On the influence of head motion on the swimming kinematics of robotic fish

Up to now bio-inspired fish-mimicking robots fail when competing with the swimming performance of real fish. While tail motion has been studied extensively, the influence of the head motion is still not fully understood and its active control is challenging. In this experimental study, we show that head yawing strongly impacts on the propulsion force and determines the optimal fin actuation amplitude and tail beat frequency when aiming for a maximal propulsion force. In a parametric experimental study on a tethered 367 mm long fish robot the pivot point location of the head yaw has been varied along with tail beat frequency and actuation amplitude. The experiments took place in a still water tank and the swimming force has been measured with a single axis load cell. The robot is actuated with non-conventional area actuators based on micro fiber composites. 105 parameter sets have been investigated while the highest pivot point distance of roughly 0.36 body length from the nose tip provided the highest propulsion force of 500 mN with the lowest actuation frequency of 2.5 Hz and the highest head motion amplitude of a magnitude of 0.18 body length. Even though the pivot point location on a free swimming robot is a consequence of the complex fluid-structure interactions of fish and fluid, the results provide valuable information for the design of fish mimicking robots and questions the paradigm that head yaw is a simple recoil effect from tail motion and has to be minimized for an effective propulsion.


Introduction
Fish perform complex motion patterns for efficient and powerful propulsion. The type of these patterns depends on their ecological niche or, more specifically their swimming requirements due to their habitat, such as still water or fast flowing mountain rivers, but also their role, such as predator or prey. Breder [1] introduced a classification comprising the group of body and caudal fin (BCF) swimmers and separated them according to their propulsion strategies into four locomotion modes A) anguilliform like eels; B) subcarangiform, comprising among others the salmonides; C) carangiform, such as mackerel; and D) thunniform, used by the most efficient and powerful swimmers such as tunas. These four discrete locomotion modes differ in the role of body motion in relation to caudal fin motion when BCF swimmers are cruising. While anguilliform employ the entire body to generate thrust with significant involvement of the head, thunniform, the most efficient swimmers, only move the caudal fin and the slender tail penducle, which leads to minimal head motion as pointed out by Sfakiotakis et al [2]. This head motion is seen as a recoil mechanism from the lateral forces acting on the caudal fin and should be minimized in order to gain thrust for biomimicking robots, as stated by Lou et al [3]. According to Xie et al [4], this head or recoil motion is one of the remaining research gaps in bioinspired robotics and also not fully understood in the swimming kinematics of real fish. Recent publications suggest that the common classification, as first introduced by Breder [1] and based on body geometry and stiffness, is not reflected by the real motion patterns observed in high speed video recordings. Di Santo et al [5] show in their observations that there is only one unique motion pattern for all BCF swimmers in cruising mode. This motion can be expressed by a second-order polynomial. The key difference between all BCF motion patterns is: (A) the wavelength and (B) the number of waves passing the body per stroke. As a consequence and in full contradiction to common mechanical concepts, they found the tunniform swimmers to provide the highest head to tail motion ratio of all swimming modes investigated. This calls into question the aforementioned assumptions on the role of head motion as a simple recoil compensation mechanism which has to be suppressed as much as possible.
According to Liao et al [6], trout position their head differently than their tail relative to vortices rolling over their body. Liu et al [7] studied unsteady bio-fluid dynamics with a focus on locomotion costs. According to their findings, using a Kármán gaiting locomotion in bio-inspired flexible underwater robots allows for the exploitation of energy from the surrounding turbulence. There have been numerous studies on the locomotion patterns of underwater robotic systems with extensive experimental studies on the effects of tail flapping motion, the length of the tail and even complex kinematic motion patterns for robotic fishes, but these have always concentrated on control of the posterior body, such as [8][9][10][11][12][13][14][15][16], to name a few of these studies.
Some recent research, like Akanyeti et al [17], Scaradozzi et al [18] and Lou et al [3], both published in 2017, consider the head motion or a head shaking respectively, in the motion pattern of a swimming robotic fish to be key in achieving thrust. A holistic study on the performance effects of the tail fin's size and stiffness of different amplitudes and frequencies, as well as on head traction steering methods on the robotic swimming speed, was conducted by Tong et al [19]. Akanyeti et al [20] performed a detailed study on Kármán gaiting for fish and robots, analyzing the parameters for head motion, lateral translation, body rotation and body bending. This study recommends concentrating on the head and the anterior body motion for thrust generation.
