Robotic manipulation of a rotating chain

This paper considers the problem of manipulating a uniformly rotating chain: the chain is rotated at a constant angular speed around a fixed axis using a robotic manipulator. Manipulation is quasi-static in the sense that transitions are slow enough for the chain to be always in "rotational" equilibrium. The curve traced by the chain in a rotating plane -- its shape function -- can be determined by a simple force analysis, yet it possesses complex multi-solutions behavior typical of non-linear systems. We prove that the configuration space of the uniformly rotating chain is homeomorphic to a two-dimensional surface embedded in $\mathbb{R}^3$. Using that representation, we devise a manipulation strategy for transiting between different rotation modes in a stable and controlled manner. We demonstrate the strategy on a physical robotic arm manipulating a rotating chain. Finally, we discuss how the ideas developed here might find fruitful applications in the study of other flexible objects, such as elastic rods or concentric tubes.Comment: 12 pages, 9 figure

On the Statics, Dynamics, and Stability of Continuum Robots: Model Formulations and Efficient Computational Schemes

This dissertation presents advances in continuum-robotic mathematical-modeling techniques. Specifically, problems of statics, dynamics, and stability are studied for robots with slender elastic links. The general procedure within each topic is to develop a continuous theory describing robot behavior, develop a discretization strategy to enable simulation and control, and to validate simulation predictions against experimental results.Chapter 1 introduces the basic concept of continuum robotics and reviews progress in the field. It also introduces the mathematical modeling used to describe continuum robots and explains some notation used throughout the dissertation.The derivation of Cosserat rod statics, the coupling of rods to form a parallel continuum robot (PCR), and solution of the kinematics problem are reviewed in Chapter 2. With this foundation, soft real-time teleoperation of a PCR is demonstrated and a miniature prototype robot with a grasper is controlled.Chapter 3 reviews the derivation of Cosserat rod dynamics and presents a discretization strategy having several desirable features, such as generality, accuracy, and potential for good computational efficiency. The discretized rod model is validated experimentally using high speed camera footage of a cantilevered rod. The discretization strategy is then applied to simulate continuum robot dynamics for several classes of robot, including PCRs, tendon-driven robots, fluidic actuators, and concentric tube robots.In Chapter 4, the stability of a PCR is analyzed using optimal control theory. Conditions of stability are gradually developed starting from a single planar rod and finally arriving at a stability test for parallel continuum robots. The approach is experimentally validated using a camera tracking system.Chapter 5 provides closing discussion and proposes potential future work

Manipulation and mechanics of thin elastic objects

In this thesis, multiple problems concerning the equilibrium and stability properties of thin deformable objects are considered, with particular focus given to the analysis of thin elastic rods. The problems considered can be divided into two related categories: manipulation and mechanics. First, a few results concerning symmetries in geometric optimal control theory are derived, which are later used in the analysis of thin elastic objects. Then the problem of quasi-statically manipulating an elastic rod from an initial configuration into a goal configuration is considered. Based upon an analysis of symmetries, geometric and topological characterizations of the set of all stable equilibrium configurations of an elastic rod are derived. Specifically, under a few regularity assumptions, it is shown that the set of all stable equilibrium configurations without conjugate points of an extensible, shearable, anisotropic, and uniform Cosserat elastic rod subject to conservative body forces is a smooth six-dimensional manifold parameterized by a single global coordinate chart. Furthermore, in the case of an inextensible, unshearable, anisotropic, uniform, and intrinsically straight Kirchhoff elastic rod without body forces, this six-dimensional manifold is shown to be path-connected. In addition to their applications to manipulation, the geometric and topological results described above can be used to answer questions concerning the mechanics of elastic rods and other deformable objects. For an inextensible, unshearable, isotropic, and uniform Kirchhoff elastic rod, it is shown that the closure of the set of all stable equilibria with helical centerlines is star-convex, and this property is used to compute and visualize the boundary between stable and unstable helical rods. Finally, two applications of geometric optimal control theory to the analysis of constitutive equations for thin elastic objects are considered. In the first application, the Pontryagin maximum principle is used to analyze curvature discontinuities observed in inextensible surfaces. In the second application, the Pontryagin maximum principle is used to derive constitutive equations for an elastic rod subject to a local injectivity constraint, and the use of this model for analyzing highly flexible helical springs with contact between neighboring coils is considered

