Soft dorsal/anal fins pairs for roll and yaw motion in robotic fish

Willam Coral; Claudio Rossi

doi:10.1088/1748-3190/aca132

1. Introduction 1.介绍

Various artificial fins for generating propulsion and turning (i.e., moving on the horizontal plane) in robotic fishes have been proposed in the literature. However, the bibliography on more complex motions, like roll and pitch, is still very limited.
用于产生推进力和转向的各种人造鳍（即，在水平面上移动）。然而，更复杂的运动，如滚动和俯仰的参考书目仍然非常有限。

This paper presents our recent work on 3D motion for underwater robots mimicking live fish. To this purpose, we have designed and developed soft actuated fins for controlling motion around the longitudinal axes of the robot (roll) and along the vertical axis (yaw), which mimic the function of a pair of soft dorsal/anal fins in live fish.
本文介绍了我们最近在模仿活鱼的水下机器人3D运动方面的工作。为此，我们已经设计和开发了软驱动鳍控制运动的机器人（滚动）和沿着垂直轴（偏航），模仿的功能，一对软背鳍/臀鳍在活鱼的纵轴周围。

Fish use this pair of fins, in combination with others, to maintain and change the orientation of their bodies, for vertical stabilization, and changes their orientation along the three axes. Such changes are needed in many fish, for instance to reach food or produce fast vertical movements, rolling their bodies and then performing fast C-start or S-start movements to escape from predators.
鱼用这对鳍，结合其他鳍，来保持和改变它们身体的方向，以实现垂直稳定，并沿着沿着改变它们的方向。许多鱼类都需要这样的变化，例如，为了达到食物或产生快速的垂直运动，滚动身体，然后执行快速的C启动或S启动运动以逃避捕食者。

Compared to forward locomotion and turning in the horizontal plane, few examples of artificial fins for maneuvering can be found in the literature. Rigid fins acting as paddles with a single degree of freedom that produces the roll movement in an autonomous underwater vehicle are presented in previous studies [1, 2]. Another study used a rigid anal fin for improving yaw stability [3].
与向前运动和在水平面内转向相比，在文献中可以找到用于操纵的人工鳍的例子很少。在以前的研究中提出了刚性鳍作为桨，具有单自由度，在自主水下航行器中产生滚动运动[1，2]。另一项研究使用刚性尾鳍来提高偏航稳定性[3]。

Most of the fins developed are actuated by wires [4, 5] and require large assemblies of several motors pulling each of these cables to achieve bending of the fins. Using a similar mechanics, Curet et al [6] reproduced the attitude control of the Knifefish by means of an undulating fin.
开发的大多数鳍片由线[4，5]致动，并且需要拉动这些线缆中的每一个的多个电机的大型组件来实现鳍片的弯曲。使用类似的机制，Curet等人 [6]通过波浪形鳍再现了鱼的姿态控制。

Flexible fins for a manta ray-inspired robotic fish that allow forward locomotion and pitch maneuvers are proposed in an earlier study [7]. In this work, the movements of the fins are produced by servomotors.
在早期的研究中提出了一种灵活的鳍，用于曼塔射线启发的机器鱼，允许向前运动和俯仰机动[7]。在这项工作中，鳍的运动由伺服电机产生。

Few studies have focused on soft dorsal and anal fins using smart materials. In particular, two studies have obtained important conclusions [8, 9]. In the first study, soft fluidic elastomer–based dorsal and anal fins are used. It is shown that the linear acceleration rate can be controlled by erecting and folding down the fins. In the second study, a sensory feedback system in each of the fin rays was added that allowed to mimic the biological fin nerves. The main difference between these works and the work presented here is that they concentrate on the sagittal plane (they can be erected and folded down) and do not have lateral motion. Here, we focus on lateral bending to produce transversal motion.
很少有研究集中在软背鳍和臀鳍使用智能材料。特别是，两项研究获得了重要结论[8，9]。在第一项研究中，使用了基于软流体生物的背鳍和臀鳍。结果表明，通过鳍的竖立和折叠，可以控制线加速度。在第二项研究中，在每个鳍条中添加了一个感觉反馈系统，可以模仿生物鳍神经。这些作品和这里介绍的作品之间的主要区别是，它们集中在矢状面（它们可以直立和折叠），没有横向运动。在这里，我们专注于横向弯曲产生横向运动。

In this work, we present a novel soft fin design, consisting of shape memory alloys (SMAs) embedded in silicone. Because actuators are embedded in the structure, the proposed design is lightweight, compact, and completely waterproof. The simplicity and compactness of the design is a distinguishing feature of the proposed fins (the comparison between figure 1 and figure 1 from [10] is advised).
在这项工作中，我们提出了一种新的软鳍设计，包括形状记忆合金（SMA）嵌入在硅树脂。由于致动器嵌入在结构中，因此所提出的设计重量轻、紧凑且完全防水。设计的简单性和紧凑性是所提出的翅片的显著特征（建议将图1与[10]中的图1进行比较）。

**Figure 1.** Left, center: Schematics of the SMA-based fin. Right: The fins bend because of the contraction of the SMA wire.
**图1.** 左中：基于SMA的鳍板示意图。右图：翅片由于SMA丝的收缩而弯曲。
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SMAs are not new to robotic fish. A number of different prototypes inspired by different animals, such as the manta ray, the lamprey, the tuna and the jellyfish, adopting SMAs as an actuation mechanism for propulsion have been proposed [11]. In [12, 13], SMAs have been used to generate body undulations for locomotion and turning in the horizontal plane. In [14], a robot imitating a manta ray is proposed that uses SMAs for 'flapping' movement. This work also focuses on forward locomotion and turning (yaw) maneuvers.
SMA对机器鱼来说并不新鲜。已经提出了许多不同的原型，这些原型受到不同动物的启发，例如曼塔、七鳃鳗、金枪鱼和水母，采用SMA作为推进的致动机构[11]。在[12，13]中，SMA已被用于生成用于在水平面中运动和转向的身体波动。在[14]中，提出了一种模仿曼塔的机器人，该机器人使用SMA进行“拍打”运动。这项工作还侧重于向前运动和转向（偏航）机动。