During its gliding phase, the fish does not move and no propulsive momentum transfer occurs. Neglecting any deceleration of its motion, the fish will only encounter drag forces on the surfaces of its body. This drag is mainly based on the kinetic pressure forces as result of the deviation of the surrounding water and shear in the boundary layer due to friction and velocity gradients as a result of the relative swimming speed of the fish. Some further pressure losses result from the generation of vortices when the flow detaches from the body.
During active swimming, the body will be subject to unsteady changes in the pressure field and the resulting lift and drag forces from the relative velocity of the body. It also interacts with mass-driven inertial forces from its own mass and the mass distribution over the body. Additionally, the displaced water masses, the so called added mass effects, provide momentum and are crucial to the mechanical system. The added mass depends on the acceleration of the body, the fluid density and viscosity and on the shape and the size of the body. The summarized hydrodynamic reaction forces F R act on the entire body always in the direction opposite to the fish's motion (subsequently calledḋ for the head andḊ for the caudal fin motion) and provide a lateral F L and a thrust force F T component (see figure 1). The latter represents the overall propulsion force as a sum of all small shares over the entire body [2] and is from major interest for the study at hand, subsequently called F swim .
The added mass effects from fluid-structure interactions of a rigid body moving in a fluid can be expressed by the Kirchhoff equations. They provide a simplified model based on a single tensor which comprises all of the constants necessary to estimate the hydrodynamic load. It can be computed numerically and provides a reasonable trade off between computational effort and accuracy, as shown by Weissmann and Pinkall [21]. A drawback of the model is that it neglects any viscous or boundary layer effects. Zong et al [14] used this approach for a model with simplified physics to simulate his fish robot. It is derived from a combination of the rigid body motion of the head using the Kirchoff equations and the Morison equations for the compliant body and tail region of the robot. Some of the coefficients had to be determined experimentally. The model quantitatively overestimated the average static thrust of his tethered fish robot by 87.5% compared to his subsequent experiments. This underlines the complexity of the underlying real physics. Even though, these models allow for a reasonable estimation of the forces acting on a swimming body and successfully predict the quality of different motion laws with negligible computational effort compared to classical computational fluid dynamics (CFD), experiments are necessary to generate precise and reliable results.
A fundamental analysis of the kinematics and mechanics of oscillatory swimmers is given by Smits [22] and from highest interest for the subsequent study. Smits uses the analogy of heaving and pitching hydrofoils for the oscillatory motion of a fish tail to assess its flow characteristics. He neglects the influence of body motion as he assumes that the propulsion originates by far from the caudal fin. We fully follow the subsequent analysis by Smits [22] for caudal fin propulsion, even though the authors are convinced that both, the head and caudal fin motion are key for swimming kinematics.
For the heaving part of the motion, thrust generation will be dominated by the lift forces, while added mass effects dominate thrust generation in the pitching motion. However, when averaged over a cycle, both components of thrust generation scale by the product of added mass per length (ρs) and the square of the lateral (tail) motion velocity (Ḋ orḋ respectively in our notation). This shows that the swimming speed itself (the longitudinal velocity) does not affect the generalized swimming mechanics and motion pattern. This finding is of high importance for our study as it shows that the experimental setup using a tethered fish robot in order to measure the propulsion force can provide meaningful insights into the general kinematics of untethered swimming. It is also supported by the findings of Di Santo et al [5]. They found no change in the swimming modes or pattern for various cruising speeds for all species examined experimentally but a variation in the stroke frequency proportional to the swimming velocity.
