This article presents a novel solution for the range-only simultaneous localization and mapping (RO-SLAM) that builds a map of the ultrawideband (UWB) anchors affixed to the unknown positions of the environment and localizes an autonomous mobile robot (AMR) at the same time. Delays in the initial positioning of the anchors are common in the RO-SLAM. To solve the delay problem, three UWB sensors named as the trilateration UWB tag (TUT) system on AMR are proposed. The TUT system makes it possible to estimate the range and bearing of each anchor without any delay at the first observation of three distance measurements from the UWB tags. However, the concentration of three UWB tags at a local point on AMR may degrade the performance of the range and bearing estimations. In order to improve the accuracy of the estimation, a constrained least-squares (CLS) algorithm is proposed in this article. The range and bearing estimations serve as the observational measurements for the proposed SLAM algorithm in this article, enabling accurate localization with computational efficiency. The performance of the proposed method is analyzed and demonstrated through simulations and experiments.