Hybrid LSTM-UKF Framework for Gait Analysis

• The paper demonstrates significant error reduction by combining a two-layer LSTM and UKF to estimate ankle angle and ground reaction force with lower RMSE.
• It explains a multimodal sensor fusion approach that leverages force-plate, IMU, and knee angle inputs to generate context-aware embeddings via sliding window analysis.
• Comprehensive validation on subject-specific gait data shows increased R² values and robust performance, supporting applications in prosthetics and exoskeleton control.

The Hybrid LSTM-UKF framework is a computational architecture designed for accurate estimation of ankle angle and ground reaction force (GRF) in human gait analysis under varying walking speeds. It integrates multimodal biomechanical sensor inputs through @@@@1@@@@ (LSTM) deep networks and recursive uncertainty estimation using the unscented Kalman filter (UKF). The framework achieves subject-specific, robust predictions with lower error compared to standalone LSTM or UKF models, supporting applications in biomechanics, prosthetics, and exoskeleton control (Narasimhappa et al., 10 Jan 2026).

1. Framework Structure and Mathematical Foundations

The hybrid architecture couples a two-layer LSTM—processing temporal sensor-fusion features—with a UKF that recursively estimates the state of biomechanical variables. The LSTM receives sliding windows of multimodal input signals, including force-plate-derived GRF, shear forces, vertical acceleration, and joint angles, and outputs a context-aware embedding. This embedding is employed as either a pseudo-measurement or an augmentation in the UKF update step.

LSTM Component:

• Input: $x_t \in \mathbb{R}^d$ ($d$ varies according to fused sensor signals).
• Network: Two stacked LSTM layers, each with 50 hidden units, interleaved with a 0.2 dropout regularization.
• Final Layer: Dense (linear output for regression of continuous variables $\theta$ and/or $F_z$).
• Gate equations:
  • $f_t = \sigma(W_f [h_{t-1}, x_t] + b_f)$
  • $i_t = \sigma(W_i [h_{t-1}, x_t] + b_i)$
  • $\tilde{C}_t = \tanh(W_C [h_{t-1}, x_t] + b_C)$
  • $C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t$
  • $o_t = \sigma(W_o [h_{t-1}, x_t] + b_o)$
  • $h_t = o_t \odot \tanh(C_t)$

Loss is computed as mean squared error (MSE), with optional uncertainty regularization:

$$L = \mathrm{MSE}(\hat{y}_t, y_t) + \lambda \, \mathrm{KL}(N_t\, \| \, N_\mathrm{prior})$$

UKF Component:

• Discrete-time state-space modeling:

$$x_t = f(x_{t-1}, u_{t-1}) + w_{t-1},\quad w \sim \mathcal{N}(0, Q)$$

$y_t = h(x_t) + v_t,\quad v \sim \mathcal{N}(0, R)$

• State vector: $x_t = [\theta_t; \dot{\theta}_t; F_{z,t}]$
• Sigma-point propagation and weighting via:
  • $\lambda = \alpha^2 (n + \kappa) - n$
  • $W_0^m = \lambda/(n + \lambda)$; $$W_0^c = \lambda/(n + \lambda) + (1 - \alpha^2 + \beta)$$; $W_i^m = W_i^c = \frac{1}{2(n+\lambda)}$
• Nonlinear mapping $f(\cdot)$ arises from ankle biomechanics discretized with precomputed $A$, $B$, $C$, $D$ matrices.

Pseudo-measurements or uncertainty-adaptive noise covariances ($R_t$) can be introduced from the LSTM.

2. Sensor Fusion Methodology

Multimodal sensor integration in the framework improves biomechanical context and prediction reliability. Input features include:

• Force-plate signals ($F_z$, $F_x$)
• IMU-derived vertical acceleration ($\ddot{z}$)
• Measured knee joint angle ($\phi_\mathrm{knee}$)

Inputs are structured as $$x_t = [F_\text{plate channels}; F_z; \phi_\mathrm{knee}]$$. The LSTM processes sliding windows of these features, outputting an embedding $z_t$ representing the fused temporal context. This $z_t$ is used directly in the UKF update, as a pseudo-measurement or supplement to missing data. Measurement noise covariance $R_t$ can be adaptively tuned based on the uncertainty output of the LSTM, enhancing robustness.