On the Elastic Stability of Folded Rings in Circular and Straight States

Single-loop elastic rings can be folded into multi-loop equilibrium configurations. In this paper, the stability of several such multi-loop states which are either circular or straight are investigated analytically and illustrated by experimental demonstrations. The analysis ascertains stability by exploring variations of the elastic energy of the rings for admissible deformations in the vicinity of the equilibrium state. The approach employed is the conventional stability analysis for elastic conservative systems which differs from most of the analyses that have been published on this class of problems, as will be illustrated by reproducing and elaborating on several problems in the literature. In addition to providing solutions to two basic problems, the paper analyses and demonstrates the stability of six-sided rings that fold into straight configurations

Forward dynamics of continuum and soft robots: a strain parametrization based approach

soumis à IEEE TROIn this article we propose a new solution to the forward dynamics of Cosserat beams with in perspective, its application to continuum and soft robotics manipulation and locomotion. In contrast to usual approaches, it is based on the non-linear parametrization of the beam shape by its strain fields and their discretization on a functional basis of strain modes. While remaining geometrically exact, the approach provides a minimal set of ordinary differential equations in the usual Lagrange matrix form that can be solved with standard explicit time-integrators. Inspired from rigid robotics, the calculation of the matrices of the Lagrange model is performed with a continuous inverse Newton-Euler algorithm. The approach is tested on several numerical benches of non-linear structural statics, as well as further examples illustrating its capabilities for dynamics

A Coarse-to-Fine Framework for Dual-Arm Manipulation of Deformable Linear Objects with Whole-Body Obstacle Avoidance

Manipulating deformable linear objects (DLOs) to achieve desired shapes in constrained environments with obstacles is a meaningful but challenging tasks. Global planning is necessary for such a highly-constrained task; however, accurate models of DLOs required by planners are difficult to obtain owing to their deformable nature, and the inevitable modeling errors significantly affect the planning results, probably resulting in task failure if the robot simply executes the planned path in an open-loop manner. In this paper, we propose a coarse-to-fine framework to combine global planning and local control for dual-arm manipulation of DLOs, capable of precisely achieving desired configurations and avoiding potential collisions between the DLO, robot, and obstacles. Specifically, the global planner refers to a simple yet effective DLO energy model and computes a coarse path to guarantee the feasibility of the task; then the local controller follows that path as guidance and further shapes it with closed-loop feedback to compensate for the planning errors and guarantee the accuracy of the task. Both simulations and real-world experiments demonstrate that our framework can robustly achieve desired DLO configurations in constrained environments with imprecise DLO models. which may not be reliably achieved by only planning or control

Model Based Control of Soft Robots: A Survey of the State of the Art and Open Challenges

Continuum soft robots are mechanical systems entirely made of continuously deformable elements. This design solution aims to bring robots closer to invertebrate animals and soft appendices of vertebrate animals (e.g., an elephant's trunk, a monkey's tail). This work aims to introduce the control theorist perspective to this novel development in robotics. We aim to remove the barriers to entry into this field by presenting existing results and future challenges using a unified language and within a coherent framework. Indeed, the main difficulty in entering this field is the wide variability of terminology and scientific backgrounds, making it quite hard to acquire a comprehensive view on the topic. Another limiting factor is that it is not obvious where to draw a clear line between the limitations imposed by the technology not being mature yet and the challenges intrinsic to this class of robots. In this work, we argue that the intrinsic effects are the continuum or multi-body dynamics, the presence of a non-negligible elastic potential field, and the variability in sensing and actuation strategies.Comment: 69 pages, 13 figure

Energy Shaping Control of a CyberOctopus Soft Arm

This paper entails application of the energy shaping methodology to control a flexible, elastic Cosserat rod model. Recent interest in such continuum models stems from applications in soft robotics, and from the growing recognition of the role of mechanics and embodiment in biological control strategies: octopuses are often regarded as iconic examples of this interplay. Here, the dynamics of the Cosserat rod, modeling a single octopus arm, are treated as a Hamiltonian system and the internal muscle actuators are modeled as distributed forces and couples. The proposed energy shaping control design procedure involves two steps: (1) a potential energy is designed such that its minimizer is the desired equilibrium configuration; (2) an energy shaping control law is implemented to reach the desired equilibrium. By interpreting the controlled Hamiltonian as a Lyapunov function, asymptotic stability of the equilibrium configuration is deduced. The energy shaping control law is shown to require only the deformations of the equilibrium configuration. A forward-backward algorithm is proposed to compute these deformations in an online iterative manner. The overall control design methodology is implemented and demonstrated in a dynamic simulation environment. Results of several bio-inspired numerical experiments involving the control of octopus arms are reported

A numerical study of elastica using constrained optimization methods

Master'sMASTER OF ENGINEERIN