However, the drag and, by consequence, efficiency obviously depends on the swimming velocity and tail beat frequency as a function of the Strouhal (St) and Reynolds (Re) numbers, as shown numerically by Senturk and Smits [23]. In fact, the thrust generally rises along with St and Re. Nevertheless, the efficiency only rises continuously as a function of Re until it becomes asymptotic at high Re because of the viscous effects and drag forces. The St features an optimum which was found by Triantafyllou et al to be between St ≈ 0.25-0.35 [24]. Senturk and Smits reported a much wider range and a dependence to Re. The most efficient cruising speed is therefore a tradeoff between thrust and drag forces and also strongly depends on the shape and size of the propulsor. For the study at hand these effects were neglected: in order to reduce the complexity of the system a still water tank was employed for the experiments.
Obviously locomotion strategies for robots are associated with many constraints, such as materials, a rigid head or body and the adjustment of the center of gravity, as pointed out by Lou et al [3]. As a consequence it remains challenging to implement and adjust an optimized head motion in a robotic device to generate the maximum propulsion and highest efficiency. In contrast to real fish, robotic motion patterns are generally simplistic and of low complexity. Additionally, the head motion turns out to be dependent not only on the mass distribution but also on complex fluid-structure or better said fluid-fish interactions. Added mass effects, a result of the displacement of water masses originating from the head motion, depend not only on the geometry, such as the dimensions of the surface which interacting with the water, but also the head shape and the motion's acceleration. However, due to the reduced complexity in the sensing and motion of robotic devices compared to real animals, fully suppressing or adaptively controlling the head motion for robots seems challenging, while real fish naturally make use of their head motion while cruising but also in maneuvering.
Until now, all fish inspired underwater robots known to the authors failed to reach the swimming performance of real fish and to reproduce real fish locomotion patterns. According to Scaradozzi et al [18], this is due to the complexity of real fish's motion and a completely different mass distribution in the body of a real fish compared to a robot. In the opinion of the authors, an additional crucial point is the significantly lower power relative to the average system density for artificial actuation systems.
Real fish use their entire body for propulsion. Analytical models for the locomotion patterns of fish in different reference frames describe a substantial motion of the head for BCF swimmers such as Akanyeti et al [20] and Scaradozzi et al [18]. Di Santo et al even found that the most efficient swimmer had the highest head motion amplitude relative to the caudal fin motion [5]. It therefore seems necessary to consider the head motion in bio-inspired propulsion strategies. In this context, the location of the pivot point has to be known (see figure 1), the pivot point allows the distinction between either head or tail motion of a the swimmer and can be considered as a crucial parameter in an efficient and effective motion pattern for biomimetic locomotion in the opinion of the authors.
Over the long term, the robotic device subsequently described in this study is dedicated to replacing live fish experiments due to the risk of injury to fish in hydraulic structures. Therefore, the project's biologists defined quite challenging, and for roboticists rather utopic, specifications of a real brown trout in shape, size (not longer than a common brown trout, e.g. 300 mm), swimming capacity (up to 3 m s −1 at a sprint) and neutral buoyancy. The targeted specifications clearly describe a small, powerful and neutrally buoyant robot with a flexible and soft tail to reproduce a real fish's body stiffness for the impact assessment. Later on, the robot shall be equipped with sensors and perform rheotaxis, which means that it will align itself with the head in the direction counter to the flow.
For the design of small, neutrally buoyant fish robots with good swimming capacities, conventional design approaches based on electromagnetic actuators were found to be inadequate because of their relative weight by Abbaszadeh et al [25]. Different unconventional approaches such as piezoelectric based actuators, also called micro fiber composites (MFCs), introduced by [26][27][28][29], as well as dielectric elastomer actuators presented by [30,31] promise to allow for higher actuation power densities under those constraints. However, the optimization of the propulsion system of a bio-mimicking device requires a thorough investigation not only of the actuator technology itself, but also the key parameters for the motion pattern which includes the optimal head motion.
This research investigates the effect of the pivot point location on the propulsive force F swim with the aim to determine the head's optimal pivot point for further improvement and finally a systematic optimization of the motion pattern, which will be the next research step. In this study, an experimental setup (see figure 2) to investigate the interdependence of the aforementioned parameters by evaluation of the propulsion force in a parametric study will be introduced. After this, it will be shown how the pivot point location for head motion becomes key to an efficient locomotion pattern and how it interacts with other parameters such as tail beat frequency and amplitude.

