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Abstract The motion averaging problem arises when recovering the motions for a set of scans in three‐dimensional (3D) reconstruction. Motion averaging is taken as an optimisation problem, whose solution can be obtained via matrix decomposition in principal component analysis. The latest motion averaging methods utilize the Cauchy kernel function or overlap rate to construct the weighted matrix in the cost function, yielding weighted low‐rank sparse matrix decomposition solutions. Nevertheless, the over‐reliance on the initial value and high computational complexity remain two major problems. Here, we aim to improve the motion averaging performance via a fast and robust method based on the angle constraints of multiple scans. The additional term representing the angle constraint is added into the original cost function. By doing this, two benefits are facilitated: 1) the iteration times for the optimisation process are reduced; and 2) it effectively mitigates the over‐reliance on the initial value and thus caters for outliers and random noise of initial motions. Compared with state‐of‐the‐art motion averaging methods, it is indicated that the proposed method is more efficient and robust for numerical simulations of using both simulated motions and real 3D scanning data, given a large range of angle constraint values.