Swimming Performance of a Tensegrity Robotic Fish

Abstract

We described a tensegrity robotic fish and detailed its overall structure, stiffness, and mechatronics. The main flexible structure of the robotic fish body was composed with a series of rigid segments linked with tensegrity joints by means of tension elements. Each rigid segment can rotate around tensegrity-compliant joint and have no direct contact with each other. The dominant vibrational mode of the tensegrity robotic fish can be excited by a single harmonic input to mimic the desired kinematics of locomotion. For our tensegrity robotic fish, the experimental results showed that its maximum stride length was about 0.5 body length per cycle; its Strouhal number was roughly between 0.45 and 0.55 near the biological data of carangiform swimmers. Two different vibrational modes that can be achieved would be demonstrated by the harmonic analysis technique. The results indicated that the swimming performance can be improved by using tensegrity joints.

Introduction

In nature, fish are remarkably versatile swimmers, due to their fast speed and exceptional maneuverability. Their evolved physical structures and outstanding performance have attracted researchers' interests, who hope to design better underwater vehicles.^1–4 The biological swimming modes were divided into two types: the body and/or caudal fin (BCF) propulsion and the median and/or paired fin propulsion.⁵ Since the greatest number of the fish families use BCF mode as their propulsive means,⁵ many scholars have paid close attention to it.

Research has indicated that the structures adopted by robotic fish based on BCF can be roughly classified into two categories: the standard discrete and stiff mechanism,^6–11 and the continuous compliant mechanism.^10–16

The former design used hyper-redundant serial or serial-parallel mechanisms, composed of rigid segments connected by multiple rigid joints, as robotic fish spine. This simulated swimmer is inspired from the biological evidence that fish can be represented by a series of parallel mechanisms.^17–19 By actuating each joint independently to mimic the body wave of fish, such robotic fish is expected to achieve the superior swimming performance of fish. However, the actuation and the control techniques are complex when trying to achieve the lateral flexibility observed in the fish. The complexity of the resultant structure of the robotic fish can diminish internal mechanical efficiency, mostly through frictional losses.^9,14 For example, Barrett et al. measured transmission losses of rigid joints already in the order of 10% for his hyper-redundant tuna.^9,14 However, if the robotic fish adopted hyper-redundant parallel mechanisms, there exists a potential advantage on variable stiffness since the stiffness of parallel mechanisms can be altered by an internal load distribution.^20–23

As for the second method, the continuous compliant robotic fish made of viscoelastic materials,^12–16 such as silicon and rubber, can achieve the required motion by exploiting the natural modes of vibration of the biomimetic device.²⁴ Comparing to the discrete mechanism, the continuous compliant mechanism can be designed to store energy to minimize energy consumption. The actuation of this mechanism by exploiting several vibrational modes excited by a single input force can simplify its design.¹⁴ The continuous compliant mechanism is relatively simple and efficient, however, the body stiffness generally is difficult to be adjusted as needed. This is a limitation, since the stiffness distribution and variable stiffness of fish play an important role in fish swimming.^25–28

It is important to find an alternative mechanism to combine these advantages and overcome these limitations of the above two approaches. The key is how to transform redundant serial-parallel mechanisms into a compliant structure with lower frictional losses.

We presented a tensegrity robotic fish, an alternative to the standard discrete mechanism and the continuous compliant mechanism, which is to exploit not only the potential on variable stiffness of redundant serial-parallel mechanisms but also the natural modes of vibration to mimic the body wave of fish, based on the compliant fish body. The tensegrity robotic fish has three characteristics.

First, we adopt tensegrity joints as robotic fish's intervertebral joints to link rigid segments to construct the compliant fish body. Each rigid segment can achieve relative rotation around tensegrity joints. Rigid segments are in a network of tension such that no two rigid segments are in contact. This is inspired by studies of some kinds of animal joints,²⁹ while the bones of animal joints have no touching with each other, only with flexible connective tissue in between. In addition, previous studies have developed some tensegrity models for animal joints,^30,31 like Flemon's tensegrity saddle joint³² and Levin's biological wire-spoken wheel model.^33,34 Furthermore, tensegrity joints, composed of tensional flexible elements, can achieve the structural compliance. In fact, a tensegrity joint can be considered a compliant joint, and it may reduce friction losses to increase transmission efficiency.

Research has produced some tensegrity soft manipulators, including elbow, arm, shoulder,^35–37 and vertebrae,^32,38,39 using tensegrity joints; among them, the vertebrae model can achieve the range of movement as well as degrees of freedom (DOFs) of human body due to compliant tensegrity saddle joints in that.³² The tensegrity structure has been used to design bionic soft robots, such as Orki et al.'s caterpillar crawl⁴⁰ and Bliss et al.'s swimmer⁴¹. Bliss et al. also leveraged tensegrity principles to make robotic fish,⁴¹ although the structure of his robotic fish and ours is different, while his swimmer was built by a hinged fin and three cell, class 2 tensegrity beam.

