Underwater vehicles: a surprising non time-optimal path
|
|
- Marsha Gray
- 5 years ago
- Views:
Transcription
1 Underwater vehicles: a surprising non time-optimal path M. Chyba Abstract his paper deals with the time-optimal problem for a class of underwater vehicles. We prove that if two configurations at rest can be joined by a horizontal or a vertical translation in the body frame coordinates, then there exists a shorter path. Our computations are based on the maximum principle and use that translations in the body frame coordinates are time-equivalent to paths formed by concatenations of 2-singular extremals that are proved to be non time-optimal. ntroduction One of the most important problems in robotics is the motion planning problem. t adresses the question of finding a path between an initial and a final position for a given system such as for instance a wheeled mobile robot, an articulated arm or an underwater vehicle. his problem is a particularly challenging one and has been widely studied during the past few years. Among all the possible paths, the optimal ones with respect to some given criteria are in many ways interesting. We want for instance to minimize the duration of a mission, the fuel comsumption of our vehicle or the energy spent to attain the desired goal. Notice also that when considering the motion planning problem in a cluttered environment, most of the existing planners are based on a steering method coupled with a global geometric obstacle avoidance scheme. Optimal paths satisfy, in general, a particular topological property that makes the algorithm converge and therefore allows one to steer the system even in the presence of obstacles, see [7]. n this paper, we focus on the time-optimal problem for a specific class of underwater vehicles. We consider a simplified model moving in the vertical plane and whose actuation is realized through thrusters with magnitude limits. Mathematically, the evolution of such a vehicle is described by an affine control system Department of Mathematics, 2565 McCarthy Mall, University of Hawaii, Honolulu, H ẋ = f(x) m i= g i(x)u i and the question we address is that of: given a set of initial and final states, determine among all possible trajectories joining them a time-optimal one. Optimization problems are known as highly difficult problems. A great deal of energy has been spent in the past decades to find a systematic way to characterize optimal trajectories. n particular, differential geometric techniques have been efficiently used to extract information from the maximum principle. However, due to technical and computational difficulties it is still impossible in most cases to find the optimal trajectories for a given criteria, the hard part being to reduce the set of candidate extremals for optimality all the way to a single optimal extremal. n this paper, we are concerned with a more modest goal: given a specific trajectory, we discuss its optimality status. Our motivation for such a study is the incorporation of our methods into a working vessel and is not only based on theoretical aspects. n a long term basis, our main objective is to be part of a general effort to provide enhanced research tools to other scientists. hen, we have to consider the optimal control problem with new eyes and take into account practical issues as for instance the fact that controls with discontinuities cannot be realized by the robot. Using theoretical tools as the maximum principle and differential geometric techniques with the utlimate goal of solving concrete problems is a new direction mathematicians and engineers have been taken recently. o get some insight on the problem, we consider in this paper as a first approach an idealized model. We then have to be careful not to draw to fast any practical conclusion for our underwater vehicles. However, in the conclusion section we include a brief discussion on why the results obtained under our assumptions seem to reflect the reality and hence are a good starting point. n [2], we address the general question of the time-optimality for the important class of controlled mechanical systems. We derive conditions on the Lie brackets of the vector fields describing the system for a trajectory to be optimal. An application on underwater vehicle is considered, and based on the maximum
2 as follows (see [5] for the details): z x v v 3 θ ẋ ż θ v v 3 Ω = cos θv sin θv 3 cos θv 3 sin θv Ω v 3 Ω m3 v Ω m v v 3 Figure : he model principle we analyze the structure of the singular extremals. Moreover, in [] we include a discussion on the time-optimality status of some specific trajectories and conjecture the time-optimality for the translations in the body frame coordinates. More precisely, we conjecture that if two configurations at rest can be joined by a horizontal or a vertical translation in the body frame coordinates, with one switching for the non singular control, then it is the time-optimal motion. Surprisingly, at least for me in a first time, this conjecture happens to be false. ndeed, in this paper we provide a time-equivalent path to the translations in the body frame coordinates that does not satisfy the necessary conditions of the maximum principle. 2 he model We make the following assumptions on the vehicle: it is of ellipsoidal form, neutrally buoyant and uniformly distributed. he motion of the vehicle is restricted in the vertical plane. he absolute position of the vehicle is described by its horizontal position x and its vertical position z, while the orientation is represented by the angle θ, see Figure. hen, the configuration variable corresponding to our vehicle is the triple q = (x, z, θ). Let us introduce the horizontal and vertical velocities v, v 3 of the vehicle in the body frame coordinates (see Figure ) and Ω the scalar angular rate in the plane. hen, neglecting the viscosity effects, the equations of motion of such a vehicle under the assumptions that it is submerged in an infinitely large volume of incompressible, irrotational and inviscid fluid at rest at infinity are described as a conservative mechanical system where, ( ) are the body-fluid mass terms in the body horizontal and vertical directions and is the body-fluid moment of inertia in the plane. he fundamental reason why we consider the velocities in the body frame coordinates instead of having a mechanical system defined on the state variable (q, q), as it is done usually, is that this formulation makes our result much easier to interpret from a pratical point of view. Let us now describe the inputs we use to control our vehicle. As mentioned previsouly, we consider a vehicle actuated by thrusters. We assume the thrusters to be such that one control denoted u is a force in the body -axis, one denoted u 2 is a force in the body 2-axis and the third one u 3 is a pure torque in the plane. t follows that the equations of motion of our fully actuated underwater vehicle are described by an 3 affine control system: ẇ = f(w) g i (w)u i, where i= the drift f is given by the equations of motion of the conservative system described above, the g i are the following constant vector fields g =, g 2 =, g 3 = and the inputs u i are measurable bounded functions. o reflect the fact that the thrusters have limited power, we assume the following constraints on the inputs: { u i }, i =, 2, 3 (our results can be easily generalized to domains of control of the form {α i u i β i ; α i, β i R}, α i < < β i, i =, 2, 3). n a forthcoming article [3], we prove that fully actuated controlled mechanical systems are controllable even if we assume bounds on the controls. hus, it follows that our system is controllable.
