A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

A Constraint-Based Approach to Manipulator Kinematics and Singularities

<p>In this thesis, a differential-geometric approach to the kinematics of multibody mechanisms is introduced that enables analysis of singularities of both serial and parallel manipulators in a flexible and complete way. Existing approaches such as those of Gosselin and Angeles [1], Zlatanov et al. [2] and Park and Kim [3] make use of a combination of joint freedoms and constraints and so build in assumptions. In contrast, this new approach is solely constraint-based, avoiding some of the shortcomings of these earlier theories. The proposed representation has two core ingredients. First, it avoids direct reference to the choice of inputs and their associated joint freedoms and instead focuses on a kinematic constraint map (KCM), defined by the constraints imposed by all joints and not requiring consideration of closure conditions arising from closed loops in the design. The KCM is expressed in terms of pose (i.e. position and orientation) variables, which are the coordinates of all the manipulator’s links with respect to a reference frame. The kinematics of a given manipulator can be described by means of this representation, locally and globally. Also, for a family of manipulators defined by a specific architecture, the KCM will tell us how the choice of design parameters (e.g. link lengths) affects these kinematic properties within the family. At a global level, the KCM determines a subset in the space of all pose variables, known as the configuration space (C-space) of the manipulator, whose topology may vary across the set of design parameters. The Jacobian (matrix of first-order partial derivatives) of the KCM may become singular at some specific choices of pose variables. These conditions express a subset called the singular set of the C-space. It is shown that if a family of manipulators, parametrised by a manifold Rd of design parameters, is “well-behaved” then the pose variables can be eliminated from the KCM equations together with the conditions for singularities, to give conditions in terms of design parameters, that define a hypersurface in Rd of manipulators in the class that exhibit C-space singularities. These are referred to as Grashof-type conditions, as they generalise classically known inequalities classifying planar 4-bar mechanisms due to Grashof [4]. Secondly, we develop the theory to incorporate actuator space (A-space) and workspace (W-space), based on a choice of actuated joints or inputs and on the manipulator’s end-effector workspace or outputs. This will facilitate us with a framework for analysing singularities for forward and inverse kinematics via input and output mappings defined on the manipulator’s C-space. This provides new insight into the structure of the forward and inverse kinematics, especially for parallel manipulators. The theory is illustrated by a number of applications, some of which recapitulate classical or known results and some of which are new.</p>

Indicative conditionals: probabilities and relevance

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional$$\varphi \rightarrow \psi$$φ→ψto the extent that (i) the conditional probability$$\mathrm{p}(\psi |\varphi )$$p(ψ|φ)is high, provided that (ii)$$\varphi$$φis relevant for$$\psi$$ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals.

News on Baer–Nunziato-type model at pressure equilibrium

A six-equation Baer–Nunziato model at pressure equilibrium for two ideal gases is derived from a full non-equilibrium model by applying an asymptotic pressure expansion. Conditions on the interfacial pressure are provided that ensure hyperbolicity of the reduced model. Closure conditions for the relaxation terms are given that ensure consistency of the model with the second law of thermodynamics.

Computational Fluid Dynamic Analysis of Different Velopharyngeal Closure Patterns

Objective: Velopharyngeal (VP) closure has high impact on the quality of life, especially in patients with cleft palate. For better understanding the VP closure, it is important to understand the airflow dynamics of different closure patterns, including circular, coronal, sagittal, and circular with a Passavant’s ridge. The purpose of this study was to demonstrate the airflow characteristics of different velopharyngeal closure patterns. Methods: Sixteen adults with no notable upper airway abnormality who needed multislice spiral computed tomography scans as part of their clinical care. Airways were reconstructed. A cylinder and a cuboid were used to replace the VP port in three models of VP port patterns. Flow simulations were carried using computational fluid dynamics. Airflow pressures in the VP orifice, oral cavity and nasal cavity, as well as airflow velocity through the velopharyngeal orifice, were calculated. Results: The airflow dynamics at the velopharynx were different among different velopharyngeal patterns as the area of the velopharyngeal port increased from 0 to 25 mm2. The orifice areas of different closure conditions in four velopharyngeal closure patterns were significantly different. The maximal orifice area for adequate velopharyngeal closure was 7.57 mm2 in the coronal pattern and 6.21 mm2 in the sagittal pattern. Conclusions: Airflow dynamics of the velopharynx were correlated to the velopharyngeal closure patterns. Different closure patterns had different largest permitted orifice areas for getting the appropriate oral pressures for normal speech.

