12/21/2023 0 Comments Compute method mapThe feasibility of these ideas was investigated by applying these techniques to the problem of locating stable walking motions for a disc-foot passive walking machine. Rather than providing explicit formulas for these derivatives, automatic differentiation can be employed to compute these quantities efficiently during the course of a simulation. The acceleration is computed most easily in a redundant set of coordinates giving the spatial positions of each body: since the acceleration is just the second derivative of these positions. Much of this difficulty stems from computing the acceleration of each rigid body in an inertial reference frame. Formulating mechanically correct equations of motion for systems of interconnected rigid bodies, while straightforward, is a time-consuming error prone process. The first is to provide a simple yet powerful framework for studying periodic motions in mechanical systems. There are two different yet related goals that motivate the algorithmic choices listed above. A metric is defined for computing the distance between two given periodic orbits which is then minimized using a trust-region minimization algorithm to find optimal fits of the model to a reference orbit. Numerical solutions to the shooting equations are then estimated by a Newton process yielding an approximate periodic orbit. The boundary value problem representing a periodic orbit in a hybrid system of differential algebraic equations is discretized via multiple-shooting using a high-degree Taylor series integration method. These techniques make extensive use of automatic differentiation to accurately and efficiently evaluate derivatives for time integration, parameter sensitivities, root finding and optimization. This paper describes techniques for computing periodic orbits in systems of hybrid differential-algebraic equations and parameter estimation methods for fitting these orbits to data. Improved computational tools for exploring parameter space and fitting models to data are clearly needed. Modelers are often faced with the difficult problem of using simulations of a nonlinear model, with complex dynamics and many parameters, to match experimental data. In addition, mathematical models more » of biological processes frequently contain many poorly-known parameters, and the problems associated with this impedes the construction of detailed, high-fidelity models. Modeling these processes with high fidelity as periodic orbits of dynamical systems is challenging because: (1) (most) nonlinear differential equations can only be solved numerically (2) accurate computation requires solving boundary value problems (3) many problems and solutions are only piecewise smooth (4) many problems require solving differential-algebraic equations (5) sensitivity information for parameter dependence of solutions requires solving variational equations and (6) truncation errors in numerical integration degrade performance of optimization methods for parameter estimation. Rhythmic, periodic processes are ubiquitous in biological systems for example, the heart beat, walking, circadian rhythms and the menstrual cycle. For this case the two dimensional symmetry planes give a cross section of the resonances and a visualization of the Arnold web. A pair of coupled standard maps provides a four dimensional example. more » As a parameter varies, the symmetric orbits are observed to undergo symmetry breaking bifurcations, creating a pair nonsymmetric orbits. There is ''dominant'' symmetry plane on which the minimizing orbit is never observed to occur. omega., and 2/sup N/ /minus/ 1 other ''minimax'' orbits. Furthermore, we conjecture there is an orbit which minimizes the periodic action for each. We conjecture that such maps have at least 2/sup N/ symmetric periodic orbits for each frequency. Reversible maps of this form have 2/sup N+1/ invariant symmetry sets topologically equivalent to N dimensional planes. The q's are assumed to be angle variables. = ,Ī 2N dimensional symplectic mapping which satisfies the twist condition is obtained from a Lagrangian generating function F(q,q').
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |