Propulsive Ground Reaction Forces

• Propulsive ground reaction forces are the forward component of contact forces that generate thrust in both biological and engineered systems.
• Modern measurement techniques—including force plates, pressure insoles, and vision-based models—provide normalized data for accurate push-off dynamics analysis.
• Data-driven and physics-informed models integrate sensor fusion, deep learning, and Newtonian mechanics to enhance real-time GRF estimation for rehabilitation and robotics.

Propulsive ground reaction forces (GRFs) are the components of contact force exerted by the ground on a moving body that contribute to forward acceleration or thrust across a range of biological, engineered, and simulated systems. In human biomechanics, propulsive GRF is traditionally quantified along the anterior–posterior axis (denoted $F_x$), distinguishing forward propulsion from rearward braking. This metric is central to locomotor performance analysis, motor control, rehabilitation, computer graphics, sports science, and robotics. Modern measurement approaches leverage force plates, pressure insoles, inertial sensors, and vision-based pipelines, with both data-driven and physics-informed models enabling direct estimation in laboratory and field settings.

1. Measurement and Definition in Human Locomotion

Propulsive GRF in human gait is conventionally defined as $F_{x}(t)$, the anterior–posterior component obtained from tri-axial force plates embedded in the ground. Under standard conventions ($x$: anterior–posterior, $y$: medial–lateral, $z$: vertical), forward propulsive push is indicated by $F_x>0$, while $F_x<0$ marks the braking phase. GRF signals are typically normalized by body weight ($F^*_x(t) = F_x(t)/mg$), yielding dimensionless time series for inter-subject comparison. High-frequency GRF signals are low-pass filtered (e.g., 20 Hz Butterworth, zero-lag) to eliminate noise while retaining stance and push-off dynamics (as in the GroundLink protocol) (Han et al., 2023).

Peak propulsive GRF and its integral over stance ($$I_{\rm prop} = \int_{t_{\rm start}}^{t_{\rm end}} F_x(t)\,dt$$) are used to quantify push-off capacity, muscular output, and kinetic impulse. The center of pressure (CoP), computed $CoP_x(t) = -M_y(t)/F_z(t)$ (moment about the mediolateral axis, vertical force), tracks load transfer from heel to toe; maximal $CoP_x$ excursion coincides with $F_{\rm prop}$ peaks, revealing the timing and locus of propulsive effort (Han et al., 2023).

2. Data-Driven Estimation Methods

Contemporary estimation of propulsive GRFs exploits large-scale datasets indexed by full-body kinematics (e.g., GroundLink) (Han et al., 2023), wearable sensor kinetics (Johnson et al., 2019, Song et al., 2023), and advanced deep learning models. Temporal convolutional networks (GroundLinkNet: 4x 1D conv + 3x FC layers, ELU activation) regress joint-level kinematics to GRF and CoP, attaining $F_x$ RMSE $<$0.5% BW ($\sim$0.01–0.05 mg) and correlation $\rho>0.9$.

Vision-based methods (PoseFormer transformer variant) (Louis et al., 2022) regress GRFs from RGB video or pose keypoints using spatial-temporal attention and multi-task heads, with gated-MSE loss functions that emphasize sharp propulsive peaks and reduce RMSE by $\sim$20% over LSTM baselines. IMU-based approaches (SER: SVD Embedding Regression) (Song et al., 2023) leverage low-rank latent mappings between time blocks of sensor acceleration/gyro and GRFs for fast, accurate inference (rRMSE $\sim$5–7%), outperforming LSTMs when subject-specific data are available. CNN-based pipelines trained on global PCA-aligned accelerations can recover running and sidestep propulsive forces with $r(F_y)\sim0.95$–0.96, $rRMSE\sim$15–17% (Johnson et al., 2019).