Second, after achieving the structural compliance of tensegrity robotic fish, we adopt the method presented by Alvarado and Youcef-Toumi¹⁴ to achieve the body wave of fish by excitation of modes of vibration. As Alvarado illustrated, this actuation method can simplify the design and the control technique, and it is also identified by the biological evidence that musculoskeletal measurements confirm the approximate correlation of body natural frequencies with locomotion frequencies.^14,42 Research has developed some soft tensegrity robots by exploiting the vibration as a means of control.⁴³

Third, since the tensegrity robotic fish is a serial-parallel structure, there exists a potential for variable stiffness compared to the continuous compliant fish using viscoelastic materials. Based on the stiffness theory of parallel mechanisms,^17–19 the stiffness of parallel mechanisms can be adjusted by changing internal forces of their legs. The tensegrity robotic fish can be used to study the role of the stiffness distribution and variable stiffness on fish locomotion.

As the first step for establishing this alternative approach, we study the structure and performance of our tensegrity robotic fish. We will explain the structural principle of tensegrity fish body. After illustrating the relationship between stiffness anisotropy and DOFs, we will show how the tensegrity joints have one rotational DOF by differing stiffness with the aid of stiffness formulations. We will confirm that the compliance center can be approximately the virtual rotational center of tensegrity joints. With the complete tensegrity fish robot, we will measure the swimming performance and show the different vibrational modes that can be achieved.

The Structure of Tensegrity Robotic Fish Body

The structural principle of tensegrity fish body

The overall structure of our robotic fish body is shown in Figure 1. This is a serial mechanism composed by seven rigid platforms, which are linked with tension-loaded springs (elastic strings in this article). Although the platforms are rigid and discrete, this mechanism can achieve the structural compliance due to the elastic properties of tension elements.

FIG. 1.

The robotic fish body. (A) The entire fish body. (B) Tensegrity joint. (C) The explosive view of a tensegrity joint. 1: rigid segments. 2: tension elements. 3: cable outlets. 4: bolts and nuts. The tensegrity fish body has seven rigid segments. The tensegrity joint consists of two rigid segments connected through tension elements, with the help of the cable outlets. The cable outlets are fixed by the bolts and nuts. Color images are available online.

The flexibility of tensegrity body can be attributed to tensegrity joints (Fig. 2). Generally, tensegrity structures are composed of two main components, compression elements and tension elements.³⁰ The compression elements of our tensegrity joint are two platforms, including base and follower. They can be described by their two parts, the plate and the foot with a “Y” shape connector, respectively. The tension elements are springs, and they can be divided into two groups: the horizontal tension elements (in green) and the axial tension elements (in blue) in Figure 2. The horizontal tension elements are used to connect the follower foot and the base plate, and provide a horizontal tension network to support the follower in suspension, above the base. In addition, this arrangement of springs is similar to the Levin's wire-spoken wheel model.^33,34 The axial tension elements are used to connect the follower plate and the base plate and it can provide an axial tension network to tense the structure. Relative to the base, the follower is exposed to the downward tension resulting from the axial tension network. Also the horizontal tension network can provide the upward tension and resist the downward tension. The resistance would expand the horizontal tension network to the overall structure of tensegrity joints and distribute the load. Due to this expanded tension network, the follower can achieve rotation around the base without direct contact. This is the principle of tensegrity joints.

FIG. 2.

The structure of a tensegrity joint. (A) Combination of two platforms. (B) Axially tensed structure. Color images are available online.

Similarly, tensegrity joints can be connected in series as shown in Figure 1.

Unlike Flemon's tensegrity model, we adopt the configuration in Figure 2 to construct tensegrity joints. The first reason for this arrangement is the biological structure of animals can be a series of parallel mechanisms.^17–19 Besides, for our compliant joints, there exists a pseudo DOF at the lower stiffness direction, and a higher stiffness can be considered the constraint. The tensegrity joint would have one rotational DOF to achieve the required motions of fish by the stiffness anisotropy. This structure is simple and convenient to differ stiffness at each DOF direction, while the horizontal tension network mainly provides the translational stiffness, and the axial tension network mainly provides the rotational stiffness. The tensegrity rotational joint has a virtual rotational center, and it is near the center of horizontal tension network.

The geometry of tensegrity joints

The geometry of tensegrity joint is as shown in Figure 3, and the tensegrity joint has 3 DOFs (y, z, φ). The elastic limbs i = 1–6 are connected to the lower joint at B₁ − B₂ and C₁ − C₄ on the base and to the upper joint at A₁ − A₂ and D₁ − D₂ on the follower, while the elastic limb is elastic string, and have the identical stiffness k. Although the base and follower are oval according to the shape of fish, the arrangement of joint points can be designed and located at circles. In detail, the joint points A₁ − A₂, B₁ − B₂, C₁ − C₄, D₁ − D₂, and E₁ − E₂ are distributed on five circles with the radius, r_a, r_b, r_c, r_d, and r_e, respectively. A, B, D, and E are the centers of four circles among them. As shown in Figure 3A, the tensegrity joint is symmetric and these four circles lie on parallel planes when unloaded and limb 1 and limb 2 are identical, and limbs i = 3–6 are identical, respectively. This position is called the neutral pose, where the pose has zero orientation and rotation.

