est to the vehicle is provided as a reference. The reference posture is formally computed as ( ()), where () =arg min R k ()k P is a projection mappin


 Chastity Wilkinson
 10 days ago
 Views:
Transcription
1 PROCEEDINGS OF MED'  9 T H MEDITERRANEAN CONFERENCE ON CONTROL AND AUTOMATION, DUBROVNIK, CROATIA, JUNE. Combined trajectory tracking and path following for marine craft P. Encarnac~ao A. Pascoal Institute for Systems and Robotics and Dept. Electrical Eng. Instituto Superior Tecnico Av. Rovisco Pais,, 49 Lisboa, Portugal fpme,antoniogisr.ist.utl.pt Abstract The paper presents a solution to the problem of combined trajectory tracking and path following for marine craft. This problem is motivated by the practical need to develop control systems for marine vehicles that can yield good trajectory tracking performance while keeping some of the desired properties associated with path following. The solution described builds on and extends previous work by Hindman and Hauser on socalled maneuver modied trajectory tracking. The theoretical tools used borrow from Lyapunov stability theory and backstepping techniques. Simulations with a nonlinear model of an underactuated marine craft illustrate the performance of the combined trajectory tracking and path following controller derived. Keywords Trajectory Tracking, Path Following Systems, Nonlinear Control, Backstepping Techniques, Autonomous Marine Crafts. I. Introduction Over the last few years, there has been considerable interest in the development of powerful methods for motion control of autonomous vehicles. The problems of motion control addressed in the literature can be roughly classied in three groups: point stabilization, where the goal is to stabilize a vehicle at a given point, with a desired orientation; trajectory tracking, where the vehicle is required to track a time parameterized reference, and path following, where the vehicle is required to converge to and follow a desired path, without any temporal speci cations. Point stabilization presents a true challenge to control system designers when the vehicle has nonholonomic constraints since, as pointed out in the celebrated result of Brockett [], there is no smooth (or even continuous) constant statefeedback control law that will do the job. To overcome this diculty two main approaches have been proposed: smooth timevarying control laws [], [3] and discontinuous feedback laws [4], [5], [6], [7]. The trajectory tracking problem for fully actuated systems is now well understood and satisfactory solutions can be found in standard nonlinear control textbooks [8]. However, when it comes to underactuated vehicles, this is, when the vehicle has less actuators then state variables to be This work was supported in part by the Commission of the European Communities under contract no. MAS3CT979 (ASI MOV) and the Portuguese PRAXIS and PDCTM programs under projects CARAVELA and MAROV. The rst author beneted from a PRAXIS XXI Graduate Student Fellowship. tracked, the problem is still a very active topic of research. Linearization and feedback linearization methods [9], [], as well as Lyapunov based control laws [], [], [] have been proposed. Applications to underactuated surface vessels can be found in [3], [4], [5]. Path following control has received relatively less attention than the other two problems. See for example [], [6] and the references therein. Path following systems for marine vehicles have been reported in [7], [8], [9], []. While on a path following mode, the vehicle forward speed need not be controlled accurately since it is sucient to act on the vehicle orientation to drive it to the path. Typically, smoother convergence to a path is achieved in this case when compared to the performance obtained with trajectory tracking controllers, and the control signals are less likely pushed to saturation []. These three classes of problems have traditionally been addressed separately. In fact, Brockett's condition implies that the point stabilization problem cannot be simply solved by designing a trajectory tracking control law and applying it to a trajectory that degenerates into a single point. Conversely, it is not clear how to extend stabilization methodologies now available to trajectory tracking problems. However, a paper by Hindman and Hauser [] that has not received much attention in the literature shows clearly how to go from a trajectory tracking to a path following controller. In their work, the authors assume that a trajectory tracking controller is available and that a Lyapunov function is known that yields asymptotic stability of the resulting control system about a desired trajectory. The Lyapunov function captures the distance between the vehicle's posture (position and orientation) and the desired posture along the trajectory as measured in some P metric, that is, V = k (t) (t)k P ; where (t) is the vehicle's posture, (t) is the desired posture, and kxk P = x T P x with P >. The key idea in the work of Hindman and Hauser can now be simply explained as follows. To execute a path following maneuver, the vehicle should look at the "closest point on the path" and adopt the corresponding hypothetical vehicle's posture as a reference to which it should converge. Thus, instead of feeding a desired posture (t) to the tracking controller, the posture corresponding to the path point that is clos