3 3 Adjoint equations Our analysis is based on the maximum principle [6]. t provides necessary conditions for a trajectory to be time-optimal. For our problem, the maximum principle states that along a time-optimal trajectory w defined on the interval [, ] and with u being the corresponding optimal control, there exists an absolutely continuous function λ = (λ,, λ 6 ) defined on [, ] that never vanishes and such that it maximizes the Hamiltonian H(w, λ, u) = λ t( 3 ) f (w) g i (w)u i i= over the set of all possible controls. Moreover, λ is a solution of the following equations: λ 3 = λ (v sin θ v 3 cos θ) λ = () λ 2 = (2) λ 2 (v cos θ v 3 sin θ) (3) λ 4 = λ cos θ λ 2 sin θ λ 5 Ω λ 6 v 3 α (4) λ 5 = λ sin θ λ 2 cos θ λ 4 Ω λ 6 v α (5) λ 6 = λ 3 λ 4 v 3 λ 5 v (6) where α = m3 m is a nonzero constant. he function λ is called the adjoint vector and the equations ()-(6) are called the adjoint equations. A triple (w, λ, u) that solves the maximum principle is called an extremal. Since the proof of our result is based on the necessary conditions of the maximum principle let us introduce some definitions and notations. Remark first that the Hamiltonian takes the form H(w, λ, u) = λ t f (w) λ 4 u λ 5 u 2 λ 6 u 3. hen, since along an optimal path w with control u we have the maximization condition: max v H(w(t), λ(t), v) = H(w(t), λ(t), u(t)), it follows that: u i 3 (t) = sign(λ i (t)) if λ i (t), i = 4, 5, 6. As a consequence, the structure of the optimal paths is governed by the zeroes of the functions λ 4, λ 5, λ 6. ndeed, when the sign of one of these functions changes, we have to switch the corresponding component of the control from one extreme to the other. We call these functions the switching functions. Notice that these functions are given by λ i3 = λ t g i (w). f there exists a nonempty interval such that a given switching function is identically zero, no information is provided directly by the maximum principle on the component u i. We then say that the extremal is u i -singular on that interval. A time t s such that u i is not almost everywhere constant on any interval of the form ]t s ε, t s ε[, ε > is called a switching time for u i. 4 Results As mentioned previously, our main result concerns the time-optimality status of some specific trajectories, namely horizontal and vertical translations in the body frame coordinates between two configurations at rest. heore A horizontal or vertical translation in the body frame coordinates between two configurations at rest is not time-optimal. o prove our result, several steps are necessary. First, let us restrict ourself to horizontal translations in the body frame coordinates. n a previous paper [], we completely characterize the 2-singular trajectories, i.e. the extremals such that 2 components of the control are singular at the same time. We show that for such extremals there exists a bound on the number of switchings for the nonsingular component of the control and that this bound is in fact. Moreover, we also show that if along a trajectory we have Ω and v 3 (i.e. along a horizontal translation in the body frame coordinates), the components u 2, u 3 of the control must be identically zero and hence are singular. We can then conclude that if a horizontal translation in the body frame coordinates between two configurations at rest is optimal, it a 2-singular extremal and, due to the form of the constraints on the domain of control, there is exactly one u -swicthing to be performed at the half-time of the trajectory. n other words, we accelerate as much as possible for the first half and then decelerate at maximum speed for the second half. As this trajectory satisfies the necessary conditions of the maximum, it is a candidate for time-optimality. Surprisingly, as stated in heore, it happen not to be. Let us start by considering the simple situation when the initial and final configurations of the underwater vehicle are such that θ = and differ only by their horizontal component. his means we assume the initial and final states of the vehicle to be (x, z,,,, ) and (x 2, z,,,, ) where x 2 > x. he horizontal translation between these two configurations with exactly one u -swithing at half-time
4 is defined on the time interval = 8 (x 2 x ) and corresponds to the inputs u (t) = if t [, 2 ], u (t) = if t [ 2, ] and u 2, u 3 identically zero, see Figure 2. Figure 2: Horizontal translation n coordinates, we have that w(t) = ( t2 2 t x, z,,,, ) if t [, t2 2 ] and w(t) = ( 2 t (4x 3x 2 ), z,, t,, ) if t [ 2, ]. As mentioned previously, this is a 2-singular extremal, the two singular components of the control being u 2, u 3. For this simple situation we can explicitely compute an adjoint vector. ndeed, using the fact that along this trajectory we must have λ 5 = λ 6, the adjoint equations provide the information that λ has to be of the form (λ,,, λ t λ 2,, ) where λ is any constant greater than. heore contradicts the conjecture that it is a time-optimal path. n other words, we claim that there exists another trajectory defined on a time interval ˆ linking these two configurations at rest and such that ˆ < 8 (x 2 x ). o prove our result, we will show that there exists a time-equivalent path to the horizontal translation that does not satisfy the necessary conditions of the maximum principle and hence that cannot be time-optimal. he key idea is the following remark. By adding a vertical motion to the horizontal translation previously studied we do not alter the time of the trajectory, see Figure 3. Figure 3: ime- equivalent trajectory More precisely, the trajectory defined on = 8m (x 2 x ) and given by w(t) = ( t2 2 t x, 2 t t 2,,,, ) if t [, t2 2 ] and w(t) = ( 2 t (4x 3x 2 ), t2 2 2 t z,, t, t, ) if t [ 2, ] also links the initial and final states: (x, z,,,, ) and (x 2, z,,,, ). he corresponding control is u (t) = u 2 (t) = if t [, 2 ] and u (t) = u 2 (t) = if t [ 2, ] while u 3 is deduced from the fact that along this trajectory we have Ω, which gives u 3 (t) = v (t)v 3 (t) m3 m. Depending on the positions x, x 2 and on the structural constants of the vehicle,,, the component u 3 of the control might not be admissible. n other words, we might not have u 3 (t) along the whole trajectory.o make this trajectory admissible, we should introduce a time reparametrization but then we would loose timeequivalence between these two paths and comparaison would not make any sense. So instead of reparametrizing this trajectory, notice that our remark can be generalized to any vertical motion with a finite number of switchings on the component u 2 of the control. he reason is that as long as we keep the angular velocity Ω identically zero along the trajectory (which is equivalent to saying that the angle θ is constant) the variables x and z are decoupled, x depending only on u and z on u 2. Along all these motions we have max(v ) = 2, hence we can choose a trajectory with enough switchings on u 2 such that max(v 3 ) will satisfy max(v ) max(v 3 ) m3 m < and make the control an admissible one. More precisely, if 2n is the number of switchings on u 2 (it has to be an odd number in order to satisfy both the initial and final velocities v 3 to be zero), since the maximum of the velocity v 3 is given by: max(v 3 ) = 2(n), to get an admissible time-equivalent trajectory we have to choose n such that (n) <. o summarize, we found that to the horizontal translation in the body frame coordinates with θ and one u -switching there exists a time-equivalent trajectory with no angular velocity, with at least 3 switchings on the component u 2 of the control (if the control is admissible with only one switching, then it will be admissible with any number of switchings) and such that u 3 (t) < along the trajectory. We now claim that this time-equivalent trajectory is not timeoptimal. ndeed, if it was an optimal trajectory it has to be a solution of the maximum principle. n other words, there should exist a nonzero adjoint vector such that the adjoint equations are satisfied and such that it maximizes the Hamiltonian. As the componnent of the control u 3 belongs strictly to the inside of the do-
5 main of control: u 3 (t) <, it is a consequence of the maximization condition that this component of the control has to be a singular one. t follows that along our time-equivalent trajectory we have λ 6 since we pointed out in Section 3 that λ 6 is the switching function corresponding to the control u 3. As a consequence, from the adjoint equations (4), (5) we have that λ 4 = λ and λ 5 = λ 2 (recall that there is no angular velocity along our path and that θ ). Since λ, λ 2 are cyclic coordinates, or in other words are constant along an extremal, we deduce that the switching functions λ 4, λ 5 are linear functions with respect to t: λ 4 = λ t λ 4 (), λ 5 = λ 2 t λ 5 (). hen, as the switchings of the controls u and u 2 happen when λ 4 and λ 5 respectively vanish we deduce that both nonsingular controls u, u 2 have at most one swicthing. t contradicts the fact that along our time-equivalent trajectory u 2 has at least three switchings. o generalize our result to any horizontal translation in the body frame( coordinates, simply ) remark cos θ sin θ that the rotation matrix applied to sin θ cos θ the coordinates x, z transforms the horizontal-vertical translations with θ = to horizontal-vertical translations in the body frame coordinates whith θ as the motion angle of direction. his close the proof that horizontal translations are not time-optimal trajectories. Finally, notice that due to the form of the equations of motions all the computations we made can be duplicated for vertical translations in the body frame coordinates. n that case, we add horizontal motions keeping the scalar rate Ω identically zero. 5 Conclusion Since we can prove that a car-like robot moving in the plane cannot go faster than along the straight line with its orientation in the direction of motion, the results presented here were at first surprising. However, there is a crucial difference between a car-like robot and an underwater vehicle. ndeed, the first one is a nonholonomic system with 2 controls while the second one is a fully controlled mechanical system with 3 controls. t follows that for the underwater vehicle a faster strategy to the one presented here would be to use the full power of 2 controls instead of only one (for instance u, u 2 in the case of a horizontal motion) by rotating first the vehicle to an optimal angle. his result is presented in [4] where we emphasize that our nonoptimality result is in fact very natural. Notice that the problem of finding a time-optimal trajectory between any two configurations at rest, including the ones on Figure 2, is still an open problem. First, remark that for this last situation if we consider the horizontal translation coupled with some rotation but no vertical motion, in other words we consider v 3 along the trajectory, then the trajectory is not optimal neither. ndeed, along such a trajectory the velocity v is not affected with respect to the horizontal