A recently derived local preconditioning of the Euler equations is shown to be useful in developing multistage schemes suited for multigrid use. The effect of the preconditioning matrix on the spatial Euler operator is to equalize the characteristic speeds. When applied to the discretized Euler equations, the preconditioning has the effect of strongly clustering the operator's eigenvalues in the complex plane. This makes possible the development of explicit marching schemes that effectively damp most high-frequency Fourier modes, as desired in multigrid applications. The technique is the same as developed earlier for scalar convection schemes: placement of the zeros of the amplification factor of the multistage scheme in locations where eigenvalues corresponding to high-frequency modes abound.

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This work extends the concepts and methodologies of Shen et al (1) and Al-Smadi et al (2) to synthesis of spatial RRSS motion generators with applied coupler loads. A general four-bar mechanism constraint that includes a coupler static load and a driver static torques is formulated using the principal of virtual work. This constraint is combined with the conventional RRSS motion generation model by Suh and Radcliffe (13) to form a nonlinear optimization problem from which RRSS mechanism solutions are calculated that approximate prescribed coupler poses and satisfy prescribed driver static torques for given coupler loads.

While the concepts of RSSR-SS motion generation, order defect elimination and branch defect elimination have all been well-researched over the years (individually and in combination), a major challenge exists when considering a general RSSR-SS optimization model for motion generation in the mathematical analysis software Matlab. To overcome this challenge, this work presents a small-scale nonlinear equation system for RSSR-SS motion generation. The equation system also includes inequality constraints to eliminate order defects and branch defects.

Background: A transfemoral prosthetic knee is an artificial knee used by above-the-knee amputees. There are two major categories of transfemoral prosthetic knee designs: pin joint-based and polycentric designs. While pin joint-based knee designs only allow pure rotation of the knee, polycentric knee designs allow a combination of rotational and translational knee motion which is exhibited in natural knee motion. Objective: This work presents both the recently-patented design process and the resulting design of a polycentric transfemoral prosthetic knee that approximates natural spatial human knee motion during flexion and extension. Methods: The design process includes tibial motion acquisition, Revolute-Revolute-Spherical-Spherical linkage (or RRSS) motion generation, RRSS linkage axode generation and circle fitting. The polycentric transfemoral prosthetic knee design produced from this process includes a gear joint with a specific spatial orientation to approximate natural spatial human knee motion. Results: Using the design process, a polycentric transfemoral prosthetic knee was designed to replicate a group of five tibial positions over 37.5° of knee flexion (the amount of knee flexion in a standard human gait cycle) with a minimal structural error. Conclusion: The circular gear-based knee design accurately replicated natural spatial knee motion over the tibial position data given for a standard human gait cycle. The knee design method must be implemented over a broader sampling of tibial position data to determine if a circular gear-based knee design is consistently accurate.