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docs: align markdown tables
Apply markdown-table-formatter's layout to the tables this tier adds or edits, as suggested by MegaLinter. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Q53m7MZDmHwocmKbhhd2o5
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‎README.md‎

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## Documentation
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| Category | Description |
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|--------------------------------------------------------------------|----------------------------------------------------------------------|
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| [Analysis](doc/analysis/README.md) | FFT, Real-Input FFT (RFFT), Power Spectral Density, DCT, Discrete Wavelet Transform (Haar/Daubechies), Window Functions, Signal Detectors, Convolution & Correlation, Goertzel Algorithm, Decibels, Hilbert Transform / Analytic Signal, Mel Filterbank / MFCC |
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| [Control Analysis](doc/control_analysis/README.md) | Frequency Response, Root Locus, Controllability/Observability Matrices & Gramians, Continuous-to-Discrete, Transfer Function ↔ State Space |
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| [Controllers](doc/controllers/README.md) | Bang-Bang/Hysteresis, Deadbeat Control, PID, LQR, LQI (Integral/Servo State Feedback), MPC, Saturation, Rate Limiter, Slew-Limited Saturation, Feedforward/2-DOF, Gain-Scheduled Controller, Lead-Lag Compensator, Luenberger Observer, Friction Compensation (Coulomb + Viscous + Stribeck) |
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| [Estimators](doc/estimators/README.md) | Linear Regression, Polynomial Fitting, Total Least Squares, Yule-Walker (offline), Recursive Least Squares, Recursive Instrumental Variables, LMS / NLMS Adaptive Filter (online), Consistency Metrics / NEES / NIS |
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| Category | Description |
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|--------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
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| [Analysis](doc/analysis/README.md) | FFT, Real-Input FFT (RFFT), Power Spectral Density, DCT, Discrete Wavelet Transform (Haar/Daubechies), Window Functions, Signal Detectors, Convolution & Correlation, Goertzel Algorithm, Decibels, Hilbert Transform / Analytic Signal, Mel Filterbank / MFCC |
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| [Control Analysis](doc/control_analysis/README.md) | Frequency Response, Root Locus, Controllability/Observability Matrices & Gramians, Continuous-to-Discrete, Transfer Function ↔ State Space |
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| [Controllers](doc/controllers/README.md) | Bang-Bang/Hysteresis, Deadbeat Control, PID, LQR, LQI (Integral/Servo State Feedback), MPC, Saturation, Rate Limiter, Slew-Limited Saturation, Feedforward/2-DOF, Gain-Scheduled Controller, Lead-Lag Compensator, Luenberger Observer, Friction Compensation (Coulomb + Viscous + Stribeck) |
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| [Estimators](doc/estimators/README.md) | Linear Regression, Polynomial Fitting, Total Least Squares, Yule-Walker (offline), Recursive Least Squares, Recursive Instrumental Variables, LMS / NLMS Adaptive Filter (online), Consistency Metrics / NEES / NIS |
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| [Filters](doc/filters/README.md) | Kalman, Extended Kalman, Unscented Kalman, Square-Root Kalman, Alpha-Beta/Alpha-Beta-Gamma, FIR, IIR, Exponential Moving Average, Moving Average, Complementary, Median Filter, CIC (Cascaded Integrator-Comb), Notch/Comb Filter, Savitzky-Golay Filter, Biquad/Second-Order-Section Cascade, IIR Filter Design (Butterworth/Chebyshev-I + Bilinear Transform), Madgwick/Mahony AHRS |
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| [Optimization](doc/optimization/README.md) | Gradient Descent, SGD / Adam Step Optimisers |
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| [Regularization](doc/regularization/README.md) | L1 (Lasso), L2 (Ridge) |
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| [Math](doc/math/README.md) | CORDIC, Quaternion, MatrixNorms, Step Response Metrics, MatrixExponential, Angle Utilities, Tridiagonal (Thomas) Solver |
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| [Solvers](doc/solvers/README.md) | Gaussian Elimination, Levinson-Durbin, Durand-Kerner, Cholesky, DARE, Runge-Kutta ODE Integrators (RK4 + Dormand-Prince), Spectral Radius & Discrete Stability Margin, QR Decomposition (Householder / Givens), LU Decomposition with Partial Pivoting, Singular Value Decomposition (Golub-Kahan), Symplectic Integrators (Semi-Implicit Euler, Velocity Verlet) |
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| [Nonlinear Control](doc/nonlinear_control/README.md) | Feedback Linearization, Backstepping Control, Model Reference Adaptive Control (MRAC) |
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| [Robust Control](doc/robust_control/README.md) | Active Disturbance Rejection Control (ADRC + ESO), Sliding Mode Control (SMC), Disturbance Observer (DOB), H∞ State-Feedback Control |
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| [Performance Optimization](doc/performance-optimization/README.md) | Compiler optimizations |
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| [Optimization](doc/optimization/README.md) | Gradient Descent, SGD / Adam Step Optimisers |
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| [Regularization](doc/regularization/README.md) | L1 (Lasso), L2 (Ridge) |
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| [Math](doc/math/README.md) | CORDIC, Quaternion, MatrixNorms, Step Response Metrics, MatrixExponential, Angle Utilities, Tridiagonal (Thomas) Solver |
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| [Solvers](doc/solvers/README.md) | Gaussian Elimination, Levinson-Durbin, Durand-Kerner, Cholesky, DARE, Runge-Kutta ODE Integrators (RK4 + Dormand-Prince), Spectral Radius & Discrete Stability Margin, QR Decomposition (Householder / Givens), LU Decomposition with Partial Pivoting, Singular Value Decomposition (Golub-Kahan), Symplectic Integrators (Semi-Implicit Euler, Velocity Verlet) |
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| [Nonlinear Control](doc/nonlinear_control/README.md) | Feedback Linearization, Backstepping Control, Model Reference Adaptive Control (MRAC) |
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| [Robust Control](doc/robust_control/README.md) | Active Disturbance Rejection Control (ADRC + ESO), Sliding Mode Control (SMC), Disturbance Observer (DOB), H∞ State-Feedback Control |
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| [Performance Optimization](doc/performance-optimization/README.md) | Compiler optimizations |
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Each category page lists its algorithms with a brief description and links to the detailed documentation.
