In this paper, we use the derivative of the exponential map to derive the exact evolution of the logarithm of the tracking error for mixed-invariant systems. Following correct-by-construction software paradigm, we propose an invariant control law for mixed-invariant systems, with application to Unmanned Aerial Systems (UASs), that is designed for efficient safety verification. We derive the nonlinear distortion matrix in the transformed differential equation in the Lie algebra and express the distortion matrix in a series form for any matrix Lie group and in a closed-form for the SE(2) Lie group. Given the input distortion, we employ dynamic inversion to linearize the evolution of error dynamics and apply a linear control strategy. We employ Linear Matrix Inequalities (LMIs) to bound the tracking error given a bounded disturbance amplified by the distortion matrix and leverage the tracking error bound to create flow pipes for the creation of a Polyhedral Invariant Hybrid Automaton (PIHA) model. We demonstrate the usefulness of our method by applying it to a simplified holonomic aircraft and nonholonomic rover with polynomial-based path planning methods.