Body and propulsion system design
The robotic fish has an overall length BL = 367.5 mm with a rigid 3D-printed head of 100 mm length. All relevant dimensions and components are shown in figures 1 and 2. The head is followed by a flexible body based on a thin (0.2 mm) fiber-reinforced polymer composite plate, which ensures a low lateral bending resistance for maximum flexibility (for propulsion), but simultaneously provides high tensile strength (caudal) and rigidity (dorso-ventral), replacing a fish's spine and comprising a passive tail fin.
In a later stage of the project, the fish mimicking body shape will be provided by silicone bodies which ensure an overall neutral or slightly positive buoyancy. For constructive simplicity and with the aim of being able to access actuators and position feedback sensors, they were not used in this study. The robot's body geometry is a simplified model of a 3D scanned brown trout with the fins except the caudal fin were removed, as well as small features like the eyes. In future, all of the sensitive electronic components will be placed in the head of the robot to protect them from possible impact. In the current state, placeholders and cable ducts were provided for the battery, MFC drivers and micro-controller unit (MCU), which are placed outside the water. This also applies to subsequently installed sensing system, which was provided by collaboration partners from the Environmental Sensing Group at Taltech University. For reproduction of the design, the specifications of the materials employed can be found in table 1. More details can be found in the vendorsupplied technical data sheets.
The entire robotic mass is 167-172 g with the distribution as follows: the head with head mount has a weight of 92-97 g depending on the length of the mount required to vary the pivot point. The composite mass, consisting of the four MFC and the artificial spine as well as the coating and cables, is 75 g.
The actuation of the body consists of four piezoelectric MFC actuators. These actuators were also deployed by Cen and Erturk [26], Chen et al [27] and Tan et al [29] as non-conventional actuation systems in biomimicking fish robots. They were arranged in two pairs of different sizes and performances. For actuation these piezoceramic composites are bound to a thin composite plate, forming an actuated structure with two pairs of so-called bimorphs which behave like artificial muscles. The entire active part  [32]. The four MFC are driven by a DC peak-to-peak voltage of 2000 V from two amplifiers purchased from the actuators' vendor. A maximum deflection of 100 mm were achieved with the robot in air as the surrounding fluid. The deformation was determined with optical measurements using a high speed recordings and deformation tracking based on image segmentation, as shown in Abbaszadeh et al [25]. However, the maximum tail deflection is expected to be significantly lower in water as a result of the higher hydrodynamic loads from its 800 fold higher fluid density. It was not reassessed in an additional experiment in water, however the kinematic model developed in the aforementioned study provides the position feedback of the actuated artificial spine of the robot. The deviation from the optical measurements in air was less than 0.1 mm. Due to the setup using two independent actuator groups, each of the two artificial muscles can be controlled separately. At this stage, four sinusoidal PWM signals provided by an MCU (TI-F28069) govern two dual channel high voltage power amplifiers. The power amplifiers provide separate driving voltages of −500 V to +1500 V. The actuators are controlled using a sinusoidal excitation, with two independent normalized amplitudes A 1 and A 2 and two independent frequencies f 1 and f 2 , respectively, for the first and second artificial muscle. Additionally, the phase shift ϕ between the two muscles can be set. In total, this leads to a five freely adjustable parameters for the propulsion control. A more detailed description of the actuation system and its control system, as well as for the kinematic model, can be found in [25].

Experimental design 2.2.1. General experimental setup
As mentioned earlier, aim of this study is to build a meaningful setup for the investigation of the influence of the pivot point location on the locomotion's performance and efficiency of a fish mimicking robot. We also strive to provide an experimental setup that allows for the automatized measurement of the propulsion force for different locomotion patterns. A direct force measurement of a freely swimming robot with six DoF is impossible as each load cell requires constraining the robot in at least one DoF. Optical measurement techniques will require a parallel model to extract the underlying mechanics, which suffers from additional uncertainties.
In order to simplify the setup and to provide reproducible conditions for the measurements, the robot was mounted on a fixed support in a still water tank with a channel width of 500 mm. In order to investigate possible effects of the boundaries such as wave reflections, the measurements were repeated with a set of arbitrarily selected parameters in the laboratory flume channel with a width of 1200 mm with negligible deviations in the range of the measurement uncertainty.
