A scaled robotic endoscopy platform (REP) was previously developed to efficiently test new control schemes in a simulated colon environment. This article presents the derivation and tuning of a nonlinear model of the REP operating on various substrates. The modeling technique and novel empirical friction profiling demonstrated here are useful for a wide variety of devices interacting with unconventional substrates. The model is first derived from the REP drivetrain inertial characteristics, and then the interaction with synthetic tissue is quantified by an automated traction measurement system for multiple substrates. The resulting model is then used with ground-truth VICON and sensor data to optimize uncertain parameters by minimizing pose error over a variety of tests and substrates. The results show an average error reduction of 67% over all tests and substrates, with a worst-case 10% open-loop final position error. The success of these results proves a robust dynamic model of the REP and its tissue interactions without the need to model complex and computationally expensive viscoelastic material properties or discrete/nonlinear events such as stalling. The resulting model will be used to develop model-based feedback control for estimation, disturbance rejection, and autonomy for the REP in an actuated colon simulator.