repulsive forces (but they are different from zero) is equal to zero This can be viewed graphically in Figure the steady state, we have 12 since 1 and 2 The equation 12 has two roots q12 q12 f , q12 f q1 f q2 f and q12 q12c , q12c q1c q2c Therefore (18) implies that q12 approaches either q12 f or q12c Since substituting q12 q12 f y12 or q12 q12c into the equations 1 and 2 results in q1 q1 f , q2 q2 f and q1 q1c , q2 q2c , we just need to investigate behavior of the system (19) near the equilibrium points q12 f and q12c Before going further, it is noted that q12c has a T property that the term q12 is strictly c q12 f negative, i.e the point at which q12 [0 0]T locates between the equilibrium point q12c and the equilibrium point q12 f This is y12 f y12c O x12 f x12 x12c Figure Illustrating location of equilibrium points We will show that the equilibrium point q12 f is asymptotically stable while the equilibrium point q12c is saddle The general gradient of 12 (q12 , q12 f ) with respect to q12 is given by 2 4 x12 y12 / 123 12 1 2 1/ 12 f 1/ 12 4 x12 / 12 (20) q12 4 x12 y12 / 123 2 1/ 122 f 1/ 122 4 y12 / 123 T where x12 and y12 are defined from q12 [ x12 y12 ] To show that the equilibrium point q12 f is asymptotically stable, we need to show that the matrix Aq12 f 12 q12 is positive definite q12 q12 f Substituting q12 q12 f into (20) yields 4 x12 f / 12 f Aq12 f 4 x12 f y12 f / 12 f Since 4 x12 f / 12 f and 4 x12 f y12 f / 123 f 4 y12 f / 12 f (21) det( Aq12 f ) 4 / 122 f where det() denotes the determinant of , the matrix Aq12 f is positive definite, i.e the equilibrium point q12 f is asymptotically stable On the other hand at the equilibrium point q12c , we have 12 q12 78 q12 q12 c 1 2 1/ 122 f 1/ 122 c 4 x12 4 x12 c y12 c / 123 c c / 12 c Aq 12 c 4 x12 c y12 c / 123 c 2 1/ 122 l 1/ 122 c 4 y12 / c 12 c (22) http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology 192(16): 73 - 86 where x12c and y12c are defined from q12c [ x12c y12c ]T and 12c 0.5 || q12c ||2 The determinant of the matrix Aq12 c is given by det( Aq12 c ) 2 1/ 122 f 1/ 122 c 1 2 1/ 12 f Since at the equilibrium point q12c , we have 12c where 12c is 12 being evaluated at q12 q12c Multiplying both sides of T T 12c with q12 c , we have q12 c 12 c T Expanding q12 c 12 c gives T 2 1/ 122 f 1/ 122 c q12 c ( q12 c q12 f ) 12 c (24) Substituting (24) into (23) yields T q12 c q12 f det( Aq12 c ) 2 1/ 122 f 3/ 122 c 212c (25) T Since q12 c q12 f is strictly negative, we have det( Aq12 c ) , which implies that the equilibrium point q12c is saddle FORMATION STABILIZATION OF N AGENTS In this section, we extend the results obtained for the simple system presented in the previous section to a more complex system of N mobile agents Problem statement We consider a group of N mobile agents, of which each has the following dynamics qi ui , i 1, , N (26) where qi n and ui n are the state and control input of the agent i We assume that n and N In this paper, we treat each agent as an autonomous point The assumption that each agent is represented as a point is not as restrictive as it may seem since various shapes can be mapped to single points through a series of transformations 0, 0, Our task is to design the control input ui for each agent i that forces the group of N agents to stabilize with respect to the fixed coordinate in a particular formation specified T T ] by a desired vector q f [q1Tf , q2T f , , qNf while avoiding collisions between http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 3/ 122 c (23) themselves The control objective is formally stated as follows: Control objective: Assume that at the initial time t0 each agent starts at a different location, and that each agent has a different desired location, i.e there exist strictly positive constants and such that || qi (t0 ) q j (t0 ) || 1 (27) || qif q jf || , i, j {1,2, N } Design the control input ui for each agent i such that each agent (almost) globally asymptotically approaches its desired location while avoids collisions with all other agents in the group, i.e limt ( qi (t ) qif ) || qi (t ) q j (t ) || , i, j {1,2, N }, t t0 (28) where is a strictly positive constant The fixed desired formation can be represented by a labeled directed