An Asymmetric Instability Threshold for Fixed-Gain Sliding Mode Control Under Power-Law Contact Stiffness Mismatch

Fixed-gain sliding mode controllers are routinely designed against an assumed value of an uncertain plant parameter, such as the contact stiffness of an object grasped by a robotic gripper. This paper shows, for a boundary-layer (saturation-smoothed) sliding mode controller regulating a single-degree-of-freedom plant with power-law contact force F_contact = k*x^n -- which includes both the linear spring (n=1) and Hertzian contact (n=3/2, the standard model for compliant and biological tissue contact) as special cases -- that the consequence of a mismatched stiffness coefficient is fundamentally asymmetric: underestimating the true coefficient produces a bounded, benign steady-state tracking error, while overestimating it beyond a sharp threshold causes the closed-loop trajectory to be ejected from the sliding-mode boundary layer and settle at a saturation-locked false equilibrium far from the target. We derive the exact threshold for general n, k*_assumed = k_true + eta / (x_d + Phi/lambda)^n, where eta is the switching gain, lambda the sliding-surface slope, Phi the boundary-layer width, and x_d the target displacement, and show it reduces exactly to a simpler, previously derived linear-contact result at n=1. We verify the threshold against direct nonlinear simulation across three independently swept controller parameters for both n=1 and n=3/2, finding exact agreement in 62 of 63 and 63 of 63 tested configurations respectively, with the sole exception in the linear case matching to floating-point precision. The result gives a simple, actionable design rule that holds across contact-law nonlinearity: when a plant parameter entering the equivalent control is uncertain, a controller should be biased toward underestimation, since underestimation degrades performance gracefully while overestimation risks a discontinuous safety failure. Verification code available at: https://github.com/kingwdalzain-glitch/asymmetric-smc-threshold