Geometric Control of Mechanical Systems: Modeling, Analysis, by Francesco Bullo

By Francesco Bullo

 

Control idea for mechanical platforms is a subject that has received considerable curiosity some time past decade, influenced via purposes akin to robotics and automation, self sufficient automobiles in marine, aerospace and different environments, flight regulate, difficulties in nuclear magnetic resonance, micro-electro mechanical structures (MEMS), and fluid mechanics. This ebook provides a unified method for studying these structures utilizing differential geometry. The emphasis is on figuring out in a geometrical method all elements of a mechanical regulate process: configuration house, the Lagrangian, exterior forces, and nonholonomic constraints. This publication is well-suited to be used as a textbook, with a number of examples and routines incorporated to assist consolidate the idea.

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Thus, the correction term, H(x('»)/(h,/(x(i»)), is the time it would take the trajectory to travel from x(') to ~ if the the vector field were (locally) constant with value /(x('»). Henon's method This clever method is due to Henon [1982]. The technique hinges on his observation that if one wants to know the state of a system at time t, one simply integrates the system to time t. It follows that if the independent variable of the system is X n , the next crossing with the hyperplane Xn = a is calculated by integrating to aj there is no interpolation or time-step halving.

5) where h E IRn is a vector normal to ~, XI: E IRn is some point lying on the hyperplane, and (u, v) := uT v is the inner product. , as in Fig. 3. , (h,f( o. , (h,f( o. ~ in a P±: ~ --+~. --+ ~. --+ ~. P±(x) is the first point where o. P+ and P_ are called one-sided Poincare maps while Pi is called the two-sided Poincare map. , an x E ~ for which (h, f(x)) = 0) satisfies the direction criterion of each of the three maps.

The main drawback of the higher-order Poincare map is that it requires an enormous amount of input data to produce a reasonable amount of output. If one assumes that each sampling reduces the amount of data by a factor of 100, then a third-order Poincare map CHAPTER 2. 0 -5 . 11: Higher-order Poincare map of a three-periodic trajectory with frequency base {I, V2, va}. (a) First-order Poincare map sampled at rate 1; (b) the second-order Poincare map sampled at rates 1 and V2. 4. ALGORITHMS requires 10000 more data points than a first-order map.

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