The setup in our study is inspired by and aims for an improved version of the system presented by Costa et al [33], who investigated the propulsive performance of an ostraciiform swimming robot. Zhong et al [14] investigated multiple control strategies for his robot in a similar setup. As mentioned previously, the tethered setup can provide meaningful insight into a free swimming robot as the quality of a locomotion pattern depends on the lateral velocity of the motion [22], a condition which is fully satisfied by the setup at hand.
A single axis bending beam force sensor (RB-Phi-203) built inside the head holder with custom signal amplifier was used to record the propulsion force. The setup of the head and the mounting support was constructed with 3D-printed parts. The latter was in the shape of a NACA0036 profile to reduce drag and to minimize wake vortices from the support when used in the flowing environment of the laboratory's flume. The experimental setup allows for the head to rotate as the robot moves. Figure 2 shows an overview of the design of the robot and the measurement system. Figure 1 provides a sketch of the references and lengths to describe the fish's motion. The nonactuated, uninstrumented and flexible caudal fin is not part of the kinematic model and was not considered further in this study.

Sensing system
The body motion is captured by strain gauges installed in two full bridges on the MFC area actuators. This allows for very accurate measurement of the driving strain of the piezoceramics and is translated into a deflection function of the composite spline. Parallel optical calibration tests found a deviation of <0.1 mm from the model. However, the passive, flexible dorsal fin deflection is not assessed by the strain gauges or the kinematic model and remains a constant parameter which is not assessed further. For thrust measurements, a micro load cell featuring a measurement range of ±1 N and a rated output of 0.6 mV V −1 was built into the mounting support, and the interchangeable head holder measures a one directional force F LC . The mounting support is equipped with a sleeve consisting of two polymer bearings with glass balls in order to fix the shaft so that a low friction based attenuation of the rotational motion of the head is possible. The lower part of the load cell is clamped in the head holder of the fish. As a consequence, the load cell follows the head motion, and the thrust has to be calculated from the head motion angle α and F LC (see figure 1). For this purpose, the load cell is connected to a rotary encoder via the shaft in order to capture the synchronized rotation angle of the head. The average thrustF swim is calculated from the instantaneous (sampling rate 200 Hz) force components in the swimming direction, which is obtained from the force measured with the load cell F LC and the head angle α. A customised signal amplifier with a gain factor of 4000 was built for the setup in order to generate a linear output voltage from the bending beam load cell for the MCU with a good signal to noise ratio. The force depicted in the subsequent sections is the net propulsion force. Prior to each measurement the offset was determined by an averaged force measurement for 3 s (600 samples) while capturing without actuation. Then 4 s (800 samples) of time were given to establish the flow with actuation but without measurements. Finally 10 s of measurements (2048 samples) were taken with actuation and subtraction of the offset force to achieve the net propulsion force.
In this study all results are presented as their ensemble averages over at least five cycles for 0.5 Hz, with rising number of cycles for higher frequencies, achieved from a measurement duration of 10 s for each point. Measurements on 6 arbitrarily selected parameter sets have been explicitly repeated for five times to quantify the uncertainty and the reproducibility of the findings. The standard deviation varied in a bandwidth of 1.86 Nm (minimum) for a median F swim of 104.36 mN and 16.89 Nm (maximum) for a median F swim of 179.54 mN. The system noise was determined with repeated zero actuation measurements with a median of 0.01 mN with a standard deviation of 0.22 mN (see table 2 for details).

Calibration and uncertainty
For the purpose of calibrating the customized force sensing setup, defined loads of ±1 N were applied to the system. The load magnitudes were determined with the help of a second load cell (Digital Force Gauge Sauter FC10 with a minimum accuracy of ±0.3% F.S. for ±10 N) serving as reference. The gain in the voltage read (in binary format) from the MCU to the actual force was determined by a linear regression on the measured load points. As shown in figure 3, the customized sensor features very accurate and linear characteristics in the range from −1000 to +750 mN. The deviation in the aforementioned range was 3.2 mN (median) with a standard deviation of 6.3 mN after outlier removal. This corresponds to an average relative deviation of 1.7%. The rotation angle measurement is based on a hall sensor which showed significant non linear characteristics. With the aim of determining the uncertainty in this second sensor, the indexing head of a milling machine was employed. The sensor delivered satisfying results in the expected range of −14 • to +14 • with deviations of less than 0.5 • .