FIG. 3.

The geometry of the tensegrity joint. (A) Front view. (B) Top view 1. (C) Top view 2. In fact, we describe Top view 1 and Top view 2 from the same view point to show the geometrical information clearly. Color images are available online.

To describe the motion of the follower, we can define the base frame, {B}: O − XYZ, and the body frame, {M}: O₁ − X₁Y₁Z₁, for the base and follower. In this study, we assumed that the virtual rotational center would be located at the origin O₁ of {M}. The architecture of the mechanism can be fully defined by the remained parameters (β, h, H, and H₀), where β is the distributing angle for the joint points C₁ − C₄, h is the height of the origin O of the coordinate systems with respect to the point A of the follower's plate, H is the height of the follower, and H₀ is the height of the point D on the follower foot with respect to the base.

The lengths of limb 1 and 2 can be described by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$l = \sqrt {{{ \left( {{r_a} - {r_b}} \right) }^2} + {{ \left( {H - {H_0}} \right) }^2}}$$ \end{document} , and the lengths of limb 3, 4, 5, and 6 can be described by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_0} = \sqrt {r_c^2 + r_d^2 - 2{r_c}{r_d} \sin \beta + H_0^2}$$ \end{document} .

Recalling the principle of tensegrity joints, our tensegrity joints can achieve rotation and such rotational DOF can be used to achieve the required motion of fish. In detail, the BCF fish always undulate their backbones to achieve locomotion.⁴⁴ Many scholars used a series of joint-linked parallel mechanisms to represent the biological structures of fish.^17–19 For one segment of fish body, the body bending resulted from the relative rotation of each parallel mechanism. For our tensegrity fish, the rotation (φ) of tensegrity joints in Figure 3 can be used to achieve this bending and fish locomotion, and the translation of the direction (y, z) should be restricted to be small when applied the hydrodynamic forces (the reactive force and the resistive force^45,46) or the actuator forces.

Stiffness anisotropy and DOF

For our compliant joints composed of tensional flexible elements, zero stiffness means a DOF and a higher stiffness can be regarded as constraints. Moreover, lower stiffness can be considered a pseudo DOF. If actuated by the same force, the parts with lower stiffness would easily generate a larger motion compared to the parts with higher stiffness.⁴⁷ The greater the stiffness difference between them, the greater difference of motions is generated.

Our tensegrity joint should have one rotational DOF, that is, the DOF of φ, by the stiffness anisotropy. In detail, the rotational stiffness would be near zero, and the translational stiffness should be higher.

Next, to achieve the stiffness anisotropy, we would formulate the stiffness of tensegrity joints at the neutral pose. The reasons for adapting the method at the neutral pose were as follows: (a) the rotation of fish body is relative to the neutral pose, this position is the most studied. (b) It does not affect its generality to achieve the stiffness anisotropy. (c) According to the Lighthill's elongated-body theory,⁴⁶ we expected to obtain the large amplitude of tail tip. For our tensegrity fish, the large amplitude is accumulated by six tensegrity joints, and despite the small rotation of each joint. For our tensegrity fish, if the tail amplitude is required to achieve 20% body length, and we assumed that each tensegrity joint has the same angular displacement, it can be roughly estimated that the rotation angle of ±2 degree is enough. Such small rotation and its symmetry of tensegrity joints at the neutral pose allow us to obtain analytical solution about the stiffness and help us to qualitatively analyze the effect of kinematic parameters on the stiffness anisotropy. This method can provide a theoretical guidance for the design of the tensegrity robotic fish.

Generally, the stiffness of the parallel mechanisms is a 6 × 6 stiffness matrix.⁴⁸ Since the stiffness of directions of x, θ, and ψ cannot affect the planar swimming of robotic fish, we only focused on the directions of y, z, and φ in Figure 3, and then simplified its stiffness matrix to a 3 × 3 stiffness matrix corresponding to the directions of y, z, and φ. In this study, for ease, we focused on the passive stiffness of the tensegrity joints, which is only influenced by the stiffness of springs and kinematic parameters, and the effect of the external wrench and the internal force on stiffness is not taken into account as the first step for establishing this approach for tensegrity robotic fish. According to the stiffness theory of parallel mechanisms,^20–23 the stiffness matrix of the tensegrity joint at the neutral pose can be written as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \bf{K}} = \left[ { \begin{matrix}{{K_y}} & 0 & {{K_{y \varphi }}} \\ 0 & {{K_z}} & 0 \\ {{K_{y \varphi }}} & 0 & {{K_ \varphi }} \\ \end{matrix} } \right] \tag{1} \end{align*} \end{document}

where K_y represents the stiffness of the y DOF, K_z represents the stiffness of the z DOF, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_ \varphi }$$ \end{document} represents the stiffness of the φ DOF, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_{y \varphi }}$$ \end{document} represents the coupling stiffness of the y DOF and the φ DOF.

