This article presents a new approach for constructing convex enclosures of reachable sets of parametric ordinary differential equations (ODEs), for use in deterministic methods for global dynamic optimization. In our new approach, we modify an established ODE relaxation framework by Scott and Barton (2013), using kinematic intuition to replace certain discontinuous transitions between discrete modes with tighter, smoother transitions, and ultimately producing tighter, smoother relaxations of the original ODE solution that are more amenable to integration by off-the-shelf numerical ODE solvers. We refer to our new relaxation approach as “kinematic smoothing”. Our new ODE relaxations are straightforward to construct automatically based on established tools, and we present several numerical examples based on a proof-of-concept implementation in Julia.

Monitoring the changes in gait patterns is important to individuals’ health. Gait analysis should be taken as early as possible to prevent gait impairments and improve gait quality. Accurate stride-length estimation and gait rehabilitation activity recognition are fundamental components in gait monitoring, gait analysis, and long-term gait care. This article proposes a novel multimodality deep learning architecture to investigate the applications of stride length (SL) estimation and rehabilitation activity recognition. In order to verify this architecture, we have conducted the data collection and data labeling with our customized wearable sensing system. The sensing system can provide sensor readings from 96 sensors-based pressure array and 3-channels accelerometer and gyroscope. Many experiments with multiple perspective analysis are implemented to evaluate the models’ precision, robustness, and reliability. The multimodality deep learning architecture can map multiple sensor readings to the resulting SL with a mean absolute error of 3.89 cm and accurately detect the gait activity with an accuracy of 97.08%. It correlates the step length estimation and gait activity recognition to fulfill comprehensive long-term gait information statistic. The proposed applications’ implementation enriched our previous gait study and brought insights for clinically relevant wearable gait monitoring and gait analysis.

With the development of Internet of Things (IoT), indoor localization has been a research focus in recent years. For inertial measurement unit (IMU)-based indoor localization method, zero velocity update (ZUPT) uses the known velocity at stationary epoch as a benchmark to calibrate the velocity drift. However, stationary epoch only takes up 24% of a whole gait cycle time, and the velocity drift at the remaining 76% time is usually estimated according to an assumption that velocity has a linear drift over time, which would introduce errors. In this paper, a two-step velocity calibration method was proposed based on human gait characteristics with Smart Insole: known velocity update (KUPT) and double-foot position calibration (DFPC). KUPT could measure the velocity from heel-strike to toe-off based on the recorded real-time foot angle and the shoe dimensions, which increases the time period when the velocity could be measured from 24% to 62% of a whole gait cycle time. DFPC method could fuse the position information of both feet based on the symmetrical characteristic of human gait to further increase the reliability of the localization results. The statistical result of a 20 times 20-m walking experiment showed that KUPT method was more accurate and reliable than ZUPT method for both feet, and DFPC method could further improve the result of KUPT method. Another experiment about walking in an indoor environment for 91 m showed that the proposed KUPT+DFPC method had an error of about 0.78 m which is acceptable for most IoT applications.