2 est to the vehicle is provided as a reference. The reference posture is formally computed as ( ()), where () =arg min R k ()k P is a projection mapping that computes the trajectory time that minimizes the distance k ()k P. Under some technical conditions on path shape and path parametrization, it is shown that this strategy actually drives the vehicle to the path. Interestingly enough, the work of Hindman and Hauser goes even one step further. In fact, it shows how to combine in a single control law trajectory tracking and path following behaviors, thus achieving smooth spatial convergence to the trajectory as well as time convergence. This is accomplished by modifying the projection function () through the addition of a time dependent penalty term to obtain (; t) = arg min R h ( ) k ()k P + (t ) i : The constant > weighs the relative importance of convergence in time over spatial convergence to the path. If = is chosen, pure path following is achieved ( (; t) = ()). If =, (; t) = t since = t is always a minimizer of (t ), and the original trajectory tracking controller is recovered. When < <, the system displays attributes of both schemes. Since the Lyapunov function that is used to prove stability of the trajectory tracking control system is also used to dene the distance to the path, the methods proposed by Hindman and Hauser are intrinsically suitable for vehicles with negligible dynamics. This follows from the fact that in practical applications the distance to the path should only take into account kinematic variables such as vehicle position and possibly orientation, thus avoiding the need to feedforward dynamic variables. With the objective of applying similar methods to the control of marine craft, this paper extends the key results of Hindman and Hauser to encompass systems that capture the motion of vehicles with non negligible dynamics. This is done by resorting to backstepping techniques after modication of a standard tracking control law for marine craft. Using the methodology developed, the distance to the path is evaluated by taking into consideration the kinematic variables only, namely the vehicle position. The paper is organized as follows: Section II derives a control law that enables balancing combined requirements of trajectory tracking and path following for vehicles with non negligible dynamics. The application to an autonomous marine craft is described in Section III. Finally, Section IV presents the main conclusions. II. Combined trajectory tracking and path following control Consider the following general nonlinear control system _ = f () + g () () where R n is the state and R m is the control input. The functions f : D! R n and g : D! R n+m are smooth in a domain D R n that contains = and f () =. Let (t) be a reference trajectory for and let (; (t)) a trajectory tracking control law for () such that, with = (; (t)),!. Assume that local asymptotic stability of the trajectory tracking system can be proven using the Lyapunov function V (; t) = k (t)k P ; () where P is a positive denite weighting matrix (notice that V can be simply viewed as the tracking error measured in the P metric). Assume also that _V (; t) = h (t) ; _ _ (t)i P k V (; t) ; where hx; yi P = x T P y is the inner product relative to P and k is a positive scalar. The condition V _ k V can be weakened if one is able to prove stability with V _ negative semidenite by resorting to LaSalleYoshizawa or Barbalat's lemmas [8]. To convert the trajectory tracking control law into a combined trajectory tracking and path following control law, let the square of the distance from the path point ( () ; ) to the point (; t) be given by the function U (; t; ) = ( ) V (; ) + (t ) = ( ) k ()k P + (t ) : Given U (; t; ), dene the mapping (; t) =arg min U (; t; ) = arg min R R h ( ) k ()k P + (t ) i ; (3) where is a positive scalar that allows balancing the relative importance of time convergence to the trajectory against spatial convergence to the path. If =, (; t) =arg min R k ()k P gives the trajectory time that minimizes the distance k ()k P and a pure path following controller is obtained. If =, (; t) = t (since = t is always the minimizer of (t ) ), yielding the original trajectory tracking controller. When < <, the mapping gives the trajectory time of the path point ( () ; ) that is closest to (; t) in the metric induced by U (; t; ), and a mixed trajectory tracking and path following behavior is achieved. Under some mild technical conditions on the reference path shape and parameterization (see [] for details), () is unique and continuous. Note that a necessary condition for = (; t) to be a minimizer of U (; t; ) is that U (; t; ) = ( ) ( (; t)) ; d d ( (; t)) (t (; t)) = : (4)
3 Consider now the candidate Lyapunov function Y (; t) =min R U (; t; ) = U (; t; (; t)) (5) = ( ) k ( (; t))k P + (t (; t)) : Dierentiating Y with respect to time gives _Y (; t) = ( ) D ( (; t)) ; _ d d ( (; t)) d E dt (; t) P (; t) (t (; t)) d dt = ( ) D ( (; t)) ; _ d E d ( (; t)) + P d E d d ( (; t)) h dt (; t) ( ) D ( (; t)) ; i + (t (; t)) : From (4), the second term of this expression is zero, hence _Y (; t) = ( ) V _ (; (; t)) ( ) k V (; (; t)) : To conclude asymptotic time convergence to the path one must resort to Barbalat's Lemma [8] since Y _ is not negative denite in t. Assuming that the rst and second derivatives of () (the velocity and the acceleration along the reference path, respectively) are bounded, it is easy to prove that Y is bounded. Thus, Y _ is uniformly continuous. Since Y _ and Y is bounded below by zero, Y (; t)! Y as t!, where Y is positive and nite. Now, Barbalat's Lemma allows for the conclusion that _Y (; t)! as t!. This implies that! ( (; t)) as t!. Using (4), it follows that (; t)! t as t!. Therefore,! (t) as t!, that is, the system tracks the reference path asymptotically. In order to apply the method described to motion control of autonomous vehicle, some modications are required. This stems from the fact that the equations of motion of these vehicles exhibit a clear split between kinematics and dynamics and the specication of a trajectory to be tracked includes the kinematic variables only. For example, in the case of a marine craft a reference trajectory may simply include the desired evolution of the center of mass (and possibly the orientation) of that craft. In this case, however, the methodology proposed by Hindman and Hauser still provides a powerful tool to obtain a path following controller for the craft assuming a trajectory tracking control law exists and its stability has been proved using Lyapunov theory. In fact, the work of Hindman and Hauser provides a clear recipe to suitably modify the Lyapunov function for trajectory tracking so as to yield a new Lyapunov function for path following stability. Suppose for example that () captures