Here, we focus on soft dorsal/anal fin pairs for the 3D maneuvering of bio-inspired underwater robots, focussing on roll and yaw motions. As pointed out by Lauder, 'Dorsal fins play an active role in generating off-axis forces during maneuvering' [15]. The anal fin has a similar role, acting in coordination with the dorsal fin.
在这里，我们专注于软背鳍/臀鳍对生物启发的水下机器人的3D操纵，专注于滚动和偏航运动。正如Lauder所指出的那样，背鳍在机动过程中产生离轴力方面发挥着积极的作用' [15]。尾鳍也有类似的作用，与背鳍协调行动。

Pioneer studies on the role of fish fins can be found in [16–18]. More advanced studies using advanced technological tools can be found in [15, 19–24]. More specifically, fundamental biological studies that provide information and data about the function of the paired fins during locomotion of the fish can be found in [25–29].
关于鱼鳍作用的先驱研究可以在[16-18]中找到。使用先进技术工具的更先进的研究可以在[15，19-24]中找到。更具体地说，基础生物学研究提供了有关鱼类运动过程中成对鳍功能的信息和数据，可以在[25-29]中找到。

This paper is organized as follows. In the remainder of this section, we analyze related works on 3D maneuvering and fins for robot fish, as well as on SMA-based soft actuators. Section 2 describes the construction of the fins and the experimental setup. In section 3, we analyze the kinematics of the dorsal fin in water and report the experimental results obtained with the proposed prototype. Section 4 concludes the paper.
本文的组织结构如下。在本节的其余部分，我们分析了机器鱼的3D操纵和鳍，以及基于SMA的软致动器的相关工作。第2节描述了鳍的构造和实验装置。在第三节中，我们分析了背鳍在水中的运动学，并报告了所提出的原型所获得的实验结果。第四节是论文的总结。

2. Materials and methods 2.材料和方法

In this work, we focus on developing soft dorsal and anal fins that can move on both sides (left and right) of the fish body.
在这项工作中，我们专注于开发柔软的背鳍和臀鳍，可以在鱼体的两侧（左右）移动。

2.1. Mechanical design 2.1.机械设计

We base our design of fish fins on the largemouth bass (Micropterus salmoides): a carnivorous freshwater gamefish in the Centrarchidae (sunfish) family (figure 2). This species of fish reaches a maximum length of between 30 and 50 cm in adulthood. We base the size and shape of the artificial dorsal and anal fins on a largemouth bass with a length of 50 cm.
我们的鱼鳍设计基于大口黑鲈（Micropterus salmoides）：一种肉食性淡水斗鱼，属于太阳鱼科（图2）。这种鱼在成年期的最大长度为30至50厘米。我们根据人工背鳍和臀鳍的大小和形状，在一个大嘴鲈鱼与50厘米的长度。

**Figure 2.** Picture of the two prototypes. Two SMAs are positioned inside the fin. On each side, one SMA is folded a number of times corresponding to the number of fin rays in a dorsal and anal fin.
**图2.** 两个原型的照片。两个SMA位于鳍内。在每一侧，一个SMA折叠的次数对应于背鳍和臀鳍中鳍条的数量。
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In our design, two SMA wires are embedded within a fin structure. When the SMA wires are heated by means of electrical current, they contract, causing the bending of the structure. Figure 3 shows the principle.
在我们的设计中，两个SMA线嵌入在鳍结构。当SMA线通过电流加热时，它们收缩，导致结构弯曲。图3显示了原理。

**Figure 3.** Largemouth bass (*Micropterus salmoides* have). The fins are named as follows (from left to right): SPlf—Soft Pelvic Fin, SPcf—Soft Pectoral Fin, SpDf—Spiny Dorsal Fin, SoDF—Soft Dorsal Fin, SAnf—Soft Anal Fin, SCaf—Soft Caudal Fin.
**图3.** 大口黑鲈（*Micropterus salmoides*）。鳍的名称如下（从左到右）：SPlf-软骨盆鳍，SPcf-软胸鳍，SpDf-多刺背鳍，SoDF-软背鳍，SAnf-软肛鳍，SCaf-软尾鳍。
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Thus, contrary to other designs (e.g. [4]), the proposed fins do not use external cable-pulled mechanisms or classical servo-motors. The thickness of the fins is set at 3 mm to avoid unnecessary hydrodynamic disturbances in the steady swimming and maneuvers of the fish and also to reduce their stiffness, because too much stiff fins would need too much force to be bent.
因此，与其他设计（例如[4]）相反，所提出的鳍不使用外部电缆牵引机构或经典伺服电机。鳍片的厚度设定为3mm，以避免在鱼的稳定游动和机动中不必要的流体动力学扰动，并且还减小它们的刚度，因为太硬的鳍片将需要太大的力来弯曲。

To produce the movement of the fins in both directions, two SMA wires have been included on the two sides of the fins, so that they work in an antagonistic configuration. The arrangement of the SMAs is the same on both sides: one single SMA is arranged in 'zigzag', turning number times corresponding to the number of fin rays in a dorsal and anal fin (seven and six, respectively [22]). The bending direction of the fin is controlled by powering one of the two SMAs.
为了产生翅片在两个方向上的运动，在翅片的两侧上包括两个SMA线，使得它们以对抗配置工作。SMA的排列在两侧是相同的：一个单一的SMA被安排在“”中，转动次数对应于背鳍和臀鳍中的鳍条数量（分别为7和6）。鳍片的弯曲方向通过向两个SMA中的一个供电来控制。