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‎doc/controllers/FrictionCompensation.md‎

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## Complexity Analysis
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| Case | Time | Space | Notes |
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|------|------|-------|-------|
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| Case | Time | Space | Notes |
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|------|--------|--------|--------------------------------------------------------------------------------------------------------------------------------------------|
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| All | $O(N)$ | $O(N)$ | $N$ axes; per axis one exponential, one power, one hyperbolic tangent, two divisions and a clamp; $N$ parameter sets stored, no allocation |
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Cost is dominated by the three transcendental calls; there is no iteration and no dependence on history, so the latency is constant per sample.
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Parameters: $F_c = 0.5$, $F_s = 0.8$, $F_v = 0.1$, $v_s = 0.05$, $\delta = 2$, $\varepsilon = 10^{-3}$, with a non-binding cap $\tau_{max} = 1$.
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| $v$ | $(\lvert v\rvert/v_s)^2$ | Level $L$ | $\tanh(v/\varepsilon)$ | $F_v v$ | $\tau$ |
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|-----|---------------|-----------|------------------------|---------|--------|
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| 1 | 400 | $0.5 + 0.3\,e^{-400} \approx 0.5$ | 1 | 0.1 | 0.6 |
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| 0.01 | 0.04 | $0.5 + 0.3\,e^{-0.04} \approx 0.788$ | $\tanh 10 \approx 1$ | 0.001 | 0.789 |
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| 0.05 ($=v_s$) | 1 | $0.5 + 0.3/e \approx 0.610$ | $\tanh 50 \approx 1$ | 0.005 | 0.615 |
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| $v$ | $(\lvert v\rvert/v_s)^2$ | Level $L$ | $\tanh(v/\varepsilon)$ | $F_v v$ | $\tau$ |
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|---------------|--------------------------|--------------------------------------|------------------------|---------|--------|
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| 1 | 400 | $0.5 + 0.3\,e^{-400} \approx 0.5$ | 1 | 0.1 | 0.6 |
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| 0.01 | 0.04 | $0.5 + 0.3\,e^{-0.04} \approx 0.788$ | $\tanh 10 \approx 1$ | 0.001 | 0.789 |
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| 0.05 ($=v_s$) | 1 | $0.5 + 0.3/e \approx 0.610$ | $\tanh 50 \approx 1$ | 0.005 | 0.615 |
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At high speed the Stribeck term has vanished and only Coulomb plus viscous drag remain. At $v = 0.01$ the axis is still deep in the Stribeck region, so the level is close to $F_s$. At $v = v_s$ the stiction excess has decayed to a fraction $1/e$ of its initial value. At $v = 0$ the output is exactly zero, and $v = -0.01$ gives $-0.789$ by symmetry.
