Robust watermarking is an effective method and a promising solution for securing and protecting the copyright of digital images. Moments and moment invariants have become popular tools for robust watermarking due to their geometric invariance and favorable capability of image description. Many moments-based robust watermarking schemes have been proposed. However, there is a challenging problem of these schemes that should be addressed. One of these problems is to improve both imperceptibility and robustness. In contrast, the other problem, most of these schemes used inefficient, traditional computation methods of the moments, resulting in an inaccurate and inefficient performance of the watermarking schemes. To overcome these challenges, in this paper, we propose a novel robust color image-watermarking algorithm based on new multiple fractional multi-channel orthogonal moments, fractional-order exponent moments (MFrEMs), fractional-order polar harmonic transforms (MFrPHTs), and fractional-order radial harmonic Fourier moments (MFrRHFMs). Firstly, highly accurate fractional new multi-channel orthogonal moments are computed for the host color images. Then, more stable and accurate coefficients of fractional new multi-channel orthogonal moments are selected. Finally, a robust color image watermarking approach for multiple watermarks images is proposed based on MFrEMs, MFrPHTs, and MFrRHFMs using a 1D Sine chaotic map. The experimental results demonstrate that the proposed approach provides robustness against various attacks and better imperceptibility than the existing methods.