Octopus-inspired multi-arm robotic swimming

The SIMUUN computational environment [36], which is based on the SimMechanics toolbox of Simulink, is used to set up a lumped-element model of an eight-arm system, whose configuration and mechanical parameters are specified to reflect those of the robotic prototype described in section 5. The arms, whose length is denoted by L, are attached to the rear of a mantle-like body shaped as a half ellipsoid with diameter D and half-length D. They are arranged axisymmetrically at 45° intervals, and oriented so that each pair of diametrically opposite arms moves in the same plane (figure 2). Each arm is modeled as a kinematic chain of n = 10 cylindrical rigid segments, interlinked by 1-degree-of-freedom planar revolute joints, whose axis is normal to the arm's motion plane. The joint for the base (i = 1) segment, which connects the arm to the main body in the model, is assumed to be actively driven by an appropriate actuator that enables prescribing a specific angular trajectory for it. Unlike our previous studies [29–31], the segment interlinking joints (i.e., for i = 2..10 in figure 2(b)) are here considered to be unactuated, and feature rotary linear spring-and-damper elements, the parameters of which can be appropriately specified to describe the flexibility of different materials used for the arms (note that active control of the arms' stiffness is not accounted for in this work). For the present study, the arm compliance involved relatively large stiffness for the passive arm joints, which resulted in limited bending motions of the arm, mainly near its tip, during swimming. More specifically, the spring coefficient of the unactuated joints was specified in the range of 0.63 − 0.015 Nm rad⁻¹, gradually decreasing from the arm base toward its tip. Regarding the viscous damping coefficient, a uniform value of $1.2\cdot {{10}^{-4}}\;{\rm Nm}\cdot {\rm s}\;{\rm ra}{{{\rm d}}^{-1}}$ was specified for all of the mechanism's unactuated joints. The preceding values were empirically determined, on the basis of achieving a close match of the simulated arm kinematics with those exhibited by the polyurethane (PU) arms used in the robotic prototype (see section 5).

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In the developed lumped-element model, the fluid interactions of the mechanical system are described through a simplified resistive fluid drag model. In particular, the modeled system is assumed to move within quiescent fluid so that hydrodynamic forces acting on a single arm segment result only from its motion (i.e., they are not influenced by the motion of neighboring segments). Further, the system is considered to move at speeds at which the generated fluid forces are predominantly inertial ($400\lt \operatorname{Re}\lt 4\cdot {{10}^{5}},$ where the non-dimensional Reynolds number Re characterizes the flow regime). Lastly, it is assumed that the three components of the total flow-induced force (tangential, F_T, normal, F_N, and lateral, F_L) are decoupled. Under these assumptions, the fluid forces describing the interaction of individual arm segments with the surrounding fluid can be expressed as follows:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{F}_{i,{\rm dir}}} & = & -{{\lambda }_{i,{\rm dir}}}\;{\rm sgn} ({{v}_{i,{\rm dir}}})\cdot {{({{v}_{i,{\rm dir}}})}^{2}}, \\ \ \ {\rm dir} & = & \{T,N,L\}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where v_i,T, v_i,N, and v_i,L are the tangential, normal, and lateral velocity components, respectively, of the ith segment, and λ_i,T, λ_i,N, and λ_i,L are the fluid drag coefficients on the ith segment (i = 1..10), corresponding to each fluid force component. Such resistive fluid drag models originate from studies of swimming animals [48] and are commonly adopted in the robotics literature on bio-inspired elongated underwater systems (e.g., [33–35]). Prior studies by our group, involving detailed CFD analysis of rotating octopus-like arms [29, 49–52], indicate that the underlying assumptions of this fluid drag model are, to a large extent, valid. In addition, the results of these CFD studies have been used to specify the fluid drag coefficients in (1) for the arm segments. The numerical values of λ_i,dir (listed in mN per m²/s²) are provided in table 1. The distribution of hydrodynamic forces acting along the length of the arm is non-uniform and differs from base to tip [29, 49]. It is also noted that, due to the axial symmetry of the arm segments, the normal and lateral coefficients are equal (i.e., λ_i,N = λ_i,L). In addition, the equivalent standard non-dimensional hydrodynamic coefficients C_i,dir are provided in table 2. These were obtained as:

$$\begin{eqnarray}&&{{C}_{i,{\rm dir}}}=\frac{{{\lambda }_{i,{\rm dir}}}}{\frac{1}{2}\rho {{S}_{i,{\rm dir}}}},\end{eqnarray} \tag{ 2 }$$

where ρ is the density of water, while S_i,dir denotes the projected area of the ith segment for the corresponding velocity component. Since each segment is a cylinder of height h_i = L/n and radius r_i, the projected areas are calculated as:

$$\begin{eqnarray}&&{{S}_{i,N}}={{S}_{i,L}}={{h}_{i}}{{r}_{i}}\quad {\rm and}\quad {{S}_{i,T}}=\pi r_{i}^{2}.\end{eqnarray} \tag{ 3 }$$

