This paper investigates an FPGA architecture whose primary function is to accelerate parallel computations involved in the rapid-exploring random tree (RRT) algorithm. The RRT algorithm is inherently serial, while in each computing step there are many computations that can be executed simultaneously. Nevertheless, how to carry out these parallel computations on an FPGA so that a high degree of acceleration can be realized is the key issue. Membrane computing is a parallel computing paradigm inspired from the structures and functions of eukaryotic cells. As a newly proposed membrane computing model, the generalized numerical P system (GNPS) is intrinsically parallel; so, it is a good candidate for modeling parallel computations in the RRT algorithm. Open problems for the FPGA implementation of the RRT algorithm and GNPS include: (1) whether it possible to model the RRT with GNPS; (2) if yes, how to design such an FPGA architecture to achieve a better speedup; and (3) instead of implementing GNPSs with a fixed-point-number format, how to devise a GNPS FPGA architecture working with a floating-point-number format. In this paper, we modeled the RRT with a GNPS at first, showing that it is feasible to model the RRT with a GNPS. An FPGA architecture was fabricated according to the GNPS-modeled RRT. In this architecture, computations, which can be executed in parallel, are accommodated in different inner membranes of the GNPS. These membranes are designed as Verilog modules in the register transfer level model. All the computations within a membrane are triggered by the same clock impulse to implement parallel computing. The proposed architecture is validated by implementing it on the Xilinx VC707 FPGA evaluation board. Compared with the software simulation of the GNPS-modeled RRT, the FPGA architecture achieves a speedup of a 104 order of magnitude. Although this speedup is obtained on a small map, it reveals that this architecture promises to accelerate the RRT algorithm to a higher level compared with the previously reported architectures.