Position Loop — Friction Compensation Feedforward

Theory document: Explains the mathematical and engineering principles behind a component or algorithm. This document is descriptive — it records the why and how at a scientific level, independent of any specific implementation. Equations use KaTeX (\(inline\) and \[block\]). Block diagrams use Mermaid or ASCII art.

Not implemented. No such algorithm exists in the firmware: foc::PositionAlgorithm offers pid, cascadeP, lqr, lqi and twoDof, and nothing else is selectable over CAN or the CLI. This chapter is design groundwork for a law that has not been built, kept for the theory rather than as a description of shipped behaviour. Read it as a proposal, not as a specification.

Overview

Friction compensation is not a selectable algorithm — it is an independently enabled feedforward that stacks on top of any speed or position feedback controller. Viscous friction (\(B_f \cdot \omega_m\)) is estimated by the mechanical RLS and compensated implicitly by the feedback controllers. Coulomb and Stribeck friction are nonlinear: at low speeds the motor sticks until the drive overcomes the static breakaway torque, then slips. This creates position hunting at rest and velocity ripple during slow moves — the classic precision servo failure mode that no linear feedback controller can fully eliminate.

Prerequisites

Parameters \(T_c\), \(T_s\), \(\omega_{st}\) are identified by a dedicated friction sweep — separate from the mechanical RLS that estimates only the linear viscous coefficient \(B_f\).

Mathematical Foundation

Friction Model (Coulomb + Stribeck)

\[ \boxed{T_f(\omega_m) = \Bigl[T_c + (T_s - T_c)\,e^{-(\omega_m/\omega_{st})^2}\Bigr]\cdot\mathrm{sgn}(\omega_m)} \]

where: - \(T_c\): Coulomb (kinetic) friction torque — the flat plateau at moderate speed - \(T_s\): static (breakaway) torque, \(T_s > T_c\) — the peak at near-zero speed - \(\omega_{st}\): Stribeck velocity — speed at which kinetic friction reaches its minimum

Current Feedforward

Converting friction torque to \(i_q\) and injecting before the current loop:

\[ i_{ff}(\omega_m) = \frac{T_f(\omega_m)}{K_t} \] \[ i_q^{*,total} = i_q^{*,ctrl} + i_{ff}(\omega_m) \]

Dead-Zone Smoothing

The \(\mathrm{sgn}(\omega_m)\) discontinuity at \(\omega_m = 0\) must be replaced with a linear ramp over \(\pm\omega_{dead}\) to prevent chattering during direction reversals and holding. Typical \(\omega_{dead} = 0.05\)–\(0.2\) rad/s.

Parameter Identification

\(T_c\), \(T_s\), and \(\omega_{st}\) require a dedicated friction sweep (ramp velocity at steady state while logging \(i_q^*\)). This is separate from the mechanical RLS, which estimates only the linear viscous coefficient \(B_f\).

Tuning: - \(T_c\) (N·m): read from the \(i_q^* \times K_t\) plateau at constant low speed. - \(T_s\) (N·m): read from the breakaway \(i_q^*\) at near-zero speed. - \(\omega_{st}\) (rad/s): speed at which the \(i_q^*\) vs. \(\omega_m\) curve reaches its kinetic minimum.

Numerical Properties

Limitations & Assumptions

• Over-compensation (\(T_s\) or \(T_c\) over-estimated) causes position overshoot at zero crossing. Prefer slight under-compensation and let the feedback controller handle the residual.
• Friction parameters drift with temperature and wear. Re-identification is recommended when low-speed precision degrades.
• The \(\mathrm{sgn}(\omega_m)\) dead-zone must be smoothed to prevent chattering; too wide a dead-zone reintroduces hunting in the smoothed region.

References

1. Armstrong-Hélouvry, B., Dupont, P. & De Wit, C.C. — “A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction”, Automatica, 30(7):1083–1138, 1994.
2. Krishnan, R. — Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press, 2010.