the kinematics of a vehicle, and that (after a suitable coordinate transformation) the dynamics can be cast in the form _ = ; (6) where is the actual input vector. Suppose also that a stabilizing trajectory tracking controller = (; (t)) for () and a corresponding Lyapunov function V (; t) have been found. The vehicle dynamics can be included in the control design through a backstepping step [] as follows. Consider the change of variables z = : The variable z can be interpreted as the dierence between the virtual control variable and the virtual control law. Thus, if z tends to zero, the system approaches the manifold corresponding to the "kinematic system". This motivates the choice of V = Y (; t) + zt z as a candidate Lyapunov function. Dierentiating V with respect to time yields Setting _V = Y Y () [f () + g () ] + () g () z + zt _ ( ) k V (; (; t)) + Y () g () z + zt _ = _ g T () Y T () K dyn z; where K dyn is a positive denite matrix, it follows that _V ( ) k V (; (; t)) z T K dyn z : Using the same arguments as before, convergence to the path can be concluded. In practice, the modications required to deal with more realistic cases may warrant further complexity of the resulting control laws, as the next section will show. However, the key recipe provided by Hindman and Hauser still applies. III. Application to a surface craft The methodology described in Section II is now applied to the design of a combined trajectory tracking and path following controller for the DELFIM autonomous surface craft developed at the Instituto Superior Tecnico of Lisbon, Portugal. See [3], [4] for details on the construction, modeling, and control of the DELFIM vehicle. A. Vehicle model Following standard practice, the equations of motion of a marine craft can be developed using a global coordinate frame fug and a bodyxed coordinate frame fbg that moves with the vehicle. The following notation is required [5], [6]: U p Borg := (x; y) T  position of the origin of fbg measured in fug; B v Borg := (u; v) T  velocity of the origin of fbg relative to fug, expressed in fbg (i.e., bodyxed linear velocity); B! B := r  angular velocity of fbg relative to fu g, expressed in fbg (i.e., bodyxed angular velocity). The symbol U BR ( ) denotes the rotation matrix from : 3
4 fbg to fug, parameterized by the yaw angle. With this notation, the dynamics and kinematics of the marine craft can be cast in the form Dynamics: M RB B _v Borg + C RB B v Borg B v Borg = RB (7) Kinematics: I z _r = N (8) U _p Borg = U BR ( ) B v Borg (9) _ = r; () where RB denotes the vector of external forces and N is the external yawing torque. The symbols M RB and I z denote the rigid body mass matrix and the moment of inertia about the z B axis respectively, and C RB represents the matrix of Coriolis and centripetal terms. The vector RB can further be decomposed as RB = add ( _; ) + body () + ; () T where add is the added mass term and = B vb T org ; r. The term body consists of the hydrodynamic forces acting on the vehicle's body. The vector = (T c ; ) T denotes the forces applied to the vehicle by the propulsion system, where T c is the force exerted on the vehicle's body by the two back propellers working in common mode. A simplied model for the yawing torque N is [4], [7] N = N v uv + N r ur + T d ; () where T d represents the torque generated by the two back propellers in dierential mode. The symbols N v and N r denote hydrodynamic torque coecients. Combining equations (7)(8) and ()(), the body xed dynamic model of the AUV can be written as [4], [7] M B _v Borg + rjm B v Borg + D B v Borg ; r B v Borg = Tc (3) I z _r N v uv N r ur = T d ; (4) where M includes the rigid body mass matrix and added mass terms, and J is the skew symmetric matrix J = The term rjm B v Borg captures Coriolis and centripetal, as well as added mass terms, and D B v Borg ; r includes the hydrodynamic damping terms. For the DELFIM vehicle, a simplied model of the hydrodynaminc damping terms is Xu Xuuu D BvBorg ; r X = uuuu X vvv Y rr Y vu : ; where X () and Y () denote hydrodynamic force coecients. The D () matrix can be further decomposed as D B v Borg ; r = D B v Borg + D r, where D BvBorg = D r = Xu Xuuu X uuuu X vvv Y vu Y r Observe that the DELFIM model was written in such a way as to separate the surge and sway dynamics from the yaw dynamics. This is because control design will resort to two backsteppinglike steps, the rst to include the surge and sway dynamics, and the second to include the yaw dynamics. B. Trajectory tracking controller As explained in Section II, the starting point for the derivation of a combined trajectory tracking and path following controller for the surface craft is the availability of a trajectory tracking controller for the same vehicle. In what follows, the general structure of the trajectory tracking controller for underactuated vehicles described in [4] is adopted. In his work, Godhavn presents both a nonadaptive and an adaptive controller where the latter aims at estimating and rejecting constant ocean currents. Both schemes are applicable to surface vehicles that are only actuated in surge and yaw. In this paper, only the nonadaptive version will be used since, to derive the adaption laws for the adaptive case, the estimation errors are included in the Lyapunov functions, thus preventing the use of the Hauser and Hindman methods. The controllers developed by Godhavn are applicable to low speed slender ships for which decoupling the surge from the steering (sway and yaw) dynamics is natural. Since that is not the case for the DELFIM vehicle, a slight modication in the design method is necessary. The presentation that follows borrows considerably from the work in [4]. Let (t) = (x ref (t) ; y ref (t)) T be a reference for the position U p Borg and let ~p = U p Borg (t) be the corresponding position error. Notice how in this case the reference trajectory only includes the desired position of the vehicle, thus avoiding specifying its orientation explicitly. Consider the Lyapunov function : V = ~pt ~p: (5) Dierentiating V with respect to time yields Let _V = ~p T U R B B v Borg _ (t) : U R B B v Borg where k is a positive scalar. Then Following backstepping tech is negative denite in ~p. niques, dene = _ (t) k ~p; _V = k ~p T ~p z = U R B B v Borg ( _ (t) k ~p) 4
