This study proposes a new frictional algorithm that implements angular increment. The proposed algorithm is used to solve the numerical solutions of dynamic problems in two-dimensional frictional systems. It can accurately obtain the motion responses of a lumped mass under time-varying external forces, and it can compensate for the shortcomings of the numerical frictional algorithm that implements a time step. Specifically, the proposed algorithm 1) overcomes the difficulties encountered when the angles between resultant tangential forces and slip motion are infinitely close, 2) provides accurate solutions for two-dimensional systems under fierce planar motions, and 3) calculates the responses of the mass within a reasonable period. We compare the computation accuracy, efficiency, and robustness of the proposed frictional algorithm and the previous frictional algorithm [1] through several representative scenarios. We reveal that the proposed algorithm has superior computation accuracy, efficiency, and robustness for two-dimensional frictional problems involving slip/stick transitions and sharp bending.