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

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寄件者: Mathematics@CUHK

寄件日期: 2017年11月8日 23:16

收件者: yiuyu4@hotmail.com

主旨: [New post] Euler’s formula e^ix = cos x + i sin x – a geometric approach

KKK posted: ” Today I mentioned the famous Euler’s formula briefly in my Calculus class: where is a”

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寄件者: Mathematics@CUHK

寄件日期: 2017年5月20日 15:24

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主旨: [New post] The Cauchy-Schwarz inequality and the Lagrange identity

KKK posted: ” The classical Lagrange identity is the following: $latex \displaystyle \begin{array}{rl} \displaystyle \left(\sum_{i=1}^n a_ib_i\right)^2+\sum _{1\le i<j\le n} \left(a_ib_j-a_jb_i\right)^2 =\left(\sum_{i=1}^{n}a_i^2\right)\left(\sum_{j=1}^{n}b_j^2\rig"

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