graph in the following definition Definition The formation graph, G {V , E , L} is a directed labeled graph consisting of: -a set of vertices (nodes), V {v1 , , v N } indexed by the mobile agents in the group, -a set of edges, E {(vi , v j ) V V } , containing ordered pairs of vertices that represent inter-agent position constraints, and -a set of labels, L { dij | dij || qi q j lij ||2 , (vi , v j ) E} , lij qif q jf n indexed by the edges in E Indeed, when the control objective is achieved, the edge labels become || qi q j lij ||2 , (vi , v j ) E , i.e the relative distance between the agents i and j is lij qif q jf Control design The example in Section motivates us to use the following local potential function 79 Do Khac Duc et al i i i , i 1, , N TNU Journal of Science and Technology (29) where are positive tuning constants, the functions i and i are the goal and related collision avoidance functions for the agent i specified as follows: -The goal function i is designed such that it puts penalty on the stabilization error for the agent i , and is equal to zero when the agent is at its final position (30) -The related collision function i should be chosen such that it is equal to infinity whenever any agents come in contact with the agent i , i.e a collision occurs, and attains the minimum value when the agent i is at its desired location with respect to other group members belong to N i , which are adjacent to the agent i This function is chosen as follows: i || qi q jf ||2 ijk i (31) k k ij jN i ijf where k is a positive constant to be chosen later, ij and ijf are collision and desired collision functions chosen as 1 2 It is noted from (32) that ij ji and ij || qi q j ||2 , ijf || qif q jf ||2 (32) ijf jif Remark The above choice of the potential function i given in (29) with its components specified in (30)-(32), has the following properties: 1) it attains the (unique) minimum value when the agent i is at its final position qif , and 2) it is equal to infinity whenever any two or more agents come in contact with the agent i , i.e when a collision occurs The potential function (29) is different from the one proposed in and in the sense that the ones in and are centralized and does not put penalty on the distance between 80 192(16): 73 - 86 the agent and its final position, i.e does not include the goal function i Therefore, the controllers developed in and not guarantee the formation converge to a specified configuration but to any configuration that minimize the potential function Our potential function (29) is also different from the navigation functions proposed in 0, and in the sense that our potential function is in the form of sum of collision avoidance functions while those navigation functions in the form of product of collision avoidance functions and This feature makes our potential function “more decentralized” Our potential function is equal to infinity while those in 0, and is equal to a finite constant when a collision occurs Moreover, our potential function puts penalty on stabilization error between the agent and its final position, hence, guarantees the formation will be stabilized with respect to a fixed coordinate system instead of “loosing” in space as in 0, However, those in 0, and also cover obstacle and work space boundary avoidance We not include these issues in our present paper for clarity Including these issues is possible and is the subject of the future research Our potential function does not have problems like local minima and nonreachable goal as listed in Error! Reference source not found To design the control input ui , we differentiate both sides of (29) along the solutions of (26) to obtain i Ti ui Tij u j (33) where 1 ij k k k ijk 1 (qi q j ) ijf ij i qi qif ij (34) jNi From (33), we simply choose the control ui for the agent i as follows: ui Ci (35) http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology where C nn is a symmetric positive definite matrix Substituting (35) into (33) yields i Ti Ci Tij u j (36) jNi Substituting (35) into (26) results in the closed loop system qi Ci , i 1, , N (37) N ( i 0.5 i ) we show that all other equilibrium(s) of (37) are either unstable or saddle Step Proof of no collision