Figures 1 and 3 show the two prototypes and the arrangement of the SMAs.
图1和图3显示了两个原型和SMA的布置。

The shape and size of the dorsal and anal fins were taken from an adult fish with a size of approximately 50 cm. After extracting the shape and size from the real fish fins, we converted the 2D drawings into a 3D model using the CAD software Autodesk Inventor $^{\circledR}$ $^{®}$ . Then, we created a 3D-printed mould for the fins. To make the fin bend on both sides upon the contraction of SMAs, it is necessary that each of the SMAs is precisely placed at 0.5 mm from the middle of the fin (figure 3, center.)
背鳍和臀鳍的形状和大小取自一条大约50厘米大小的成鱼。从真实的鱼鳍中提取形状和尺寸后，我们使用CAD软件Autodesk Inventor $^{\circledR}$ $^{®}$ 将2D图纸转换为3D模型。然后，我们为鳍创建了一个3D打印模具。为了使翅片在SMA收缩时在两侧弯曲，必须将每个SMA精确地放置在距翅片中部0.5 mm处（图3，中心）。

It is also necessary that the 'turning points' of the SMAs have the support to prevent it from slipping inside the fin when it is contracted. To this end, we used 3D-printed 'brackets' with very specific shape that allowed maintaining the wires in position.
SMA的“转折点”也必须具有支撑，以防止其在收缩时在鳍内滑动。为此，我们使用了3D打印的“支架”，其形状非常特殊，可以将电线保持在适当的位置。

The SMA wires used are NiTi alloys from Flexinol $^\circledR$ $^{®}$ , with a diameter of 150 µm and a nominal maximum current of 400 mA.
所使用的SMA线是来自Flexinol $^\circledR$ $^{®}$ 的NiTi合金，直径为150µ m，标称最大电流为400 mA。

The material used to make the fin was silicone rubber PlatSil $^{\circledR}$ $^{®}$ FS-20 brand Polytek Development Corp. The Shore Hardness of these silicone rubbers is A20 with a working time of 8 min and a cure time of 25 min. The molds for the fin castings were printed using the Anycubic $^{\circledR}$ $^{®}$ Photon Mono X UV Resin SLA 3D Printer.
用于制造翼片的材料是硅橡胶PlatSil $^{\circledR}$ $^{®}$ FS-20品牌Polytek Development Corp。这些硅橡胶的海岸硬度为A20，工作时间为8分钟，固化时间为25分钟。使用Anycubic $^{\circledR}$ $^{®}$ Photon Mono X UV Resin SLA 3D打印机打印散热片铸件的模具。

2.2. SMA control 2.2. SMA控制

A low-level control has been designed to control stroke speed and timing.
设计了一个低水平控制来控制冲程速度和定时。

To control the electronics that regulate the contraction of the SMA is a motor driver breakout board (MAX14870 Single-Brushed DC Motor Driver Carrier) adapted to drive the SMA through a PWM signal. This driver allows the measurement of the voltage and current applied to the SMA. In this way, by using the Ohm's law, we can estimate the value of the electrical resistance of the SMA, which depends on the actual length of the SMA. The resistance value is actually a position measure, which can be used for closed-loop control in the contraction of the SMA. The voltage and current signals are measured using an ADS1015 (12-bit, 3.3-kSPS, 4-channel, delta-sigma ADC with I2C) analog-to-digital converter. The entire system is controlled by the Teensy $^{\circledR}$ $^{®}$ 4.0 development board, which has an ARM Cortex-M7 microcontroller at 600 MHz that implements a relatively simple proportional-integral-derivative controller that can be experimentally tuned. The stroke time is determined by the SMA activation signal produced by the control electronics.
为了控制调节SMA收缩的电子器件，电机驱动器分线板（MAX 14870单刷直流电机驱动器载体）适于通过PWM信号驱动SMA。该驱动器允许测量施加到SMA的电压和电流。通过这种方式，通过使用欧姆定律，我们可以估计SMA的电阻值，这取决于SMA的实际长度。电阻值实际上是一个位置测量，可用于SMA收缩中的闭环控制。电压和电流信号使用ADS 1015（12位、3.3 kSPS、4通道、带I2C的Δ-Σ ADC）模数转换器测量。整个系统由Teensy $^{\circledR}$ $^{®}$ 4控制。0开发板，该开发板具有600 MHz的ARM Cortex-M7微控制器，该微控制器实现了一个相对简单的比例-积分-微分控制器，可以通过实验进行调整。行程时间由控制电子设备产生的SMA激活信号确定。

2.3. Experimental setup 2.3.实验装置

To measure and analyze the behavior of the anal and dorsal fins, we use spherical reflective markers of different colors (see figure 1). This allows us to follow the movement of the fin when it bends. To capture the movement of the fin and each of the markers, we used three ELP OmniVision OV4689 USBFHD08S-L36 high-speed cameras, operating at a sampling rate of 300 frames per second (1/300 s shutter speed) and 720P resolution. These three cameras allow us to make a 3D reconstruction of the movement of each marker of the fin. As a result, we have a realistic graphical representation of the movement of the fin (figure 4).
为了测量和分析臀鳍和背鳍的行为，我们使用不同颜色的球形反射标记（见图1）。这使我们能够在鳍弯曲时跟踪它的运动。为了捕捉鳍和每个标记的运动，我们使用了三台ELP OmniVision OV 4689 USBFHD 08 S-L36高速摄像机，以每秒300帧的采样率（1/300 s快门速度）和720 P分辨率运行。这三个摄像机使我们能够对鳍的每个标记的运动进行3D重建。因此，我们有一个现实的图形表示的运动鳍（图4）。

