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of Proceedings the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 MoB03.4 Formation Control of Marine Surface Craft using Lagrange Multipliers Ivar-Andr F. Ihle1 , J r me Jouffroy1 and Thor I. Fossen1 2 Abstract We propose a method for constructing control laws for formation control of marine craft using classical tools from analytical mechanics. The control law is based on applying inter-vessel constraint functions which again impose forces on the individual vessels that maintain the given constraints on the total system. In this way, a formation can be assembled and stay together when exposed to external forces. A brief comparison with other control designs for a group of marine craft is done. Further, control laws for formation assembling (with dynamic positioning), and formation keeping during maneuvering is derived and simulated to illustrate the properties of the proposed method. I. INTRODUCTION The elds of coordination and formation control with applications towards mechanical systems, ships, aircraft, unmanned vehicles, spacecraft, etc., have been the object of recent research efforts in the last few years. This interest has gained momentum due to technological advances in the development of powerful control techniques for single vehicles, the increasing computation and communication capabilities, and the ability to create small, low-power and low-cost systems. But coordinated control of several systems also means control of a more complicated system with challenges such as environmental information, communication between systems, etc [1]. Researchers have also been motivated by formation behaviors in nature, such as ocking and schooling, which bene ts the animals in different ways [2], [3], [4], [5]. Many models from biology have been developed to give an understanding of the traf c rules that govern sh schools, bird ocks, and other animal groups, which again have provided motivation for control synthesis and computer graphics, see [6], [7] and the references therein. The interest in cooperative behavior in biological systems, and the mixture with the control systems eld, have led to observations and models that suggest that the motion of groups in nature applies a distributed control scheme where the members of the formation are constrained by the position, orientation, and speed of their neighbors [2]. In this paper, we focus on the formation control aspect in cooperative behavior of marine craft. There exists a large number of publications on the elds of cooperative and formation control recent results can be found in [5], [8], [9], [10] and [11]. See the papers and references therein for a thorough overview. A brief introduction is given here. This project is sponsored by The Norwegian Research Council through the Centre for Ships and Ocean Structures (CeSOS), Norwegian Centre of Excellence at NTNU. 1 Centre for Ships and Ocean Structures, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway. 2 Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. E-mails: ihle@ntnu.no, jouffroy@ntnu.no, fossen@ieee.org. There exists roughly three approaches to vehicle formation control in the literature: leader-following, behavioral methods and virtual structures. Brie y explained, the leader-following architecture de nes a leader in the formation while the other members of the formation follows that leader. The behavioral approach prescribes a set of desired behaviors for each member in the group, and weight them such that desirable group behavior emerges. Possible behaviors include trajectory and neighbor tracking, collision and obstacle avoidance, and formation keeping. In the virtual structure approach, the entire formation is treated as a single, virtual, structure. Virtual structures have been achieved by for example, having all members of the formation tracking assigned nodes which move through space in the desired con guration, and using formation feedback to prevent members leaving the formation [12]. In [13] each member of the formation tracks a virtual element, while the motion of the elements are governed by a formation function that speci es the desired geometry of the formation. Mechanical constraint forces which cause independent bodies to act in accordance with given geometric constraints, are well known from the early days of analytical mechanics [14] and has been used with success, e.g. in computer graphics applications [15], [16]. The main idea in this paper is to show how classic and powerful tools from analytical mechanics for multi-body dynamics [17] can be used for formation control issues. A collection of independent bodies/vehicles can be controlled as a virtual structure by introducing functions that describe a vehicles behavior with respect to the others. In this