From Single 2D Depth Image to Gripper 6D Pose Estimation: A Fast and Robust Algorithm for Grabbing Objects in Cluttered Scenes

In this paper, we investigate the problem of grasping previously unseen objects in unstructured environments which are cluttered with multiple objects. Object geometry, reachability, and force-closure analysis are considered to address this problem. A framework is proposed for grasping unknown objects by localizing contact regions on the contours formed by a set of depth edges generated from a single-view 2D depth image. Specifically, contact regions are determined based on edge geometric features derived from analysis of the depth map data. Finally, the performance of the approach is successfully validated by applying it to scenes with both single and multiple objects, in both simulation and experiments. Using sequential processing in MATLAB running on a 4th-generation Intel Core Desktop, simulation results with the benchmark Object Segmentation Database show that the algorithm takes 281 ms on average to generate the 6D robot pose needed to attach with a pair of viable grasping edges that satisfy reachability and force-closure conditions. Experimental results in the Assistive Robotics Laboratory at UCF using a Kinect One sensor and a Baxter manipulator outfitted with a standard parallel gripper showcase the feasibility of the approach in grasping previously unseen objects from uncontrived multi-object settings.

Leading-Edge Vortices: Mechanics and Modeling

The leading-edge vortex (LEV) is known to produce transient high lift in a wide variety of circumstances. The underlying physics of LEV formation, growth, and shedding are explored for a set of canonical wing motions including wing translation, rotation, and pitching. A review of the literature reveals that, while there are many similarities in the LEV physics of these motions, the resulting force histories can be dramatically different. In two-dimensional motions (translation and pitch), the LEV sheds soon after its formation; lift drops as the LEV moves away from the wing. Wing rotation, in contrast, incites a spanwise flow that, through Coriolis tilting, balances the streamwise vorticity fluxes to produce an LEV that remains attached to much of the wing and thus sustains high lift. The state of the art of vortex-based modeling to capture both the flow field and corresponding forces of these motions is reviewed, including closure conditions at the leading edge and approaches for data-driven strategies.

Closure conditions for a one temperature non-equilibrium multi-component model of baer-nunziato type

A class of non-equilibrium models for compressible multi-component uids is investigated. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production and thermodynamical stability as well as mathematical properties such as hyperbolicity. For the relaxation of chemical potentials a three-component model gas-water-vapor is considered.

Inductive definitions and proofs

An inductive definition of a predicate R characterizes the Rs as the smallest class which satisfies a basis clause of the form (β(x)→Rx), telling us that certain things satisfy R, together with one or more closure clauses of the form (Φ(x,R)→Rx), which tell us that, if certain other things satisfy R, x satisfies R as well. ’Smallest’ here means that the class of Rs is included in every other class which satisfies the basis and closure clauses. Inductive definitions are useful because of inductive proofs. To show that every R has property P, show that the class of Rs that have P satisfies the basis and closure clauses. The closure clauses tell us that if certain things satisfy R, x satisfies R as well. Thus satisfaction of the condition Φ(x,R) should be ensured by positive information to the effect that certain things satisfy R, and not also require negative information that certain things fail to satisfy R. In other words, the condition Φ(x,R) should be monotone, so that, if R ⊆S and Φ(x,R), then Φ(x,S); otherwise, we would have no assurance of the existence of a smallest class satisfying the basis and closure conditions. While inductive definitions can take many forms, they have been studied most usefully in the special case in which the basis and closure clauses are formulated within the predicate calculus. Initiated by Yiannis Moschovakis, the study of such definitions has yielded an especially rich and elegant theory.