translation and moreover, using the equations of motions for our fully actuated underwater vehicle, we get that ẋ(t) = cos θv (t) which is less or equal to the horizontal speed v (t) of the horizontal translation. We can deduce that to be optimal the path as to leave the horizontal line and needs to have a nonzero angular velocity. n other words, none of the velocities v, v 3, Ω can be identically zero along the time optimal path. Going further into this study, a first important result would be to give a bound on the number of switchings for each component of the control for the optimal path. ndeed, in case there is an accumulation point of zeroes for one of the swicthing functions it is not realistic from a practical point of view and going along the straight line might still have to be considered. his join our remark made in the introduction that practical issues may interfere with the optimal strategy provided by the theory. References [] M. Chyba, N.E. Leonard, E.D. Sontag, Optimality for underwater vehicles, Proceedings of the 4th Conference on Decision on Control, pp., 2 [2] M. Chyba, N.E. Leonard, E.D. Sontag, Singular trajectories in multi-input time-optimal problems. Application to controlled mechanical systems, Journal on Dynamical and Control Systems, Vol. 9 (), pp , 23 [3] M. Chyba, E.D. Sontag, Controllability of dynamic extensions with bounded controls: Application to controlled mechanical systems, in preparation [4] M.Chyba, R.N. Smith, G.R. Wilkens, S.K. Choi, Underwater vehicles: how to go faster?, in preparation [5] N.E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, Vol 33, pp , 997.
6 [6] L.S. Pontryagin and B. Boltyanski and R. Gamkrelidze and E. Michtchenko, he Mathematical heory of Optimal Processes, nterscience, 962, New York [7] S. Sekhavat, J.P. Laumond, opological Property for Collision-Free Nonholonomic Motion Planning: the Case of Sinusoidal nputs for Chained Form Systems, EEE Rans. on Autom. and Control, Vol. 4(5), pp , 998
An Explicit Characterization of Minimum Wheel-Rotation Paths for Differential-Drives
An Explicit Characterization of Minimum Wheel-Rotation Paths for Differential-Drives Hamidreza Chitsaz 1, Steven M. LaValle 1, Devin J. Balkcom, and Matthew T. Mason 3 1 Department of Computer Science
More informationExtremal Trajectories for Bounded Velocity Differential Drive Robots
Extremal Trajectories for Bounded Velocity Differential Drive Robots Devin J. Balkcom Matthew T. Mason Robotics Institute and Computer Science Department Carnegie Mellon University Pittsburgh PA 523 Abstract
More informationExtremal Trajectories for Bounded Velocity Mobile Robots
Extremal Trajectories for Bounded Velocity Mobile Robots Devin J. Balkcom and Matthew T. Mason Abstract Previous work [3, 6, 9, 8, 7, 1] has presented the time optimal trajectories for three classes of
More informationCONTROL OF THE NONHOLONOMIC INTEGRATOR
June 6, 25 CONTROL OF THE NONHOLONOMIC INTEGRATOR R. N. Banavar (Work done with V. Sankaranarayanan) Systems & Control Engg. Indian Institute of Technology, Bombay Mumbai -INDIA. banavar@iitb.ac.in Outline
More informationRobotics, Geometry and Control - A Preview
Robotics, Geometry and Control - A Preview Ravi Banavar 1 1 Systems and Control Engineering IIT Bombay HYCON-EECI Graduate School - Spring 2008 Broad areas Types of manipulators - articulated mechanisms,
More informationThe time-optimal trajectories for an omni-directional vehicle
The time-optimal trajectories for an omni-directional vehicle Devin J. Balkcom and Paritosh A. Kavathekar Department of Computer Science Dartmouth College Hanover, NH 03755 {devin, paritosh}@cs.dartmouth.edu
More informationTime optimal trajectories for bounded velocity differential drive vehicles
Time optimal trajectories for bounded velocity differential drive vehicles Devin J. Balkcom Matthew T. Mason June 3, 00 Abstract This paper presents the time optimal trajectories for differential drive
More informationA motion planner for nonholonomic mobile robots
A motion planner for nonholonomic mobile robots Miguel Vargas Material taken form: J. P. Laumond, P. E. Jacobs, M. Taix, R. M. Murray. A motion planner for nonholonomic mobile robots. IEEE Transactions
More informationEnergy-based Swing-up of the Acrobot and Time-optimal Motion
Energy-based Swing-up of the Acrobot and Time-optimal Motion Ravi N. Banavar Systems and Control Engineering Indian Institute of Technology, Bombay Mumbai-476, India Email: banavar@ee.iitb.ac.in Telephone:(91)-(22)
More informationL 2 -induced Gains of Switched Systems and Classes of Switching Signals
L 2 -induced Gains of Switched Systems and Classes of Switching Signals Kenji Hirata and João P. Hespanha Abstract This paper addresses the L 2-induced gain analysis for switched linear systems. We exploit
More informationMinimum Wheel-Rotation Paths for Differential-Drive Mobile Robots
Minimum Wheel-Rotation Paths for Differential-Drive Mobile Robots Hamidreza Chitsaz, Steven M. LaValle, Devin J. Balkcom, and Matthew T. Mason August 7, 007 Abstract The shortest paths for a mobile robot
More informationA note on stability of spacecrafts and underwater vehicles
A note on stability of spacecrafts and underwater vehicles arxiv:1603.00343v1 [math-ph] 1 Mar 2016 Dan Comănescu Department of Mathematics, West University of Timişoara Bd. V. Pârvan, No 4, 300223 Timişoara,
More informationOptimization techniques for autonomous underwater vehicles: a practical point of view