and existence of solutions: We consider the following common potential function given by (38) i 1 The function is indeed a proper (positive definite, radially unbounded and equal to infinity when a collision occurs) function since substituting the functions i and i given in (29) and (31) into (38) results in ijk ( i 0.5 i ) i k k ij i 1 i 1 i 1 j i 1 ijf (39) which is essentially sum of all goal functions and a combination of all possible related collision functions Differentiating both sides of (38) along the solutions of (36) and the closed loop system (37), or (39) along the solutions of the closed loop system (37) yields N Theorem Under the assumptions stated in the control objective, see (27), the control for each agent i given in (35) with an appropriate choice of the tuning constants and k , solves the control objective Proof We prove Theorem in two steps At the first step, we show that there are no collisions between any agents and the solutions of the closed loop system (37) exist At the second step, we prove that the equilibrium point of the closed loop system (37), at which qi qif , is asymptotically stable Finally, 192(16): 73 - 86 N N 1 N Ti Ci N 1 N i 1 i 1 i (t ) (40) i 1 From (40), we have Integrating both sides of results in (t ) (t0 ) From definition of given in (39) we can write (t ) (t0 ) as N 1 N k ijk (t ) ij (t0 ) N ( t ) k i k k k ij (t ) i 1 ij (t0 ) j i 1 ijf i 1 j i 1 ijf N N (41) where 1 || qi (t ) qif ||2 , ij (t ) || qi (t ) q j (t ) ||2 , 2 (42) 1 i (t0 ) || qi (t0 ) qif || , ij (t0 ) || qi (t0 ) q j (t0 ) ||2 2 From (27) we have ij (t0 ) and ijf are strictly larger than some positive constants Therefore the i (t ) right hand side of (41) is bounded by some positive constant depending on the initial conditions Boundedness of the right hand side of (41) implies that the left hand side of (41) must be also bounded As a result, ij (t ) must be strictly larger than some positive constant denoted by for all t t0 From definition of ij (t ) , see (42), || qi (t ) q j (t ) || must be larger than some strictly positive constant denoted by , i.e there are no collisions Boundedness of the left hand side of (41) also implies that of || qi (t ) || for all t t0 , i.e the solutions of the closed loop system (37) exist Furthermore, applying Barbalat’s lemma to (40) gives (43) limt i (t ) Step Behavior near equilibrium points http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 81 Do Khac Duc et al TNU Journal of Science and Technology 192(16): 73 - 86 At the steady state, the equilibrium points are found by solving 1 k 1 i qi qif k (44) k k ij ( qi q j ) 0, i 1, , N ij jN i ijf It is directly verified that q q f where q and q f are stack vectors of qi and qif , respectively, is one root of the equation (44) In addition there is (are) another root(s) denoted by qc of (44) different from q f satisfying i where ijc k 1 (45) ijc ( qic q jc ) 0, i 1, , N ijc2 k jN i 0.5 || qic q jc ||2 Moreover, since the collision avoidance functions are specified in q qc qic qif k 2k ijl terms on relative distances between the agents, we write the closed loop system of the inter-agent dynamics from the closed loop system (37) as (46) qij C(i j ), (i, j ) {1, , N }, i j where qij qi q j Defining q and q f are stack vectors of qij and qijf with qijf qif q jf , T T T T T T T , q13 , , qijT , , qNT 1, N ]T and q f [q12 respectively, i.e q [q12 f , q13 f , , qijf , , qN 1, Nf ] , we can write the closed loop system of the inter-agent dynamics (46) as q CF ( q , q f ) C diag(C , (47) , C ) with E the number of edges of the formation graph, and E F ( q , q f ) [1T T2 , 1T T3 , , Ti Tj , , TN 1 TN ]T (48) In the followings, we will show that the equilibrium point q q f is asymptotically stable, and the equilibrium point(s) q qc is (are) unstable or saddle Since (43) holds for all i 1, , N , at the steady state we have i j 0, (i, j ) {1, , N }, i j Therefore the equilibrium points q q f and q qc are also the equilibrium points of (47) The general gradient of F ( q , q f ) with respect to q is given by 12 12 q q13 12 ij F ( q , q f ) ij q12 qij q N 1, N q 12 12 qN 1, N ij , i j , (i, j ) {1, , N }, i j (49) qN 1, N ij N 1, N qN 1, N It can be checked that 1 2k I nn 2 k k k