The Numerical Control Design for a Pair of Dubin’s Vehicles  Heru Tjahjana 1   , Iwan Pranoto2   , Hari Muhammad   3   , J. Naiborhu 4   , and Miswanto 5 1,2,4,5   Mathematics  Institut Teknologi Bandung, Indonesia 1   Permanent address: Mathematics  Diponegoro University, Indonesia e-mail: 1   heru_tjahjana@students.itb.ac.id 2   pranoto@math.itb.ac.id 4   janson@math.itb.ac.id 5   miswanto@students.itb.ac.id  3   Aeronautics and Astronautics Institut Teknologi Bandung, Bandung, Indonesia e-mail: harmad@ae.itb.ac.id  Abstract  In this paper, a model of a pair of Dubin’s vehicles is considered. The vehicles move from an initial position and orientation to final position and orientation. A long the motion, the two vehicles are not allowed to collide however the two vehicles cann’t to far each other.   The optimal control   of   the vehicle   is   found   using   the   Pontryagin’s Maximum   Principle   (PMP).   This   PMP   leads   to   a Hamiltonian system consisting of a system of differential equation and its adjoint. The originally differential equation has initial and final condition but the adjoint system doesn't have one.   The classical difficulty is solved numerically by the   greatest   gradient   descent   method.   Some   simulation results are presented in this paper.  1   Introduction  The swarm behavior in nature is interesting by itself, but in this   current   paper   modeling   the   multi-vehicle   systems together with their designed optimal control is considered. Multi-agent   system   and   swarm   phenomena   have   been studied   extensively.   Many   studies   regarding   these   two topics have been done and published, but all of them focus on the local swarm behavior or something very far from transportation situations[6]-[9]. They cannot be applied to the problems such as aircraft or ship convoy. In this paper, the   optimal   control   of   a   pair   of   Dubin’s   vehicles   is designed. The previous research about swarm modeling through   optimal   control   can   be   found   in   [1]-[5].   The previous paper which expos Dubin’s vehicles can be found in [10]-[12].  2   A Pair of Dubin’s Vehicles Model  Consider a pair of Dubin’s vehicles model. The model of first vehicles is given as follows 1   3   1 2   3   1 3   2 (sin   ) (cos   )  x   x   u x   x   u x   u  =  = =      (1) And the second vehicle is given as 1   3   1 2   3   1 3   2 (sin   ) (cos   )  y   y   v y   y   v y   v  =  = =      (2)  Figure 1: Model of Dubin’s Vehicle  The functional cost that must be minimized is  (   )   (   )  2   2   2   2 1   2   1   2 0 2   2   2   2 1   2   1   2 2   2 1   1   2   2 1 2 +   (   )   (   ) +  T J   u   u   v   v x   x   y   y dt x   y   x   y  δ   δ   δ   δ β α ρ =   +   +   + +   +   + −   +   − ∫  (3) The Hamiltonian function is ICIUS 2007 Oct 24-25, 2007 Bali, Indonesia ICIUS2007-C003 ISBN 978-979-16955-0-3 335 © 2007 ICIUS
