# Design of a Carangiform Swimming Robot through a Multiphysics Simulation Environment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{j}is the absolute angle of link j with respect to a reference frame attached to the robot head; a

_{j}is the oscillation amplitude; f is its frequency, which is the same for all the links; and φ

_{j}is the constant phase shift between the harmonic motion laws of the links. Position control is then required to comply with the non-linear function (1) and to synchronize the servomotors in order to maintain the constant angular position difference between the links. Furthermore, each servomotor along the tail mechanism must be properly sealed, while watertight connections must be installed on both the joint shafts and on the electrical cables, leading to increases of the structure’s inertia, encumbrance, and possibility of failure.

## 2. Materials and Methods

#### 2.1. Tail Kinematics and Optimization Criteria

_{TW}is the lateral displacement; x is the distance from the connection point (Figure 4a); k and f are the wave number and frequency, respectively; and c

_{1}and c

_{2}are kinematic parameters called linear and quadratic amplitude envelopes, respectively. In order to approximate the continuous bending deformation expressed by Equation (2) by means of a piecewise flexible mechanism, the authors propose a discretization method widely used in former prototypes [1,10]: the tail motion function (2) is discretized in M postures h

_{T}(x,i), (i = 0 … M−1) over time, as shown in Figure 4b; then, each posture is approximated by a pose of the multijoint mechanism. The number of links of the mechanism is limited to 3 in this work, because the links must be long enough to house the components of the transmission system, and a minimum length of 100 mm is imposed on each link for an overall length of 300 mm. The first link is hinged to the fish head at the connection point (Figure 4a), while the third link is the caudal fin. Figure 5 shows the approximation of a tail posture, where (x

_{k},y

_{k}) (k = 0 … 2) are the coordinates of joint k, (x

_{3},y

_{3}) is the endpoint of the third link, and p

_{ij}(i = 1 … M−1, j = 1 … 3) are the absolute angles of link j relative to the head in the ith tail posture. Hence, the pose of the mechanism can be represented by the piecewise linear function g(x,i), defined as:

_{T}(x,i), a root mean square error criterion is usually adopted [1]. However, since the main purpose is to approximate Equation (2) while staying true to the real fish dynamics, the added mass method in [10] is preferred and implemented to solve the optimal pose of the mechanism and to compute the unknown angles p

_{ij}. In carangiform locomotion thrust is generated, since the propulsive wave travels along the body of the fish—momentum is transferred by the tail motion to the surrounding water, which in turn develops a reaction force pushing the swimmer. The mass of the water passing backwards is called added mass and the magnitude of the thrust force depends on it. Therefore, when using a multijoint serial mechanism, the optimal configuration is computed by minimizing the difference between the added mass pushed by the linkage and by the real tail, which leads to the following expression [10]:

_{i}is the error function, which depends on the absolute angle p

_{ij}in the ith posture. Expression (4) can then be minimized as a function of p

_{ij}for each value of i, thus computing the optimal pose of the tail mechanism, as shown in Figure 6. As stated before, each link of the kinematic chain oscillates following a harmonic function of time. Thus, the optimal values of the angles p

_{i}

_{1}, p

_{i}

_{2}, and p

_{i}

_{3}(i = 0 … M−1) can be interpolated to closely resemble the sinusoidal functions p

_{j}(t) (j = 1 … 3) stated by (1), as shown in Section 3.

#### 2.2. Elements of the Cam-Joint-Based Transmission System

_{0}shown in Figure 7b.

#### 2.3. Computational Fluid Dynamics Analysis

#### 2.4. Multibody Model

_{b}, O

_{b}− x

_{b}y

_{b}z

_{b}is attached to the body. The velocity of the origin O

_{b}is expressed in the body frame Σ

_{b}by the vector

**ν**= [u v w]

_{1}^{T}; likewise, vector

**ν**= [p q r]

_{2}^{T}represents its angular velocity [19]. In this paper, the authors have focused their analysis on plane motion. In this case and according to the Newton–Euler formulation, the dynamics equations can be written as:

_{z}is the z principal moment of inertia, which is computed under the hypothesis that the frame Σ

_{b}is coincident with the body central frame. The right side of Equation (7) accounts for the hydrodynamic loads applied to a rigid body moving in the surrounding fluid. A rigorous analysis of an incompressible flow would require the solution of the Navier–Stokes equations; however, if the velocities are reasonably low, most of the hydrodynamic effects have no significant influence on the resulting motion. Moreover, if the body features three planes of symmetry, the terms in the right side of Equation (7) can be linearized [19,20], leading to the following expression:

_{b,i}(i = 0 … 3) are attached to the rigid bodies of the model; the zero index refers to the robot head, while non-zero indexes identify the tail links. Each body of the assembly, except the caudal fin, is approximated by a cylinder and is subjected to hydrodynamic forces coming from Equation (8). The propulsive forces and torque F

_{T}, F

_{L}, and M, are applied to the fin. Following the convention widely adopted in other works [3], the thrust force component is aligned with the swimming direction, coincident with the free stream velocity vector in the CFD analysis.