Base setup without head motion
In a first experimental setup with the robot, the head is directly clamped to a bending beam load cell which suppresses any kind of head motion in particular the yaw. This setup deserves as a baseline study for reference. Only the flexible part of the body can move as a consequence of the tail actuation. In our experiment, regardless of the amplitudes or beat frequencies of the actuators, only negative net thrust with small magnitude has been measured, which leads to backwards propulsion, as shown in figure 4. The diagram provides the data of two independent runs. The net force is the difference of instantaneous propulsion force to the averaged force from zero actuation (offset). A 3rd order B-spline regression has been performed to generate the trend curve. The gray area depicts the uncertainty of the load cell as described in section 2.2.2. This experimental failure shows the importance of the head rotation not only to compensate for the recoil (which was provided by the clamped mount) but also to generate thrust. This finding is in line with Akanyeti and Liao [20], which stated that it is more important to control the head anterior body and the head angle of the robot than the posterior part of it. We hypothesize that in order to generate thrust in the given actuation system, the head has to be able to turn. Moreover, due to these results we hypothesize that the location of the head motion pivot point, constrained by the mount point, has a direct impact on the propulsive force. For the subsequent main study, the setup has been redesigned using ball bearings to allow for yaw motion, i.e. rotation of the head. Other recent research such as [34,35] report a positive thrust for a clamped head configuration. We hypothesize that this contradiction is due to our different actuation design using MFC. However we are convinced that the head motion has more influence on the swimming kinematics than a simple recoil function.

Parametric study design with head yaw
A Python script running on a PC was used to automate the experiment. It was used to communicate from the PC to the MCU, to provide the set points for the MCU-based control of the robot and to save and process the feedback from the sensors (see figure 5). The experiment was conducted as follows: I. The propulsion force was assessed for each of the five pivot point locations ranging from a distance to the robots head tip from 91.5 mm (K/BL = 0.25) to 131.5 mm (K/BL = 0.36) in a step size of 10 mm. The interchangeable head holders are shown in figure 6. Each of the holders was evaluated at the maximum normalized tail beat amplitude (A 1 = A 2 = 1) with unified actuation frequencies ranging in bandwidth from 0.5 to 7 Hz with a step size of 0.5 Hz and no phase lag between the actuators. This allows for the study  of the influence of the frequency for each pivot point location and will deliver the best frequency for each individual point with highest thrust for each distance. II. For the five best points of each head (the frequencies with the maximum propulsion force), a set of five different amplitudes ranging from a normalized value from 0.2 to 1.0 with a step size of 0.1 were evaluated for 11 frequencies ranging from 2.0 to 7.0 Hz with a step size of 0.5 Hz were tested for all head holders.
These tests led to a set of 105 parameter variations in total. An overview of all of the parameter variations tested can be found in table 3. Figure 7 depicts the results of the first experiment using fixed normalized amplitudes (A 1 = A 2 = 1) and unified actuation frequencies for both artificial muscles ranging from 0.5 to 7.0 Hz plotted on the abscissa of the diagram. The ordinate shows the propulsion force as a function of the actuation frequency. Each line depicts one of the interchangeable head holders. Curves were generated from the measurements using radial basis functions from the scipy toolkit [36]. Generally, the curves show good agreement in their qualitative shape characteristics  with a lower slope in the ascending branch of the curve before the thrust peak is reached, compared to almost abrupt falling propulsion force for actuation frequencies higher than optimal. The curves also feature broader, smoother peaks for smaller pivot point distances.

