SDRE BASED EADER-FOOWER FORMAION CONRO OF MUIPE MOBIE ROBOS Caio Igor Gonçalves Chinelato, uiz S. Martins-Filho Universidade Federal do ABC - UFABC Av. dos Estados, 5001, Bangu, 09210-971, Santo André, SP, Brasil E-mail: caio_i_c@hotmail.com, luiz.martins@ufabc.edu.br Abstract: Formation control of multiple mobile robots is relatively a new area of robotics and increase the control performance and advantages of multiple mobile robots systems. In this work we present a study concerning the modeling and formation control of a robotic system composed by two mobile robots, where one robot is the leader and the other is follower. he system is a nonlinear dynamical system and cannot be controlled by traditional linear control techniques. he control strategy proposed is the SDRE (State-Dependent Riccati Equation) method. Simulations results with the software Matlab show the efficiency of the control method. Keywords: Formation Control, Multi-Robot Systems, Mobile Robots, Nonlinear Dynamical Systems, SDRE Control. 1. Introduction Formation control of multiple robots have drawn an extensive research attention in robotics and control community recently. he objective of formation control of multiple mobile robots is maintain a desired orientation and distance between two or more mobile robots. In this work we study two mobile robots. his area has a wide range of applications like transportation of large objects, surveillance, exploration, etc. he main advantages of formation control are reliability, adaptability, flexibility and perform complex missions and tasks that would be certainly impracticable for a single mobile robot. he main approaches and strategies proposed in the literature for the formation control are Virtual structure, behavior based and leader-follower [3,11,12]. he virtual structure treats the entire formation as a single virtual rigid structure. By behavior based approach, several desired behaviors are prescribed for each robot, and the final action of each robot is derived by weighting the relative importance of each behavior. In the leader-follower approach, one of the robots is designated as the leader, with the rest being followers. he follower robots need to position themselves relative to the leader and maintain a desired relative position with respect to the leader. he strategy analyzed in this work is the leader-follower approach. he system is a nonlinear dynamical system [10] and there are several control methods to control the system presented in literature like backstepping [4], direct lyapunov method [11], feedback linearization [7], variable structure [8], sliding mode [6], neural network [5] and Fuzzy [13]. In this work, the control method to realize the leader-follower formation control is the SDRE (State-Dependent Riccati Equation). 2. System Modeling he configuration of the system analyzed is showed in fig.1 [11]. X-Y is the ground coordinates and x-y is the Cartesian coordinates fixed of the leader robot. (X,Y ) and (X F,Y F ) are global positions of the leader and follower respectively in which the subscripts '' and 'F' represent leader and follower respectively. v and v F are leader's and follower's linear velocities; θ and θ F are their orientation angles; w and w F are leader's and follower's angular velocities. And l and φ are follower's relative distance and angle with respect to the leader. 1 707
Figure 1: Configuration of the system analyzed [11]. he modeling of the nonlinear dynamical system is [11]: e w e v cose f (1) x y F e w e v sine f (2) y x F e w F w (3) f l 1 d dsin dw l d sin dv (4) f l cos w l cos (5) 2 d d d d 1 2 d where e x = l xd - l x, e y = l yd - l y and e θ = θ F - θ. Given v, w, l d and φ d (d means desired), we need to find the control inputs v F and w F in order to make l x l xd, l y l yd and e θ stable. 3. SDRE Control Method SDRE (State-Dependent Riccati Equation) control method have drawn an extensive research attention in control community recently [1]. his strategy is very efficient for nonlinear feedback controllers. he method represents the nonlinear system in a linear structure that have state-dependent matrices and minimizes a quadratic performance index. he algorithm solves, for each point in the state space, a algebraic Riccati equation and statedependent. Because of this the method calls State-Dependent Riccati Equation. Given the nonlinear system (1) to (5): X f ( X ) g( X ) U (6) he system needs to be transformed in following form: X A( X ) X B( X ) U (7) 2 708