^{®}(Newport Beach, CA, USA). The complete modeling procedure and the computed results are shown in Section 3.3.

## 3. Results

_{i}

_{1}, p

_{i}

_{2}, p

_{i}

_{3}(i = 0 … M−1), which describe the orientation of the links in M postures, can then be interpolated, leading to the following expressions:

_{j}and phase shifts Δ

_{j}are presented in Table 2. The curves showing the results of the interpolation are gathered in Appendix A.

#### 3.1. Functional Design of the Transmission System

_{1}, θ

_{2}, and θ

_{3}are the output angles of the cam joint mechanisms shown in Figure 10; λ

_{j}and δ

_{j}are their functional parameters and initial rotations, respectively; whereas ω is the motor velocity. The approximated expression of the output angles (6) is used. By solving the first row of (10), this immediately results in:

_{2}and δ

_{2}, from Equation (10) it can be found that:

_{2}, Δ

_{2}, and λ

_{1}, leading to:

_{3}and δ

_{3}as functions of the known quantities λ

_{1}, λ

_{2}, δ

_{2}, A

_{3}, and Δ

_{3}:

_{j}are computed, the geometric parameters of each cam joint, h

_{j}and L

_{j}, can be chosen according to Expression (5), which states that λ

_{j}= h

_{j}/L

_{j}; moreover, the radius of the drive spheres shown in Figure 7a,b must be large enough to house the transmission input shaft and prevent interferences with the rectangular grooves of the joint output member. However, as the sphere radius grows, the encumbrance of the cam joints rises as well. In order to overcome this constraint, the drive spheres are drawn with a cut shape, leaving only the small surface astride the contact line with the driven member in an oscillation cycle, whereas the remaining material, i.e., the part not involved in torque transmission, is ideally removed. As a matter of fact, contact occurs on the line resulting from the intersection of the drive sphere surface and plane Π, as shown in Figure 11b, where angle θ

_{0j}is equal to the output oscillation amplitude [23]. Table 3 summarizes the geometric parameters of the cam joints designed in this paper.

_{j}(t), expressed by (9), with their approximations obtained by using the exact expressions (5) of the cam joints output angles θ

_{j}in Equation (10). The curves show a good agreement between the target trends and those achieved with the transmission system proposed here.

#### 3.2. Propulsive Performance of the Caudal Fin

_{3}(t), whereas the translation motion law can be calculated as the lateral displacement s of the revolute joint that connects the fin to the former linkage member:

_{0}measures 57 mm and Δ

_{y}is equal to −0.43. This follows a constant phase shift Λ of about 51 degrees.

^{4}and 10

^{6}, while the remaining kinematic parameters remain constant. The simulations show that both the efficiency and the thrust coefficients are poorly related to the Reynolds number, as reported in Table 4. The simulations confirm that beyond the critical Re for a roto-translating foil, the flow becomes turbulent and propulsive forces stabilize close to a steady-state value. Still, a slightly growing trend is observed for both the efficiency and the thrust coefficient as the Reynolds number increases. In fact, the higher the Re, the stronger the turbulent boundary layer surrounding the foil, thus preventing flow separations, even at the highest incidence angles. The low dependency on the Reynolds number allows the authors to focus their investigation on the other kinematic parameters. Indeed, the robotic fish designed in this paper should swim at Re values varying between 10

^{5}and 10

^{6}, a range that is dependent on its speed, which in turns derives from the tail undulation frequency, as shown in Section 3.3.

_{0}[24]. Figure 13 shows the propulsive efficiency and the average thrust coefficient as a function of the Strouhal number. The four lines refer to two different foils obtained by slicing the fin with two horizontal planes at 25% and 75% of its span, respectively, as explained in Section 2.2 and Figure 8. Despite the tapered shape of the fin, since both foils share the same translation amplitude due to the heaving rigid motion component, they are also characterized by the same values of St.

_{pl}is a propulsive load coefficient, K

_{pl}is a proportionality factor, and φ

_{pl}is a phase constant. As a matter of fact, the CFD simulations proved that the trends in an oscillation cycle of both foil load coefficients can be approximated by harmonic functions, whose amplitudes depend on the square of St, whereas their phase shifts are independent from the aforementioned parameter. Since both foils share the same Strouhal number, the propulsive forces and torque resulting from the three-dimensional fin can be finally approximated by the following expression, which is widely used in aircraft wing design:

_{T}, F

_{L}, and M are the propulsive forces and torque already introduced in the multibody model in Section 2.4; ρ is the water density; whereas c

_{25%}and c

_{75%}are the foil chord lengths. Table 5 shows the values of the geometric quantities used in (20).