5 as the dierence between the virtual control variable U R B B v Borg and the virtual control law _ (t) k ~p, and let V be the Lyapunov function V = V + zt z : (6) Finally, consider the Lyapunov function with time derivative V 3 = V + z The time derivative of V is given by where k >, and with _V = k ~p T ~p k z T z + z T U R B M Tc a = h r + h a ; _V 3 = k ~p T ~p k z T z z T U R B M + a z h I z (T d + N v uv + N r ur) + h = k ~p T ~p k z T z + z T U R B M z z h I z (T d + N v uv + N r ur) + h : The control signal + Choosing yields h = (JM MJ + D ) B v Borg h = D B v Borg B v Borg k M B v Borg M U R T B (~p (t) k _ (t) + k z ) : T c = a (7) _V = k ~p T ~p k z T z U R B M z T To include the yaw dynamics, consider now z = a; : a with >. The time derivative of a can be computed to give _a = h _r + h ; where h = (JM MJ + D ) B _v Borg r+ _D B v B Borg v Borg + D B v B Borg _v Borg k M B _v Borg U _R B T (~p (t) k (t) + k z ) M U RB T ~p _ (3) (t) k (t) + k _z : Therefore, h _z = h : I z (T d + N v uv + N r ur) In order to simplify the expressions, dene With these denitions, h h = h = h h = h : _z = h I z (T d + N v uv + N r ur) + h : T d = N v uv N r ur I z h h + z T U R B M where k 3 is a positive scalar, renders _V 3 = k ~p T ~p k z T z k 3 z + k 3 z ; (8) negative semidenite. In [4] it is shown, using LaSalle arguments, that the control law (7)(8) provides exponential tracking of z as long as saturation of the inputs is avoided, and the vehicle surge velocity u is kept nonzero. It is now straight forward to derive a combined trajectory tracking and path following controller for the surface craft. To do this, one starts by identifying the kinematic equation (9) with the general model () by making = (x; y) T and as the vehicle inertial velocity, that is, = U R B B v Borg. Hence, f () = and g () = I, where I denotes the identity matrix. The Lyapunov function (5) should then be modied according to (5). The last step consists of redening the Lyapunov function (6) as V = Y (; t) + zt z ; where Y is dened in (5). The details are omitted. C. Simulation results Simulations with the model of the autonomous surface craft DELFIM were performed choosing k =, k = k 3 = :, and = :7, as in [4]. In the simulations, the initial posture of the vehicle was set to (x; y; )j t= = 35 m; ; and the desired velocity along the reference path was chosen as ms. The performance achieved with pure trajectory tracking ( = ), pure path following ( = ), and for combined trajectory tracking and path following ( = :) was assessed. Figure shows the vehicle following a path consisting of two straight line segments joined together by a cubic spiral, for the case when =. Figure depicts the actuator signals, that is, the force T c and the torque T d generated by the common and dierential thruster modes, respectively, and Figure 3 shows how the tracking errors x (t) x ref (t) and y (t) y ref (t) converge to zero. The 5
6 x y vehicle trajectory for = (pure path following case) is presented in Figure 4. Figure 5 shows the actuator signals. The trajectory tracking errors x (t) x ref (t) and y (t) y ref (t), and the modied errors x (t) x ref ( (; t)) and y (t) y ref ( (; t)) are plotted in Figures 6 and 7, respectively. From Figure 7 it is possible to see that the vehicle actually converges to the path. However, from Figure 6 it is clear that it is always about : m ahead of the "reference" vehicle. Finally, Figures 8 to refer to the combined trajectory tracking and path following controller that is obtained by setting = :. Figure 8 shows the x y vehicle trajectory, while Figure 9 displays the actuator signals. From the tracking errors in Figure and from the modied errors in Figure, the combined behavior can be interpreted as follows: the vehicle converges quickly to the desired path and only then will it react to achieve zero trajectory tracking error. Notice that in this particular example the overall spatial approach to the desired path does not show drastic dierences in the three dierent simulations. However, the control activity is considerably smoother in the cases where 6=, thus showing the benets of path following. y y ref x Fig. 3. Tracking errors (t) (t) for =. x y Fig. 4. Pure path following: =. T c [N] y Fig.. Pure trajectory tracking: = modied trajectory tracking and allows for its application to the control of vehicles with non negligible dynamics. The methodology described has been used to develop a combined trajectory tracking and path following controller for a fully actuated underwater vehicle and also applied to the coordinated control of autonomous marine craft [8], [9]. Further work will address the problem of controlling surface crafts in the presence of parameter uncertainty and external disturbances, as well as reduced motion sensor information. References [] R. W. Brockett, \Asymptotic stability and feedback stabilization," in Dierential Geometric Control Theory, R. W. Brock T d [Nm] Fig.. Actuator signals for = : force T c and torque T d. IV. Conclusions The paper presented a solution to the problem of combined trajectory tracking and path following for marine craft. The solution described builds on and extends previous work by Hindman and Hauser on socalled maneuver T c [N] T d [Nm] Fig. 5. Actuator signals for = : force T c and torque T d. 6