The stiffness in Equation (1), can be written as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { K_y } = { \frac { 2k { { \left( { { r_a } - { r_b } } \right) } ^2 } } { { l^2 } } } + { \frac { 4kr_c^2 { { \cos } ^2 } \beta } { l_0^2 } } , \tag { 2 } \end{align*} \end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { K_z } = { \frac { 2k { { \left( { H - { H_0 } } \right) } ^2 } } { { l^2 } } } + { \frac { 4kH_0^2 } { l_0^2 } } , \tag { 3 } \end{align*} \end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { K_ { y \varphi } } = - { K_y } h - { \frac { 4kr_c^2 \ { { \cos } ^2 } \beta H } { l_0^2 } } - { \frac { 2k { r_a } \left( { { r_a } - { r_b } } \right) \left( { H - { H_0 } } \right) } { { l^2 } } } . \tag { 4 } \end{align*} \end{document}

From Equations (2) and (3), the translational stiffness K_y and K_z are not functions of h. The proportionality coefficient of the coupling stiffness \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_{y \varphi }}$$ \end{document} , about the variable h, in Equation (4) is negative. From the mathematical aspect, when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_{y \varphi }} = 0$$ \end{document} , the DOFs (y, z, and φ) can be decoupled. This implies that the direction of independent eigenvectors of the stiffness matrix can be designed to be consistent with the corresponding DOF in the Cartesian coordinate. With the aid of the decoupling condition, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_{y \varphi }} = 0$$ \end{document} , we can obtain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { h^ * } = - H + { \frac { 2 \left( { H { r_b } - { H_0 } { r_a } } \right) \left( { { r_b } - { r_a } } \right) } { { { 4 { l^2 } r_c^2 \ { { \cos } ^2 } \beta } \mathord { \left/ { \vphantom { { 4 { l^2 } r_c^2 \ { { \cos } ^2 } \beta } { l_0^2 + 2 { { \left( { { r_a } - { r_b } } \right) } ^2 } } } } \right. \kern- \nulldelimiterspace } { l_0^2 + 2 { { \left( { { r_a } - { r_b } } \right) } ^2 } } } } } . \tag { 5 } \end{align*} \end{document}

When the height h = h*, this point is the compliance center^47,48 at which the 3 DOFs (y, z, and φ) are decoupled. A pure force along a certain DOF of parallel mechanisms at the compliance center will only produce a pure displacement, and a pure torque about the compliance center will give rise to a pure rotation.^47,48 With regard to the expression h*, the compliance center can be designed by changing kinematic parameters. For example, if r_a = r_b, the compliance center h* = −H.

With the aid of Equations (2) and (5), the rotational stiffness K_φ can be written as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { K_ \varphi } = { K_y } { \left( { h - { h^ * } } \right) ^2 } + { \frac { 8 { k^2 } { { \left( { H { r_b } - { H_0 } { r_a } } \right) } ^2 } \ r_c^2 \ { { \cos } ^2 } \ \beta } { { K_y } { l^2 } l_0^2 } } . \tag { 6 } \end{align*} \end{document}

The stiffness K_φ in Equation (6) has a quadratic form about (h − h*) and has a minimum rotational stiffness at h = h*.

With the aid of Equations (1) to (6), we can achieve the stiffness anisotropy at the compliance center. The rotational stiffness K_φ could be designed to be near zero and the translational stiffness should be higher. The rotation of φ would be main motion and our tensegrity joint can be equivalent to a rotational joint. Next, its virtual rotational center would be used to further explain this equivalence, since for each traditional rigid rotational joint, there exists a rotational center to describe its rotation.

The virtual rotational center of tensegrity joints

The compliance center (the point at h = h*) was regarded as the virtual rotational center of our tensegrity joint. We illustrated it from three aspects.

(1)

The rotation always appears at the compliance center, since \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${K_ \varphi }$$ \end{document} has a minimum rotational stiffness from Equation (6).

(2)

Although the translational stiffness of the tensegrity joint would be designed to be higher, there still exists a small translation caused by the translational stiffness at the compliance center when exposed to an external force. Thus, the actual rotational center would have a small deviation with respect to the compliance center, and is close to the compliance center. For ease, we still used the compliance center to describe the rotational center. For our tensegrity fish, the method is acceptable and helpful to describe the motion of the tensegrity joint.