setting, imposing constraints on a system lead to control laws, which can be combined with the formation constraints that maintain the virtual structure, and together form control laws that govern the movement of the entire formation. The rest of this paper is organized as follows. In Section II, we model a system with constraints, present stabilizing methods, and show how constraints can be imposed for control purposes. Section III presents examples of formation control of marine craft with constraints, and Section IV contains some concluding remarks. II. MODELING, CONTROL, AND CONSTRAINTS A. Motivating Example To introduce the main idea of the method given in this section and illustrate how the constraint function affects the motion of independent systems, we look at a formation R2 with kinetic energy of two point masses 1 2 >> T = 1 > where = [ 1 2 ]> and = > 0 is 2 the mass matrix; = diag ( 1 2 2 2 ). The distance 0-7803-9568-9/05/$20.00 2005 IEEE 752 2.5 2 1.5 No force 1 0.5 0 0.5 1 1.5 0.5 0 0.5 1 1.5 where is the Lagrangian multiplier(s). To obtain the equations of motion, we apply the Euler-Lagrange differential equations with auxiliary conditions for = 1 , L which implies L L + C( ) + C( )= constraints adds potential energy to the system, so we end up with the following, modi ed, Lagrangian L=T U + C( ) 2.5 2 1.5 With force 1 0.5 0 0.5 1 1.5 0.5 0 0.5 1 1.5 L + C( ) = (4) Fig. 1. Position plot of 1 and 2 starts in (0 0) and (1 1) respectively. = 0 5 The lower part includes the initial force disturbances. where is the generalized external force associated with coordinate . Equation (3) constrains the systems motion to a subset, M R2 of the state space where C ( ) = 0. Since we want to keep the systems on M , neither the velocity nor the acceleration should violate the constraints. This gives the additional conditions C( ) = C( ) = ( ) =0 ( ) + ( ) = 0 (5) C( ) = 0 where 0 is the desired distance between 1 and 2 . The procedure in the Section II-B and II-C gives the following equations of motion (on the manifold where (1) is satisi ed) (2) = ( )> where ( ) is the Jacobian of the constraint function and is the Lagrangian multiplier with stabilizing feedback from the constraints. The system has initial conditions that violate the constraint (1), but the masses converge such that (1) is met. The system converges to a position between the initial position of 1 and 2 where the distance between 1 and 2 is . If we apply some damping and initially perturb both integrators with a small amount of force, the motion of the integrators will still satisfy the constraint function, eventually. As seen in Figure 1 the two points move such that (1) is satis ed in both cases they assemble into a formation de ned by the constraint function. B. System Modeling Consider systems of order with kinetic and potential energy, T and U respectively. The Lagrangian of the total system is then L=T U= X =1 between the point masses shall satisfy the constraint function > 2 C( )=( 1 =0 (1) 2) ( 1 2) where () R is the Jacobian of the constraint function, i.e. ( ) = C( ) . We combine (4) and (5) to obtain an expression for the Lagrangian multiplier. The main focus in this paper is marine craft, so we introduce the equations of motion for a marine vessel in the body- xed frame, derived analytically in [18] using an energy approach, + = ( ) + () ( ) + ( )= T U (3) where =[ ]> is the Earth- xed position vector, ( ) is the position on the ocean surface and is the heading angle (yaw), and =[ ]> is the body xed velocity vector. The model matrices , , and denote inertia, Coriolis plus centrifugal and damping, respectively, while is a vector of generalized forces and = () (3) is the rotation matrix between the body and Earth coordinate frame1 . For more details regarding ship modeling, the reader is suggested to consult [18] and [19]. Consider a formation of vessels with position , and so on. We collect given by , inertia matrix the vectors into new vectors, and the matrices into new, >> ], block-diagonal, matrices by de ning = [ > 1 }, and so on. The addition of the = diag { 1 potential energy from the constraints gives + ( ) + = ( ) ( ) + ( )= > Suppose there exists kinematic relations between the coordinates which restricts the state space to a constraint manifold M with less than 2 dimensions. We denote C ( ) the constraint function, where R contains all generalized positions, , of the systems. We know, from [17], that the forces that maintain the kinematic C( )=0 C( ) R + constraint where constraint = () is the expression for the constraint forces which maintain the kinematic constraint. The equations of motion can be transformed to the Earth xed frame by the kinematic transformation in [18, Ch. 1 Note that this approach is also valid for mechnical systems such as ( ) + ( ) = (robot manipulator). 