寄件者: Mathematics@CUHK

寄件日期: 2014年12月8日 11:43

收件者: yiuyu4@hotmail.com

主旨: [New post] Hopf-Cole transformation and the Burgers’ equation

KKK posted: ” The Burgers equation is the following: Here can be regarded as the (scalar) velocity of a fluid and can be regar”

]]>since a continuous linesr transformtion is differrentiable at each point open mapping gheorem of functional nalaysis follows. a great contribution of noniner functional analysis to linear functional analysis . please d corresond with me on this topic anilped@hotmail.com ]]>

寄件者: Mathematics@CUHK

寄件日期: 2018年2月26日 1:08

收件者: yiuyu4@hotmail.com

主旨: [New post] A simple proof of the Gauss-Bonnet theorem for geodesic ball

KKK posted: ” In this short note, we will give a simple proof of the Gauss-Bonnet theorem for a geodesic ball on a surface. The only prerequisite is some knowledge of Jacobi field, in particular how its second derivative (or the second derivative of the Jacobian) is r”

]]>LAGRANGIAN WITH VIRTUAL NON HOLONOMIC CONSTRAINTS

Author Horia Orasanu

ABSTRACT

The idea of virtual holonomic constraints is particularly a useful concept for control of oscillations

We will in this section show how this approach can be used to solve the path following control problem of snake robots. In particular, we will show how, by designing the joint reference trajectories i

INTRODUCTION

Our main motivation for using this approach is the fact that while performing the gait pattern lateral undulation which consists of fixed periodic body motions, all the solutions of the snake robot dynamics have inherent oscillatory behaviour ]]>

SIMULATION RESULTS ON LAGRANGIAN AND FOURIER INTEGRAL

Author Horia Orasanu

ABSTRACT

. There are many instances in which the basic physics is known (or postulated), and the behavior of a complex system is to be determined. A typical example is that in which there are too many particles for the problem to be tractable in terms of single-particle equations, and too few for a statistical analysis to apply. In such situations, use of a computer may furnish information on enough specific cases for the general behavior of the system to be discernable. If the basis physics is postulated but not known, a computer simulation can relate the theory to observations on complex systems and thus test the theory.

1 INTRODUCTION

A wide range of numerical techniques are available, from simple searches to sophisticated methods, such as annealing, an algorithm for finding global minima which was inspired by an actual physical process. Some of the techniques are repetitive applications of deterministic equations. Others invoke stochastic processes (using “random” numbers), to focus on the important features.theoretical and experimental physics have been joined over the last three decades by that of computational physics

While most applications of such simulations yield expected results, surprises do occur. This is analogous to an unexpected result from an experiment. Either the simulation/experiment went wrong (usual) or a new aspect of nature has been uncovered (rare). Examples of the latter are the identification of constants of motion in chaotic systems and the discovery of runaway motion in the drift and diffusion of ions in gas. Such discoveries are followed by “proper” theories and “proper” experiments, but the computer plays a vital role in the research.

The research of Professor Gatland involves data analysis and the mathematical modeling and simulation of microscopic physical processes. These activities encompass both research and instruction.

In this lecture we recall the definitions of autonomous and non autonomous Dynamical Systems as well as their different concepts of attractors. After that we introduce the different notions of robustness of attractors under perturbation (Upper semicontinuity, Lower semicontinuity, Topological structural stability and Structural stability) and give conditions on the dynamical systems so that robustness is attained. We show that enforcing the appropriately defined virtual holonomic constraints for the configuration variables implies that the robot converges to and follows a desired geometric path. Numerical simulations and experimental rMethods

2 THE GEOMETRY OF THE PROBLEM

The (N+2)-dimensional configuration space of the snake robot is denoted as 𝒬 = 𝒮 × 𝒢, which is composed of the shape space and a Lie group which is freely and properly acting on the configuration space. In particular, the shape variables, i.e. qa=(q1,…,qN−1), which define the internal configuration of the robot and which we have direct control on, take values in . Moreover, the position variables, i.e. qu=(θN,px,py), which are passive DOF of the system, lie in . The velocity space of the robot is defined as the differentiable (2N+4)-dimensional tangent bundle of as T𝒬 = 𝕋N × ℝN+4, where 𝕋N denotes the N-torus in which the angular coordinates live. The free Lagrangian function of the robot ℒ:T𝒬 → ℝ is invariant under the given action of on . The coupling between the shape and the position variables causes the net displacement of the position variables, according to the cyclic motion of the shape variables, i.e. the gait pattern. Note that for simplicity of presentation, in this paper, we consider local representation of T𝒬 embedded in an (2N+4)-dimensional open subset of the Euclidean space ℝ2N+4. To this end, we separate the dynamic equations of the robot given by (11) into two subsets by taking x = [qa,qu]T ∈ ℝN+2, with qa ∈ ℝN−1 and qu ∈ ℝ3 which were defined in the subsection describing the geometry of the problem:

m11(qa)q¨a+m12(qa)q¨u+h1(x,x˙)=ψ∈RN−1

(20)

m21(qa)q¨a+m22(qa)q¨u+h2(x,x˙)=03×1∈R3

(21)

where m11 ∈ ℝ(N−1)×(N−1), m12 ∈ ℝ(N−1)×3, m21 ∈ ℝ3×(N−1), and m22 ∈ ℝ3×3 denote the corresponding submatrices of the inertia matrix, and 03×1 = [0,0,0]T ∈ ℝ3. Furthermore, h1(x,x˙)∈RN−1 and h2(x,x˙)∈R3 include all the contributions of the Coriolis, centripetal, and friction forces. Moreover, ψ ∈ ℝN−1 denotes the non-zero part of the vector of control forces, i.e. B(x)τ = [ψ,03×1]T ∈ ℝN+2. From (21), we have

q¨u=−m−122(h2+m21q¨a)∈R3

References

1. Liljebäck P, Stavdahl Ø, Beitnes A (2006) SnakeFighter – development of a water hydraulic fire fighting snake robot. In: Proc. IEEE international conference on control, automation, robotics, and vision ICARCV, Singapore.

