Estimating Ground Reaction Forces (GRF) Solely from Kinematic Pose Data and Dynamic Contact Points (Technischer Leitfaden und Architektur)

### Technical Architecture: Estimating Ground Reaction Forces (GRF) Solely from Kinematic Pose Data and Dynamic Contact Points Real-time spatial computing, 3D anatomical skeletal tracking, and biomechanical human-machine interfaces require low-latency processing, robust sensor fusion, and mathematically constrained kinematics. Within FitRecognition, this subsystem resolves the fundamental trade-offs between tracking frame rate, keypoint jitter, and multi-user occlusion. #### 1. Sensor Ingestion & Mathematical Formulation High-precision motion capture relies on accurate coordinate transformation from 2D pixel space $(u, v)$ to metric 3D camera coordinates $(X, Y, Z)$. The projection geometry is formalized through camera intrinsic calibration matrices: $\begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \frac{1}{Z} \begin{bmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} X \\ Y \\ Z \end{bmatrix}$ Where: - $f_x, f_y$ represent the focal lengths along orthogonal sensor axes. - $c_x, c_y$ denote the optical principal point coordinates on the sensor plane. - $Z$ represents metric depth derived from stereoscopic disparity or time-of-flight phase shifts. #### 2. Deep Neural Landmark Regression Keypoint estimation utilizes anchor-free fully convolutional neural networks trained on millions of multi-view skeletal images. Feature maps extracted via spatial feature pyramids are processed through depthwise separable deconvolution heads, generating volumetric heatmaps: $H_k(x, y, z) = \exp\left(-\frac{(x - x_k)^2 + (y - y_k)^2 + (z - z_k)^2}{2\sigma^2}\right)$ Sub-pixel coordinates $(\hat{x}_k, \hat{y}_k, \hat{z}_k)$ are recovered via soft-argmax operations, ensuring end-to-end differentiability and sub-millimeter anatomical precision. #### 3. Kinematic Constraint Solving & Jitter Filtering Raw neural predictions exhibit high-frequency jitter caused by lighting variance and sensor noise. FitRecognition implements adaptive dual-stage Kalman filtering, dynamically scaling process noise covariance based on Euclidean joint velocity: - At low velocities (quasistatic gestures), aggressive filtering dampens high-frequency jitter to under 0.2 mm. - At high velocities (rapid flick gestures), process noise bounds expand, eliminating phase lag and maintaining responsive 120 FPS interaction.