Kinematic Analysis of Spherical Four-Bar Linkages via the Inflection Cubic and Thales Ellipse

The subject of this paper is the formulation of a specific algorithm for the kinematic analysis of spherical four-bar linkages via the inflection spherical cubic and spherical Thales ellipse by devoting particular attention to the crossed four-bar linkage (anti-parallelogram). Moreover, both the inflection and the elliptic cones, which represent the equivalent of the Bresse cylinders of the planar case in three-dimensions, are obtained by showing the particular properties of the spherical motion in terms of the curvature of a coupler curve and both the velocity and acceleration vector fields. Of special interest are also the cases in which the three acceleration poles coincide at one unique point or in two plus one, which depends on the intersections of two spherical curves of third and second degree.

Decentralized Algorithm for Force Distribution With Applications to Cooperative Transport

We consider a system of cooperating robots transporting an object in which each individual robot is small and resource constrained and the number of robots is large precluding architectures that support a centralized coordination and control system. In nature, a swarm of ants can transport large payloads that are beyond the capability of individual organisms by using simple strategies that only require coordination with their immediate neighbors. We adopt a similar paradigm in which robots communicate with their immediate neighbors although the wireless communication links that are typical in small robots can be noisy and therefore not robust. We derive decentralized coordination algorithms that allow robots to converge to a load distribution system that is optimal in the sense of minimizing interaction forces while achieving the desired resultant wrench to allow manipulation and transport. We demonstrate the key ideas using simulations for 100 robots.

Analytic Determination of Wrench Closure Workspace of Spatial Cable Driven Parallel Mechanisms

This paper proposes an efficient way of determining analytically the Wrench Closure Workspace (WCW) of spatial redundant cable-driven parallel mechanisms (CDPM). The method builds upon the boundary surface equations obtained from the null space of the structure matrix of CDPM. The set of feasible solutions is obtained that satisfies positive tension in the cables. This method was applied to characterize the WCW of spatial CDPM which has redundancy of 1 or 2. A simulation study was carried out to validate the accuracy and efficiency of the method. Several advantages over conventional approaches for determining the WCW were identified through simulation.

Selecting Kinematic Structures of Parallel Manipulators Using Symmetry and Connectivity

This paper presents an application of symmetry and connectivity to select kinematic structures of parallel manipulators. One kinematic chain can originate several mechanisms and each mechanism can originate several parallel manipulators and, in early stages of conceptual design, it is difficult to decide what is the most promising one. Hunt [1] introduced the concept of connectivity and, since then, the connectivity has been used as an important parameter to select the most appropriated parallel manipulators to develop determined task. However, it is difficult to analyze non isomorphic parallel manipulators from the connectivity matrix. In this sense, in this paper, we apply symmetry to reduce the set of parallel manipulators to a manageable few with the desired connectivity. As a result, all promising parallel manipulators originating from a kinematic chain can be analyzed without isomorphisms.

Design of Spatial Non-Anthropomorphic Articulated Systems Based on Arm Joint Constraint Kinematic Data for Human Interactive Robotics Applications

The design of human interactive robotic systems requires additional considerations compared to conventional robotic designs to take into account human factors. In this paper, a recently developed linkage kinematic synthesis incorporating higher order motion constraints is utilized to the synthesis of a five degree of freedom serial TS linkage for human interactive applications. The T represents a universal two degrees-of-freedom shoulder, while the S defines a spherical three degrees-of-freedom wrist joint. The desired hand kinematics and its time derivatives can be obtained by a motion capture system as well as from the hand-object/environment contact geometries at two task locations. In order to determine the design parameters (i.e., locations of the base/shoulder and moving/wrist pivots, as well as the link length connecting these joints), position, velocity and acceleration constraint equations of the TS linkage are solved in the vicinity of the initial and the final reaching locations. The entire robotic joint trajectories are formulated via minimum jerk theory to closely approximate human natural hand profile with an elbow joint constraint. In this manner, the TS linkage system can be designed to guarantee to reproduce the natural human hand kinematics with the minimum amount of information about the desired hand kinematic specifications. The applicability of the proposed technique was verified by designing a TS linkage system from a captured human data, and then comparing the generated end-effector trajectory with the human hand motion trajectory, which show promising results.

Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness

The mobility of a linkage is determined by the constraints imposed on its members. The constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The instantaneous motions are determined by the constraints, rather than by the c-space geometry. Shaky linkages are prominent examples that exhibit a higher instantaneous than finite DOF even in regular configurations. Inextricably connected to the mobility are kinematic singularities that are reflected in a change of the instantaneous DOF. The local analysis of a linkage, aiming at determining the instantaneous and finite mobility in a given configuration, hence needs to consider the c-space geometry as well as the constraint system. A method for the local analysis is presented based on a higher-order local approximation of the c-space adopting the concept of the tangent cone to a variety. The latter is the best local approximation of the c-space in a general configuration. It thus allows for investigating the mobility in regular as well as singular configurations. Therewith the c-space is locally represented as an algebraic variety whose degree is the necessary approximation order. In regular configurations the tangent cone is the tangent space. The method is generally applicable and computationally simple. It allows for a classification of linkages as overconstrained and underconstrained, and to identify singularities.

Creating a Digital Homework Set for a Kinematics and Dynamics of Machinery Course

Dynamics of Machinery is a traditional engineering course; in fact kinematics was one of the earliest fields of study in engineering. The course relies on a strong combination of learning new theory and acquiring the skills to apply this theory through regular and repeated practice. This practice is commonly incorporated through homework sets, provided through a combination of book or instructor-given assignments. This homework includes testing conceptual understanding of key concepts, creating kinematic schematics, vector model construction, constructing equations, and performing mathematical analyses. The use of online, electronic tools for automating homework has been widely incorporated in K-12 education and in some college-level curriculum, but not, to the authors’ knowledge, significantly in a kinematics and dynamics of machinery class. This paper will present a framework for creating these problems, provide an overview of an entire set of problems associated with the common kinematics curriculum, and present an evaluation of this digitized coursework throughout four semesters of implementation.

Systematic Use of Instant Centers in the Dynamics Analysis of Single-DOF Planar Mechanisms

Many even complex machines employ single-dof planar mechanisms. The instantaneous kinematics of planar mechanisms can be fully understood by analyzing where the instant centers (ICs) of the relative motions among mechanism’s links are located. ICs’ positions depend only on the mechanism configuration in single-dof planar mechanisms and a number of algorithms that compute their location have been proposed in the literature. Once ICs positions are known, they can be exploited, for instance, to determine the velocity coefficients of the mechanism and the virtual work of the external forces applied to mechanism’s links. Here, these and other ICs’ properties are used to build a novel dynamic model and an algorithm that solves the dynamic problems of single-dof planar mechanisms. Then, the proposed model and algorithm are applied to a case study.

A Network of Type III Bricard Linkages

Among Bricard’s overconstrained 6R linkages, the third type has two collapsed configurations, where all joint axes are coplanar. The paper presents a one-degree-of-freedom network of such linkages. Using the two coplanar states of the constituent Bricard units, the network is able to cover a large surface with a specific outline when deployed, and can be folded compactly into a stack of much smaller planar shapes. Five geometric parameters describing each type III Bricard mechanism are introduced. Their influence on the outline of one collapsed configuration is discussed and inverse calculation to obtain the parameter values yielding a desired planar shape is performed. The network is built by linking the units, either using scissor linkage elements, if the thickness of the panels can be ignored, or with hinged parallelograms, for a thicker material. Two case studies, in which the Bricard network deploys as a rectangle and a regular hexagon, respectively, are presented, validating the analysis and design methods.