Undesired lateral and longitudinal wheel slippage can disrupt a mobile robot's heading angle, traction, and, eventually, desired motion. This issue makes the robotization and accurate modeling of heavy-duty machinery very challenging because the application primarily involves off-road terrains, which are susceptible to uneven motion and severe slippage. As a step toward robotization in skid-steering heavy-duty robot (SSHDR), this paper aims to design an innovative robust model-free control system developed by neural networks to strongly stabilize the robot dynamics in the presence of a broad range of potential wheel slippages. Before the control design, the dynamics of the SSHDR are first investigated by mathematically incorporating slippage effects, assuming that all functional modeling terms of the system are unknown to the control system. Then, a novel tracking control framework to guarantee global exponential stability of the SSHDR is designed as follows: 1) the unknown modeling of wheel dynamics is approximated using radial basis function neural networks (RBFNNs); and 2) a new adaptive law is proposed to compensate for slippage effects and tune the weights of the RBFNNs online during execution. Simulation and experimental results verify the proposed tracking control performance of a 4,836 kg SSHDR operating on slippery terrain.