(   )   (   ) (   )   (   ) (   )   (   )  1   1   3   2   1   3   3   2 4   1   3   5   1   3   6   2 2   2   2 0   1   0   2   0   1 2   2   2 0   2   0   1   0   2 2   2 0   1   0   2 0 2   2 1   1   2   2 sin   cos +   sin   cos 1   1   1 2   2   2 1   1   1 2   2   2 1   1 2   2 1 2  H   p   u   x   p   u   x   p u p   v   y   p   v   y   p v p   u   p   u   p   v p   v   p   x   p   x p   y   p   y p x   y   x   y  δ δ δ δ β β α α ρ =   +   +   + +   + +   +   + +   +   + +   + +   −   +   −  (4) The Hamiltonian system can be expressed as  (   ) (   ) (   ) (   )   (   ) (   ) (   ) (   )   (   ) (   )  1   3   1 1 2   3   1 2 3   2 3 0   1   1 1   0   1   2   2 1   1   1   2   2 0   2   2 2   0   2   2   2 2   1   1   2   2 3   1   1   3   2   1   3 3 1   3   2   3   0   1 1 3 2 sin cos 2   2 1 2 2   2 1 2 cos   sin 0   sin   cos 0  H   x   x   u p H   x   x   u p H   x   u p p   x   y H   p   p   x x   x   y   x   y p   x   y H   p   p   x x   x   y   x   y H   p   p u   x   p u   x x H   p   x   p   x   p   u u H   P u  ρ β ρ β δ ∂   =   = ∂ ∂   =   = ∂ ∂   =   = ∂ − ∂   = −   =   − ∂   −   +   − − ∂   = −   =   − ∂   −   +   − ∂   = −   =   − ∂ ∂   =   =   +   + ∂ ∂   =   =   + ∂         (   ) (   ) (   ) (   )   (   ) (   ) (   ) (   )   (   ) (   )  0   2 1   3   1 4 2   3   1 5 3   2 6 0   1   1 4   0   1   2   2 1   1   1   2   2 0   2   2 5   0   2   2   2 2   1   1   2   2 6   4   1   3   5   1   3 3 4   3   5   3   0   1 1 sin cos 2   2 1 2 2   2 1 2 cos   sin 0   sin   cos  p   u H   y   y   v p H   y   y   v p H   y   v p p   x   y H   p   p   y y   x   y   x   y p   x   y H   p   p   y y   x   y   x   y H   p   p v   y   p v   y y H   p   y   p   y   p   u v H u  δ ρ α ρ α δ ∂   =   = ∂ ∂   =   = ∂ ∂   =   = ∂ −   + ∂   = −   =   − ∂   −   +   − −   + ∂   = −   =   − ∂   −   +   − ∂   = −   =   − ∂ ∂   =   =   +   + ∂ ∂ ∂         6   0   2 2 0   P   p   v  δ =   =   +  (5) Consider the Hamiltonian system (5), the adjoint variables or co-state variables   i p   appear in the system. The original state variables are companied by the co-state variables. Here, the problem is the co-state variable don‘t have initial value or initial condition. This fundamental difficulty can be solved by greatest gradient descent method or sometime is   called   as   shooting   method.   Through   Pontryagin Maximum Principle,   the optimal control of two vehicles can   be   found.   If   the   optimal   control   of   two   vehicles substitute to the Hamiltonian system, then the differential equation system (5) equivalent with  (   ) (   ) (   ) (   )   (   ) (   ) (   ) (   )   (   ) (   ) (   )  2 1   3   1   3   2   3   3 2 2   3   1   3   2   3   3 3 3 1   1 1   1   2   2 1   1   2   2 2   2 2   2   2   2 1   1   2   2 3   1   3   1   3   2   3 1 sin   (   sin   sin   cos   ) 1 cos   (   sin   sin   cos   ) 2   2 1 2 2   2 1 2 1 cos   (   sin   cos   )  x   x   p   x   p   x   x x   x   p   x   p   x   x P x x   y p   x x   y   x   y x   y p   x x   y   x   y p   p   x   p   x   p   x  δ  δ  δ   ρ β ρ β   δ     =   +               =   +           = − −   = −   + −   +   − − −   = −   + −   +   −     −   =   +   −                 (   ) (   ) (   ) (   ) (   )   (   ) (   ) (   ) (   )   (   ) (   )  2   3   1   3   2   3 2 1   3   4   3   5   3   3 2 2   3   1   3   2   3   3 6 3 1   1 4   1   2   2 1   1   2   2 2   2 5   2   2   2 1   1   2   2 6   4 1 sin   (   sin   cos   ) 1 sin   (   sin   sin   cos   ) 1 cos   (   sin   sin   cos   ) 2   2 1 2 2   2 1 2 cos  p   x   p   x   p   x y   x   p   x   p   y   y y   x   p   x   p   x   x P y x   y p   y x   y   x   y x   y p   y x   y   x   y p   p   y  δ δ δ δ   ρ α ρ α     +               =   +               =   +           = −   + −   = −   + −   +   − −   + −   = −   + −   +   − −   =          (   ) (   )  3   4   3   5   3 5   3   4   3   5   3 1 (   sin   cos   ) 1 sin   (   sin   cos   )  p   y   p   y p   x   p   y   p   y  δ δ     +   −             +            (6) If the   initial condition for vehicle 1 and vehicle 2 are (5.5,0,0)   and   (15.5,0,0)   respectively   and   the   terminal condition are given (0,0.2,0) and (9.8,0,0) respectively, then the simulation result as follows ICIUS 2007 Oct 24-25, 2007 Bali, Indonesia ICIUS2007-C003 ISBN 978-979-16955-0-3 336 © 2007 ICIUS
Figure 2: Optimal Trajectory of Dubin’s Vehicles 3   Conclusions  The greatest gradient descent method can be used to solve the problem that the originally differential equation has initial and final condition but the adjoint system doesn't have one. Through this method, the optimal control and optimal trajectory of a pair of Dubin’s vehicles can be found.  References  [1]   Pranoto,   I.,   Tjahjana,   H.,   dan   Muhammad,   H., Simulation   of   Swarm   Modeling   Through   Bilinear Optimal   Control,   Proceeding   of   International Conference on Mathematics and Statistics ,   pp. 443- 450, 2006. [2]   Pranoto,   I.,   Tjahjana,   H.,   dan   Muhammad,   H., Simulation   of   Swarm   Modeling   Through   Optimal Control,   Proceeding of Asian Control Conference,   pp. 800-803, 2006. [3]   Tjahjana,   H.,   Pranoto,   I.,   dan   Muhamad,   H., Pemodelan perilaku   swarm   melalui kontrol optimum dengan   penalti   fungsi   eksponensial,   Prosiding Koferensi   Nasional   Matematika   XIII,   pp.   779-784, 2006 [4]   Tjahjana,   H.,   Pranoto,   I.,   Muhammad,   H.   dan Naiborhu,   J.,   Swarm   with   Triangle   Formation,  Proceeding   of   International   Conference   on Mathematics   and   Natural   Sciences ,   pp.   778-780, 2006. [5]   Tjahjana,   H.,   Pranoto,   I.,   Muhammad,   H.   dan Naiborhu, J., Perancangan Kontrol Sistem Multi Agen Dengan formasi segitiga,   Submitted to   JMS., 2007 [6]   Breder, C., Equation Descriptive of Fish Schools and Other Animal Aggregation,   Ecology   35(3), pp. 361- 370, 1954. [7]   Warburton, K., and Lazarus, J., Tendency-Distance Models of Social Cohesion In   Animal Groups,   J. Theoretical Biology   150, pp.473-488, 1991. [8]   Gazi, V. dan Passino, K.M., Stability Analysis of Swarms,   IEEE Transaction on   Automatic Control , Vol. 48, No. 4, 2003. [9]   Chu, T., Wang, L., and Chen, T., Self-Organized Motion In Anisotropic Swarm ,   J. Control Theory And Application   Vol. 1 No.1, pp. 77-81, 2003. [10]   A.   Balluchi,   A.   Bicchi,   A.   Balestrino,   and   G. Casalino,   Path   Tracking   Control   for   Dubin's   Car, P roceedings   of   the   1996   IEEE   International Conference   on   Robotics   and   Automation , Minneapolis, MN, April 1996, pp.3123-3128, 1996. [11]   A.   Kelly   dan   B.   Nagy,   Reactive   Nonholonomic Trajectory Generation via Parametric Optimal Control,  The International   Journal of   Robotics   Research , July–August 2003, pp.583-600, 2003. [12]   K.H. Johannson,   Hybrid system , Lecture note, UC Berkeley, Spring 2002 . ICIUS 2007 Oct 24-25, 2007 Bali, Indonesia ICIUS2007-C003 ISBN 978-979-16955-0-3 337 © 2007 ICIUS