753 3.3.1] which results in ( ) + ( ) = () () > the constraint function (6) We rewrite (9), using 1 ( ) + ( ) + ( ) and where ( ) = = (). By replacing with , the combination of (5) and (6) gives a differential algebraic equation (DAE). Solving for and substituting gives () 1 C= C( ) C( ) =C( ) = 1 2 (9) =C( ) 1 2 2 =C( ) 0 3 such that () () 1 () > = ( )} + ( ) () Then, by using in (6), we ontain the equations of motions for the systems subject to the constraint (3). Assumption A1: The mass matrix is positive de nite, > ( )= () 0 by [18]. i.e. = > 0, hence Assumption A2: The Jacobian ( ) has full rank, i.e., the constraints are not con icting or redundant, and is bounded by a linear growth rate, 1 | | 2 | |, for some positive 1 2 R. Note that redundant or con icting constraints arise when one, or more, row (column) in C is a linear combination of other rows (columns), or when the functions are contradicting. 1 Assumptions A1 and A2 guarantees that 1 exists since is positive de nite, hence exists and 1 > is nonsingular. C. Stabilization of constraints If the system starts on the constraint manifold M , that is, the initial conditions ( (0) (0)) = ( 0 0 ) satisfy ( 0 0 ) M such that and the force does not perturb the system so that ( ) leaves M , then the system is well-behaved and ( ) M for all times. However, if the initial conditions are not in M , or the system is perturbed s.t. ( ) 6 M , feedback must be used to stabilize the constraint. We want to investigate stability of the constraint, and in terms of set-stability we look at stability of the set Consider the case when 6= 0 in (6), and suppose that ( 0 0 ) 6 M . If we use the equations from Section II-B, then C( )=0 (7) M = {( ) : C ( ) = 0 ( ) = 0} C ( 0 ) = 0 and C ( 0 ) = 0 ( ){ = = 1 2 2 > (10) R2 By appropriate choice of and , Then, by choosing a design matrix = nd a = > 0 such that + Then () = ( ) = > > > is Hurwitz. 0, we can = 0 0 6= 0 6= 0 (11) (12) Theorem 1: The set M is a globally exponentially stable (GES) set of equilibrium points of the system ( ) + ( ) = C( )=0 () () > (13) which is unstable if C ( ) is a scalar function it contains two poles at the origin. Hence, if C ( ) = 0 is not ful lled initially, the solution might blow up in nite time. Even with C ( 0 ) = 0, this might happen if there is measurement noise on . This unstability is, in fact, an inherent property of higher-index DAEs [20]. However, if we introduce feedback from the constraints in the expression for the Lagrangian multiplier, () ( where 1 under Assumptions A1 and A2. Proof: The proof follows from (11) and (12). By applying feedback from the constraints and converting (7) to (9) the condition (9) has become more robust, and is stable in the cases when the initial values do not ful ll the constraint and when there is measurement noise present. Remark 1: Consider the case in the motivating example = 2 and = 2 , so C + 2 C + 2 C = 0, and set 0. In the numerical scienti c society the last equation is referred to as the Baumgarte stabilization technique [21] an algortihm used for numerical stabilization in simulations of multi-body and constrained systems. Remark 2: Given a position control law 1 for a single vessel that renders the equilibrium points of the system 1 ( 1 ) 1 + ( 1 1 1 ) = 1 UGAS. In combination with constraint forces that maintain the con guration, > ( ) + ( ) = ( ) ( ) , the total, closed-loop, system can be treated as a cascade see Figure 2. Further, the constraint stabilization dynamics can be shown to be Input-to-State Stable [22] with respect to the position errors in the control law 1 . This implies that the system s equilibrium points remain UGAS, and the combination of the formation constraint stabilization and the position control laws yields a system that moves according to both controllers. This is a topic of ongoing research. D. Using Constraints for Control So far we have talked about systems with constraints without saying anything about how the constraints arise. In the control literature, the main focus has been on constrained robot manipulators with physical contact between the end effector and a constraint surface. This occurs () R () () > = C( )+ () 1 )} + ( ) + ( ){ and positive de nite, we stabilize C( ) (8) 754 Fig. 2. Combination of constraints and control law as a cascade. in many tasks, including scribing, writing, grinding, and others as described in [23], [24] and references therein. These constraints are inherently in the system as they are all based on how the model or environment constrain the system. However, if constraints are imposed on the system, the framework for stabilization of constraints described in Section II-C can be used to design control laws with feedback from the constraints applicable for a wide variety of purposes. Example 1: Consider a single, unforced, double integrator subject to the constraint C ( ) = = 0, R = where ( )=1 = 1 and 2 () is given as in (8): R = =2 + The constraint manifold is now the origin, i.e. M {(0 0)}. The dynamics are = 2 2 which can be shown to be GES for 0. Given an inital condition (0) = 0 6 M the trajectory ( ) converges exponentially to the origin. The constraint function corresponds in this case to a proportional-derivative (PD) controller. III. CASE STUDIES To illustrate how constraints can be used in a formation setup control scheme, we will compare some schemes from formation [25] and synchronization [26] control of marine craft and see how constraints are used implicitly. Secondly, we will focus on how a set of constraints can help us to control a formation during the initial phase the formation assembling phase. Further, we will show that by using constraints to maintain a formation of vessels, we can use a path following design for one vessel to achieve formation control. A. Control Plant Ship Model The control plant model of a supply vessel is used in the case study [27]. We consider a vessel model for low-speed applications (up to 2-3 m/s) and station-keeping where the kinetics can be described accurately by a linear model. Extension to maneuvering at higher speeds can be done using the nonlinear model in [28]. Furthermore, the surge is mode decoupled from the sway and yaw mode in the ship model. The body- xed equations of motion are given as + () = = ( ) (14a) (14b) B. Formation Control Schemes and Constraints This section will show that, under some assumptions, constraints of the form dicussed in this paper appear implicitly in some previous schemes for coordinated control of a group of ships. In the formation maneuvering design given in [25] based on the maneuvering design in [29] the control objective is to make an error vector go to zero in particular, the design establishes UGES of the set M = {( ) : = 0}. The error vector contains information about both position and velocity, where the position for = 1 2, where error is de ned as 1 = R is the position and R is the desired 1 location for the i th member of the formation. We consider a formation with two members where = + ( ( )) , R3 and we assume that ( ) = . This corresponds to a desired motion where the formation moves parallel to the -axis in the inertial frame. When the systems have reached = M, = 0 and the position errors are 1 = =0 Combining 11 and 12 , we get 1 1= 2 2, (1 or 1 2) = 1 12 = 0. Similar with 2 2 the velocity errors on M, 2 = 1 , where 1 is a virtual control law. For the two systems we get 2 = ) = 0, where is the velocity and 1 is a 1( control design matrix. A combination of 21 and 22 gives, assuming that 11 = 12 , 1 2 = 11 ( 11 12 ) = 0. Hence, on M, the error variable gives constraints on the form C( ) = 1 2 12 ( )= + ( ) and are nondiwhere mensional system matrices from [27] which are Bis-scaled coef cients, identi ed by full-scale sea trials in the North ( ) contains nonlinear damping the Sea. The matrix term in surge, i.e., ( ) = diag | | | | 0 0 . The model will be transformed corresponding to Section II-B. =0 =0 With the above assumptions we see that in the formation assembling phase, the set M corresponds to M . In [26] the authors use synchronization techniques to develop a control law for rendezvous control of ships. In a case study with two ships the control objective is to control the supply ship to a position relative to the main ship. The desired con guration is reached when the errors = and = are zero, where the subscripts and stands for supply- and main ship, respectively. Both error functions t into the framework for constraints in Section II-B. For a replenishment operation, we can de ne the constraint to depend on the lateral position coordinate only. Then, the supply vessel converge to a position paralell to the course of the main ship, and this position can be maintained during forward speed for replenishment purposes. C. Case 1: Assembling of Marine Craft We consider a formation of three vessels where the control objective is to assemble the craft into a prede ned con guration, e.g. in order to be in position to tow a barge C( ) = 1 2 = 755 Constraints or another object. We assume there is no external force or control law acting on the formation, i.e. = 0 and the purpose is to show that assembling of the individual vessels into a formation can be done by imposing constraints Consider the constraint function C1 ( ) = 2 Formation Assembling 200 C 11 C12 C13 150 100 ( ( ( 1 2 3 2) ( 3) ( > 1) ( > > 1 2 3 2) 3) 1) 2 12 2 23 2 31 50 =0 (15) 0 0 5 10 15 20 25 R is the reduced position vector without the where R is the distance between vessel orientation and and . This constraint enables us to specify the positions of each vessel with respect to the others. The constraint manifold is now equivalent to the formation con guration, and by the previous sections we know that we can stabilize M by using feedback from the constraints. Hence, the constraints method gives control laws for formation assembling. From the ship model and the constraint, we have > + = () () where = ( ) = diag ( 1 ), = 2 > 3 > > > ( ) = diag ( 1 =123, 2 3 ), and, so on. The Lagrangian multiplier is obtained from 1 > 5 0 Derivatives 5 10 15 20 25 0 5 10 15 20 25 Fig. 3. Time response of formation constraints during assembling. C1 corresponds to the th row in C1 . 