Appendix A: Proofs Associated with Chapter 3

Journal of Intelligent and Robotic Systems, 2011

We consider a specific type of singularities for kinematic chains, so-called point singularities. These were characterized in 2005 by Borcea and Streinu. We give a new proof for this result in the framework of the Exterior Algebra. As an illustration we give an exhaustive list of the point singularities of a specific robot manipulator.

2005

Résumé: Synthesis of robots may be decomposed into two processes:{\ em structural synthesis}(determine the general arrangement of the mechanical structure such as the type and number of joints and the way they will be connected) and {\ em dimensional synthesis}(determine the length of the links, the axis and location of the joints, the necessary maximal joint forces/torques, $\ ldots $). The performances that may be obtained for a robot are drastically dependent on both synthesis.

Geometriae Dedicata, 2009

In this paper we derive stratifications of the Euclidean motion group, which provide a complete description of the singular locus in the configuration space of a family of parallel manipulators, and we study the adjacency between the strata. We prove that classically known cell decompositions of the flag manifold restricted to the open subset parameterizing the affine real flags are still stratifications, and we introduce a refinement of the classical Ehresmann-Bruhat order that characterizes the adjacency between all the different strata. Then we show how, via a four-fold covering morphism, the stratifications of the Euclidean motion group are induced.

International Journal of Architectural Engineering Technology, 2021

Given five pairs of attachment points of a planar platform, there exists a sixth point pair so that the resulting planar architecturally singular platform has the same solution for the direct kinematics. This is a consequence of the Prix Vaillant problem posed in 1904 by the French Academy of Science. The theorem discusses the displacements of certain or all points of a rigid body that move on spherical paths. Borel and Bricard awarded the prizes for two papers in this regard, but they did not solve the problem completely. In this paper, the theorem is extended to the elliptic paths in order to determine the displacements of certain or all points of a rigid body that move on super-ellipsoid surfaces. The poof is based on the trajectories of moving points which are intersections of two implicit super-ellipsoid surfaces.

Neuroscience, 1988

Biological systems are hypothesized to control behavior with reference. to invariants, because this would allow the variable but robust accomplishment of tasks observed in biological behaviors. Invariants for legged locomotion are specified. Combined with observed properties of locomotion, they lead to predictions of forms of control for legged locomotion.

2000

Discrete Applied Mathematics, 1984

Applied Mathematics and Computation, 1995

The purpose of this study is to simulate the motion of the lower extremity of a human being, a biped mechanism, walking along a straight path and to suggest a control strategy for minimizing the deviation from the linear path. A "gait" function is defined as a control that ensures that the biped walks along a straight path. By varying some parameters associated with the "gait" functions, which is chosen in such a manner as to simulate the motion of one member of the biped relative to an adjoining member, the most suitable combinations of such parameters for the specified geometry is subsequently determined. The study contributes to a better understanding in the design of robots, humanoids, and other artificial intelligence (A.I.) systems.

1994

A unit-modular robot is a robot that is composed of modules that are all identical. In this thesis we study the design and control of unit-modular dynamically recon gurable robots. This is based upon the design and construction of a robot called Polypod. We further choose statically stable locomotion as the task domain to evaluate the design and control strategy. The result is the creation of a number of unique locomotion modes. I w ould also like to thank the other members of my reading committee Professors Mark Cutkosky and Oussama Khatib, and the members of my defense committee Professors Bernard Roth and Greg Kovacs. Professor Cutkosky was also my program advisor and gave insightful suggestions throughout my time at Stanford. I also thank Professor Khatib for his unwavering advocacy. Many thanks must also go to Professor Ed Carryer and his Smart Product Design Lab class which in large part led to the creation of Polypod and thus this thesis. Likewise the student machine shop and its caretakers were invaluable in the prototyping and construction phase of Polypod. I am also grateful to Meggy Gotuaco, Rocky Kahn and Dan Arquilevich for their help in the design and construction of Polypod.

Abstract Rapid development of humanoid robots brings about new shifts of the boundaries of Robotics as a scientific and technological discipline. In relation to this, the work raises some new fundamental questions concerning the necessary anthropomorphism of humanoid robots, and how to achieve sufficiently high degree of anthropomorphism with a reasonable number of degrees of freedom. The paper describes a study of the role of hands and twolink trunk in the synthesis of anthropomorphic gait is investigated.

2011

This thesis presents a hierarchical geometric control approach for fast and energetically efficient bipedal dynamic walking in three-dimensional (3-D) space to enable motion planning applications that have previously been limited to inefficient quasi-static walkers. In order to produce exponentially stable hybrid limit cycles, we exploit system energetics, symmetry, and passivity through the energy-shaping method of controlled geometric reduction. This decouples a subsystem corresponding to a lower-dimensional robot through a passivity-based feedback transformation of the system Lagrangian into a special form of controlled Lagrangian with broken symmetry, which corresponds to an equivalent closed-loop Hamiltonian system with upper-triangular form. The first control term reduces to mechanically-realizable passive feedback that establishes a functional momentum conservation law that controls the "divided" cyclic variables to set-points or periodic orbits. We then prove extensive symmetries in the class of open kinematic chains to present the multistage application of controlled reduction. A reduction-based control law is derived to construct straightahead and turning gaits for a 4-DOF and 5-DOF hipped biped in 3-D space, based on the existence of stable hybrid limit cycles in the sagittal plane-of-motion. Given such a set of asymptotically stable gait primitives, a dynamic walker can be controlled as a discrete-time switched system that sequentially composes gait primitives from step to step. We derive "funneling" rules by which a walking path that is a sequence of these gaits may be stably followed by the robot. The primitive set generates a tree exploring the action space for feasible walking paths, where each primitive corresponds to walking along a nominal arc of constant curvature. Therefore, dynamically stable motion planning for dynamic walkers reduces to a discrete search problem, which we demonstrate for 3-D compass-gait bipeds. After reflecting on several connections to human biomechanics, we propose extensions of this energy-shaping control paradigm to robot-assisted locomotor rehabilitation. This work aims to offer a systematic design methodology for assistive control strategies that are amenable to sequential composition for novel progressive training therapies.

2007

IEEE Transactions on Robotics, 2016

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group SE(3) admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of SE(3) can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of SE(3), symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of SE(3); 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.

Chinese Journal of Mechanical Engineering

1999

This chapter is a reproduction of a paper by Mariano Garcia, Anindya Chatterjee, Andy Ruina, and Michael Coleman entitled \The Simplest Walking Model: Stability, Complexity, and Scaling. It was published in the ASME Journal of Biomechanical Engineering Vol. 120, April 1998, pp. 281 { 288. Some additional gures and text have been added in Section 3.7.2. Sentences which refer to these gures, as well as this paragraph, are shown in italics to denote material which did not appear in the original text. My role in this paper was as follows: I concocted the model and its equations, and did all of the simulation and data collection, including nding gait cycles and analyzing them. The stability results suggested the possibility of period-doubling, and Anindya Chatterjee prodded me to look for it. Anindya and I also observed the scaling results and he formulated an analytic approach which we then implemented together. While implementing the approach, I realized that the higher-period solution...