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## Variants & Generalizations
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| Variant | Key Difference |
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|---------|----------------|
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| **Coulomb + viscous only** | $F_s = F_c$ removes the Stribeck term; the level is constant |
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| **Asymmetric friction** | Separate parameters per direction of motion; handles direction-dependent drag |
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| **Karnopp model** | Replaces the smooth layer with a velocity dead-zone and a stick state |
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| **Dahl / LuGre** | Dynamic models with internal bristle state; capture presliding and hysteresis at the cost of state and a stiffer ODE |
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| **Inverse-dynamics term** | Same static model folded into a robot inverse-dynamics computation |
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| Variant | Key Difference |
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|----------------------------|----------------------------------------------------------------------------------------------------------------------|
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| **Coulomb + viscous only** | $F_s = F_c$ removes the Stribeck term; the level is constant |
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| **Asymmetric friction** | Separate parameters per direction of motion; handles direction-dependent drag |
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| **Karnopp model** | Replaces the smooth layer with a velocity dead-zone and a stick state |
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| **Dahl / LuGre** | Dynamic models with internal bristle state; capture presliding and hysteresis at the cost of state and a stiffer ODE |
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| **Inverse-dynamics term** | Same static model folded into a robot inverse-dynamics computation |
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## Applications
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## Connections to Other Algorithms
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| Algorithm | Relationship |
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|-----------|--------------|
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| [Feedforward / 2-DOF](Feedforward2Dof.md) | Same additive-feedforward structure; friction is a velocity-driven feedforward term |
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| [PID Controller](Pid.md) | The feedback loop this term is added to; compensation reduces integrator work and stick-slip hunting |
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| [Bang-Bang / Hysteresis](BangBangHysteresis.md) | The $\varepsilon \to 0$ limit is a relay on velocity, with the same chattering mechanism |
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| [Saturation / Rate Limiter](SaturationRateLimiter.md) | The output clamp is the same saturation, applied to the compensation term alone |
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| [QR Decomposition](../solvers/QrDecomposition.md) | Backbone of the least-squares fit of the linear friction coefficients |
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| Algorithm | Relationship |
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|-------------------------------------------------------|------------------------------------------------------------------------------------------------------|
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| [Feedforward / 2-DOF](Feedforward2Dof.md) | Same additive-feedforward structure; friction is a velocity-driven feedforward term |
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| [PID Controller](Pid.md) | The feedback loop this term is added to; compensation reduces integrator work and stick-slip hunting |
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| [Bang-Bang / Hysteresis](BangBangHysteresis.md) | The $\varepsilon \to 0$ limit is a relay on velocity, with the same chattering mechanism |
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| [Saturation / Rate Limiter](SaturationRateLimiter.md) | The output clamp is the same saturation, applied to the compensation term alone |
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| [QR Decomposition](../solvers/QrDecomposition.md) | Backbone of the least-squares fit of the linear friction coefficients |
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## References & Further Reading
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‎doc/controllers/README.md‎

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## Algorithms
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| Algorithm | Description |
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|--------------------------|-----------------------------------------------------------------------------------------|
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| [Bang-Bang / Hysteresis Controller](BangBangHysteresis.md) | Two-state relay controller with Schmitt-trigger dead-band to prevent chatter |
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| [Deadbeat Control](DeadbeatControl.md) | Discrete-time N-step deadbeat controller — exact convergence in N samples via reachability pseudo-inverse |
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| [PID Controller](Pid.md) | Proportional-Integral-Derivative controller using a discrete recursive formulation |
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| [LQR Controller](Lqr.md) | Linear Quadratic Regulator — optimal state-feedback control minimizing a quadratic cost |
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| [LQI / Servo Controller](IntegralStateFeedbackLqi.md) | Integral state feedback (LQI) — LQR augmented with integrators on the tracking error for zero steady-state offset |
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| [LQG Controller](Lqg.md) | Linear Quadratic Gaussian — output-feedback optimal control via LQR + Kalman Filter |
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| [MPC Controller](Mpc.md) | Model Predictive Controller — receding-horizon optimal control with constraint handling |
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| [Linear Time-Invariant Model](LinearTimeInvariant.md) | Discrete-time state-space plant model (A, B, C, D) shared across controllers and filters |
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| [Feedforward / 2-DOF Controller](Feedforward2Dof.md) | Feedforward + feedback wrapper for decoupled reference tracking and disturbance rejection |
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| [Luenberger Observer](LuenbergerObserver.md) | Deterministic state estimator using pole placement (Ackermann's formula) to reconstruct unmeasured states |
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| [Friction Compensation](FrictionCompensation.md) | Coulomb + viscous + Stribeck feedforward with a tanh boundary layer and output clamp, per axis |
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| Algorithm | Description |
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|------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------|
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| [Bang-Bang / Hysteresis Controller](BangBangHysteresis.md) | Two-state relay controller with Schmitt-trigger dead-band to prevent chatter |
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| [Deadbeat Control](DeadbeatControl.md) | Discrete-time N-step deadbeat controller — exact convergence in N samples via reachability pseudo-inverse |
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| [PID Controller](Pid.md) | Proportional-Integral-Derivative controller using a discrete recursive formulation |
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| [LQR Controller](Lqr.md) | Linear Quadratic Regulator — optimal state-feedback control minimizing a quadratic cost |
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| [LQI / Servo Controller](IntegralStateFeedbackLqi.md) | Integral state feedback (LQI) — LQR augmented with integrators on the tracking error for zero steady-state offset |
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| [LQG Controller](Lqg.md) | Linear Quadratic Gaussian — output-feedback optimal control via LQR + Kalman Filter |
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| [MPC Controller](Mpc.md) | Model Predictive Controller — receding-horizon optimal control with constraint handling |
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| [Linear Time-Invariant Model](LinearTimeInvariant.md) | Discrete-time state-space plant model (A, B, C, D) shared across controllers and filters |
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| [Feedforward / 2-DOF Controller](Feedforward2Dof.md) | Feedforward + feedback wrapper for decoupled reference tracking and disturbance rejection |
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| [Luenberger Observer](LuenbergerObserver.md) | Deterministic state estimator using pole placement (Ackermann's formula) to reconstruct unmeasured states |
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| [Friction Compensation](FrictionCompensation.md) | Coulomb + viscous + Stribeck feedforward with a tanh boundary layer and output clamp, per axis |

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