(3)

The translation variation caused by the coupling stiffness can be represented as a rotation at the compliance center. From Equations (1) to (6), the variation of the translation at h \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\ne$$ \end{document} h* caused by the coupling stiffness, with respect to the translation caused by the translational stiffness, is related to a quadratic term about (h − h*). For our tensegrity joints, the contribution of the translation would be limited and the rotation is mainly due to the stiffness anisotropy design. Thus, it is acceptable that this translation variation can be approximately linear to the term (h − h*), and the linear approximation would make the translation variation be equivalent to a rotation. As shown in Figure 4, the force F is applied at arbitrary h and curve 1 represents the translation variation with the point of force application, with respect to the compliance center. A rotation at the compliance center can be used to represent these translations at arbitrary h. Therefore, when meeting the linear approximation, it is suitable and right to regard the compliance center as the equivalent rotational center of the tensegrity joint.

FIG. 4.

Schematic of the translation caused by the coupling stiffness. Curve 1 represents the actual translational variation by changing the position h of applied force. Color images are available online.

The compliance center, that is, the virtual rotational center of the tensegrity joint, was calculated at the neutral pose. The reasons for adopting this method include the reasons (a)–(c) for formulating the stiffness at the neutral pose. Moreover, the compliance center changes a little since the rotation of each joint is small.

Design of the Tensegrity Fish

After introducing the structure of tensegrity fish body, we would introduce the overall structure, the stiffness design, and the mechatronics design of our tensegrity robotic fish, while the detailed dimensions would be given.

Overall structure

The schematic of tensegrity robotic fish is shown in Figure 5A, and the tensegrity robotic fish is composed of three parts in Figure 5B. The first one (from left to right) is a rigid head; the middle part is a fish body, which is composed of six tensegrity joints; and a passive caudal fin attached to fish body. Similar to the arrangement of the mechanism in Figure 3, Platform i and Platform (i + 1) are connected with tensegrity joints. Platform 1, an oval plate, supports the robotic fish head.

FIG. 5.

Tensegrity robotic fish. (A) Schematic; (B) the overview; (C) schematic of actuation; (D) the fish body at neutral state; (E) the fish body at deformed state. Color images are available online.

The nominal length of the compliant fish body is 178 mm and that of the head is 138 mm. However, for each tensegrity joint, there exists H₀ due to the resistance of tension-loaded limbs (elastic strings) in the tensegrity joints. It would reduce the length of the fish body, thus the actual length of fish body is reduced from 178 to 148 mm. The actual length of compliant fish is reduced from 420 to 390 mm.

In Figure 5C, a servomotor in the rigid head actuates the compliant body by pulling two cables attached to the sixth tensegrity joint located at a nominal distance of 275 mm from the snout.

Stiffness design

We exemplified the parameter definition with the follower in Figure 3, showing the parameters of Platform i = 1 − 7. The architecture of each platform can be fully defined by the parameters (k, R_a, R_a₀, R_b, R_b₀, H, H₀, r_a, r_e, and r_d). The stiffness of limb, k, is about 0.025 N/mm. In Figure 3C, R_b and R_b₀ are the external and internal radius at semimajor axis (x-axis) of the platform, and R_a and R_a₀ are the external and internal radius at semiminor axis (y-axis). These four parameters (R_a, R_a₀, R_b, and R_b₀) are roughly designed according to the biological parameters of subcarangiform and carangiform fish. According to the relationship between stiffness anisotropy and DOF, the rest parameters (H, r_a, r_e, and r_d) are determined. Finally, H₀ is a result after tensing the tensegrity joint by the axial tension network, and it can be measured.

In summary, their parameters are shown in Supplementary Table S1.

Figure 5D shows the tensegrity fish body at neutral pose and Figure 5E shows a deformed state of the tensegrity fish body. Our tensegrity fish can achieve a wide range of motion.

According to Supplementary Table S1, we calculated the stiffness at the corresponding compliance center, at h = h*. The stiffness and h* are given in Supplementary Table S2.

It should be noted that the translational stiffness is not high due to small stiffness of limbs. This can be solved since the tensegrity joints can be in tension and produce the internal force to increase the translational stiffness. In detail, the translational stiffness would be increased by the preload of the limbs at the neutral pose.

Mechatronics design

The rigid head and platforms of tensegrity joints were fabricated with acrylonitrile butadiene styrene (ABS) material by the 3D printer. The fin was cast using silicon of average elasticity. A rigid head accommodated the electronics and a waterproof servomotor Jx6221 capable of 2.74 Nm torque at 7.4 V. The electronics, in a seal bag for isolating them from the water, included a microcontroller unit STC89C52, a wireless transmitter PT2272, power regulators, and two 7.4 V rechargeable batteries. To control the servomotor, an open-loop controller was designed. The controller was coded into the microcontroller unit, which could communicate with wireless transmitter and receiver. The microcontroller unit and the servomotor were powered by batteries, respectively. A compliant artificial skin covered the robot to displace the water during fish swimming. The fish body parts were flooded with water and kept at neutral buoyancy.

Results and Discussions

We varied different driving frequencies f (Hz) and driving amplitudes A₀ (degree) of the servo motor to study the swimming performance of tensegrity robotic fish, including its tail amplitude, swimming velocity, stride length, Strouhal number, and so on.

Video processing

Our robotic fish was put into a still water tank to do experiments, and the tank has a calibration ruler at the right side of Figure 6A. An OV7725 camera was used to capture these experiments at shutter speed of 30 frames per second and send data to a computer. The video was then transformed into individual frames and imported into Matlab where the images could be analyzed. The image in Figure 6A was converted to black and white by the binary progress (white background and black fish in Fig. 6B). After that, we can obtain the position where the tensegrity robotic fish is on the image. As shown in Figure 6C, the midline of the robotic fish from the snout point to the tail can be traced. The code* recorded the position of the midline points (30 points) and obtained the body wave of the tensegrity robotic fish (in Fig. 11), which reflects the relationship between its midline motions and the position along midline. Finally, the time-averaged swimming performance of the tensegrity fish can be calculated.

FIG. 6.

Descriptions of video processing. (A) The original snapshot. (B) The binary image. (C) The fitting curve. (D) The processed image. The x and y indicate the directions of coordinates. Color images are available online.

The tail amplitude and swimming velocity of robotic fish

The tail amplitude A (BL, i.e., body length) was defined as a peak-to-peak distance of midline motions⁴⁵ at the tail tip. The relationship between the driving frequencies and tail amplitudes for different driving amplitudes of the servomotor is shown in Figure 7A. The tail amplitudes A decreased with increasing driving frequency, and the tail amplitudes increased as the driving amplitudes increased. When the driving frequency was 1.1 Hz and the driving amplitude was 63°, the maximum tail amplitude was about 0.26 BL.

FIG. 7.

The swimming performance of the tensegrity robotic fish with varying driving frequencies f and driving amplitudes A₀. (A) The tail amplitude. (B) The swimming velocity. U is average swimming velocity data. There exists different individual trials for same frequency, and the mark of lines represents means. Color images are available online.

Figure 7B shows average swimming velocity data U (BL/s) for different driving amplitudes A₀ and frequencies f. The swimming velocity first increased and then decreased with the increase of driving frequency. The swimming velocity was larger for larger driving amplitude, except for driving amplitude of 63°. As shown in Figure 7A and B, the swimming velocity and the tail amplitude of driving amplitude of 63° are almost the same as the ones of driving amplitude of 54°. It has indicated that once the driving amplitude exceeds 54°, increasing the driving amplitude will not further increase the tail amplitude and the swimming velocity. At the frequency of 2.35 Hz and driving amplitude of 63°, the maximum swimming velocity of the robotic fish reaches about 0.7 BL/s. In addition, at the frequency of 1.72 Hz and driving amplitude of 54°, the swimming velocity of the robotic fish also reaches about 0.7 BL/s. This raw video is offered in Supplementary Movie S1.

The stride length

The stride length U* (BL/cycle), U* = U/f,⁴⁹ which is the distance travelled during one tail beat period, can be used to characterize the swimming velocity⁵⁰ and can reflect the swimming performance of our tensegrity robotic fish. As shown in Figure 8, the stride length of the tensegrity fish decreased with increasing driving frequency after an initial increase, and it increased with increasing driving amplitude of servomotor. At the driving frequency of 1.2 Hz and driving amplitude of 54°, the maximum stride length U* of the robotic fish reached about 0.5 BL/cycle and it is close to the biological data of 0.71 BL/cycle.⁵¹ A sequence of snapshots of the tensegrity fish for the frequency of 1.2 Hz and driving amplitude of 54° are demonstrated in Figure 9A–I for one cycle period, while the snapshot interval was about \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ \frac { 1 } { 8 } $$ \end{document} cycle. This raw video is offered in Supplementary Movie S2.

FIG. 8.

The stride length U* of the tensegrity fish with varying driving frequencies f and driving amplitudes A₀. Color images are available online.

FIG. 9.

Sequences depicting the tensegrity fish at 1.2 Hz and A₀ = 54° performing one cycle (A–I).

The Reynolds number and the Strouhal number

In Figure 10A, the Reynolds number, Re = LU/v, was roughly between 1 × 10⁵ and 2.8 × 10⁵ where L is a characteristic length, that is, the length of fish body, and v = 10⁻⁶ m²·s⁻¹ is the kinematic viscosity of water at room temperature. For BCF swimmers, the inertial force plays a leading role and the reference range of Reynolds number⁵ is 10³–10⁶. Our results fitted the biological data.

FIG. 10.

Swimming performance of the tensegrity robotic fish with varying driving frequencies f and driving amplitudes A₀. (A) The Reynolds number. (B) The Strouhal number. Re indicates the Reynolds number, and St indicates the Strouhal number. Color images are available online.