7 4 T c [N] 3 y y ref Fig. 6. Tracking errors (t) (t) for =. T d [Nm] Fig. 9. Actuator signals for = :: force T c and torque T d. (π λ ) y y ref (π λ ).5.5 y y ref Fig. 7. Modied error (t) ( (; t)) for = Fig.. Tracking errors (t) (t) for = :. ett, R. S. Millman, and H. J. Sussman, Eds., Boston, 983, pp. 8{9, Birkhauser. [] C. Canudas de Wit, H. Khennouf, C. Samson, and O. J. Srdalen, \Nonlinear control design for mobile robots," in Recent Trends in Mobile Robotics, Yuan F. Zheng, Ed. 993, vol., pp. {56, World Scientic Series in Robotics and Automated Systems. [3] J. M. Godhavn and O. Egeland, \A Lyapunov approach to exponential stabilization of nonholonomic systems in power form," IEEE Transactions on Automatic Control, vol. 4, no. 7, pp. 8{3, July 997. [4] A. Aguiar and A. Pascoal, \Stabilization of the extended nonholonomic double integrator via a logicbased hybrid control," in Proceedings of SYROCO', 6 th IFAC Symposium on Robot Control, Vienna, Austria, Sept.. [5] A. Astol, \Exponential stabilization of a wheeled mobile ro x y Fig. 8. Combined trajectory tracking and path following: = :. bot via discontinuous control," Journal of Dynamic Systems, Measurement, and Control, vol., pp. {6, Mar [6] J. P. Hespanha, \Stabilization of nonholonomic integrators via logicbased switching," in Proceedings of IFAC'96, 3 th World Congress of IFAC, S. Francisco, CA, USA, July 996, vol. E, pp. 467{47. [7] C. Canudas de Wit and O. J. Srdalen, \Exponential stabilization of mobile robots with nonholonomic constraints," IEEE Transactions on Automatic Control, vol. 37, no., pp. 79{ 797, Nov. 99. [8] H. K. Khalil, Nonlinear Systems, PrenticeHall, New Jersey, Second edition, 996. [9] G. Walsh, D. Tilbury, S. Sastry, R. Murray, and J. P. Laumond, \Stabilization of trajectories for systems with nonholonomic constraints," IEEE Transactions on Automatic Control, vol. 39, no., pp. 6{, Jan [] E. Freund and R. Mayr, \Nonlinear path control in automated vehicle guidance," IEEE Transactions on Robotics and Automation, vol. 3, no., pp. 49{6, Feb [] D. H. Kim and J. H. Oh, \Globally aymptotically stable tracking control of mobile robots," in Proceedings of CCA'98, 998 IEEE International Conference on Control Applications, Trieste, Italy, Sept [] R. Fierro and F. L. Lewis, \Control of a nonholonomic mobile robot: Backstepping kinematics into dynamics," in Proceedings of CDC'95, 34 th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec. 995, pp. 385{38. [3] J. M. Godhavn, \Nonlinear tracking of underactuated surface vessels," in Proceedings of CDC'96, 35 th IEEE Conference on Decision and Control, Kobe, Japan, Dec. 996, pp. 987{99. [4] J. M. Godhavn, Topics in Nonlinear Motion Control: Non 7
8 y y ref (π λ ) (π λ ) holonomic, Underactuated, and Hybrid Systems, Ph.D. thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway, 997. [5] K. Y. Pettersen and H. Nijmeijer, \Tracking control of an underactuated surface vessel," in Proceedings of CDC'98, 37 th IEEE Conference on Decision and Control, Tampa, Florida, USA, Dec. 998, pp. 456{4566. [6] Z. Jiang and H. Nijmeijer, \A recursive technique for tracking control of nonholonomic systems in chained form," IEEE Transactions on Robotics and Automation, vol. 44, no., pp. 65{79, Feb [7] P. Encarnac~ao, A. Pascoal, and M. Arcak, \Path following for marine vehicles in the presence of unknown currents," in Proceedings of SYROCO', 6 th IFAC Symposium on Robot Control, Vienna, Austria, Sept.. [8] G. Indiveri, M. Aicardi, and G. Casalino, \Nonlinear timeinvariant feedback control of an underactuated marine vehicle along a straight course," in Proceedings of MCMC', IFAC Conference on Manoeuvring and Control of Marine Craft, Aalborg, Denmark, Aug.. [9] P. Encarnac~ao and A. Pascoal, \3D path following for autonomous underwater vehicles," in Proceedings of CDC', 39 th IEEE Conference on Decision and Control, Sydney, Australia, Dec.. [] M. Aicardi, G. Casalino, G. Indiveri, A. Aguiar, P. Encarnac~ao, and A. Pascoal, \A planar path following controller for underactuated marine vehicles," in Proceedings of MED', 9 th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia, June. [] R. Hindman and J. Hauser, \Maneuver modied trajectory tracking," in Proceedings of MTNS'96, International Symposium on the Mathematical Theory of Networks and Systems, St. Louis, MO, USA, June 99. [] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc, New York, 995. [3] A. Pascoal, P. Oliveira, C. Silvestre, L. Sebasti~ao, M. Runo, V. Barroso, J. Gomes, G. Ayela, P. Coince, M. Cardew, A. Ryan, H. Braithwaite, N. Cardew, J. Trepte, N. Seube, J. Champeau, P. Dhaussy, V. Sauce, R. Moitie, R. Santos, F. Cardigos, M. Brussieux, and P. Dando, \Robotic ocean vehicles for marine science applications: the european ASIMOV project," in Proceedings of OCEANS' MTS/IEEE, Rhode Island, Providence, USA, Sept.. [4] M. Prado, \Modeling and control of an autonomous ocean vehicle," M.S. thesis, Department of Electrical Engineering, Instituto Superior Tecnico, Lisboa, Portugal,. [5] D. Fryxell, P. Oliveira, A. Pascoal, C. Silvestre, and I. Kaminer, \Navigation, guidance, and control of AUVs: An application to the MARIUS vehicle," IFAC Journal of Control Engineering Practice, vol. 4, no. 3, pp. 4{49, Mar [6] C. Silvestre and A. Pascoal, \Control of an AUV in the vertical and horizontal planes: System design and tests at sea," Transactions of the Institute of Measurement and Control, vol. 9, no. 3, pp. 6{38, 997. [7] T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, Chichester, England, 994. [8] P. Encarnac~ao and A. Pascoal, \Combined trajectory tracking and path following for underwater vehicles," Accepted for presentation at CAMS', IFAC Conference on Control Applications in Marine Systems, Apr.. [9] P. Encarnac~ao and A. Pascoal, \Combined trajectory tracking and path following control: an application to the coordinated control of autonomous marine craft," Submitted to CDC', 4 th IEEE Conference on Decision and Control, Mar Fig.. Modied error (t) ( (; t)) for = :. 8
Modelling and Identification of Underwater Robotic Systems
University of Genova Modelling and Identification of Underwater Robotic Systems Giovanni Indiveri Ph.D. Thesis in Electronic Engineering and Computer Science December 1998 DIST Department of Communications,
More informationA Lagrangian Framework to Incorporate Positional and Velocity Constraints to Achieve PathFollowing Control
211 5th IEEE Conference on Decision and Control and European Control Conference (CDCECC) Orlando, FL, USA, December 1215, 211 A Lagrangian Framework to Incorporate Positional and Velocity Constraints
More informationMOBILE ROBOT TRACKING OF PREPLANNED PATHS. Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.