Representative quantitative performance table

Method (Dataset)	RMSE (%BW)	Correlation ($\rho$)
GroundLinkNet	0.1–0.5	>0.90
SER (IMU, "everyone")	0.06	~0.90
CNN (running)	15–17	0.95–0.96
Transformer (video)	~77.95 N	—

3. Physics-Informed Modeling Approaches

Physics-informed propulsive GRF estimation applies Newtonian mechanics and feedback control. The whole-body is modeled as a point-mass governed by

$$\mathbf{F}(t) + \mathbf{G} = m\,\ddot{\mathbf{x}}(t),$$

with propulsive component $F_{\rm AP}(t)=m\,\ddot{x}(t)$. A proportional–derivative (PD) controlled law

$$\mathbf{F}_{\rm PD}(t) = \kappa_P[\mathbf{x}(t+1)-\mathbf{x}(t)] - \kappa_D\,\dot{\mathbf{x}}(t)$$

is forward-simulated via explicit Euler integration to regularize total GRF estimates from motion capture (Le et al., 2 Jul 2025). The associated loss couples plate supervision ($\lambda_1$) and physics regularization ($\lambda_2$), driving AP RMSE down to 0.12–0.13 N/kg, $r>0.87$ across varied motions. This approach enforces biomechanical consistency in learning pipelines and post-processing.

Optimization-free observers (e.g., conjugate momentum observer in multi-modal aerial–legged robots) reconstruct $\hat{\bm u}_g$ (ground wrench) via

$$\dot{\bm p} = \bm u_g + \bm B_t\,\bm u_t - \bm\beta,$$

yielding instantaneous estimates of propulsive and normal GRFs for real-time control and slip avoidance (Krishnamurthy et al., 18 Nov 2024).

4. Propulsive Forces in Non-Biological Systems and Granular Media

In bio-inspired hydrofoils and robotic locomotion, propulsive ground-reaction forces emerge from interactions between moving bodies and both solid and particulate substrates. For pitching foils operating near walls, ground effect manifests in increased added mass (scaling as $1/(D^*)^2$) and reduced circulatory thrust/power from wake–image vortex cancellation (Mivehchi et al., 2020). Scaling laws for thrust coefficient ($C_T$) and power coefficient ($C_P$) incorporate these corrections for accurate prediction and design optimization.

On granular surfaces, the resistive force model (RFM) partitions a limb into infinitesimal planar segments, each experiencing direction–orientation–depth-dependent local stresses $a_x(\theta,\phi), a_z(\theta,\phi)$ that combine across the limb surface:

$$F_x(t) = b\int_0^L a_x(\theta(s,t),\phi(s,t))|z(s,t)|ds$$

to yield the instantaneous propulsive force. RFM accurately predicts locomotor speed and average $F_x$ in both bio-inspired robots and animal models, outperforming simple penetration–drag laws (Li et al., 2019).

5. Clinical and Applied Contexts: Gait Rehabilitation and Motor Adaptation

Propulsive GRF is a critical endpoint in rehabilitation targeting deficits after stroke or injury. Push-off force (POF) during terminal stance is quantified as $$\mathrm{POF}_i = \max_{t\in[\mathrm{HS}_i,\mathrm{TO}_i]}\hat F_z(t)$$ (normalized to body weight), with intervention-induced changes tracked as $$\Delta\mathrm{POF}_i = \mathrm{POF}_i - \overline{\mathrm{POF}_{\rm BL}}$$ (Hobbs et al., 7 Dec 2025). Multimodal training—combining visual real-time GRF biofeedback with programmable treadmill compliance (e.g., 25 kN/m)—yields sustained $\Delta\mathrm{POF}$ increases and lasting neuromechanical adaptations (↑ RF/hamstring activation, joint flexion, stance duration).

These findings underscore the importance of both sensory modalities and physical substrate in motor learning. Protocol recommendations include alternating baseline and push-off blocks, compliance perturbations, and extended observation phases for retention analysis.

6. Integration, Limitations, and Current Directions

Contemporary research emphasizes the synergy between direct measurement (force plates, insoles), data-driven prediction models (deep nets, transformers, embedding regression), and physics-informed constraints. Key limitations include reliance on laboratory equipment, subject calibration, and simplifying biomechanical assumptions (single-mass models, restricted contact). However, robust estimation on wearable sensors, enhanced by personal data, and physics-informed learning strategies effectively mitigate overfitting and generalize across activities.

Future directions focus on scalable, real-time propulsive GRF estimation across unconstrained environments, improved substrate modeling (soft, granular, compliant), and transferable protocols for rehabilitation and robotics. Multimodal datasets, open-source benchmarks, and physics-based architectures will continue to anchor advances in propulsive ground reaction force quantification and application.