**Figure 4.** Examples of the bending obtained, dorsal fin. PWM duty cycle of 70% (current of 753 mA), stroke times of 0.033, 0.034, 0.035, 0.034 s, respectively.
**图4.** 弯曲的例子获得，背鳍。PWM占空比为70%（电流为753 mA），冲程时间分别为0.033、0.034、0.035、0.034 s。
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Additionally, forces and torques have been measured using a six-axis Nano17 IP65/IP68 force sensor from ATI Industrial Automation, Inc. installed directly at the base of the fins. The setup is shown in figure 5.
此外，还使用ATI Industrial Automation，Inc.的六轴Nano 17 IP65/IP68力传感器测量力和扭矩。直接安装在翅片的底部。设置如图5所示。

**Figure 5.** Rear view of the dorsal fin (a) unbent fin (b) maximally bent fin.
**图5.** 背鳍的后视图（a）未弯曲的鳍（B）最大弯曲的鳍。
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3. Experimental results 3.实验结果

To assess the performance of the fins, we performed several tests by varying the amplitude and speed of the strokes in still water. The amplitude of the stroke is determined by the duration of the activation signal, and the speed is determined by the intensity of the current fed to the SMA. The latter parameter is controlled by the PWM duty cycle of the power electronics driver. Four different values were used for the amplitude, and four values were used for the speed. Tables 1 and 2 summarize the parameters used in the tests. The terms Small, Mid, Big and Max refer to amplitudes of $10,20,30,40$ $10, 20, 30, 40$ degrees, respectively.
为了评估鳍的性能，我们通过改变静水中划水的幅度和速度进行了几次测试。冲程的幅度由激活信号的持续时间确定，并且速度由馈送到SMA的电流的强度确定。后一个参数由电力电子驱动器的PWM占空比控制。四个不同的值用于幅度，四个值用于速度。表1和表2总结了测试中使用的参数。术语“小”、“中”、“大”和“最大”分别指 $10,20,30,40$ $10, 20, 30, 40$ 度的振幅。

Table 1. Amplitude (time), current (mA), and speed (seconds) parameters applied in the tests to the soft dorsal fin.
表1. 在测试中应用于软背鳍的幅度（时间）、电流（mA）和速度（秒）参数。

			Stroke time (section) 行程时间（节）
	PWM duty PWM占空	Current 电流	Small 小	Mid 中期	Big 大	Max
Speed (mm s⁻¹) 速度（mm s⁻¹）	cycle (%) 周期（%）	(mA) （mA）	(10^∘) (10）	(20^∘) (20）	(30^∘) (30）	(40^∘) (40）
Slow (50) 慢（50）	60	633	0.034	0.036	0.039	0.034
Mid-Slow (100) 中慢（100）	70	753	0.033	0.034	0.035	0.034
Mid-Fast (150) 中快（150）	80	875	0.036	0.039	0.039	0.064
Fast (200) 快速（200）	90	1062	0.043	0.041	0.04	0.042

Table 2. Amplitude (time), current (mA), and speed (seconds) parameters applied in the tests to the soft anal fin.
表2. 振幅（时间）、电流（mA）和速度（秒）参数在测试中应用于软臀鳍。

Speed (mm s⁻¹) 速度（mm s⁻¹）	PWM duty cycle (%) PWM占空比（%）	Current (mA) 电流（mA）	Stroke time (section) 行程时间（节）
Speed (mm s⁻¹) 速度（mm s⁻¹）	PWM duty cycle (%) PWM占空比（%）	Current (mA) 电流（mA）	Small 小	Mid 中期	Big 大	Max
Slow (50) 慢（50）	60	577	0.133	0.165	0.167	0.168
Mid-Slow (100) 中慢（100）	70	691	0.111	0.114	0.124	0.152
Mid-Fast (150) 中快（150）	80	808	0.104	0.111	0.112	0.122
Fast (200) 快速（200）	90	975	0.080	0.083	0.086	0.105

It is important to note that due to the way the SMAs are positioned inside the fin, the way it bends is not completely straight from the top of the fin ray toward the center of the fish's axis. Instead, the fin is bent from the outer end toward the fin base. In this way, the fin rolls into itself. This effect is shown in figure 6.
重要的是要注意，由于SMA位于鳍内的方式，它弯曲的方式并不是从鳍条顶部向鱼的轴中心完全直的。相反，翅片从外端朝向翅片基部弯曲。以这种方式，鳍卷入自身。图6显示了这一效果。

**Figure 6.** Experimental system. (a) Applied Force and Torque Vector for the Nano 17 a six-Axis Force and Torque transducer. (b) Soft Dorsal Fin attached to the Nano17 transducer. For this fin, we use six coloured markers, which are located on the edge of the fin. With the help of high-speed cameras, we use these markers to trace the trajectory of the fin. (c) Experimental Setup. We use a 42 x 21 x 25 cm water tank where we submerge the fin attached to the force transducer. (d) Block diagram of the control and sensing system.
**图6.** 实验系统。(a)Nano 17六轴力和扭矩传感器的作用力和扭矩矢量。(b)连接到Nano17探头的软背鳍。对于这个鳍，我们使用六个彩色标记，它们位于鳍的边缘。在高速摄像机的帮助下，我们用这些标记来追踪鳍的轨迹。(c)实验设置。我们使用一个42 x 21 x 25 cm的水箱，我们将鳍片连接到力传感器上。(d)控制和传感系统框图。
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3.1. Kinematics 3.1.运动学

