Real-time accurate visual object tracking (VOT) for quadrupedal robots is a great challenge when the scale or aspect ratio of moving objects vary. To overcome this challenge, existing methods apply anchor-based schemes that search a handcrafted space to locate moving objects. However, their performances are limited given complicated environments, especially when the speed of quadrupedal robots is relatively high. In this work, a newly designed VOT algorithm for a quadrupedal robot based on a Siamese network is introduced. First, a one-stage detector for locating moving objects is designed and applied. Then, position information of moving objects is fed into a newly designed Siamese adaptive network to estimate their scale and aspect ratio. For regressing bounding boxes of a target object, a box adaptive head with an asymmetric convolution (ACM) layer is newly proposed. The proposed approach is successfully used on a quadrupedal robot, which can accurately track a specific moving object in real-world complicated scenes.