2. Wang Z, Appleton E (2003) The concept and research of a pipe crawling rescue robot. Adv Robot 17.4: 339–358.

3. Fjerdingen SA, Liljebäck P, Transeth AA (2009) A snake-like robot for internal inspection of complex pipe structures (PIKo). In: Proc. IEEE/RSJ international conference on intelligent robots and systems, St. Louis, MO, USA.

4. Dacic DB, Nesic D, Teel AR, Wang W. Path following for nonlinear systems with unstable zero dynamics: an averaging solution. IEEE Trans Automatic Control. 2011;56:880–886. doi: 10.1109/TAC.2011.2105130. [Cross Ref]

5. Hirose S. Biologically inspired robots: snake-like locomotors and manipulators. Oxford, England: Oxford University Press; 1993.

6. Matsuno F, Sato H (2005) Trajectory tracking control of snake robots based on dynamic model. In: Proc. IEEE international conference on robotics and automation, 3029–3034. 18-22 April 2005.

7. Date H, Hoshi Y, Sampei M (2000) Locomotion control of a snake-like robot based on dynamic manipulability. In: Proc. IEEE/RSJ international conference on intelligent robots and systems, Takamatsu, Japan.

8. Tanaka M, Matsuno F (2008) Control of 3-dimensional snake robots by using redundancy. In: Proc. IEEE international conference on robotics and automation, 1156–1161, Pasadena, CA.

9. Ma S, Ohmameuda Y, Inoue K, Li B (2003) Control of a 3-dimensional snake-like robot. In: Proc. IEEE international conference on robotics and automation, vol. 2, 2067–2072, Taipei, Taiwan.

10. Tanaka M, Matsuno F (2009) A study on sinus-lifting motion of a snake robot with switching constraints. In: Proc. IEEE international conference on robotics and automation, 2270–2275. 12-17 May 2009.

11. Prautsch P, Mita T, Iwasaki T (2000) Analysis and control of a gait of snake robot. Trans IEE J Ind Appl Soc 120-D: 372–381.

12. McIsaac K, Ostrowski J. Motion planning for anguilliform locomotion. IEEE Trans Robot Automation. 2003;19:637–652. doi: 10.1109/TRA.2003.814495. [Cross Ref]

13. Hicks G, Ito K. A method for determination of optimal gaits with application to a snake-like serial-link structure. IEEE Trans Automatic Control. 2005;50:1291–1306. doi: 10.1109/TAC.2005.854583. [Cross Ref]

14. Ma S, Ohmameuda Y, Inoue K (2004) Dynamic analysis of 3-dimensional snake robots. In: Proc. IEEE/RSJ international conference on intelligent robots and systems, 767–772. 28 Sept.-2 Oct. 2004.

15. Ma S. Analysis of creeping locomotion of a snake-like robot. Adv Robot. 2001;15(2):205–224. doi: 10.1163/15685530152116236. [Cross Ref]

16. Liljebäck P, Pettersen KY, Stavdahl Ø, Gravdahl JT (2013) Snake robots – modelling, mechatronics, and control. Advances in industrial control. Springer.

17. Liljebäck P, Haugstuen IU, Pettersen KY. Path following control of planar snake robots using a cascaded approach. IEEE Trans Control Syst Technol. 2012;20:111–126.

18. Rezapour E, Pettersen KY, Liljebäck P, Gravdahl JT (2013) Path following control of planar snake robots using virtual holonomic constraints. Paper presented at the IEEE international conference on robotics and biomimetics, Shenzhen, China. [PMC free article] [PubMed]

19. Liljebäck P, Pettersen KY, Stavdahl Ø, Gravdahl JT. Controllability and stability analysis of planar snake robot locomotion. IEEE Trans Automatic Control. 2013;56(6):1365–1380. doi: 10.1109/TAC.2010.2088830. [Cross Ref]

20. Westervelt ER, Grizzle JW, Chevallereau C, Choi JH, Morris B. Feedback control of dynamic bipedal robot locomotion. Boca Raton: CRC press; 2007.

21. Maggiore M, Consolini L. Virtual holonomic constraints for Euler-Lagrange systems. IEEE Trans on Automatic Control. 2013;58(4):1001–1008. doi: 10.1109/TAC.2012.2215538. [Cross Ref]

22. Consolini L, Maggiore M (2010) Control of a bicycle using virtual holonomic constraints. In: Proc. 49th IEEE conference on decision and control, Atlanta, Georgia, USA, December 15-17, 2010.

23. Shiriaev A, Perram JW, Canudas-de-Wit C. Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach. IEEE Trans Automatic Control. 2005;50(8):1164–1176. doi: 10.1109/TAC.2005.852568. [Cross Ref]

24. Freidovich L, Robertsson A, Shiriaev A, Johansson R. Periodic motions of the Pendubot via virtual holonomic constraints: theory and experiments. Automatica. 2008;44(3):785–791. doi: 10.1016/j.automatica.2007.07.011. [Cross Ref]

25. Spong MW, Hutchinson S, Vidyasagar M. Robot modeling and control. New York: John Wiley and Sons; 2006.

26. Bullo F, Lewis A (2005) Geometric control of mechanical systems. Springer.

27. Fossen TI. Marine control systems: guidance, navigation and control of ships, rigs and underwater vehicles. Marine Cybernetics: Trondheim, Norway; 2002.