ijk 1 I nn 2 k ( k 1) k k ijk 2 k 2 qij qijT ijf ij ijf ij qij ij ij 2k T k k k cdk 1 I nn k (k 1) k k cdk 2 k qcd qcd , cdf cd cdf cd qcd c d ij 82 H ij (50) http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology 192(16): 73 - 86 where ( c, d ) {1, , N }, ( c, d ) (i, j ), c d , and or 1 depending on value of c, d , i and j However, we not need to specify the sign of for our next task We now investigate properties of the equilibrium points q q f and q qc based on the general gradient F ( q , q f ) / q evaluated at those points Step 2.1 Proof of q q f being the asymptotic stable equilibrium point: At the equilibrium point q q f , we have ij qij I nn q q f 4 k k 2 ijf T qijf qijf , ij qcd q q f 2 k k 2 cdf T qcdf qcdf , where cdf 0.5 || qcdf || , qcdf qcf qdf With (51), let T F (q , q f ) q q q f nE (51) we have 4 k En max( qijfa ) T 1 , (i, j ) {1, , N }, i j min( ijfk ) (52) where qijfa is the a th element of qijf Therefore, for any given constant k if we choose the tuning constant such that 4 k En max(qijfa ) min( ijfk ) 1 0 , (i, j ) {1, , N}, i j min( ijfk ) 4k En max(qijfa ) then the matrix F ( q , q f ) / q q q f (53) is positive definite, which in turn implies that the equilibrium point q q f is asymptotically stable Step 2.2 Proof of q qc being the unstable/saddle equilibrium point(s): The idea is to consider block matrices on the main diagonal of the matrix F ( q , q f ) / q q qc and show that there exists at least one block matrix whose determinant is negative Define Hijc ij / qij and let a and b be the a th and bth elements of qijc , q qc ( a, b) {1, , n}, a b We form the matrices H ijcab from the matrix H ijc as follows 1 2 k ijc ijck 1 2 k[(k 1)ijc ijck 2 2k / ijck 2 ]a2 2 k[(k 1) ijc ijck 2 2k / ijck 2 ]ab H ijcab k 2 k 2 k 1 k 2 k 2 2 2 k[(k 1)ijc ijc 2k / ijc ]ab 2 k ijc ijc 2 k[( k 1) ijc ijc 2k / ijc ]b (54) where ijc 1/ ijf2 k 1/ ijc2 k The determinant of H ijcab is given by ab det( H ijcab ) (1 2 k ijc ijck 1 )ijc (55) ab ijc 2 k ijc ijck 1 2 k [(k 1)ijc ijck 2 2k / ijck 2 ](a2 b2 ) (56) where Let us calculate the sum: n 1 n a 1 b a 1 ab ijc n(n 1) 2 k (n 1)(2(k 1) n) ijck 1 / ijf2 k 2 k ( n 1)(2( k 1) n) / ijck 1 (57) Since n , picking 2( k 1) n k n / (58) http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 83 Do Khac Duc et al n 1 TNU Journal of Science and Technology n ensures that a 1 b a 1 ab ijc 192(16): 73 - 86 Therefore, there exists at least one pair ( a , b) {1, , n} denoted by ab Now for all (i, j ) {1, , N }, i j let us consider the sum: (a* , b* ) such that ijc * * N 1 * * N det( H ijca b ) ab ijc * * i 1 j i 1 N 1 N ijc ( ijc 2 k ijc ijck ) (59) i 1 j i 1 On the other hand, multiplying both sides of F ( qc , q f ) with qcT results in qcT F ( qc , q f ) , which is expanded to N 1 N (q T ijc i 1 j i 1 ( qijc qijf ) 2 kN ijc ijck ) (60) Substituting (60) into (59) results in * * N 1 N det( H a b ) N 1 N N 1 N ijc T ( N 2) qijc qijf ijc ijc a*b* N ijc i 1 j i 1 i 1 j i 1 i 1 j i 1 (61) q12 qN 1, N q13 The point where all attractive and repulsive forces are zero F qN 1,1 C O q1N q21 q2 N The point where sum of attractive and repulsive forces are zero Figure Illustration of location of critical points N 1 N The term q T ijc qijf i 1 j i 1 is strictly negative since at the point where q ij qijf (the point F in Figure 2) all attractive and repulsive forces are equal to zero while at the point where qij qijc (the point C in Figure 2) the sum of attractive and repulsive forces is equal to zero (see Section for discussion of a simple case) Therefore the point qij (the point O in Figure 2) must locate between the points qij qijf and qij qijc , see Figure Furthermore if we write (60) as N 1 N 1 N N T ijc kN ( ijck / ijf2 k 1/ ijck ) qijc qijf i 1 j i 1 (62) i 1 j i 1 we can see that deceasing results in decrease in ijc since ijf is a bounded constant and the right hand side of (62) is negative Therefore, choosing a sufficiently small ensures that the right hand of (61) is strictly negative since ijc 0.5 || qijc ||2 That is N 1 N i 1 j i 1 * * det( H ijca b ) ab ijc * * ijc (63) which implies that there exists at least one pair (i* , j* ) {1, , N} such that 84 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn Do Khac Duc et