Master Equations Reference¶
Every formula from across this system, gathered in one place. None of these are meant to be solved on court — they're here so the underlying logic of each engine and drill is traceable back to a single equation when you need to check it. See Constants and KPIs for the target values these formulas are measured against.
Energy and Transfer¶
Total energy (DET)
$$E_{total} = F_{muscle} + I_{neuro} + GRF_{ground}$$
Vertical force (serve)
$$F_v = m_{player}(g + a_z)$$
Elastic energy (fascia)
$$E_{stored} = \frac{1}{2} k \Delta x^2$$
Fascial elastic force
$$F_{elastic} = k_{fascia} \cdot \Delta L$$
Synchronization and the Nervous System¶
Synergy (NKF)
$$\text{Combined force} = \sum_{k=1}^{K} A_k \cdot e^{-\frac{(t-t_k)^2}{2\sigma_{sync}^2}}$$
EMD (electromechanical delay)
$$\Delta t_{EMD} = t_{signal} + t_{Ca} + t_{co}$$
Inertia¶
Moment of inertia
$$I = \int \rho(x)\, x^2 \, dx$$
Momentum
$$p = m \times v$$
DIT (dynamic inertia)
$$DIT = m_{player} \cdot I_{swing} \cdot \Delta\omega$$
Lead tape: position $x$ from the swing axis contributes to swingweight proportional to $x^2$ — small changes near the tip matter far more than the same weight near the handle.
Internal Rotation and the Whip¶
Angular velocity
$$\omega = \frac{d\theta}{dt}$$
Angular acceleration
$$\alpha = \frac{\Delta\omega}{\Delta t}$$
Internal rotation (IR) parameters: $\omega_{IR} \approx 2{,}800°/s$, $\alpha_{IR} \approx 112{,}000°/s^2$, over a window $\Delta t_{IR} \approx 25\text{ms}$.
Wrist stiffness
$$K_{wrist} = \frac{\Delta F_{impact}}{\Delta \theta_{deformation}}$$
| Threshold | Meaning |
|---|---|
| $\geq 0.85$ | Stiff enough for VE |
| $\geq 0.95$ | Needed for volley / DL |
Angular momentum (deceleration)
$$\Delta L_{arm} = I_{racket} \cdot (\omega_{pre} - \omega_{post})$$
Stability and Footwork¶
DSI (Dynamic Stability Index)
$$DSI = e^{-\lambda \Delta\theta_{slide}}$$
$\lambda$ scales with court surface friction $\mu$ — small on clay, large on hard court.
GSM (gravitational boost)
$$S_{gravity} = 1 + \alpha \left( \frac{h_{player}}{h_{drop}} \right)$$
Dynamic braking
$$F_{brake} = \mu \times m_{player} \times g \times \cos(\theta)$$
Torque
$$\tau = F_{impact} \times L_{lever}$$
Direct Load / Return¶
Load impulse
$$J_{load} = m_{player} \times v_{drop}$$
Reaction time budget
$$t_{total} = t_{recognition} + t_{load(DL)} + t_{contact} < t_{ball\ flight}$$
Interception point
$$\text{Intercept coordinate} = (v_{ball} \times t_{flight}) - (v_{player} \times t_{run})$$
Volley¶
Reflection
$$V_{out} = V_{in} + \text{Structural stiffness}\ (K)$$
Reflection angle
$$\text{Reflection angle} = 180° - (\text{Incoming ball angle} + \text{Racket face angle})$$
Strategy¶
$$P_{error}(x,y) = \frac{N_{error}(x,y)}{N_{total}(x,y)}$$
Smart Index
$$\text{Smart Index} = \frac{\text{Scoring potential}}{P_{error_self}}$$
Bio-Recovery¶
HRV readiness
$$\text{Readiness Score} = \frac{HRV_{morning}}{HRV_{baseline}} \times \text{Sleep quality}$$
Hybrid performance
$$\text{Real-world performance} = (\text{Skill}) \times (\text{AI accuracy}) \times (\text{Adaptability})$$