Results and discussion
With increasing distance, slopes increase and become more homogeneous for both, the ascending and descending branches of the propulsion force. Therefore, the thrust peak becomes sharper, along with propulsion force peaks increasing with distance. At a distance of 91.5 mm to the robot head's tip, a maximum propulsion force of 332.6 mN with a very high actuation frequency of 6.5 Hz is generated. With increasing distance of the pivot point location, the optimum point frequency decreases continuously, while the thrust peak magnitude increases. Unfortunately, this trend could not be broken in the range of the parameters evaluated in this study. However, the optimal pivot point distance featuring decreasing thrust with increasing distance is to be expected. The experiments showed a propulsion force of about 457.9 mN at a frequency of 2.5 Hz for the greatest distance of 131.5 mm.
These characteristics are in accordance with the experiments by Akanyeti et al [17] conducted on an artificial fish model which was not actuated itself, e.g. by the tail fin, but externally driven by the rotational oscillation of the fish head via a drive system located above the fish that actuated its mounting rod. Their maximum thrust was found at an oscillation frequency of roughly 2.5 Hz and a phase shift of 225 • between the yaw of the head and the tail motion. However, the pivot point location was neither mentioned nor further investigated. Figure 7 depicts the influence of the head angle motion, tail beat frequency, maximum tail deflection and maximum thrust on an effective motion pattern. All parameters show linear correlations with the pivot point distance in the optimum propulsion pattern and are presented in figure 8. The pivot point distance is placed on the abscissa while the aforementioned parameters are depicted on individual ordinates in a shared diagram. Linear regression was performed with accurate coefficients of determination. The tail beat frequency decreases with increasing pivot point distance which is in contrast to all other parameters. Generally speaking it can be stated that a large pivot point distance will lead to slow motion with high deflection and head motion amplitudes, which lead to better thrust compared to small pivot point distances. The latter comes with high frequencies in its extreme, almost jerky movements at low motion amplitudes and reduced effective thrust. Figure 9 shows a quadratic dependency for the tail deflection extracted from the kinematic model and strain gauge measurements relative to the head motion for each pivot point distance. At the maximum propulsion, the head motion increases nonlinearly compared to the tail with the distance of the pivot point. This trend remains unbroken over the range of the experiments. However, an outlier for the lowest pivot point distance of 91.1 mm is also clearly visible.
Considering the best point locations, a second set of experiments was conducted to investigate the influence of amplitude in the interplay between pivot point location and frequency. This set was done for all five pivot point distances. As it can be seen for the highest pivot point distance in figure 10, the thrust fully depends on the actuation amplitude, which is proportional to the tail beat amplitude but also depends on the frequency (which reaches maximum at around 2.5 Hz for this specific case). For all pivot point distances, the highest thrust correlates to highest actuation amplitude regardless of the beat amplitude at the optimum tail beat frequency, as shown in figure 7. Other distances feature exactly the same general characteristics and therefore are not explicitly depicted.
Zhong et al found the tail stiffness to be an important parameter for swimming power and efficiency. They performed a series of experiments on a fish mimicking robot with an adjustable body stiffness on their flexible tails. According to their study, adjustable tail stiffness is key to achieve high efficiency when using high-frequency actuation [34]. We consider that stiffness of the flexible tail has to increase for high frequencies as a too soft characteristic will hinder the transmission of the actuation to the tail fin when keeping the motor amplitudes constant. In other words, if the system is too soft the actuation will not reach out to the fin because the loads increase with rising frequency. In the extreme a hyper flexibility will result in a motion of the tail but not anymore in a fin motion. In our case the lever of the tail fin loads increases for low pivot point distances while the flexibility of the system remains the same. However the longer lever and therefore the higher load will decrease the possible amplitudes of the actuation for a specific frequency. Therefore in our case the actuation with small amplitudes is more beneficial for higher frequencies at low pivot point distances and amplitudes of the best point rise along with lower frequencies for higher pivot point distances.  Regarding the motion pattern, in particular the influence of the head motion, the results are also in accordance with Di Santo et al [5], which showed that the head motion is key for the kinematics of cruising fish. The best performing species were tuna (Thunnus albacares) with a cruising speed of up to 25 BL s −1 . However, contrary to expectations when following the commonly made assumptions based on Breder [1], these best swimming capacities came along with the highest mean head motion amplitude in relation to the BL ( AH BL = 0.04) and tail motion amplitude ( AT BL = 0.17). Figure 11  The results presented here do not allow for a direct comparison with a free swimming fish or robot. It must be noted that even though A H , the yaw motion amplitude of the head is not actively controlled in this study, all degrees of freedom of the robot and the pivot point location are controlled. In consequence A H strongly depends on the pivot point location. It is a result of the moments and forces. As a consequence, higher mass in the head region, leading to problems balancing the robot in the water, or a larger projected head surface with a stronger added mass effect from the surrounding fluid, would lead to higher tail to head motion ratios AT AH than were found for the free swimming tuna. This point should be considered during the design process for a wellperforming swimmer. Additionally, a cruising fish will obviously search for the most efficient propulsion strategy while only thrust magnitude without consumption was investigated in this study.

Conclusions
In this study, an experimental setup was developed in order to investigate the influence of the pivot point location of the motion pattern on a biomimicking fish robot in a still water tank. The measurement of the static thrust force allows for the determination of the best operating and mounting point for the device. An underlying kinematic model provides accurate position feedback for the actuated tail motion. Together with synchronized head angle measurements, characterization of the motion pattern is possible. During the experiments, two parametric studies were performed. Experiment set I clearly shows that the head motion and therefore the pivot point location, which allows for the distinction between head and tail motion, is key for effective fishmimicking propulsion in this static case. The highest thrust was achieved with the highest head motion amplitude. This finding is in accordance with experiments on freely cruising fish and in contradiction to the common classification of BCF swimmers, which assumes that thunniform swimmers feature only minimal head motion. This motion is not only necessary to compensate for the recoil from the caudal fin's motion but also contributes to the overall thrust.
It additionally shows that the propulsion force increases along with a increasing pivot point distance and motion amplitude of the head and tail, while the optimal frequency decreases with increasing distance of the pivot point location from the fish's head tip. As a consequence, the highest propulsion force could be achieved from the slowest motion (lowest beat frequency), with highest motion amplitude at the highest pivot point distance (131.5 mm) compared to the best operating points from other pivot point distances. The thrust was roughly 140% of the lowest best point propulsion force for the smallest distance measured (91.5 mm). Lower mounting distances were successful with higher frequencies, low motion amplitudes and a generally lower performance level. The extreme was an almost jerky motion with very low tail and head amplitudes. Generally, the head in relation to the tail motion decreased non-linearly along with the distance for the maximum propulsion. All single parameters, such as head motion angle, tail motion amplitude, frequency and propulsion force correlate linearly with the pivot point location distance.
The results of this study are key for the development of robotic fish propulsion systems and bioinspired thrusters. Fin-like propulsion systems for underwater vehicles consist of a comparable setup such as an actuated oscillating rod which moves a rigid tip and a passive tail. In this case, the results of this study could easily be extrapolated and would suggest a high distance from the fin tip to the pivot location with low actuation frequencies and high oscillating amplitudes. For fish mimicking devices, a geometrical design accounting for the added mass effects from the head motion (e.g. from the Kirchhoff equations) and a well adapted mass distribution would adjust the pivot point and allow for the optimum head motion and for an effective propulsion. The findings allow for the suggestion to build a robotic device which uses a significant yaw motion in order to maximize the propulsion force. The fish robot should be designed in such a way that the pivot point (see figure 1) appear at roughly one third of the body length (0.3 BL) or even more. The highest distance investigated is about 0.36 BL and provided highest swimming force at the lowest actuation frequency. It is obvious that the head yaw is a complex design parameter. A methodology based on simplified flow models as proposed by Zhong et al [14] or more costly but precise CFD simulations can provide the necessary tools for the design.
Future research will experimentally optimize the motion pattern of the robot and examine the instantaneous flow fields in the vicinity of the device in order to provide insights into the complex fluid-structure interactions for bio-mimicking fish robots.

Data availability statement
The underlying raw data of this study and the Python codes for post-processing can be found on the data repository server of Otto-von-Guericke University (http://open-science.ub.ovgu.de). The MCU code can be shared upon request.
The data that support the findings of this study is openly available at the following URL/DOI: 10.24352/ UB.OVGU-2022-083. Data will be available from 31 October 2023.