Dynamic Modeling for Bi-Modal, Rotary Wing, Rolling-Flying Vehicles

This paper presents a rigorous analysis of a promising bi-modal multirotor vehicle that can roll and fly. This class of vehicle provides energetic and locomotive advantages over traditional unimodal vehicles. Despite superficial similarities to traditional multirotor vehicles, the dynamics of the vehicle analyzed herein differ substantially. This paper is the first to offer a complete and rigorous derivation, simulation, and validation of the vehicle's terrestrial rolling dynamics. Variational mechanics is used to develop a six degrees-of-freedom dynamic model of the vehicle subject to kinematic rolling constraints and various nonconservative forces. The resulting dynamic system is determined to be differentially flat and the flat outputs of the vehicle are derived. A functional hardware embodiment of the vehicle is constructed, from which empirical motion data are obtained via odometry and inertial sensing. A numerical simulation of the dynamic model is executed, which accurately predicts complex dynamic phenomena observed in the empirical data, such as gravitational and gyroscopic nonlinearities; the comparison of simulation results to empirical data validates the dynamic model.

Contact-implicit trajectory optimization using variational integrators

Contact constraints arise naturally in many robot planning problems. In recent years, a variety of contact-implicit trajectory optimization algorithms have been developed that avoid the pitfalls of mode pre-specification by simultaneously optimizing state, input, and contact force trajectories. However, their reliance on first-order integrators leads to a linear tradeoff between optimization problem size and plan accuracy. To address this limitation, we propose a new family of trajectory optimization algorithms that leverage ideas from discrete variational mechanics to derive higher-order generalizations of the classic time-stepping method of Stewart and Trinkle. By using these dynamics formulations as constraints in direct trajectory optimization algorithms, it is possible to perform contact-implicit trajectory optimization with significantly higher accuracy. For concreteness, we derive a second-order method and evaluate it using several simulated rigid-body systems, including an underactuated biped and a quadruped. In addition, we use this second-order method to plan locomotion trajectories for a complex quadrupedal microrobot. The planned trajectories are evaluated on the physical platform and result in a number of performance improvements.

The Lazy Universe

Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and time. The physical and mathematical origins of the Lagrange Equations, Hamilton’s Equations, the Lagrangian, the Hamiltonian, and the Hamilton-Jacobi Equation are shown. Also, worked examples in Lagrangian and Hamiltonian Mechanics are given. However the aim is to explain physics rather than to give a technical mastery of the subject. Therefore, much of the mathematics is in the appendices. While there is still some mathematics in the main text, the reader may select whether to work through, skim-read, or skip over it: the “story-line” will just about be maintained whatever route is chosen. The work is a much-reduced and simplified version of the outstanding text, “The Variational Principles of Mechanics” written by Cornelius Lanczos in 1949. That work is barely known today, and the present work may be considered as a tiny stepping-stone toward it. A principle that underlies all of physics will have wider repercussions; it is also to be appreciated in an aesthetic sense. It is hoped that this book will lead the reader to the widest possible understanding of the Principle of Least Action. Ideas such as Variational Mechanics, phase space, Fermat’s Principle, and Noether’s Theorem are explained.

Antecedents

Early ideas about optimization principles were brought in by an eclectic group of extraordinary thinkers: the Ancients (Hero, and Princess Dido), Fermat with his Principle of Least Time, the Bernoullis, Leibniz, Maupertuis, Euler, and d’Alembert. Also, Stevin was the first to invoke the impossibility of perpetual motion in a proof, and Huygens was the first to put Galilean Relativity to a quantitative test. The Swiss family of mathematical geniuses, the Bernoullis, tackled isoperimetric problems, such as the brachystochrone, and Johann Bernoulli discovered the Principle of Virtual Velocities. The flavour of the eighteenth century is shown in the evocative tale of the König affair, and the correspondence between Daniel Bernoulli and Euler. It is shown how symmetry arguments, leading ultimately to an energy-analysis, were competing with Newton’s force-analysis. The Principle of Least Action and Variational Mechanics, proper, were developed by Lagrange, Hamilton, and Jacobi.