he feedback control law that minimizes the quadratic performance index [9]: J 0 X ( t) Q( X ) X ( t) U( t) R( X ) U( t) dt (8) is: U R 1 ( X ) B ( X ) P( X ) X (9) he matrix P(X) can be obtained by the Riccati equation: -1 P(x)A(x) A (x)p(x) Q(x) P(x)B(x)R (x)b (x)p(x) 0 (10) Q(X) e R(X) are project parameters and are positive definite. 4. Simulation Results o analyze the performance of the controller we simulate three cases with the software Matlab. In the first case w = 0, i.e., the leader's heading direction does not change. he leader moves in a constant linear speed of v = 1.5 m/s along a straight line with θ = π/6 rad and the follower keeps a constant relative distance l d = 2.0 m and a constant relative angle φ d = 5π/4 rad from the leader (l xd = l yd -1.41 m). he initial conditions are l x0 = 0.7 m, l y0 = -1.5 m and e θ = 0.65π rad. In the second case w = 0.3π rad/s and v = 0.5 m/s. he follower keeps a constant relative distance l d = 2.0 m and a constant relative angle φ d = π/2 rad from the leader (l xd = 2.0 m l yd = 0 m). he initial conditions are l x0 = 0.1 m, l y0 = 0.1 m and e θ = π/2 rad. he third case is equal to the second, the only difference is that the follower rotates around the leader at a constant relative angular speed of φ d = 0.2π rad/s. he numeric method to solve the nonlinear system is the Euler method [2]. 3 709
Figure 2: he leader moves along a straight line, and the follower keeps a constant relative distance and angle with respect to the leader. Figure 3: he leader moves goes along a circle, and the follower keeps a constant relative angle and distance with respective to the leader. 4 710
Figure 4: he leader moves goes along a circle, and the follower keeps a constant relative distance and rotates around the leader at a constant relative angular speed. Analyzing the results of the simulations we can see that the proposed controller can achieve the desired formation, and the whole system is stable. 5. Conclusions and Future Works In this work we presented a study concerning the modeling and formation control of a robotic system composed by two mobile robots, where one robot is the leader and the other is follower. he nonlinear dynamical system was controlled by the SDRE control method and the simulation results showed the efficiency of the control method. 5 711
he main future works that could be realized is modeling the system with more than two robots, try another kind of control method and considering problems like obstacle avoidance in the environment and path planning. 6. References [1] Çimen, State-Dependent Riccati Equation (SDRE) Control: a Survey, Proceedings of the 17th World Congress the International Federation of Automatic Control, 2008. [2] Chapra, "Numeric Methods for Engineers", Mcgraw Hill, 2001. [3] Desai, Modeling and Control of Formations of Nonholonomic Mobile Robots, IEEE ransactions on Robotics and Automation, vol. 17, 2001. [4] Dierks, Control of Nonholonomic Mobile Robot Formations: Backstepping Kinematics into Dynamics, International Conference on Control Applications Part of IEEE Multi-Conference on Systems and Control, 2007. [5] Dierks, Neural Network Output Feedback Control of Robot Formations, IEEE ransactions on Systems, Man, and Cybernetics, Vol.40, 2010. [6] Dongbin, Second-Order Sliding Mode Control for Nonholonomic Mobile Robots Formation, Proceedings of the 30th Chinese Control Conference, 2011. [7] Ge, "Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications" aylor and Francis, cap.11, 2006. [8] Ha, Modeling Robotic Formation Control Using Variable Structure System Approach, Proceedings of the IEEE Workshop on Distributed Intelligent Systems: Collective Intelligence and Its Applications, 2006. [9] Kirk, "Optimal Control heory: An Introduction", Princeton Hall Englewood Cliffs, 1970. [10] Khalil, "Nonlinear Systems", Prentice Hall, 2002. [11] i, Robot Formation Control in eader-follower Motion Using Direct yapunov Method, International Journal of Intelligent Control and Systems, Vol. 10, 2005. [12] Wang, eader-follower and Communications Based Formation Control of Multi-Robots, Proceedings of the 10th World Congress on Intelligent Control and Automation, 2012. [13] Yangs, A Multi-agent Fuzzy Policy Reinforcement earning Algorithm with Application to eader-follower Robotic Systems, Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006. 6 712