Optimization techniques for autonomous underwater vehicles: a practical point of view M. Chyba, T. Haberkorn Department of Mathematics University of Hawaii, Honolulu, HI 96822 Email: mchyba@math.hawaii.edu,
More informationLecture 2: Controllability of nonlinear systems
DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 2: Controllability of nonlinear systems Nonlinear Dynamical Control Systems, Chapter 3 See www.math.rug.nl/ arjan (under teaching) for info
More informationDSCC TIME-OPTIMAL TRAJECTORIES FOR STEERED AGENT WITH CONSTRAINTS ON SPEED AND TURNING RATE
Proceedings of the ASME 216 Dynamic Systems and Control Conference DSCC216 October 12-14, 216, Minneapolis, Minnesota, USA DSCC216-9892 TIME-OPTIMAL TRAJECTORIES FOR STEERED AGENT WITH CONSTRAINTS ON SPEED
More informationOptimization problems for controlled mechanical systems: Bridging the gap between theory and application
0 Optimization problems for controlled mechanical systems: Bridging the gap between theory and application M. Chyba a, T. Haberkorn b, R.N. Smith c a University of Hawaii, Honolulu, HI 96822 USA Mathematics,
More informationSome topics in sub-riemannian geometry
Some topics in sub-riemannian geometry Luca Rizzi CNRS, Institut Fourier Mathematical Colloquium Universität Bern - December 19 2016 Sub-Riemannian geometry Known under many names: Carnot-Carathéodory
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationPosition in the xy plane y position x position
Robust Control of an Underactuated Surface Vessel with Thruster Dynamics K. Y. Pettersen and O. Egeland Department of Engineering Cybernetics Norwegian Uniersity of Science and Technology N- Trondheim,
More informationRobotics. Dynamics. Marc Toussaint U Stuttgart
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory
More informationGeometric Optimal Control with Applications
Geometric Optimal Control with Applications Accelerated Graduate Course Institute of Mathematics for Industry, Kyushu University, Bernard Bonnard Inria Sophia Antipolis et Institut de Mathématiques de
More informationProof. We indicate by α, β (finite or not) the end-points of I and call
C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending
More informationTime-optimal control of a 3-level quantum system and its generalization to an n-level system
Proceedings of the 7 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 7 Time-optimal control of a 3-level quantum system and its generalization to an n-level
More informationOptimal Fault-Tolerant Configurations of Thrusters
Optimal Fault-Tolerant Configurations of Thrusters By Yasuhiro YOSHIMURA ) and Hirohisa KOJIMA, ) ) Aerospace Engineering, Tokyo Metropolitan University, Hino, Japan (Received June st, 7) Fault tolerance
More informationREMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS. Eduardo D. Sontag. SYCON - Rutgers Center for Systems and Control
REMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS Eduardo D. Sontag SYCON - Rutgers Center for Systems and Control Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
More informationOn the optimality of Dubins paths across heterogeneous terrain
On the optimality of Dubins paths across heterogeneous terrain Ricardo G. Sanfelice and Emilio Frazzoli {sricardo,frazzoli}@mit.edu Laboratory for Information and Decision Systems Massachusetts Institute
More information13 Path Planning Cubic Path P 2 P 1. θ 2
13 Path Planning Path planning includes three tasks: 1 Defining a geometric curve for the end-effector between two points. 2 Defining a rotational motion between two orientations. 3 Defining a time function
More informationLecture 9 - Rotational Dynamics
Lecture 9 - Rotational Dynamics A Puzzle... Angular momentum is a 3D vector, and changing its direction produces a torque τ = dl. An important application in our daily lives is that bicycles don t fall
More informationControl of Mobile Robots
Control of Mobile Robots Regulation and trajectory tracking Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Organization and
More informationNonholonomic Constraints Examples
Nonholonomic Constraints Examples Basilio Bona DAUIN Politecnico di Torino July 2009 B. Bona (DAUIN) Examples July 2009 1 / 34 Example 1 Given q T = [ x y ] T check that the constraint φ(q) = (2x + siny
More informationPreliminary Exam 2018 Solutions to Morning Exam
Preliminary Exam 28 Solutions to Morning Exam Part I. Solve four of the following five problems. Problem. Consider the series n 2 (n log n) and n 2 (n(log n)2 ). Show that one converges and one diverges
More informationRobotics. Dynamics. University of Stuttgart Winter 2018/19
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler, joint space control, reference trajectory following, optimal operational
More informationLine following of a mobile robot
Line following of a mobile robot May 18, 004 1 In brief... The project is about controlling a differential steering mobile robot so that it follows a specified track. Steering is achieved by setting different
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationSteering the Chaplygin Sleigh by a Moving Mass
Steering the Chaplygin Sleigh by a Moving Mass Jason M. Osborne Department of Mathematics North Carolina State University Raleigh, NC 27695 jmosborn@unity.ncsu.edu Dmitry V. Zenkov Department of Mathematics
More informationDesign and Implementation of Time Efficient Trajectories for an Underwater Vehicle
Design and Implementation of Time Efficient Trajectories for an Underwater Vehicle M. Chyba, T. Haberkorn University of Hawaii at Manoa, Mathematics Department, 565 M c Carthy Mall, Honolulu, HI 968 R.N.