8 = 1 + ( ) + Equation (15) gives the con guration of the formation, but does not provide any information about location. If the control objective is to assemble the formation and position the rst vessel in a desired location, we can add a row to (15), such that ( ( ( 1 2 3 2) C( )+ C( ) 6 4 2 C2 ( ) = ( )( 3 > 1) ( > 1 > 1 2 3 des 2) 3) 1) 2 12 2 23 2 31 0 =0 (16) 2 where des R3 is the xed desired position and orientation for the rst vessel. The closed-loop equations have the same structure as before, except that C1 is replaced with C2 . The last addition in the constraint function is equal to a PD-controller for the rst vessel, so (16) leads to a combination of a formation controller and a PD-typecontroller for dynamic positioning of ships [18]. Hence, several control laws can be handled in one step using the constraint approach. The control parameters chosen to stabilize the formation = 08 , = constraints C1 and C2 are ( = 3 3 ) 0 8 , the formation is de ned by 12 = 3, 23 = 3, 31 = 3, and the desired position for the rst vessel is des = > > [10 5 0] . The vessels starts in 10 = [8 8 0] , 20 = > > [ 2 2 0] , and 30 = [ 2 2 0] all with zero initial velocity. In Figure 3 the time-plot of the constraint function C1 and its time-derivative, 1 ( ) , are shown. The constraints and velocity terms converge to zero, and the constraint manifold is reached. The vessels have converged to the nearest positions where the constraints are ful lled, and the formation is assembled in the desired con guration, as 4 2 0 2 4 6 8 10 Fig. 4. Position response of vessels during assembling. seen in Figure 4. Figure 5 shows the position of the three vessels with the same inital conditions as before but subject to the constraint C2 . The vessels assemble according to the constraints, but this time vessel 1 is positioned at its desired position des . This forces the two other vessels to move to a different position compared to the rst case in order to satify the constraint. The additional constraint slows down convergence to the constraint manifold since vessel 1 has to be positioned at des and this forces the other vessels to move such that the constraint is ful lled. D. Case 2: Maneuvering a formation In the previous section, we showed that by imposing constraints a formation can be assembled at a desired location. Centralized and decentralized formation maneuvering has been achieved in [25], [30], respectively, where each 756 10 8 6 4 2 0 2 2 0 2 4 6 8 10 12 14 Fig. 5. Position response of vessels subject to formation constraints when vessel 1 is to be positioned at des Fig. 6. Position plots of three vessels in a formation where vessel 1 (red) is following a desired path (dashed). The resulting vessel trajectories are shown as solid lines. member of the formation follows a prede ned path. To show that constraints also can be used to maintain the structure of a formation during movement, we will use a path following scheme for one of the vessel, and apply a constraint function to assure that the distance between the vessels satis es the formation con guration. We use the constraint function C1 ( ) from (15). This constraint is used since we want only vessel 1 to follow a prede ned path the other vessels will follow while maintaining the distance. We have the closed-loop system + where 1 > = () () > = 1 ( + ) + ( ) C( )+ C( ) and the maneuvering design of the nominal system gives us the following signals, = [ 1 0 0]> , 1 1 1 1 2 1 := = 1 1 () h > > = 1 11 () h 1 2 11 > 1 := 1 + h 1 () ( 1 1 + + i + i ) = = = 1 1 > 1 + 2 1 2 + i 2 > 11 1[ 1 + = 1( ) 2 1+ 1+ 1 2 221 > 11 The control parameters in the maneuvering part are set as diag (2 2 20) ( = 3 3 ) 1 = 0 5 2= 1= 0 6 , 2 = diag (10 10 40), and = 10, while = = . The con guration of the formation is and de ned by 12 = 90, 23 = 60, 31 = 90, and the > initial conditions for the vessels are 1 (0) = [500 0 0] > 1 2 3 (0) = 2 (0) = [445 63 0] , 3 (0) = [549 29 0] 0 and (0) = 0. The speed assignment for Vessel 1, , is chosen correpsonding to a desired surge speed of 2 m/s along the path. Figure 6 shows that Vessel 1 follows the desired path accurately, and the formation stays together throughout the simulation. The distance between the vessels is constant (when the states are on the