It is known that SMA wires only contract about $4\%$ $4 %$ of their length. However, contractions of up to $6\%$ $6 %$ can be obtained using higher currents. Thanks to the configuration used, however, the bending obtained in the fins is sufficient to produce a significant movement. Figure 7 shows an example of the bending obtained in the dorsal fin using the current duty cycle of $60\%$ $60 %$ for the four amplitudes (rear view), and figure 4 shows the trajectories of the markers (front view).
已知SMA线仅收缩其长度的约 $4\%$ $4 %$ 。然而，使用更高的电流可以获得高达 $6\%$ $6 %$ 的收缩。然而，由于所使用的配置，在翅片中获得的弯曲足以产生显著的运动。图7示出了针对四个振幅使用 $60\%$ $60 %$ 的电流占空比在背鳍中获得的弯曲的示例（后视图），并且图4示出了标记的轨迹（前视图）。

**Figure 7.** Trajectories of the markers, maximum bending amplitude.
**图7.** 标记轨迹，最大弯曲幅度。
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Regarding the speed of the movement, it was already shown [13, 30] that with the proper control of SMAs can be surprisingly fast, contrary to the reputation of the SMAs as slow actuators (see also [31, 32]). In fact, in the experiment with the highest currents, the up-stroke time can be as low as 0.042 s for the dorsal fin, and for the anal fin was 0.105 s. Thanks to the elastic property of the silicone, which acts as returning spring, the fastest down-stroke achieved (returning from the maximum contraction) was of 0.27 and 0.2 s, for an overall stroke time of 0.0389 and 0.12 s for the dorsal and anal fins, respectively.
关于运动的速度，已经表明[13，30]，通过SMA的适当控制，可以惊人地快，与SMA作为缓慢致动器的声誉相反（另见[31，32]）。事实上，在最高电流的实验中，背鳍的上升冲程时间可以低至0.042 s，而尾鳍为0.105 s。由于硅树脂的弹性，它作为返回弹簧，实现的最快下冲程（从最大收缩返回）为0.27和0.2秒，背鳍和臀鳍的总冲程时间分别为0.0389和0.12秒。

3.2. Forces and torques 3.2.力和转矩

To produce roll and yaw movements, the most important parameter is the force generated by the fins for the different amplitude/speed combinations. Tables 3–6 summarize the forces measured during the strokes in the X direction (see also figure 5(c)).
为了产生滚转和偏航运动，最重要的参数是由鳍在不同幅度/速度组合下产生的力。表3-6总结了在X方向上冲程期间测得的力（也参见图5（c））。

Table 3. Peak forces measured for the dorsal fin.
表3. 测量背鳍的峰值力。

	Amplitude 振幅
Speed (mm s⁻¹) 速度（mm s⁻¹）	Small (N) 小号（N）	Mid (N) 中（N）	Big (N) 大（N）	Max (N) 最大值（N）
Slow (50) 慢（50）	0.190	0.243	0.212	0.044
Mid-slow (100) 中慢（100）	0.25	0.315	0.308	0.054
Mid-Fast (150) 中快（150）	0.329	0.419	0.378	0.097
Fast (200) 快速（200）	0.422	0.542	0.544	0.116

Table 4. Average and RMS forces measured for the dorsal fin.
表4. 测量背鳍的平均力和RMS力。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0011	0.0075	0.0016	0.0101	0.0017	0.0089	0.0026	0.0046
Mid-Slow 中慢	0.0009	0.0096	0.0010	0.0130	0.0011	0.0119	0.0005	0.0038
Mid-Fast 中快	0.0009	0.0119	0.0010	0.0162	0.0012	0.0138	0.0001	0.0048
Fast 快速	0.0025	0.0162	0.0017	0.0228	0.0013	0.0204	0.0035	0.0068

Table 5. Peak forces measured for the anal fin.
表5. 测量臀鳍的峰值力。

	Amplitude 振幅
Speed (mm s⁻¹) 速度（mm s⁻¹）	Small (N) 小号（N）	Mid (N) 中（N）	Big (N) 大（N）	Max (N) 最大值（N）
Slow (50) 慢（50）	0.029	0.050	0.075	0.116
Mid-slow (100) 中慢（100）	0.040	0.062	0.113	0.167
Mid-Fast (150) 中快（150）	0.047	0.086	0.112	0.173
Fast (200) 快速（200）	0.085	0.108	0.137	0.179

Table 6. Average and RMS forces measured for the anal fin.
表6. 测量臀鳍的平均力和RMS力。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0001	0.0043	0.0008	0.0073	0.0021	0.0084	0.0029	0.0094
Mid-Slow 中慢	0.0005	0.0049	0.0020	0.0086	0.0037	0.0105	0.0049	0.0115
Mid-Fast 中快	0.0002	0.0067	0.0016	0.0112	0.0034	0.0135	0.0037	0.0112
Fast 快速	0.0017	0.0110	0.0003	0.0148	0.0004	0.0146	0.0006	0.0099

It is interesting to note that, in general, slow and fast movements generate less force, and bigger movements generate higher forces. We can conclude that medium speed, big movements are the ones having the best performance.
值得注意的是，一般来说，缓慢和快速的运动产生较小的力，而较大的运动产生较高的力。我们可以得出结论，中速，大动作是具有最好的性能。

However, it is worth noting that for the purpose of motion, any combination of stroke speed/amplitude can be used by a high-level attitude control for choosing specific combinations of them for specific target speed and amplitude of yaw and roll maneuvers. Figures 8 and 9 show the interpolated maps of the measured forces. More details of the results can be found in the appendix.
然而，值得注意的是，为了运动的目的，行程速度/幅度的任何组合都可以由高级姿态控制来使用，以针对偏航和滚转机动的特定目标速度和幅度来选择它们的特定组合。图8和图9显示了测量力的插值图。有关结果的更多详细信息，请参见附录。

