## 1 Introduction  
2D color printing technology has made great progress, such as multicolor printing technology for planes, but 3D color printing technology is still at an exploratory stage. There are three main traditional surface printing technologies, including curved surface screen printing [1], pad printing [2], and waterless offset printing [3]. The above technologies are usually applied to cylindrical surfaces and need to be prepared with a corresponding printing plate, which limits them to specific surface patterns. Since these, 3D inkjet technology [4] and 3D rapid prototyping technology [5,6] have been developed. [PERSON] et al. [7] developed a 5-degree-of-freedom (DOF) robot and tried to print on surfaces, but it was mainly applicable to large and vertical flat surfaces. [PERSON] et al. [8,9] proposed a method by thermoforming a thin plastic sheet with texture to adhere seamlessly onto a surface. However, that technology is currently at the exploratory stage due to the complex computation and equipment limitations. Solid Terrain Modeling (STM) company [10] in the U.S. has developed a set of equipment that can make color geographical models. The system includes a carving machine and a coloring device, which can provide users with models containing 3D geographic information and surface color information. The system is mainly applied to the military sandbox and geographical models. Some devices with 3D rapid prototyping technology were launched to make colorful parts, such as Objet500 Connex3 [11] of Stratasys. However, they are not widely used because of their high price. To the best of our knowledge, there is no commercial color printer for freeform surfaces yet. However, in many rapid prototyping technologies, fused deposition modelling (FDM) is becoming more and more popular due to its simple structure and low price. Generally, FDM is mainly used for producing a monochromatic workpiece. With the increasing demand for color parts, many researchers have started to study multi-color FDM technology. There are three main ways:  
1 Multi-nozzle  
The multi-nozzle system integrates multiple nozzles, each of which can print one kind of filament with one color. Different nozzles are selected according to different times and positions. In 2014, ORD Solutions [12] in Canada developed a color 3D printer, RoVa3D, with five nozzles, which integrates five printing heads and can print models with multiple colors. However, the mechanical structure of the printing module of the device is complex, and the system does not involve the mixing of pigments, and can only print models with up to five colors. 2 Mixing filaments  
The basic principle of the mixing system is based on the three pigments with primary colors (red, green, blue). Through mixing filaments with primary colors according to different proportions, the mixed filaments of the target color can be produced to achieve colorful printing. In 2012, [PERSON] [13] developed a printing head that supported mixing two colors. Its color mixing effect is still not satisfactory. When the 3D full-color desktop printer of the Cube Series was launched by the 3 DSYSTEMS Company in 2015, the printed colorful workpiece had a good effect, but the printing effect of that printer for multi-color models with sudden color changes is still not ideal. 3 Dyeing filaments  
Its basic principle is to dye the filament while the filament is still in the melting state, thus forming a colorful model. In 2014, a research and development team at the University of Wisconsin Madison [14] developed a dye 3D printer, Spectrom. The printer realizes 3D color printing by dyeing the transparent polymer filaments. However, that printer has special requirements for materials and dyeing inks. The commonly used materials (e.g., ABS (acrylonitrile butadiene styrene) and PLA (polylactic acid)), do not meet the printing requirements, which limits the printer's promotion.