Most information in an image is contained in the edge. Sobel edge detection algorithm is a classic method to realize the edge detection of image in the applications of robot vision, including motion detection and object tracking. This paper proposes an area-efficient and energy-efficient Sobel edge detector design utilizing bit-width pruning, shift-add operation and bit-width tuning techniques, to reduce required area and computation significantly with negligible edge information loss, for mobile robot vision applications. As a result, the proposed Sobel edge detector can achieve lower hardware overhead and higher energy efficiency. FPGA implementation shows that the proposed Sobel detector design achieves 33% lower power consumption while maintaining a good detection performance in terms of CP (over 99%, $P_{co}$) and achieving a better processing performance in terms of frame rate (an increasing by 39.5%), as compared to the conventional Sobel detector design, which is suitable for resource-limited and energy-constrained mobile robots in edge IoT applications.