constraint manifold), so the control objective is satis ed and even though Vessel 2 and 3 do not follow prede ned paths, the constraints prevent collisions by maintaining the formation con guration. The constraints during the rst 25 seconds of the simulation Figure 7 shows that after the constraints converge to zero the states remain on M . E. Discussions This section have only considered constraint functions that leads to a triangular formation, but depending on the number of members in the formation, the formation can be con gured in many different ways, e.g. line- (in the transversal or longitudinal direction), circle- or box-shaped, by changing the constraint function. Different formation con gurations can be useful during changing operations, such as collision avoidance, sea-bed scanning, etc. The desired path is de ned as a circle with radius 500 " # cos () sin () = ( )= () () atan2 () = 22 ] where ( ) is the desired path, ( ) is a speed assignment, and the other signals come from the backstepping design [29]. Notice that it is possible to show that the combination of constraints and maneuvering controller is UGAS as discussed in Remark 2. 757 Assembling Constraints 3000 2000 1000 C(q) 0 1000 2000 3000 0 5 10 15 20 3000 2000 J(q)*qdot 1000 0 1000 2000 3000 0 5 10 15 20 Fig. 7. Time-plot of constraints. For simplicity, we have only considered holonomic, inter-vessel, constraints in this paper, i.e., kinematic constraints which can be expressed as nite relations between the generalized coordinates, but the proposed method can be extended to nonholonomic constraints which are linear in the generalized velocities [31]. IV. CONCLUSION In this paper, we have proposed a new method for constructing control laws based on imposing constraints functions which again impose forces that maintain the constraints. In particular, we have come up with control laws which maintain a formation structure subject to measurement errors, poor initialization, and forced motion. Furthermore, we have seen how constraints appear in many control schemes for multiple marine craft, and how our approach can be used in combination with existing control laws in order to simultaneously achieve some desired behavior and maintain formation con guration. Applications of this method have been illustrated by simulations of formation of marine craft. REFERENCES [1] D. A. Schoenwald, Auvs: In space, air, water, and on the ground, IEEE Control Systems Magazine, vol. 20, no. 6, pp. 15 18, 2000. [2] A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, ocks, and herds, Advances in Biophysics, vol. 22, pp. 1 94, 1986. [3] B. L. Partridge, The structure and function of sh schools, Scienti c American, pp. 90 99, June 1982. [4] J. K. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, vol. 284, no. 5411, pp. 99 101, 1999. [5] V. Kumar, N. Leonard, and A. S. Morse (Eds.), Cooperative Control, ser. Lecture Notes in Control and Information Sciences. Heidelberg, Germany: Springer, 2005. [6] C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Computer Graphics Proceedings, vol. 21, no. 4, pp. 25 34, 1987. [7] , Boids, 2001, <http://www.red3d.com/cwr/boids/>. [8] J. A. Fax and R. M. Murray, Information ow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465 1476, 2004. [9] S. Spry and J. K. Hedrick, Formation control using generalized coordinates, in Proc. 43rd IEEE Conf. Decision & Control, Paradise Island, The Bahamas, 2004, pp. 2441 2446. [10] R. W. Beard, J. Lawton, and F. Y. Hadaegh, A coordination architecture for spacecraft formation control, IEEE Transactions on Control Systems Technology, vol. 9, no. 6, pp. 777 790, November 2001. [11] P. gren, E. Fiorelli, and N. E. Leonard, Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment, IEEE Transactions on Automatic Control, vol. 49, 25 no. 8, pp. 1292 1302, 2004. [12] W. Ren and R. W. Beard, Formation feedback control for multiple spacecraft via virtual structures, IEE Proc. - Control Theory and Applications, vol. 151, no. 3, pp. 357 368, 2004. [13] M. Egerstedt and X. Hu, Formation constrained multi-agent control, IEEE Transactions on Robotics and Automation, vol. 17, no. 6, pp. 947 951, 2001. [14] J. L. Lagrange, M canique analytique, nouvelle dition. Acad mie des Sciences, 1811, translated version: Analytical Mechanics, Kluwer Academic Publishers, 1997. [15] D. Baraff, Linear-time dynamics using Lagrange multipliers, in Computer Graphics Proceedings. SIGGRAPH, 1996, pp. 137 146. [16] R. Barzel and A. H. Barr, A modeling system based on dynamic 25 constraints, Computer Graphics, vol. 22, no. 4, pp. 179 188, 1988. [17] C. Lanczos, The Variational Principles of Mechanics, 4th ed. New York: Dover Publications, 1986. [18] T. I. Fossen, Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Trondheim, Norway: Marine Cybernetics, 