Kinematically optimal hyper-redundant manipulator configurations

“Hyper-redundant” robots have a very large or infinite degree of kinematic redundancy. This paper develops new methods for determining “optimal” hyper-redundant manipulator configurations based on a continuum formulation of kinematics. This formulation uses a backbone curve model to capture the robot's essential macroscopic geometric features. The calculus of variations is used to develop differential equations, whose solution is the optimal backbone curve shape. We show that this approach is computationally efficient on a single processor, and generates solutions in O(1) time for an N degree-of-freedom manipulator when implemented in parallel on O(N) processors. For this reason, it is better suited to hyper-redundant robots than other redundancy resolution methods. Furthermore, this approach is useful for many hyper-redundant mechanical morphologies which are not handled by known methods

Mecánica Discreta para Sistemas Forzados y Ligados

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 10/07/2019Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with researchfrom the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problems...La mecánica clásica es un campo tan fundamental para la física como la geometría lo es para las matemáticas. Ambos están interrelacionados y su estudio conjunto así como sus interacciones forman lo que hoy se conoce como la mecánica geométrica [véase, por ejemplo, AM78; Arn89; Hol11a; Hol11 b]. Hoy es bien sabido que el concepto de simetría tiene importantes consecuencias para los sistemas mecánicos. En particular, la evolución de los sistemas mecánicos suele mostrar ciertas propiedades de preservación en forma de cantidades conservadas del movimiento o preservación de estructuras geométricas. Ser capaces de capturar estas propiedades es vital para tener una imagen fiel, tanto en términos cuantitativos como cualitativos, de cara al estudio de estos sistemas. Esto tiene gran importancia en el campo teórico y también el aplicado, como en la ingeniería. La experimentación en laboratorios y la generación de prototipos son procesos costosos y que requieren de tiempo, y para determinad os sistemas pueden no ser siquiera factibles. Con la llegada el ordenador, simular y experimentar con sistemas mecánicos de forma rápida y económica se convirtió en una realidad . Desde sencillas simulaciones balísticas para alumnos de secundaria a simulaciones de dinámica molecular a gran escala; desde la planificación de trayectorias para vehículos autónomos a la estimación de movimientos en robots bípedos; desde costosas simulaciones basadas en modelos físicos para la industria de la animación a la simulación de sólidos rígidos y deformables en tiempo real para la industria del videojuego, el tratamiento numérico de sistemas de complejidad creciente se ha convertido en una necesidad. Naturalmente surgieron nuevos algoritmos capaces de conservar gran parte de las propiedades geométricas de estos sistemas, configurando lo que a hora se conoce como integración geométrica [véase SC94; HLW1O]. En los últimos 20 a 30 años se han dado grandes pasos en esta dirección, con el desarrollo de métodos que conservan energía, métodos simplécticos y multisimplécticos, métodos que preservan el espacio de configuración y más. Aún así, la investigación en esta área está todavía lejos de acabar. Por ejemplo , los sistemas sometidos a fuerzas externas y con ligaduras ofrecen ciertas dificultades que han de ser abordadas, y esta tesis se dedica a explorar estos dos casos ofreciendo nuevos desarrollos y resultados...Fac. de Ciencias MatemáticasTRUEunpu

A Material Point Method for Simulating Frictional Contact with Diverse Materials

We present an extension to the Material Point Method (MPM) for simulating elastic objects with various co-dimensions like hair (1D), thin shells (2D), and volumetric objects (3D). We simulate thin shells with frictional contact using a combination of MPM and subdivision finite elements. The shell kinematics are assumed to follow a continuum shell model which is decomposed into a Kirchhoff-Love motion that rotates the mid-surface normals followed by shearing and compression/extension of the material along the mid-surface normal. We use this decomposition to design an elastoplastic constitutive model to resolve frictional contact by decoupling resistance to contact and shearing from the bending resistance components of stress. We show that by resolving frictional contact with a continuum approach, our hybrid Lagrangian/Eulerian approach is capable of simulating challenging shell contact scenarios with hundreds of thousands to millions of degrees of freedom. Furthermore our technique naturally couples with other traditional MPM methods for simulating granular materials. Without the need for collision detection or resolution, our method runs in a few minutes per frame in these high resolution examples. For the simulation of hair and volumetric elastic objects, we utilize a Lagrangian mesh for internal force computation and an Eulerian mesh for self collision as well as coupling with external materials. While the updated Lagrangian discretization where the Eulerian grid degrees of freedom are used to take variations of the potential energy is effective in simulating thin shells, its frictional contact response strategy does not generalize to volumetric objects. Therefore, we develop a hybrid approach that retains Lagrangian degrees of freedom while still allowing for natural coupling with other materials simulated with traditional MPM. We demonstrate the efficacy of our technique with examples that involve elastic soft tissues coupled with kinematic skeletons, extreme deformation, and coupling with multiple elastoplastic materials. Our approach also naturally allows for two-way rigid body coupling

Data-Driven Methods to Build Robust Legged Robots

For robots to ever achieve signicant autonomy, they need to be able to mitigate performance loss due to uncertainty, typically from a novel environment or morphological variation of their bodies. Legged robots, with their complex dynamics, are particularly challenging to control with principled theory. Hybrid events, uncertainty, and high dimension are all confounding factors for direct analysis of models. On the other hand, direct data-driven methods have proven to be equally dicult to employ. The high dimension and mechanical complexity of legged robots have proven challenging for hardware-in-the-loop strategies to exploit without signicant eort by human operators. We advocate that we can exploit both perspectives by capitalizing on qualitative features of mathematical models applicable to legged robots, and use that knowledge to strongly inform data-driven methods. We show that the existence of these simple structures can greatly facilitate robust design of legged robots from a data-driven perspective. We begin by demonstrating that the factorial complexity of hybrid models can be elegantly resolved with computationally tractable algorithms, and establish that a novel form of distributed control is predicted. We then continue by demonstrating that a relaxed version of the famous templates and anchors hypothesis can be used to encode performance objectives in a highly redundant way, allowing robots that have suffered damage to autonomously compensate. We conclude with a deadbeat stabilization result that is quite general, and can be determined without equations of motion.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155053/1/gcouncil_1.pd