A Discrete Geometric Optimal Control Framework for Systems with Symmetries

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d’Alembert- Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue

POINCARE-CHETAYEV EQUATIONS AND FLEXIBLE MULTI-BODY SYSTEMS

International audienceThis article is devoted to the dynamics of flexible multi-body systems and to their links with a fundamental set of equations discovered by H. Poincaré one hundred years ago [1]. These equations, called "Poincaré-Chetayev equations", are today known to be the foundation of the Lagrangian reduction theory. Starting with the extension of these equations to a Cosserat medium, we show that the two basic sets of equations used in flexible multi-body dynamics. The generalized Newton-Euler model of flexible multi-body systems in the floating frame approach and the partial differential equations of the nonlinear geometrically exact theory in the Galilean approach, are Poincaré-Chetayev equations

A Discrete Geometric Optimal Control Framework for Systems with Symmetries

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d’Alembert- Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue

Geometric discretization of nonholonomic systems with symmetries

The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of important models from science and engineering with such properties, we propose numerical methods specifically designed to account for their special geometric structure. At the core of the formulation lies a discrete variational principle that respects the structure of the state space and provides a framework for constructing accurate and numerically stable integrators. The dynamics of the systems we study is derived by vertical and horizontal splitting of the variational principle with respect to a nonholonomic connection that encodes the kinematic constraints and symmetries. We formulate a discrete analog of this principle by evaluating the Lagrangian and the connection at selected points along a discretized trajectory and derive discrete momentum equation and discrete reduced Euler-Lagrange equations resulting from the splitting of this principle. A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examples-the steered robotic car and the snakeboard. Their numerical advantages are confirmed through comparisons with standard methods

A Geometric Approach to Trajectory Planning for Underactuated Mechanical Systems

In the last decade, multi-rotors flying robots had a rapid development in industry and hobbyist communities thanks to the affordable cost and their availability of parts and components. The high number of degrees of freedom and the challenging dynamics of multi-rotors gave rise to new research problems. In particular, we are interested in the development of technologies for an autonomous fly that would al- low using multi-rotors systems to be used in contexts where the presence of humans is denied, for example in post-disaster areas or during search-and-rescue operations. Multi-rotors are an example of a larger category of robots, called \u201cunder-actuated mechanical systems\u201d (UMS) where the number of actuated degrees of freedom (DoF) is less than the number of available DoF. This thesis applies methods com- ing from geometric mechanics to study the under-actuation problem and proposes a novel method, based on the Hamiltonian formalism, to plan a feasible trajectory for UMS. We first show the application of a method called \u201cVariational Constrained System approach\u201d to a cart-pole example. We discovered that it is not possible to extend this method to generic UMS because it is valid only for a sub-class of UMS, called \u201csuper-articulated\u201d mechanical system. To overcome this limitation, we wrote the Hamilton equations of the quad- rotor and we apply a numerical \u201cdi- rect method\u201d to compute a feasible trajectory that satisfies system and endpoint constraints. We found that by including the system energy in the multi-rotor states, we are able to compute maneuvers that cannot be planned with other methods and that overcome the under-actuation constraints. To demonstrate the benefit of the method developed, we built a custom quad- rotor and an experimental setup with different obstacles, such as a gap in a wall and we show the correctness of the trajectory computed with the new method

Data-Driven Methods for Geometric Systems

The tools of geometric mechanics provide a compact representation of locomotion dynamics as ``the reconstruction equation''. We have found this equation yields a convenient form for estimating models directly from observation data. This convenience draws from the method's relatively rare feature of providing high accuracy models with little effort. By little effort, we point to the modeling process's low data requirements and the property that nothing about the implementation changes when substituting robot kinematics, material properties, or environmental conditions, as long as some intuitive baseline features of the dynamics are shared. We have applied data-driven geometric mechanics models toward optimizing robot behaviors both physical and simulated, exploring robots' ability to recover from injury, and efficiently creating libraries of maneuvers to be used as building blocks for higher-level robot tasks. Our methods employed the tools of data-driven Floquet analysis, providing a phase that we used as a means of grouping related measurements, allowing us to estimate a reconstruction equation model as a function of phase in the neighborhood of an observed behavior. This tool allowed us to build models at unanticipated scales of complexity and speed. Our use of a perturbation expansion for the geometric terms led to an improved estimation procedure for highly damped systems containing nontrivial but non-dominating amounts of momentum. Analysis of the role of passivity in dissipative systems led to another extension of the estimation procedure to robots with high degrees of underactuation in their internal shape, such as soft robots. This thesis will cover these findings and results, simulated and physical, and the surprising practicality of data-driven geometric mechanics.PHDRoboticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168033/1/babitt_1.pd