2002, <http://www.marinecybernetics.com>. [19] T. I. Fossen and . N. Smogeli, Nonlinear time-domain strip theory formulation for low-speed maneuvering and station-keeping, Modelling, Identi cation and Control, vol. 25, no. 4, 2004. [20] D. C. Tarraf and H. H. Asada, On the nature and stability of differential-algebraic systems, in Proc. of the American Control Conference, Anchorage, AK, 2002, pp. 3546 3551. [21] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Computer Methods in Applied Mechanical Engineering, vol. 1, pp. 1 16, 1972. [22] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, vol. 34, no. 4, pp. 435 443, 1989. [23] H. Krishnan and N. H. McClamroch, Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems, Automatica, vol. 30, no. 12, pp. 1885 1897, 1994. [24] N. H. McClamroch and D. Wang, Feedback stabilization and tracking of constrained robots, IEEE Transactions on Automatic Control, vol. 33, no. 5, pp. 419 426, 1988. [25] R. Skjetne, S. Moi, and T. I. Fossen, Nonlinear Formation Control of Marine Craft, in Proc. 41st IEEE Conf. Decision and Control, Las Vegas, Nevada, USA, 2002, pp. 1699 1704. [26] E. Kyrkjeb and K. Y. 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Path: Rutgers >> 640 >> 252 Fall, 2008
0949.pdf
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222.pdf
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iiss.pdf
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97ecc-clf.pdf
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1748.pdf
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bifurcation.pdf
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04cdc-circadian-final.pdf
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0068.pdf
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CD001969.PDF
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89cdc-stab.pdf
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mct-toc.pdf
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88cdcdec.pdf
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0424.pdf
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hmwPDEmodels_rev06.pdf
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CD001417.PDF
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xu_tao_noise_gene_regulation_feedback_JTB06.pdf
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moreau-sontag-arcak-bifurcations.pdf
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mancilla-garcia-wang-sontag-switched-representations-NA05.pdf
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05cdc_ejc_oct05.pdf
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0633.pdf
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0836.pdf
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comments-sharma.pdf
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0308.pdf
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1639.pdf
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0938.pdf
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kalman1.pdf
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TuP05.4.pdf
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0675.pdf
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CD001327.PDF
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kalman7.pdf
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0883.pdf
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week8.pdf
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week8.pdf
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week10.pdf
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week10.pdf
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soln10.pdf
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soln5.pdf
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soln8.pdf
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soln11.pdf
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soln-e1.pdf
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soln2.pdf
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soln1.pdf
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soln12.pdf
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soln9.pdf
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dt-iss-ifac.pdf
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0588.pdf
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ddgs.pdf
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dls.pdf
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02cdc-ingalls-sontag-wang-published.pdf
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net-ident.pdf
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516.pdf
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CD001202.PDF
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1782.pdf
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0339.pdf
Path: Rutgers >> 640 >> 336 Fall, 2008