**Figure 8.** Anal fin. Peak and total force generated for the four amplitudes and three speeds.
**图8.** 肛门鳍四种振幅和三种速度下产生的峰值力和总力。
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**Figure 9.** Dorsal fin. Peak and total force generated for the four amplitudes and three speeds.
**图9.** 背鳍。四种振幅和三种速度下产生的峰值力和总力。
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Such forces, applied to a neutrally buoyant body at the position of the fins in the real fish, can produce significant torques to produce roll motion if applied in opposite directions, and yaw motion if applied in the same direction (see figure 10). In fact, the body mass for a neutrally buoyant Largemouth bass of 50 cm would be approximately 5.5 kg. The point of application of the forces, assuming it is applied to the center of the fin, would be 26 and 27 cm from the longitudinal axis for the anal and dorsal fins, respectively, and 6 cm from the center of mass in the longitudinal axis, giving torques in the range of 7.012 Nmm for the roll movement, 2.051 Nmm for the yaw movement, and 4.609 Nmm for the roll movement.
这种力施加到真实的鱼鳍位置处的中性浮力体上，如果沿相反方向施加，则会产生显着的扭矩，从而产生滚动运动，如果沿相同方向施加，则会产生偏航运动（见图10）。事实上，体重为50厘米的中性浮力大嘴鲈鱼将约5.5公斤。假设力施加到鳍的中心，力的施加点分别距臀鳍和背鳍的纵向轴线26和27 cm，并且距纵向轴线上的质心6 cm，对于横摇运动给出的扭矩在7.012 Nmm的范围内，对于偏航运动给出的扭矩在2.051 Nmm的范围内，滚动运动为4.609 Nmm。

**Figure 10.** Application of the forces to a fish body. The red arrows indicate the distance vector of the generated thrust.
**图10.** 力对鱼体的作用。红色箭头指示所产生推力的距离矢量。
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It is worth noting that the motion of the fins generates torques in all three directions. However, one of them (Z axis, i.e. roll) is always significantly bigger than the others, with the exception of the case of anal fin for maximum bending. Forces in other directions can be balanced out or used for maneuvering in combination with other fins. In fact, in fish maneuvering, all the fins are normally employed to generate thrust in the desired directions and stabilize the movement [1].
值得注意的是，鳍的运动在所有三个方向上产生扭矩。然而，其中一个（Z轴，即滚动）总是明显大于其他的，除了最大弯曲的臀鳍的情况。其他方向的力可以被平衡或与其他鳍组合用于机动。事实上，在鱼类操纵中，所有的鳍通常用于在期望的方向上产生推力并稳定运动[1]。

Finally, table 7 shows a comparison between the forces generated by our dorsal and anal fins and the forces measured by Drucker and Lauder [33] for the dorsal fin of the Bluegill Sunfish fish, a Teleost fish like the Largemouth Bass (although from a different family). It also reports the data of the artificial dorsal fin from [10]. It must be pointed out that such a comparison is only indicative because of the different (although similar) specimens, sizes and setups used. The Sunfish data refer to a fish roughly 50% smaller than the Largemouth Bass. The size of the dorsal fin from [10] is approximately three times that of ours (6960 mm² vs. 2389 mm²). Nonetheless, the table shows how the forces generated by our dorsal fin are a single order of magnitude bigger than the live fish (see also table 3) and about five times bigger than the artificial fin from [10].
最后，表7显示了我们的背鳍和臀鳍产生的力与德鲁克和劳德[33]测量的蓝鳃太阳鱼背鳍的力之间的比较，蓝鳃太阳鱼是一种像大嘴鲈鱼一样的硬骨鱼（尽管来自不同的家庭）。并报道了文献[10]中人工背鳍的数据。必须指出的是，这种比较只是指示性的，因为不同的（虽然相似）标本，尺寸和设置使用。太阳鱼的数据指的是一种比大嘴鲈鱼小大约50%的鱼。[10]中背鳍的大小大约是我们的三倍（6960 mm² vs. 2389 mm²）。尽管如此，该表显示了我们的背鳍产生的力比活鱼大一个数量级（见表3），比[10]中的人造鳍大五倍。

Table 7. Comparison of forces between artificial and biological dorsal fins. For our fin, the data correspond to the best performance (Fast/Big speed/amplitude movement).
表7. 人工背鳍与生物背鳍受力之比较。对于我们的鳍，数据对应于最佳性能（快速/大速度/幅度运动）。

	Our 我们	Our 我们	Biological 生物	Artificial fin 人造鳍
	Dorsal fin 背鳍	Anal fin 臀鳍	fin [33] 鳍[33]	from [10] 从[10]
Duration of the movement (ms) 运动持续时间（ms）	86	168	261	175 (approx) 175（约）
Force, lateral component, swimming (mN) 力，横向分量，游泳（mN）	—	—	8.7	108.0
Force, lateral component, turning (mN) 力，横向分量，转动（mN）	544	116	11.2	—

4. Conclusions 4.结论

In this paper, we have presented two soft, actuated fins for controlling the roll and yaw motions of underwater robots that mimic the function of a pair of soft dorsal/anal fins in live fish.
在本文中，我们已经提出了两个软，驱动鳍控制的滚动和偏航运动的水下机器人，模仿一对软背鳍/臀鳍在活鱼的功能。

A key feature of our design is its mechanical simplicity. Because the actuators are embedded in the structure, there is no need for additional mechanical elements, such as pulleys and servomotors, to actuate the fins. Also, there is no need for additional mechanical sensors because SMAs can also be used as sensors.
我们设计的一个关键特点是其机械简单性。由于致动器嵌入在结构中，因此不需要额外的机械元件（例如滑轮和伺服电机）来致动鳍。此外，不需要额外的机械传感器，因为SMA也可以用作传感器。

We show how the proposed lightweight, compact and waterproof design can generate forces comparable to the one of the live fish and that such forces are indeed sufficient for maneuvering.
我们将展示如何拟议的轻量化，紧凑和防水的设计可以产生的力量与活鱼之一，这样的力量确实是足够的机动。

An additional feature of the dorsal/anal fin pair is that, when bent as rudders during steady swimming, they allow vertical stabilization, rotation and in-cruise turning. We are currently performing experiments to quantify such effects.
背鳍/尾鳍对的另一个特点是，当在稳定的游泳过程中弯曲为舵时，它们允许垂直稳定，旋转和巡航转向。我们目前正在进行实验，以量化这种影响。

The next step in our research is to use a similar concept for designing caudal fins and pectoral fins to provide a full set of bio-inspired soft appendices for complex fin-based 3D maneuvering in robot fish and other underwater robots. The forces generated for different combinations of speed and amplitude of the strokes can be used by a high-level motion controller to produce a desired 3D trajectory by means of proper sequences of fin strokes, exactly how live fish do.
我们研究的下一步是使用类似的概念来设计尾鳍和胸鳍，为机器鱼和其他水下机器人提供一整套生物灵感的软附件，用于复杂的基于鳍的3D操纵。高级运动控制器可以使用不同速度和幅度组合产生的力，通过适当的鳍击顺序产生所需的3D轨迹，就像活鱼一样。

In the long term, our aim is to let robot fish leave the 1D and 2D environments they are currently confined in.
从长远来看，我们的目标是让机器鱼离开它们目前所处的1D和2D环境。

Data availability statement
数据可用性声明

The data that support the findings of this study are openly available at the following URL/DOI: https://github.com/williamcoral/Soft-Dorsal-Anal-Fins.git.
支持本研究结果的数据可在以下URL/DOI上公开获得：https://github.com/williamcoral/Soft-Dorsal-Anal-Fins.git。

Appendix.: Details of forces and torques generated
附录：产生的力和扭矩的详细信息

In this section, we report more details and plots of the measured forces and torques.
在本节中，我们将报告测量力和扭矩的更多细节和图表。

**Figure A1.** Detail of the forces generated at *slow* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A1.** 四个振幅（从左到右：小、中、大和最大振幅）在低速下产生的力的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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**Figure A2.** Detail of the Forces generated at *mid-slow* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A2.** 四个振幅（从左到右：小、中、大和最大振幅）*中慢速*产生的力的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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**Figure A3.** Detail of the Forces generated at *mid-fast* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors
**图A3.** 四个振幅（从左到右：小、中、大和最大振幅）*中高速下*产生的力的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者
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**Figure A4.** Detail of the forces generated at *fast* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A4.** 四种振幅（从左到右：小、中、大和最大振幅）下快速产生的力的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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Table A1. Peak torques measured for the dorsal fin.
表A1. 测量背鳍的峰值扭矩。

	Amplitude 振幅
Speed (mm s⁻¹) 速度（mm s⁻¹）	Small (N mm) 小号（N mm）	Mid (N mm) 中间（N mm）	Big (N mm) 大（N mm）	Max (N mm) 最大值（N mm）
Slow (50) 慢（50）	1.998	2.725	2.590	1.761
Mid-slow (100) 中慢（100）	2.578	3.941	3.720	1.885
Mid-Fast (150) 中快（150）	3.352	4.642	4.501	2.126
Fast (200) 快速（200）	4.329	5.728	6.252	2.260

Table A2. Average and RMS forces measured for the dorsal fin, Y axis.
表A2. 测量背鳍的平均力和RMS力，Y轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0027	0.0047	0.0046	0.0071	0.0053	0.0088	0.0089	0.0126
Mid-Slow 中慢	0.0010	0.0047	0.0006	0.0065	0.0012	0.0074	0.0042	0.0100
Mid-Fast 中快	0.0048	0.0074	0.0034	0.0083	0.0019	0.0078	0.0001	0.0083
Fast 快速	0.0004	0.0071	0.0002	0.0095	0.0001	0.0094	0.0018	0.0083

Table A3. Average and RMS forces measured for the dorsal fin, Z axis.
表A3. 测量背鳍的平均力和RMS力，Z轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0018	0.0272	0.0039	0.0345	0.0060	0.0319	0.0116	0.0162
Mid-Slow 中慢	0.0073	0.0336	0.0089	0.0509	0.0046	0.0412	0.0022	0.0117
Mid-Fast 中快	0.0047	0.0426	0.0058	0.0604	0.0007	0.0500	0.0048	0.0183
Fast 快速	0.0082	0.0559	0.0125	0.0819	0.0066	0.0729	0.0025	0.0207

Table A4. Average and RMS torques measured for the dorsal fin, X axis.
表A4. 测量背鳍的平均和RMS扭矩，X轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.05 307	0.08 563	0.10 668 6	0.17 031	0.14 073	0.25 187	0.21 218	0.35 309
Mid-Slow 中慢	0.01 539	0.06 746	0.05 446 0	0.15 139	0.09 144	0.23 594	0.14 858	0.31 580
Mid-Fast 中快	0.03 613	0.09 029	0.01 416 5	0.17 043	0.05 649	0.23 590	0.08 632	0.27 136
Fast 快速	0.02 477	0.11 393	0.05 020 7	0.20 042	0.06 314	0.26 275	0.10 854	0.30 057

Table A5. Average and RMS torques measured for the dorsal fin, Y axis.
表格A5. 测量背鳍的平均和RMS扭矩，Y轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0939	0.2012	0.1514	0.3385	0.1744	0.3984	0.2091	0.4198
Mid-Slow 中慢	0.0900	0.2355	0.1256	0.3751	0.1472	0.4381	0.1813	0.4250
Mid-Fast 中快	0.0593	0.2717	0.1193	0.4496	0.1277	0.4937	0.1408	0.4167
Fast 快速	0.1484	0.3845	0.1965	0.5897	0.1916	0.6602	0.1767	0.4462

Table A6. Average and RMS torques measured for the dorsal fin, Z axis.
表A6. 测量背鳍Z轴的平均扭矩和RMS扭矩。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0597	0.3273	0.0439	0.4056	0.0364	0.3770	0.0363	0.1513
Mid-Slow 中慢	0.0414	0.3985	0.0650	0.5702	0.0732	0.5052	0.0567	0.1670
Mid-Fast 中快	0.0076	0.4997	0.0185	0.6917	0.0276	0.5697	0.0047	0.2408
Fast 快速	0.0268	0.6635	0.0609	0.9044	0.0840	0.8421	0.0183	0.2711

Table A7. Peak torques measured for the anal fin.
表A7. 臀鳍的最大扭矩。

	Amplitude 振幅
Speed (mm s⁻¹) 速度（mm s⁻¹）	Small (N mm) 小号（N mm）	Mid (N mm) 中间（N mm）	Big (N mm) 大（N mm）	Max (N mm) 最大值（N mm）
Slow (50) 慢（50）	1.062	1.486	1.414	1.130
Mid-slow (100) 中慢（100）	1.353	1.930	1.708	1.604
Mid-Fast (150) 中快（150）	1.736	2.809	2.655	1.724
Fast (200) 快速（200）	3.206	3.595	3.183	1.794

Table A8. Average and RMS forces measured for the anal fin, Y axis.
表A8. 测量臀鳍的平均力和RMS力，Y轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0004	0.0010	0.0002	0.0023	0.0004	0.0052	0.0009	0.0094
Mid-Slow 中慢	0.0022	0.0023	0.0000	0.0031	0.0020	0.0070	0.0036	0.0129
Mid-Fast 中快	0.0013	0.0019	0.0039	0.0051	0.0056	0.0084	0.0069	0.0136
Fast 快速	0.0021	0.0036	0.0036	0.0056	0.0040	0.0093	0.0037	0.0130

Table A9. Average and RMS forces measured for the anal fin, Z axis.
表A9. 在Z轴上测量臀鳍的平均力和RMS力。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0003	0.0043	0.0007	0.0109	0.0030	0.0198	0.0039	0.0299
Mid-Slow 中慢	0.0040	0.0056	0.0125	0.0176	0.0177	0.0289	0.0216	0.0435
Mid-Fast 中快	0.0022	0.0045	0.0107	0.0171	0.0149	0.0263	0.0185	0.0412
Fast 快速	0.0026	0.0078	0.0060	0.0166	0.0067	0.0271	0.0068	0.0388

Table A10. Average and RMS torques measured for the anal fin, X axis.
表A10. 测量臀鳍的平均和RMS扭矩，X轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0001	0.0355	0.0048	0.0930	0.0238	0.1831	0.0272	0.2880
Mid-Slow 中慢	0.0044	0.0325	0.0680	0.1289	0.1495	0.2731	0.1629	0.3949
Mid-Fast 中快	0.0226	0.0409	0.1070	0.1594	0.1576	0.2639	0.1764	0.3914
Fast 快速	0.0455	0.0809	0.0788	0.1552	0.0874	0.2676	0.0908	0.3918

Table A11. Average and RMS torques measured for the anal fin, Y axis.
表A11. 测量臀鳍的平均和RMS扭矩，Y轴。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0032	0.0461	0.0067	0.0978	0.0353	0.1480	0.0484	0.2051
Mid-Slow 中慢	0.0242	0.0564	0.0609	0.1211	0.1054	0.2015	0.1320	0.2785
Mid-Fast 中快	0.0029	0.0838	0.0526	0.1654	0.0830	0.2186	0.1053	0.2699
Fast 快速	0.0070	0.1424	0.0144	0.1976	0.0328	0.2467	0.0311	0.2445

**Figure A5.** Detail of the torques generated at *slow* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A5.** 四个振幅（从左到右：小、中、大和最大振幅）在低速下产生的扭矩的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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Table A12. Average and RMS torques measured for the anal fin, Z axis.
表A12. 测量尾鳍Z轴的平均扭矩和RMS扭矩。

	Amplitude 振幅
	Small (N) 小号（N）		Mid (N) 中（N）		Big (N) 大（N）		Max (N) 最大值（N）
Speed (mm s⁻¹) 速度（mm s⁻¹）	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS	Average 平均	RMS
Slow 慢	0.0099	0.1656	0.0196	0.2361	0.0285	0.2203	0.0301	0.1416
Mid-Slow 中慢	0.0138	0.2037	0.0159	0.2995	0.0112	0.2503	0.0019	0.1112
Mid-Fast 中快	0.0129	0.2709	0.0155	0.4166	0.0104	0.3972	0.0049	0.0940
Fast 快速	0.0226	0.4509	0.0294	0.5223	0.0317	0.4452	0.0125	0.0948

**Figure A6.** Detail of the torques generated at *mid-slow* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A6.** 四种振幅（从左到右：小、中、大和最大振幅）在*中低速*下产生的扭矩的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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**Figure A7.** Detail of the torques generated at *mid-fast* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A7.** 四种振幅（从左到右：小、中、大和最大振幅）*下中高速*时产生的扭矩的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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**Figure A8.** Detail of the torques generated at *fast* speed for the four amplitudes (from left to right: small, medium, big, and max amplitudes). (a) Dorsal fin. (b) Anal fin. Source: authors.
**图A8.** 四个振幅（从左到右：小、中、大和最大振幅）在高速下产生的扭矩的详细信息。(a)背鳍。(b)肛门鳍资料来源：作者。
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The figures below show the forces and torques generated in four successive strokes with four amplitudes (from left to right: small, medium, big, and max amplitudes).
下图显示了在四个连续冲程中产生的力和扭矩，具有四个振幅（从左到右：小、中、大和最大振幅）。

Soft dorsal/anal fins pairs for roll and yaw motion in robotic fish
用于机器鱼滚动和偏航运动的软背鳍/臀鳍对

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Abstract 摘要

1. Introduction 1.介绍