This study presents a set of algorithms that deal with trajectory planning of rational single-loop mechanisms with one degree-of-freedom (DoF). Benefiting from a dual quaternion representation of a rational motion, a formula for direct (forward) kinematics, a numerical inverse kinematics algorithm, and the generation of a driving-joint trajectory are provided. A novel approach using the Gauss-Newton search for the one-parameter inverse kinematics problem is presented. Additionally, a method for performing smooth equidistant travel of the tool is provided by applying arc-length reparameterization. This general approach can be applied to one-DoF mechanisms with four to seven joints characterized by a rational motion, without any additional geometrical analysis. An experiment was performed to demonstrate the usage in a laboratory setup.

Echocardiography video is a primary modality for diagnosing heart diseases, but the limited data poses challenges for both clinical teaching and machine learning training. Recently, video generative models have emerged as a promising strategy to alleviate this issue. However, previous methods often relied on holistic conditions during generation, hindering the flexible movement control over specific cardiac structures. In this context, we propose an explainable and controllable method for echocardiography video generation, taking an initial frame and a motion curve as guidance. Our contributions are three-fold. First, we extract motion information from each heart substructure to construct motion curves, enabling the diffusion model to synthesize customized echocardiography videos by modifying these curves. Second, we propose the structure-to-motion alignment module, which can map semantic features onto motion curves across cardiac structures. Third, The position-aware attention mechanism is designed to enhance video consistency utilizing Gaussian masks with structural position information. Extensive experiments on three echocardiography datasets show that our method outperforms others regarding fidelity and consistency. The full code will be released at https://github.com/mlmi-2024-72/ECM.