al TNU Journal of Science and Technology det( H ia* jb*c ) * * The inequality (64) implies that at least one eigenvalue of the matrix F ( q , q f ) / q is q qc negative This in turn guarantees that qc is unstable/saddle equilibrium point of (47) Extension to bounded control When two or more agents come very closed to each other, the control ui given in (35) can be very large in magnitude This is undesired Hence it is necessary to consider a bounded control law for ui Fortunately, this can be easily achieved by replacing the control ui given in (35) by a bounded control such as ui Ci N || || i (65) 192(16): 73 - 86 based on construction of new local potential (64) functions, and guaranteeing that all critical points, besides the desired points in formation, are either saddles or unstable points Both stabilization and tracking control problems of formation were addressed Formal analysis of the convergence and feasibility of the control solutions have also been discussed for cases when bounded controls are used It has been shown that the proposed controller design method can indeed guarantee the convergence of agents to a desired formation, which can either be stationary or moving A combination of the proposed controllers in this paper with a gradient climbing algorithm could result in potential applications such as search, selfcooperative transportation, and target seeking and attack i 1 The reason we use 1 N || i ||2 instead of REFERENCES R.T Jonathan, R.W Beard and B.J Young i1 (2003) A decentralized approach to formation maneuvers IEEE Transactions on Robotics and 1 || i || is that the former makes it easy to Automation, vol 19, pp 933-941 investigate the dynamics of inter-related T.D Barfoot and C.M Clark (2004) Motion agents Indeed, with the bounded control (65) planning for formations of mobile robots Robotics and Autonomous Systems, vol 46, pp 65–78 the derivative of the total potential function D M Stipanovica, G Inalhana, R Teo and C now becomes (instead of (40)): J Tomlina (2004) Decentralized overlapping N N control of a formation of unmanned aerial Ti Ci / || i ||2 (66) vehicles Automatica, vol 40, pp 1285 –1296 i 1 i 1 W Ren and R.W Beard (2004) Formation feedback control for multiple spacecraft via virtual On the other hand, the dynamics of interstructures IEE Proceedings-Control Theory related agents (46) is changed to Application, vol 151, pp 357-368 N H Yamaguchi (1997) Adaptive formation qij C (i j ) / || i ||2 , (i, j ) {1, , N } control for distributed autonomous mobile robot i 1 groups, in Proc IEEE Int Conf Robotics and (67) Automation, Albuquerque, NM, pp 2300–2305 P K C Wang (1991) Navigation strategies for Stability analysis can be carried out the same multiple autonomous mobile robots moving in lines as in Subsection 3.2 It is noted that with formation, J Robot Syst., vol 8, no 2, pp 177– the bounded control, the agents take longer 195 time to approach their desired locations P K C.Wang and F Y Hadaegh (1996) CONCLUSIONS Coordination and control of multiple We have presented a constructive method to microspacecraft moving in formation, J design controllers that forces a group of Astronautical Sci., vol 44, no 3, pp 315–355 J P Desai, J Ostrowski, and V Kumar (1998) N mobile agents to achieve a particular Controlling formations of multiple mobile robots,” formation in terms of shape, location and in Proc IEEE Int Conf Robotics and Automation, orientation while avoiding collisions among Leuven, Belgium, pp 2864–2869 themselves The control development was http://jst.tnu.edu.vn; 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Email: jst@tnu.edu.vn ... for formation stabilization Section concludes our paper PLANAR FORMATION STABILIZATION OF TWO AGENTS To illustrate our proposed approach to solve the problem of formation stabilization and formation. .. potential function is in the form of sum of collision avoidance functions while those navigation functions in the form of product of collision avoidance functions and This feature makes our potential. .. tracking of N mobile agents, we begin with an examination of a group of two mobile agents whose dynamics are given by qi ui (1) 75 Do Khac Duc et al where qi [ xi yi ]T TNU Journal of Science