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Abstract This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This trade-off between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPU-based numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation. 

Abstract This paper presents a design methodology for mechanisms consisting of a single continuous structure, continuum mechanisms, that blends the kinematic synthesis of rigid-body mechanisms with topology optimization for compliant mechanisms. Rather than start with a generic structure that is shaped to achieve a required force deflection task for a compliant mechanism, our approach shapes the initial structure based on kinematic synthesis of a rigid body mechanism for the required movement, then the structure is shaped using Finite Element Analysis to achieve the required force deflection relationship. The result of this approach is a continuum mechanism with the same workpiece movement as the rigid link mechanism when actuated. An example illustrates the design process to obtain an eight-bar linkage that guides its workpiece in straight-line rectilinear movement. We show that the resulting continuum mechanism provides the desired rectilinear movement. A 210 mm physical model machined from Nylon-6 is shown to achieve 21.5mm rectilinear movement with no perceived deviation from a straight-line. 

Abstract This paper presents a new two-step design procedure and preliminary kinematic evaluation of a novel, passive, six-bar knee-ankle-foot orthosis (KAFO). The kinematic design and preliminary kinematic gait analysis of the KAFO are based on motion capture data from a single healthy male subject. Preliminary kinematic evaluation shows that the designed passive KAFO is capable of supporting flexion and extension of the knee joint during stance and swing phases of walking. The two-step design procedure for the KAFO consists of (1) computational synthesis based on user's motion data and (2) performance optimization. In the computational synthesis step, first the lower leg (knee-ankle-foot) of the subject is approximated as a 2R kinematic chain and its target trajectories are specified from motion capture data. Six-bar linkages are synthesized to coordinate the angular movements of knee and ankle joints of the 2R chain at 11 accuracy points. The first step of the design procedure yields 332 six-bar KAFO design candidates. This is followed by a performance optimization step in which the KAFO design candidates are optimally modified to satisfy specified constraints on end-effector trajectory and shape. This two-step process yields an optimally designed passive six-bar KAFO that shows promising kinematic results at the knee joint of the user during walking. The preliminary prototype manufactured is cost effective, easy to operate, and suitably demonstrates the feasibility of the proposed concept. 

This paper examines the results of synthesis algorithms for four-, six-, and eight-bar linkages for rectilinear movement. Rectilinear movement is useful for applications such as suspensions that provide linear movement with out a rotation component. The algorithm yields one four-bar, seven six-bar, and 32 eightbar linkages. The synthesis strategy begins with a task guided by a multi-degree of freedom chain. The algorithm computes constraints to guide the required movement with one degree-offreedom. Each computed design is analyzed to ensure smooth movement through the specified set of task positions. Finally, we identify the design that has the least variation from a pure rectilinear movement. 

This paper presents a design procedure to achieve a flapping wing mechanism for a micro air vehicle that drives both the swing and pitch movement of the wing with one actuator. The mechanism combines a planar four bar linkage with a spatial RSSR attached to the input and output links forming a spatial Stephenson six-bar linkage. Function generation synthesis yields a planar four-bar that controls the wing swing profile. The pitch control is synthesized by inverting the movement of the combined system to isolate and compute the SS chain. In order to ensure the design achieves the specified task precision points, the SS chain was randomized within a prescribed tolerance zone. The result was 29 designs, one of which is presented in detail. 

This paper presents a design and manufacturing methodology for a mechanical system that draws trigonometric curves assembled from a series of links connected by gears. We demonstrate this technique using 11 gear-coupled links that trace a butterfly curve. The equation of a butterfly curve is converted to the relative rotations of the links of a coupled serial chain assembled so it operates with one input. We present a procedure to determine the adjustments to the gear ratios and link dimensions necessary for practical manufacture of the mechanism. The results are demonstrated by a working prototype. 

In this paper, we study the manifold of configurations of a 3SPS-1S spherical parallel manipulator. This manifold is obtained as the intersection of quadrics in the hypersphere defined by quaternion coordinates and is called its constraint manifold. We then formulate Jacobian for this manipulator and consider its singular. This is a quartic algebraic manifold called the singularity variety of the parallel manipulator. A survey of the architectures that can be defined for the 3SPS-1S spherical parallel manipulators yield a number of special cases, in particular the architectures with coincident base or moving pivots yields singularity varieties that factor into two quadric surfaces. 

This paper describes a mechanism design methodology that assembles standard components to trace plane curves that have a Fourier series parameterization. This approach can be used to approximate complex plane curves to interpolate image boundaries constructed from points. We describe three ways to construct a mechanism that generates a curve from a Fourier series parameterization. One uses Scotch yoke linkages for each term of Fourier series which are added using a belt drive. The second approach uses a coupled serial chain for each coordinate Fourier parameterization. The third method uses one constrained coupled serial chain to trace a specified plane curve. This work can be viewed as a version of the Kempe Universality Theorem that states that a linkage exists that can trace any plane algebraic curve. In our case, we include belts and pulleys, and obtain linkages that trace curves that have Fourier parameterizations.