This paper proposes a method for predicting maximum friction coefficients and optimal slip ratios as optimal control parameters for traction control or slip control of autonomous mobile robots on rough terrain. This paper focuses on strength of ground surface which indicates different characteristics depending on material types on surface. Strength of various material types can be estimated by Willoughby sinkage model and by a developed testbed which can measure forces, velocities, and displacements generated by wheel-terrain interaction. Estimated strength is collaborated on building improved Brixius model with friction-slip data from experiments with the testbed over sand and grass material. Improved Brixius model covers widespread material types in outdoor environments on predicting friction-slip characteristics depending on strength of ground surface. Thus, a prediction model for obtaining optimal control parameters is derived by partial differentiation of the improved Brixius model with respect to slip. This prediction model can be applied to autonomous mobile robots and finally gives secure maneuverability on rough terrain. Proposed method is verified by various experiments under similar conditions with the ones for real outdoor robots.