The Strouhal number⁵⁰ St = fA/U is used to dominate the hydrodynamic performance where A is the wake width (usually approximated as the tail beat peak-to-peak amplitude⁵). For the greatest number of fish, the efficiency of fish increases as the Strouhal numbers decreases, and Triantafyllou concluded that many live fish have a preferred Strouhal number 0.25–0.35 to achieve maximum efficiency.⁵² However, Eloy illustrated that the swimming performance of the swimmers, like Strouhal number, are also related to their body shape.⁵⁰ Therefore, we should concentrate more on the Strouhal numbers of the subcarangiform and carangiform fish, while our tensegrity fish was modeled by this fish family. The Strouhal numbers of Rainbow trout are 0.26, 0.38, and 0.25.^53,54 For Lake sturgeon, the Strouhal number is 0.48,⁵⁵ and for West African lungfish, the numbers are 0.75 and 1.02.⁵⁶ For Gold fish, it is 0.40, 0.44, 0.47, and 0.54.⁵⁷ The Strouhal numbers for our tensegrity robotic fish, as shown in Figure 10B, were roughly between 0.45 and 0.55. This implies that our Strouhal numbers for tensegrity fish were close to the biological data.

The vibrational mode

To demonstrate the achieved vibrational modes of the tensegrity fish, we used the reconfiguration approach advocated by Root and Long,^58,59 to decompose the whole body kinematics of fish swimming into the periodic and secular components. The periodic components are represented by Fourier series, and the secular components are represented by the term of independent velocities and accelerations of all body points.^58,59 The x and y components of raw data about body wave of the tensegrity fish should be rotated so that the average velocity vector points in the positive x direction. New axial direction is labeled as u, and new lateral direction is labeled as v. In a period of time t, the motion of a single body point, on the midline of the robotic fish during swimming, in u − v space can be described as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} { \xi \left( t \right) = { \xi _0 } + ve { l_ \xi } t + \displaystyle \frac { { ac { c_ \xi } } } { 2 } { t^2 } + \mathop \sum \limits_ { i = 1 } ^n { { A_i } } \cos \left( { i \omega t - { \phi _i } } \right) } & { \left( { \xi { \rm { = } } u , v } \right) } \end{matrix} \tag {7} \end{align*} \end{document}

where ξ₀ is the initial position of ξ(t), vel_ξ is the secular velocity, acc_ξ is the secular acceleration of body point, and w is angular frequency of robotic fish, ω = 2πf. Here, the quantity n = 2 (it can be higher order) indicates the order of Fourier series, including the fundamental and first harmonic. A₁, A₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \phi _1}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \phi _2}$$ \end{document} represent the amplitudes and phase lags for the fundamental and first harmonic frequencies, respectively. These harmonic coefficients in Equation (7) can be calculated by using a matrix H_c, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {H_c} = \left[ { \begin{matrix}1 & t & {{t^2} / 2} & { \cos \left( { \omega t} \right) } & { \sin \left( { \omega t} \right) } & { \cos \left( {2 \omega t} \right) } & { \sin \left( {2 \omega t} \right) } \\ \end{matrix} } \right] \tag{8} \end{align*} \end{document}

where H_c is a function in each column over the time span, 0…..t.

The harmonic coefficient matrix C for each body point is the least square estimates found by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} C{ \rm{ = }}{ \left[ { \begin{matrix}{{ \xi _0}} & {ve{l_ \xi }} & {ac{c_ \xi }} & {{A_1} \cos { \phi _1}} & {{A_1} \sin { \phi _1}} & {{A_2} \cos { \phi _2}} & {{A_2} \sin { \phi _2}} \\\end{matrix} } \right] ^T} = { \left( {H_c^T{H_c}} \right) ^{ - 1}}H_c^T{ \bf{ \xi }} \tag{9} \end{align*} \end{document}

where ξ represents the matrix of all the body points over the time span, 0…..t. We can use the coefficient of determination R² to show the quality of fit at each body point in u − v space.

Now, we have 60 functions (30 points on the midline, and two coordinates, u and v) of the form given in Equation (7), to describe the dominant vibrational mode of our tensegrity fish. Based on this method, we concluded that two different vibrational modes can be excited by a single harmonic input for our tensegrity robotic fish. We exemplified them with two typical cases. The first case is for the lower vibrational mode, and the driving frequency is 1.2 Hz for driving amplitude of 54°, corresponding to the frequency of the maximum stride length. The second case is for the higher vibrational mode, and the driving frequency is 2.9 Hz for driving amplitude of 63°.

The midline motions of the tensegrity robotic fish for these two cases are first shown in Figure 11A and B, respectively. The first case has 11 overlapping cycle periods in trial and the second case has 61 overlapping cycle periods in trial. The unsteadiness index (UI) was used to assess the steadiness of fish swimming.⁵⁸ It is a weighted standard deviation of the RE, where RE is the relative error between the average velocity (means of the average velocities of all body points) and the average velocity of each body point. For the first case, UI is about 0.07, and for the second case, UI is about 0.14. The swimming of robotic fish for two cases has achieved steady state since Root and Long illustrated that the fish motion is steady when UI <0.5.⁵⁸

FIG. 11.

Midline motions of two cases. (A) The first case. (B) The second case. The direction and length of red arrow, next to the initial midline, indicate the direction and amplitude of the average velocity vector for that trial. It should be noted that the superimposed midlines have been displaced laterally to demonstrate the propagation of body wave. Color images are available online.

The harmonic structure of swimming kinematics for the two cases can be plotted in Figure 12A and B, respectively, by using the reconfiguration approach presented by Root and Long.⁵⁸ The figures detailed the structure of the lateral and axial motion of each body point, using harmonic coefficients. Comparing the structures of fundamental harmonic frequency in Figure 12A and B, the mode shape of the body wave and its decomposition at frequency 2.9 Hz were different from one to another case, which can be reflected by the changes of phase and amplitude. From the head point to the tail point (1–30), for the first case, the number of wave is about 0.97 by computing total phase difference (1.95π) and dividing by 2π. For the second case, the total phase difference is about 2.72π, and the number of wave is about 1.36. From the head point to the tail point, the amplitude of the first case has only one local minimum. The amplitude of the second case has two local minimum, on which are about 1/3 and 2/3 robotic fish length.

FIG. 12.

Harmonic structures of the lateral and axial direction for two cases. (A) The first case. (B) The second case. Fund indicates fundamental harmonic frequency, and first means the first harmonic frequency of Fourier series, and they are represented by circles. Body pts indicates the body points from head to tail (1–30). Vel (black square) and Acc (black triangle) indicate the point's secular velocity and acceleration. The area of colorful circle (Fund and first), black square (Vel), and black triangle (Acc) indicates their relative amplitudes in each trial. For Fund and first frequency, the colors of circle indicate their phases. The shape of the black square (Vel) indicates each direction of a body point, positive (sides parallel square) and negative (diagonal sides square). The direction of the black triangle (Acc) indicates each direction of a body point, positive (upward) and negative (downward). R² indicates the quality of fit for each point in axial and lateral direction, and it can be read by the magnitude of the bar. If the value R² > 0.50, the bar is colored gray and read from the scale on the left. If the value R² < 0.50, the bar is colored red and read from the scale on the right. Color images are available online.

Conclusions

In this study, we developed a particular robotic fish linked with the parallel structures by the tensegrity joints, and this design approach eliminates limitations and absorbs the advantages of current approaches. The advanced actuation method by exploiting the natural modes of vibration is used in the tensegrity-compliant fish body, and simplifies the control techniques. Our tensegrity joint is composed of two Y-shaped structures of plate-foot with the horizontal tension network and the axial tension network. It can achieve a wide range of motion. The horizontal tension network in the tensegrity joint was used to support the joint in suspension, and mainly provides a high translational stiffness, while the axial tension network is used to tense the joint and reserve the antagonistic rotational stiffness at the bending direction of our robotic fish. This arrangement simplifies the design and is convenient to achieve the stiffness anisotropy. We illustrated the relationship between the stiffness anisotropy and DOF, and formulated the stiffness to have one rotational DOF for the tensegrity joint. The tensegrity joint can be regarded as a rotational joint. We confirmed that its virtual rotational center is approximately the compliance center observed in parallel mechanisms, and formulated its equations. We detailed the design of the first generation of prototype, including its overall structure, stiffness, and mechatronics.

For our tensegrity fish, the experimental results demonstrated that its preliminary swimming performances were close to real fish performances. The maximum stride length is about 0.5 BL/cycle near the biological data. The range of the Strouhal number is between 0.45 and 0.55 near the biological data. Based on the reconfiguration approach for body wave, we concluded that two different vibrational modes can be excited by a single harmonic input for our tensegrity robotic fish, and they are demonstrated by visualizing harmonic structure.

In the future, the tensegrity robotic fish body can achieve optimal stiffness distribution by changing the stiffness of each joint. This indicates that our tensegrity fish will likely improve its swimming performance after optimizing stiffness distribution. After supplementing the stiffness theory about the internal force, this tensegrity robotic fish would allow and help us to study the role of the variable stiffness on undulatory swimming of actual fish. It is interesting to understand the interactions between the variable stiffness of fish body and fluid environment. In particular, the further study of concrete role of tensegrity joints on internal damping should be in need with direct measurements of the efficiencies and damping.

Footnotes

Acknowledgments

The work presented in this article was funded by the National Natural Science Foundation of China under grant number 51275127. The authors wish to thank the anonymous reviewers for their comments, which have helped us improve the article.

Author Disclosure Statement

No competing financial interests exist.

Supplementary Material

*

The code is available on GitHub ().

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Supplementary Material

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