# Three-Dimensional Modeling of a Robotic Fish Based on Real Carp Locomotion
## 1 Introduction  
In recent years, biologically inspired behavior-based systems have been more and more popular topic in underwater vehicles. The goal of the bio-inspired approach is to adapt the biological features of underwater creatures such as fish to Autonomous Underwater Vehicle (AUV) designs and also imitate the aquatic locomotion abilities of them. Biomimetic modeling also provides enhanced performance and increased efficiency, maneuverability and acceleration for novel AUV designs [1; 2; 3; 4; 5]. As it is known, many kinds of fish perform high-efficiently locomotion and maneuvering in the water. Propulsion efficiencies of the rotary propeller AUVs are limited below 70%, while the swimming mechanism of a real fish is 20% more efficient than rotary propellers [6]. In addition, the turning radius of propellers is big and speed performances are low. Therefore, this kind of propulsion is more noisy and ineffective than bio-inspired systems [6; 7]. These advantages have great benefit for marine applications and fish-like robots have evolved to provide versatile solutions for a wide variety of underwater applications, such as undersea investigation, pollution detection, deep sea monitoring and mapping, military operations and protections, etc. [8; 9; 10; 11]. An elaborate observation of the whole biological structure of a fish is necessary to identify the principles of a bio-inspired approach to the fish. In this way, significant limitations and constraints of any biological feature are determined before modeling the robotic fish. A bio-inspired robotic fish is defined as an aquatic vehicle, which is propelled by undulatory or oscillatory body motions [10; 12]. In nature, most fish exhibit various swimming behaviors by bending their bodies and/or their caudal fins (BCF). Alternatively, some fish use their median and/or pectoral fins (MPF) [13]. Commonly, more than 85% of fish swim by BCF locomotion and on average 15% of fish swim in the world by MPF. Carangiform locomotion, the most common type of BCF, generates undulatory swimming patterns, high speed performance and low noise [11; 13]. Thus, biomimetic Carangiform design is an appropriate solution for AUVs [14; 15; 16; 17]. There have been some studies about robotic fish modeling in the literature. The first mathematical models including fish swimming were proposed by [PERSON] and [PERSON] in 1960s. [PERSON] suggested simplified two-dimensional waving plate theory and [PERSON] defined the body-traveling wave, which takes a sinusoidal form of undulatory motion [1; 18]. [PERSON] explained the fish-swimming modes for oscillatory and undulatory motions of fins and bodies [14]. [PERSON] and [PERSON] established a model for Anguilliform locomotion to give swimming gaits [19], while [PERSON] et al. derived 3D models for pectoral fins and tails [20]. Also, [PERSON] et al. applied the Central Pattern Generator (CPG)-based locomotor controller to their dynamic model. CPG is a neural circuit diagram inspired from biological musculoskeletal systems to perform gait generation of bio-inspired robots [21; 22; 23]. [PERSON] and [PERSON] built a simulator for the autonomous Carangiform motion of a four-joint robotic fish, but up-down motions were not modeled [24]. [PERSON] et al. also modeled up-down motions [25; 26]. Unlike these, [PERSON] et al. presented a dynamic backward analysis for a biomimetic Carangiform robotic fish [27]. Furthermore, [PERSON] and [PERSON] presented a dynamic model of the double-jointed robotic fish to examine self-exciting conditions and propulsive characteristics [28]. [PERSON] et al. presented the four-link motion mechanism of fish [29]. [PERSON] and colleagues modeled a four degrees of freedom (4-DoF) mathematical model including hydrodynamic forces inspired by tuna fish [10]. A multi-DoF robotic fish model was suggested by [PERSON] and [PERSON] to determine the position and orientation of the rigid body. They used a state feedback controller in the simulation [15; 30].