More informationTheory of Vibrations in Stewart Platforms
Theory of Vibrations in Stewart Platforms J.M. Selig and X. Ding School of Computing, Info. Sys. & Maths. South Bank University London SE1 0AA, U.K. (seligjm@sbu.ac.uk) Abstract This article develops a
More informationMCE 366 System Dynamics, Spring Problem Set 2. Solutions to Set 2
MCE 366 System Dynamics, Spring 2012 Problem Set 2 Reading: Chapter 2, Sections 2.3 and 2.4, Chapter 3, Sections 3.1 and 3.2 Problems: 2.22, 2.24, 2.26, 2.31, 3.4(a, b, d), 3.5 Solutions to Set 2 2.22
More information9th Bay Area Mathematical Olympiad
9th Bay rea Mathematical Olympiad February 27, 2007 Problems with Solutions 1 15-inch-long stick has four marks on it, dividing it into five segments of length 1,2,3,4, and 5 inches (although not neccessarily
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationEXPERIMENTAL COMPARISON OF TRAJECTORY TRACKERS FOR A CAR WITH TRAILERS
1996 IFAC World Congress San Francisco, July 1996 EXPERIMENTAL COMPARISON OF TRAJECTORY TRACKERS FOR A CAR WITH TRAILERS Francesco Bullo Richard M. Murray Control and Dynamical Systems, California Institute
More informationDEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS OPTION B-1A: ROTATIONAL DYNAMICS Essential Idea: The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual
More information(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.
1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of
More informationMINIMUM TIME KINEMATIC TRAJECTORIES FOR SELF-PROPELLED RIGID BODIES IN THE UNOBSTRUCTED PLANE
MINIMUM TIME KINEMATIC TRAJECTORIES FOR SELF-PROPELLED RIGID BODIES IN THE UNOBSTRUCTED PLANE A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Doctor of Philosophy
More informationMulti-Robotic Systems
CHAPTER 9 Multi-Robotic Systems The topic of multi-robotic systems is quite popular now. It is believed that such systems can have the following benefits: Improved performance ( winning by numbers ) Distributed
More informationSpacecraft Attitude Control using CMGs: Singularities and Global Controllability
1 / 28 Spacecraft Attitude Control using CMGs: Singularities and Global Controllability Sanjay Bhat TCS Innovation Labs Hyderabad International Workshop on Perspectives in Dynamical Systems and Control
More informationInterpolation on lines by ridge functions
Available online at www.sciencedirect.com ScienceDirect Journal of Approximation Theory 175 (2013) 91 113 www.elsevier.com/locate/jat Full length article Interpolation on lines by ridge functions V.E.
More informationCircular motion. Aug. 22, 2017
Circular motion Aug. 22, 2017 Until now, we have been observers to Newtonian physics through inertial reference frames. From our discussion of Newton s laws, these are frames which obey Newton s first
More informationKinematics
Classical Mechanics Kinematics Velocities Idea 1 Choose the appropriate frame of reference. Useful frames: - Bodies are at rest - Projections of velocities vanish - Symmetric motion Change back to the
More informationOptimal Control Problems in Eye Movement. Bijoy K. Ghosh Department of Mathematics and Statistics Texas Tech University, Lubbock, TX
Optimal Control Problems in Eye Movement Bijoy K. Ghosh Department of Mathematics and Statistics Texas Tech University, Lubbock, TX We study the human oculomotor system as a simple mechanical control
More informationAttitude control of a hopping robot: a feasibility study
Attitude control of a hopping robot: a feasibility study Tammepõld, R. and Kruusmaa, M. Tallinn University of Technology, Estonia Fiorini, P. University of Verona, Italy Summary Motivation Small mobile
More informationLecture 10. Rigid Body Transformation & C-Space Obstacles. CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University
CS 460/560 Introduction to Computational Robotics Fall 017, Rutgers University Lecture 10 Rigid Body Transformation & C-Space Obstacles Instructor: Jingjin Yu Outline Rigid body, links, and joints Task
More informationPhys101 Lectures 19, 20 Rotational Motion
Phys101 Lectures 19, 20 Rotational Motion Key points: Angular and Linear Quantities Rotational Dynamics; Torque and Moment of Inertia Rotational Kinetic Energy Ref: 10-1,2,3,4,5,6,8,9. Page 1 Angular Quantities
More informationPosition and orientation of rigid bodies
Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal Reference Frames RF A A p AB B RF B rigid body position: A p AB
More informationThe Rocket Car. UBC Math 403 Lecture Notes by Philip D. Loewen
The Rocket Car UBC Math 403 Lecture Notes by Philip D. Loewen We consider this simple system with control constraints: ẋ 1 = x, ẋ = u, u [ 1, 1], Dynamics. Consider the system evolution on a time interval
More informationNonlinear Tracking Control of Underactuated Surface Vessel
American Control Conference June -. Portland OR USA FrB. Nonlinear Tracking Control of Underactuated Surface Vessel Wenjie Dong and Yi Guo Abstract We consider in this paper the tracking control problem
