Accurate acceleration acquisition is a critical issue in the robotic exoskeleton system, but it is difficult to directly obtain the acceleration via the existing sensing systems. The existing algorithm-based acceleration acquisition methods put more attention on finite-time convergence and disturbance suppression but ignore the error constraint and initial state irrelevant techniques. To this end, a novel radical bias function neural network (RBFNN) based fixed-time reconstruction scheme with error constraints is designed to realize high-performance acceleration estimation. In this scheme, a novel exponential-type barrier Lyapunov function is proposed to handle the error constraints. It also provides a unified and concise Lyapunov stability-proof template for constrained and non-constrained systems. Moreover, a fractional power sliding mode control law is designed to realize fixed-time convergence, where the convergence time is irrelevant to initial states or exteal disturbance, and depends only on the chosen parameters. To further enhance observer robustness, an RBFNN with the adaptive weight matrix is proposed to approximate and attenuate the completely unknown disturbances. Numerical simulation and human subject experimental results validate the unique properties and practical robustness.