MOBILE ROBOT TRACKING OF PREPLANNED PATHS N. E. Pears Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.uk) 1 Abstract A method of mobile robot steering
More informationRobotic Ocean Vehicles for Marine Science Applications: the European ASIMOV Project*
Robotic Ocean Vehicles for Marine Science Applications: the European ASIMOV Project* António Pascoal 1, Paulo Oliveira 1, Carlos Silvestre 1, Luis Sebastião 1, Manuel Rufino 1, Victor Barroso 1, João Gomes
More informationCONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES
1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3SCNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,
More informationDesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist
DesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot
More information1996 IFAC World Congress San Francisco, July 1996
1996 IFAC World Congress San Francisco, July 1996 TRAJECTORY GENERATION FOR A TOWED CABLE SYSTEM USING DIFFERENTIAL FLATNESS Richard M. Murray Mechanical Engineering, California Institute of Technology
More informationVéronique PERDEREAU ISIR UPMC 6 mars 2013
Véronique PERDEREAU ISIR UPMC mars 2013 Conventional methods applied to rehabilitation robotics Véronique Perdereau 2 Reference Robot force control by Bruno Siciliano & Luigi Villani Kluwer Academic Publishers
More informationCFD Modelling and Realtime testing of the Wave Surface Glider (WSG) Robot
21st International Congress on Modelling and Simulation, Gold Coast, Australia, 29 Nov to 4 Dec 2015 www.mssanz.org.au/modsim2015 CFD Modelling and Realtime testing of the Wave Surface Glider (WSG) Robot
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationParameter identification of a linear single track vehicle model
Parameter identification of a linear single track vehicle model Edouard Davin D&C 2011.004 Traineeship report Coach: dr. Ir. I.J.M. Besselink Supervisors: prof. dr. H. Nijmeijer Eindhoven University of
More informationSensor Based Control of Autonomous Wheeled Mobile Robots
Sensor Based Control of Autonomous Wheeled Mobile Robots Gyula Mester University of Szeged, Department of Informatics email: gmester@inf.uszeged.hu Abstract The paper deals with the wireless sensorbased
More informationForce/position control of a robotic system for transcranial magnetic stimulation
Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationACTUATOR DESIGN FOR ARC WELDING ROBOT
ACTUATOR DESIGN FOR ARC WELDING ROBOT 1 Anurag Verma, 2 M. M. Gor* 1 G.H Patel College of Engineering & Technology, V.V.Nagar388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda391760,
More informationAwellknown lecture demonstration1
Acceleration of a Pulled Spool Carl E. Mungan, Physics Department, U.S. Naval Academy, Annapolis, MD 40506; mungan@usna.edu Awellknown lecture demonstration consists of pulling a spool by the free end
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationPath Tracking for a Miniature Robot
Path Tracking for a Miniature Robot By Martin Lundgren Excerpt from Master s thesis 003 Supervisor: Thomas Hellström Department of Computing Science Umeå University Sweden 1 Path Tracking Path tracking
More informationOperational Space Control for A Scara Robot
Operational Space Control for A Scara Robot Francisco Franco Obando D., Pablo Eduardo Caicedo R., Oscar Andrés Vivas A. Universidad del Cauca, {fobando, pacaicedo, avivas }@unicauca.edu.co Abstract This
More informationOn Motion of Robot EndEffector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot EndEffector using the Curvature
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationLecture 8 : Dynamic Stability
Lecture 8 : Dynamic Stability Or what happens to small disturbances about a trim condition 1.0 : Dynamic Stability Static stability refers to the tendency of the aircraft to counter a disturbance. Dynamic
More informationIntelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and Motion Optimization for Maritime Robotic Research
20th International Congress on Modelling and Simulation, Adelaide, Australia, 1 6 December 2013 www.mssanz.org.au/modsim2013 Intelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More informationChapter 12 Modal Decomposition of StateSpace Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More informationAdaptive Cruise Control of a Passenger Car Using Hybrid of Sliding Mode Control and Fuzzy Logic Control
Adaptive Cruise Control of a assenger Car Using Hybrid of Sliding Mode Control and Fuzzy Logic Control Somphong Thanok, Manukid arnichkun School of Engineering and Technology, Asian Institute of Technology,
More informationOnboard electronics of UAVs
AARMS Vol. 5, No. 2 (2006) 237 243 TECHNOLOGY Onboard electronics of UAVs ANTAL TURÓCZI, IMRE MAKKAY Department of Electronic Warfare, Miklós Zrínyi National Defence University, Budapest, Hungary Recent
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationNEWTON S LAWS OF MOTION
NEWTON S LAWS OF MOTION Background: Aristotle believed that the natural state of motion for objects on the earth was one of rest. In other words, objects needed a force to be kept in motion. Galileo studied
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationPhysics in the Laundromat
Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationTime Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication
Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 2168440 This paper
More informationChapter 4 Dynamics: Newton s Laws of Motion. Copyright 2009 Pearson Education, Inc.