#### 3.3. Multibody Analysis

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Interpolation of absolute link angles: the blue dots represent the values obtained by the tail postures, while the solid lines indicate the sine functions resulting from the interpolation.

## Appendix B

^{5}to obtain a fully developed turbulent flow.

**Figure A2.**Results of the sensitivity analysis of the propulsive efficiency and the thrust coefficient from the translation and rotation amplitude, plus the phase difference between the pitching and heaving components of the roto-translation (the translation amplitude s is expressed as a multiple of foil chord c).

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**Figure 1.**Body and caudal fin (BCF) swimming mode classifications [3].

**Figure 3.**Sketches of the Atlantic mackerel modeled in this work: (

**a**) side and (

**b**) upper views [15].

**Figure 4.**(

**a**) Travelling wave pattern [1]. (

**b**) Tail posture discretization (c1 = 0.2, c2 = 0; k = 7.5, f = 1, M = 18).

**Figure 8.**(

**a**) Two-dimensional mesh used in computational fluid dynamics (CFD) analysis. (

**b**) Fin geometry and foil definition.

**Figure 9.**(

**a**) Cylindrical body: hydrodynamic loads and body-frame convention [19]. (

**b**) Robotic fish: hydrodynamic loads and propulsive forces.

**Figure 11.**(

**a**) Double Cardan joint: unbent tail (left), bent tail (right). (

**b**) Spatial cam joint geometry.

**Figure 12.**Trends of the absolute angles p

_{j}(solid lines) stated by Equation (9) compared to those obtained with the cam joint angles (dotted lines), as shown in (10).

**Figure 14.**Multibody model of the carangiform swimming robot imported in Adams/View by MSC software

^{®}.

**Table 1.**Added mass and damping coefficient for a cylinder with a radius R, length L, and mass m [17].

${X}_{\dot{u}}$ | ${Y}_{\dot{v}}$ | ${N}_{\dot{r}}$ | ${X}_{u\left|u\right|}$ | ${Y}_{v\left|v\right|}$ |

$0.1m$ | $\pi \rho {R}^{2}L$ | $\pi \rho {R}^{2}{L}^{3}/12$ | $\rho {A}_{f}{c}_{D,f}/2$ | $\rho {A}_{l}{c}_{D,l}/2$ |

${N}_{r\left|r\right|}$ | ${A}_{f}$ | ${A}_{l}$ | ${c}_{D,f}$ | ${c}_{D,l}$ |

$\rho {A}_{l}{c}_{D,l}{L}^{3}/16$ | $\pi {R}^{2}$ | $2RL$ | 0.5 | [0.8–1.2] |

A_{1} | A_{2} | A_{3} | Δ_{1} | Δ_{2} | Δ_{3} |
---|---|---|---|---|---|

0.074 | 0.519 | 0.585 | 0 | −0.49 | −1.33 |

**Table 3.**Geometric parameters of the three jth cam joints; h

_{j}and L

_{j}are expressed in mm, δ

_{j}in radians.

h_{1} | L_{1} | δ_{1} | h_{2} | L_{2} | δ_{2} | h_{3} | L_{3} | δ_{3} |
---|---|---|---|---|---|---|---|---|

4 | 54 | 0 | 11.8 | 26 | −0.57 | 6.3 | 14 | −2.35 |

Reynolds Number | Propulsive Efficiency % | Thrust Coefficient |

1 × 10^{4} | 67.7 | 0.258 |

5 × 10^{4} | 68.3 | 0.261 |

1 × 10^{5} | 69.5 | 0.263 |

5 × 10^{5} | 70.1 | 0.265 |

1 × 10^{6} | 70.4 | 0.267 |

c_{25%} | c_{75%} | B | m | I_{xx} | I_{yy} | I_{zz} |
---|---|---|---|---|---|---|

0.080 | 0.060 | 0.168 | 6.147 | 0.143 | 0.012 | 0.139 |

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**MDPI and ACS Style**

Costa, D.; Palmieri, G.; Palpacelli, M.-C.; Scaradozzi, D.; Callegari, M.
Design of a Carangiform Swimming Robot through a Multiphysics Simulation Environment. *Biomimetics* **2020**, *5*, 46.
https://doi.org/10.3390/biomimetics5040046

**AMA Style**

Costa D, Palmieri G, Palpacelli M-C, Scaradozzi D, Callegari M.
Design of a Carangiform Swimming Robot through a Multiphysics Simulation Environment. *Biomimetics*. 2020; 5(4):46.
https://doi.org/10.3390/biomimetics5040046

**Chicago/Turabian Style**

Costa, Daniele, Giacomo Palmieri, Matteo-Claudio Palpacelli, David Scaradozzi, and Massimo Callegari.
2020. "Design of a Carangiform Swimming Robot through a Multiphysics Simulation Environment" *Biomimetics* 5, no. 4: 46.
https://doi.org/10.3390/biomimetics5040046