3. Model Training and Validation Protocols

Dataset and Preprocessing:

• 13 healthy subjects, speeds 1/2/3 km/h, 7 trials each.
• Each trial time-normalized to 1,001 points.
• Preprocessing: Force-plate data rectified, 4th-order Butterworth low-pass filtered (cutoff 20Hz); GRF and knee angle signals normalized to %MVC; signals row-aligned and resampled per gait cycle.

Training Details:

• LSTM: Adam optimizer (learning rate $1\times 10^{-3}$), early stopping with patience=10 epochs.
• Split: Subject-specific 80/20 train/test; four sensor-fusion scenarios:
  1. [Force-plate & GRF] 2. [Force-plate & Knee Angle]
  2. [GRF & Knee Angle] 4. [All three]

Performance Metrics:

• RMSE: $$\mathrm{RMSE} = \sqrt{ \frac{1}{T} \sum_{i=1}^T (\hat{y}_i - y_i)^2 }$$

• $$R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}$$

4. Comparative Performance Outcomes

Systematic benchmarking demonstrated the hybrid LSTM-UKF architecture outperforms both classic filters and deep models alone in ankle angle and GRF estimation.

GRF Estimation RMSE (%):

Model	Walking	Sitting	Running
KF	10.2	12.5	13.8
EKF	9.6	11.8	12.9
UKF	8.9	10.7	11.5
LSTM	7.2	9.5	10.1
LSTM+UKF	6.8	8.3	9.4

Ankle Angle Estimation RMSE (%):

Model	Walking	Sitting	Running
KF	9.8	11.2	12.6
EKF	9.1	10.5	11.7
UKF	8.5	9.6	10.9
LSTM	6.8	8.3	9.4
LSTM+UKF	6.2	7.5	8.7

• At 3 km/h, LSTM+UKF reduced GRF RMSE by 18.6% (11.5 → 9.4) compared to UKF alone.
• At 1 km/h, ankle angle RMSE dropped by 22.4% (8.5 → 6.2) versus UKF alone.
• $R^2$ values rose from approximately 0.88 (UKF) to 0.94 (LSTM+UKF).
• Reductions surpassed sensor noise and inter-subject variability thresholds.

5. Robustness, Limitations, and Generalization

UKF integration systematically corrects LSTM drift in the presence of sensor noise via recursive covariance adaptation ($Q_t$, $R_t$), with physical constraints (e.g., biological bounds $\theta$ and $F_z\geq0$) enforced at inference.

Noted limitations:

• Single-DOF ankle model; extension to multi-DOF models (knees, hips) requires higher-dimensional state spaces and UKF parameterization.
• IMU-only acceleration input exhibited excessive noise; effective fusion with force-plate or muscle EMG was critical for reliable state estimation.

Potential extensions:

• Hybridization for multi-joint kinematics/kinetics (e.g., knees, hips), high-dimensional GRF, or prosthetic/exoskeleton adaptive control.
• Incorporation of alternative deep priors (attention, GRU variants) or adaptive, learnable noise models.
• Investigation of quantum-LSTM or Transformer encoders for longer temporal dependency modeling.

6. Implementation Protocols

Implementation follows established configurations:

• LSTM: Two layers, 50 units each, dropout=0.2, Adam optimizer (lr=1e-3), implemented in TensorFlow/Keras.
• UKF: Initial parameters $\alpha=10^{-3}$, $\beta=2$, $\kappa=0$; covariance $P_0$ set from prior gait variances; process and measurement noise ($Q, R$) defined via sensor specifications.
• Precompute all continuous-time ankle-dynamics matrices ($A$, $B$, $C$, $D$) and discretize for recursive filtering.
• At each time step: assemble $x_t$, process $x_{t-k:t}$ via LSTM to obtain $z_t$, propagate UKF sigma points, use $z_t$ in measurement update, enforce anatomical constraints.
• Evaluate using RMSE and $R^2$; optionally tune $\lambda$ for hybrid loss if uncertainty quantification is included.

By adhering strictly to these methods, new gait analysis systems can replicate the observed accuracy, robustness, and adaptability across varying subjects and walking conditions (Narasimhappa et al., 10 Jan 2026).