Chapter 4 Dynamics: Newton s Laws of Motion Force Units of Chapter 4 Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the Normal
More informationHumanlike Arm Motion Generation for Humanoid Robots Using Motion Capture Database
Humanlike Arm Motion Generation for Humanoid Robots Using Motion Capture Database Seungsu Kim, ChangHwan Kim and Jong Hyeon Park School of Mechanical Engineering Hanyang University, Seoul, 133791, Korea.
More informationAutomatic Steering Methods for Autonomous Automobile Path Tracking
Automatic Steering Methods for Autonomous Automobile Path Tracking Jarrod M. Snider CMURITR0908 February 2009 Robotics Institute Carnegie Mellon University Pittsburgh, Pennsylvania c Carnegie Mellon
More informationHomework 4. problems: 5.61, 5.67, 6.63, 13.21
Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find
More informationThe Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations
The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11  NQF LEVEL 3 OUTCOME 3  ROTATING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11  NQF LEVEL 3 OUTCOME 3  ROTATING SYSTEMS TUTORIAL 1  ANGULAR MOTION CONTENT Be able to determine the characteristics
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationForce on Moving Charges in a Magnetic Field
[ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after
More informationNewton s Laws. Physics 1425 lecture 6. Michael Fowler, UVa.
Newton s Laws Physics 1425 lecture 6 Michael Fowler, UVa. Newton Extended Galileo s Picture of Galileo said: Motion to Include Forces Natural horizontal motion is at constant velocity unless a force acts:
More informationMotion Control of 3 DegreeofFreedom DirectDrive Robot. Rutchanee Gullayanon
Motion Control of 3 DegreeofFreedom DirectDrive Robot A Thesis Presented to The Academic Faculty by Rutchanee Gullayanon In Partial Fulfillment of the Requirements for the Degree Master of Engineering
More informationNewton s Laws of Motion
Section 3.2 Newton s Laws of Motion Objectives Analyze relationships between forces and motion Calculate the effects of forces on objects Identify force pairs between objects New Vocabulary Newton s first
More informationIMU Components An IMU is typically composed of the following components:
APN064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation
More information2. Dynamics, Control and Trajectory Following
2. Dynamics, Control and Trajectory Following This module Flying vehicles: how do they work? Quick refresher on aircraft dynamics with reference to the magical flying space potato How I learned to stop
More informationPHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013
PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationA Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight
eπεριοδικό Επιστήμης & Τεχνολογίας 33 A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight Vassilios McInnes Spathopoulos Department of Aircraft Technology
More informationLecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6
Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.
More informationINSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users
INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE
More informationCHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies
CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at
More informationFigure 3: Radius of Gyration Mathematical Model
The characteristics of the core s size, shape, and density play a vital role in determining how the core will influence the rotation of the bowling ball by altering the calculated RG and Differential RG.
More informationWMR Control Via Dynamic Feedback Linearization: Design, Implementation, and Experimental Validation
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002 835 WMR Control Via Dynamic Feedback Linearization: Design, Implementation, and Experimental Validation Giuseppe Oriolo, Member,
More informationSO(3) Camillo J. Taylor and David J. Kriegman. Yale University. Technical Report No. 9405
Minimization on the Lie Group SO(3) and Related Manifolds amillo J. Taylor and David J. Kriegman Yale University Technical Report No. 945 April, 994 Minimization on the Lie Group SO(3) and Related Manifolds
More informationA Python Project for Lagrangian Mechanics
The 3rd International Symposium on Engineering, Energy and Environments 1720 November 2013, Pullman King Power Hotel, Bangkok A Python Project for Lagrangian Mechanics Peter Slaets 1, Pauwel Goethals
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationOn the DStability of Linear and Nonlinear Positive Switched Systems
On the DStability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on Dstability of positive switched systems. Different
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationThe dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w
Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration
More informationdistance is equivalent to the Euclidean distance between the transformed compositions by the centered logratio function clr. The angular distance give
MEASURES OF DIFFERENCE FOR COMPOSITIONAL DATA AND HI ERARCHICAL CLUSTERING METHODS J. A. MartnFernandez (1), C. BarceloVidal (1) and V. PawlowskyGlahn (2) (1) Universitat de Girona (2) Universitat
More informationMODULE VII LARGE BODY WAVE DIFFRACTION
MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wavestructure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction.
More informationLyapunov Stability Analysis of Energy Constraint for Intelligent Home Energy Management System
JAIST Reposi https://dspace.j Title Lyapunov stability analysis for intelligent home energy of energ manageme Author(s)Umer, Saher; Tan, Yasuo; Lim, Azman Citation IEICE Technical Report on Ubiquitous
More informationDESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED CONTROL SYSTEM FOR A SERPENTINE ROBOTIC MANIPULATOR
Proceedings of the American Nuclear Society Ninth Topical Meeting on Robotics and Remote Systems, Seattle Washington, March 2001. DESIGN, IMPLEMENTATION, AND COOPERATIVE COEVOLUTION OF AN AUTONOMOUS/TELEOPERATED
More informationMultiRobot Tracking of a Moving Object Using Directional Sensors
MultiRobot Tracking of a Moving Object Using Directional Sensors Manuel Mazo Jr., Alberto Speranzon, Karl H. Johansson Dept. of Signals, Sensors & Systems Royal Institute of Technology SE 44 Stockholm,
More informationLecture L14  Variable Mass Systems: The Rocket Equation
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L14  Variable Mass Systems: The Rocket Equation In this lecture, we consider the problem in which the mass of the body changes during
More informationOUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS. You should judge your progress by completing the self assessment exercises.