More informationME 230: Kinematics and Dynamics Spring 2014 Section AD. Final Exam Review: Rigid Body Dynamics Practice Problem
ME 230: Kinematics and Dynamics Spring 2014 Section AD Final Exam Review: Rigid Body Dynamics Practice Problem 1. A rigid uniform flat disk of mass m, and radius R is moving in the plane towards a wall
More informationDISCRETE-TIME TIME-VARYING ROBUST STABILIZATION FOR SYSTEMS IN POWER FORM. Dina Shona Laila and Alessandro Astolfi
DISCRETE-TIME TIME-VARYING ROBUST STABILIZATION FOR SYSTEMS IN POWER FORM Dina Shona Laila and Alessandro Astolfi Electrical and Electronic Engineering Department Imperial College, Exhibition Road, London
More informationFinal Exam Solutions June 10, 2004
Math 0400: Analysis in R n II Spring 004 Section 55 P. Achar Final Exam Solutions June 10, 004 Total points: 00 There are three blank pages for scratch work at the end of the exam. Time it: hours 1. True
More informationControl of a Car-Like Vehicle with a Reference Model and Particularization
Control of a Car-Like Vehicle with a Reference Model and Particularization Luis Gracia Josep Tornero Department of Systems and Control Engineering Polytechnic University of Valencia Camino de Vera s/n,
More informationEN Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 2015
EN53.678 Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 25 Prof: Marin Kobilarov. Constraints The configuration space of a mechanical sysetm is denoted by Q and is assumed
More informationTHE INDIAN COMMUNITY SCHOOL,KUWAIT PHYSICS SECTION-A
THE INDIAN COMMUNITY SCHOOL,KUWAIT CLASS:XI MAX MARKS:70 PHYSICS TIME ALLOWED : 3HOURS ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ General Instructions:
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationLECTURE 10: REVIEW OF POWER SERIES. 1. Motivation
LECTURE 10: REVIEW OF POWER SERIES By definition, a power series centered at x 0 is a series of the form where a 0, a 1,... and x 0 are constants. For convenience, we shall mostly be concerned with the
More informationLECTURE 3 Matroids and geometric lattices
LECTURE 3 Matroids and geometric lattices 3.1. Matroids A matroid is an abstraction of a set of vectors in a vector space (for us, the normals to the hyperplanes in an arrangement). Many basic facts about
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationControl of Mobile Robots Prof. Luca Bascetta
Control of Mobile Robots Prof. Luca Bascetta EXERCISE 1 1. Consider a wheel rolling without slipping on the horizontal plane, keeping the sagittal plane in the vertical direction. Write the expression
More informationSliding Mode Control Strategies for Spacecraft Rendezvous Maneuvers
Osaka University March 15, 2018 Sliding Mode Control Strategies for Spacecraft Rendezvous Maneuvers Elisabetta Punta CNR-IEIIT, Italy Problem Statement First Case Spacecraft Model Position Dynamics Attitude
More informationMASTER S EXAMINATION IN MATHEMATICS EDUCATION Saturday, 11 May 2002
MASTER S EXAMINATION IN MATHEMATICS EDUCATION Saturday, 11 May 2002 INSTRUCTIONS. Answer a total of eight questions, with the restriction of exactly two questions from Mathematics Education and at most
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More information(df (ξ ))( v ) = v F : O ξ R. with F : O ξ O
Math 396. Derivative maps, parametric curves, and velocity vectors Let (X, O ) and (X, O) be two C p premanifolds with corners, 1 p, and let F : X X be a C p mapping. Let ξ X be a point and let ξ = F (ξ
More informationMATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS
MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS 1. HW 1: Due September 4 1.1.21. Suppose v, w R n and c is a scalar. Prove that Span(v + cw, w) = Span(v, w). We must prove two things: that every element
More informationRemarks on Quadratic Hamiltonians in Spaceflight Mechanics
Remarks on Quadratic Hamiltonians in Spaceflight Mechanics Bernard Bonnard 1, Jean-Baptiste Caillau 2, and Romain Dujol 2 1 Institut de mathématiques de Bourgogne, CNRS, Dijon, France, bernard.bonnard@u-bourgogne.fr
More informationPhysical Simulation. October 19, 2005
Physical Simulation October 19, 2005 Objects So now that we have objects, lets make them behave like real objects Want to simulate properties of matter Physical properties (dynamics, kinematics) [today
More informationForces of Constraint & Lagrange Multipliers
Lectures 30 April 21, 2006 Written or last updated: April 21, 2006 P442 Analytical Mechanics - II Forces of Constraint & Lagrange Multipliers c Alex R. Dzierba Generalized Coordinates Revisited Consider
More informationExistence of Optima. 1
John Nachbar Washington University March 1, 2016 Existence of Optima. 1 1 The Basic Existence Theorem for Optima. Let X be a non-empty set and let f : X R. The MAX problem is f(x). max x X Thus, x is a
More informationContinuous Curvature Path Generation Based on Bézier Curves for Autonomous Vehicles
Continuous Curvature Path Generation Based on Bézier Curves for Autonomous Vehicles Ji-wung Choi, Renwick E. Curry, Gabriel Hugh Elkaim Abstract In this paper we present two path planning algorithms based
More informationAutonomous Underwater Vehicles: Equations of Motion
Autonomous Underwater Vehicles: Equations of Motion Monique Chyba - November 18, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/2016 - Control of Unmanned Vehicles
More informationIsobath following using an altimeter as a unique exteroceptive sensor
Isobath following using an altimeter as a unique exteroceptive sensor Luc Jaulin Lab-STICC, ENSTA Bretagne, Brest, France lucjaulin@gmail.com Abstract. We consider an underwater robot equipped with an
More informationPHYSICS 220. Lecture 15. Textbook Sections Lecture 15 Purdue University, Physics 220 1
PHYSICS 220 Lecture 15 Angular Momentum Textbook Sections 9.3 9.6 Lecture 15 Purdue University, Physics 220 1 Last Lecture Overview Torque = Force that causes rotation τ = F r sin θ Work done by torque
More informationCHAPTER 1. Introduction
CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric
More informationLast Update: April 7, 201 0
M ath E S W inter Last Update: April 7, Introduction to Partial Differential Equations Disclaimer: his lecture note tries to provide an alternative approach to the material in Sections.. 5 in the textbook.
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationOptimal periodic locomotion for a two piece worm with an asymmetric dry friction model
1st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 014. Optimal periodic locomotion for a two piece worm with an asymmetric dry friction model Nak-seung Patrick Hyun
More informationOptimal Control. McGill COMP 765 Oct 3 rd, 2017
Optimal Control McGill COMP 765 Oct 3 rd, 2017 Classical Control Quiz Question 1: Can a PID controller be used to balance an inverted pendulum: A) That starts upright? B) That must be swung-up (perhaps
More informationTHE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 007 014, March 2009 002 THE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS Y. CHARLES LI Abstract. Nadirashvili presented a
More informationOPEN QUIZ WHEN TOLD AT 7:00 PM
2.25 ADVANCED FLUID MECHANICS Fall 2013 QUIZ 1 THURSDAY, October 10th, 7:00-9:00 P.M. OPEN QUIZ WHEN TOLD AT 7:00 PM THERE ARE TWO PROBLEMS OF EQUAL WEIGHT Please answer each question in DIFFERENT books
More informationGlobal stabilization of an underactuated autonomous underwater vehicle via logic-based switching 1
Proc. of CDC - 4st IEEE Conference on Decision and Control, Las Vegas, NV, December Global stabilization of an underactuated autonomous underwater vehicle via logic-based switching António Pedro Aguiar
More informationAnswers to Problem Set Number 02 for MIT (Spring 2008)
Answers to Problem Set Number 02 for 18.311 MIT (Spring 2008) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139). March 10, 2008. Course TA: Timothy Nguyen, MIT, Dept. of Mathematics,
More informationConstrained Optimal Control I
Optimal Control, Guidance and Estimation Lecture 34 Constrained Optimal Control I Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore opics Motivation Brie Summary
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationLecture PowerPoints. Chapter 8 Physics: Principles with Applications, 6 th edition Giancoli
Lecture PowerPoints Chapter 8 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the
More informationINSPIRED by biological snake locomotion, snake robots
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. XX, NO. Y, MONTH 2010 1 Path following control of planar snake robots using a cascaded approach Pål Liljebäck, Member, IEEE, Idar U. Haugstuen, and
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Input-Output and Input-State Linearization Zero Dynamics of Nonlinear Systems Hanz Richter Mechanical Engineering Department Cleveland State University
More informationStatistics 612: L p spaces, metrics on spaces of probabilites, and connections to estimation
Statistics 62: L p spaces, metrics on spaces of probabilites, and connections to estimation Moulinath Banerjee December 6, 2006 L p spaces and Hilbert spaces We first formally define L p spaces. Consider
More informationavailable online at CONTROL OF THE DOUBLE INVERTED PENDULUM ON A CART USING THE NATURAL MOTION
Acta Polytechnica 3(6):883 889 3 Czech Technical University in Prague 3 doi:.43/ap.3.3.883 available online at http://ojs.cvut.cz/ojs/index.php/ap CONTROL OF THE DOUBLE INVERTED PENDULUM ON A CART USING
More informationNon-Collision Conditions in Multi-agent Robots Formation using Local Potential Functions
2008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 2008 Non-Collision Conditions in Multi-agent Robots Formation using Local Potential Functions E G Hernández-Martínez
More information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
More informationCooperative Control and Mobile Sensor Networks
Cooperative Control and Mobile Sensor Networks Cooperative Control, Part I, A-C Naomi Ehrich Leonard Mechanical and Aerospace Engineering Princeton University and Electrical Systems and Automation University
More information