Unit 60: Dynamics of Machines Unit code: H/601/1411 QCF Level:4 Credit value:15 OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS 2 Be able to determine the kinetic and dynamic parameters of
More informationNUMERICAL SIMULATION OF REGULAR WAVES RUNUP OVER SLOPPING BEACH BY OPEN FOAM
NUMERICAL SIMULATION OF REGULAR WAVES RUNUP OVER SLOPPING BEACH BY OPEN FOAM Parviz Ghadimi 1*, Mohammad Ghandali 2, Mohammad Reza Ahmadi Balootaki 3 1*, 2, 3 Department of Marine Technology, Amirkabir
More informationLab 8: Ballistic Pendulum
Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationElgersburg Workshop 2010, 1.4. März 2010 1. PathFollowing for Nonlinear Systems Subject to Constraints Timm Faulwasser
#96230155 2010 Photos.com, ein Unternehmensbereich von Getty Images. Alle Rechte vorbehalten. Steering a Car as a Control Problem PathFollowing for Nonlinear Systems Subject to Constraints Chair for Systems
More informationQuadcopters. Presented by: Andrew Depriest
Quadcopters Presented by: Andrew Depriest What is a quadcopter? Helicopter  uses rotors for lift and propulsion Quadcopter (aka quadrotor)  uses 4 rotors Parrot AR.Drone 2.0 History 1907  BreguetRichet
More informationControl and Navigation Framework for Quadrotor Helicopters
Control and Navigation Framework for Quadrotor Helicopters Amr Nagaty, Sajad Saeedi, Carl Thibault, Mae Seto and Howard Li Abstract This paper presents the development of a nonlinear quadrotor simulation
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationTHERE is a significant amount of current research activity
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006 789 Stable Task Load Balancing Strategies for Cooperative Control of Networked Autonomous Air Vehicles Jorge Finke, Member,
More informationV(x)=c 2. V(x)=c 1. V(x)=c 3
University of California Department of Mechanical Engineering Linear Systems Fall 1999 (B. Bamieh) Lecture 6: Stability of Dynamic Systems Lyapunov's Direct Method 1 6.1 Notions of Stability For a general
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationOPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION
OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION Sérgio Pequito, Stephen Kruzick, Soummya Kar, José M. F. Moura, A. Pedro Aguiar Department of Electrical and Computer Engineering
More informationCurriculum Vitae Michael M. Zavlanos
Curriculum Vitae Michael M. Zavlanos Home Address Work Address 2114 Pine Street 3330 Walnut Street Apartment 1 Front GRASP Laboratory, Levine Hall 465 Philadelphia, PA 19103 Dept. of Electrical and Systems
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationStability Analysis of Small Satellite Formation Flying and Reconfiguration Missions in Deep Space
Stability Analysis of Small Satellite Formation Flying and Reconfiguration Missions in Deep Space Saptarshi Bandyopadhyay, Chakravarthini M. Saaj, and Bijnan Bandyopadhyay, Member, IEEE Abstract Closeproximity
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationResearch Article EndEffector Trajectory Tracking Control of Space Robot with L 2 Gain Performance
Mathematical Problems in Engineering Volume 5, Article ID 7534, 9 pages http://dx.doi.org/.55/5/7534 Research Article EndEffector Trajectory Tracking Control of Space Robot with L Gain Performance Haibo
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationThe Trimmed Iterative Closest Point Algorithm
Image and Pattern Analysis (IPAN) Group Computer and Automation Research Institute, HAS Budapest, Hungary The Trimmed Iterative Closest Point Algorithm Dmitry Chetverikov and Dmitry Stepanov http://visual.ipan.sztaki.hu
More informationMaster slave approach for the modelling of joints with dependent degrees of freedom in exible mechanisms
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2003; 19:689 702 (DOI: 10.1002/cnm.627) Master slave approach for the modelling of joints with dependent degrees of freedom
More informationCH205: Fluid Dynamics
CH05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Solutions of Mid Semester Examination Data Given: Density of water, ρ = 1000 kg/m 3, gravitational acceleration, g
More informationACCELERATION OF HEAVY TRUCKS Woodrow M. Poplin, P.E.
ACCELERATION OF HEAVY TRUCKS Woodrow M. Poplin, P.E. Woodrow M. Poplin, P.E. is a consulting engineer specializing in the evaluation of vehicle and transportation accidents. Over the past 23 years he has
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationThe Design and Application of Water Jet Propulsion Boat Weibo Song, Junhai Jiang3, a, Shuping Zhao, Kaiyan Zhu, Qihua Wang
International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 2015) The Design and Application of Water Jet Propulsion Boat Weibo Song, Junhai Jiang3, a, Shuping Zhao,
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationBasic Principles of Inertial Navigation. Seminar on inertial navigation systems Tampere University of Technology
Basic Principles of Inertial Navigation Seminar on inertial navigation systems Tampere University of Technology 1 The five basic forms of navigation Pilotage, which essentially relies on recognizing landmarks
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationCE801: Intelligent Systems and Robotics Lecture 3: Actuators and Localisation. Prof. Dr. Hani Hagras
1 CE801: Intelligent Systems and Robotics Lecture 3: Actuators and Localisation Prof. Dr. Hani Hagras Robot Locomotion Robots might want to move in water, in the air, on land, in space.. 2 Most of the
More informationTracking Moving Objects In Video Sequences Yiwei Wang, Robert E. Van Dyck, and John F. Doherty Department of Electrical Engineering The Pennsylvania State University University Park, PA16802 Abstract{Object
More informationNewton s Second Law. ΣF = m a. (1) In this equation, ΣF is the sum of the forces acting on an object, m is the mass of
Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart will be developed
More information