The preceding approach was also used to describe the fluid interactions of the mantle-shaped main body of the mechanism, according to:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{F}_{m,{\rm dir}}} & = & -{{\lambda }_{m,{\rm dir}}}\;{\rm sgn} ({{v}_{m,{\rm dir}}})\cdot {{({{u}_{m,{\rm dir}}})}^{2}}, \\ \ \ {\rm dir} & = & \{T,N,L\}, \\ \end{array}\end{eqnarray} \tag{ 4 }$$

where v_m,T, v_m,N, and v_m,L are the tangential, normal, and lateral velocity components, respectively, of the mantle, and λ_m,T, λ_m,N, and λ_m,L are the corresponding fluid drag coefficients. In prior work, the values of these coefficients had been estimated from standard hydrodynamics formulas for flow over an isolated hemisphere [32]. This approach led to considerable quantitative discrepancies between experimental data and the model's predictions, since it does not consider the effect of the arms attached at the rear of the mantle. In the present study, the numerical values of the body drag coefficients were specified on the basis of attaining a good agreement to the simulation results with the experimental data presented in section 5.

The sculling arm motion involves prescribing a two-stroke periodic angle variation ${{\varphi }_{1}}(t)$ for the actuated joint connecting each arm to the main body. In the present study, ${{\varphi }_{1}}(t)$ is obtained from the following acceleration profile

$$\begin{eqnarray}&&{{\ddot{\varphi }}_{1}}(t)=\left\{ \begin{array}{ccccccccccccccc} {{\alpha }_{0}}{\rm sin} \left( \frac{\pi t}{0.2{{T}_{r}}} \right), \\ {\rm for}0\leqslant t\lt 0.2{{T}_{r}}; \\ 0, \\ {\rm for}0.2{{T}_{r}}\leqslant t\lt 0.8{{T}_{r}}; \\ -{{\alpha }_{0}}{\rm sin} \left( \frac{\pi (t-0.8{{T}_{r}})}{0.2{{T}_{r}}} \right), \\ \quad \ \ for\;0.8{{T}_{r}}\leqslant t\lt {{T}_{r}}; \\ \beta {{\alpha }_{0}}{\rm sin} \left( \frac{\pi (t-{{T}_{r}})}{0.2{{T}_{p}}} \right), \\ \quad for0\leqslant t-{{T}_{r}}\lt 0.2{{T}_{p}}; \\ 0, \\ \quad for0.2{{T}_{p}}\leqslant t-{{T}_{r}}\lt 0.8{{T}_{p}}; \\ -{{\alpha }_{0}}{\rm sin} \left( \frac{\pi (t-{{T}_{r}}-0.8{{T}_{p}})}{0.2{{T}_{p}}} \right), \\ \quad \ \ for0.8{{T}_{p}}\leqslant t-{{T}_{r}}\lt {{T}_{p}}, \\ \end{array} \right.\end{eqnarray} \tag{ 5 }$$

integrated twice, with ${{\dot{\varphi }}_{1}}(0)=0$ and ${{\varphi }_{1}}(0)=\psi -A$, where 2A is the overall angular span and ψ is the angular offset of the sculling motion (figure 3). The preceding formulation results in a smooth motion profile, characterized by sinusoidal acceleration/deceleration phases, which is compatible with the performance characteristics of the actuators employed in the experimental platform, as will be presented in section 5.1. The power (respectively the recovery) stroke velocity is maintained at its maximum value of βω (resp. ω) for 60% of the stroke's duration, while the maximum acceleration for the power (resp. recovery) stroke is β α₀ (resp. α₀), where $\beta \gt 1$ and a₀ = 2πω²/A. The duration of the power and the recovery stroke are obtained as ${{T}_{p}}=2.5\;A/(\beta \omega )$ and T_r = βT_p, respectively. Hence, the overall period of the sculling motion is:

$$\begin{eqnarray}&&{{T}_{s}}={{T}_{p}}+{{T}_{r}}=(\beta +1){{T}_{p}}=2.5\frac{\beta +1}{\beta }\frac{A}{\omega }.\end{eqnarray} \tag{ 6 }$$

For compliant arms, the angular trajectories of the unactuated joints φ_i (i = 2..N), and hence the arm's overall shape, evolve in response to the sculling motion of the base segment, the hydrodynamic loads, the joints' flexibility (specified through the parameters of the spring-and-damper elements), and the inertial characteristics of the segments. All of these effects are accounted for by the developed computational tools through the SimMechanics substrate, which derives automatically and solves numerically the dynamic equations of the system.

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The preceding presented sculling profile can be employed as a basic motion template for the arms to generate a variety of different sculling gaits, for both forward [30] and turning [31] motions of the system, as detailed next.

Forward movement along an approximately straight line may be generated by synchronized sculling movements of the robot's arms, employing a variety of patterns of arm coordination. These give rise to patterns of swimming behavior, which, for convenience, will be termed 'gaits'. (This does not imply that transitions among these gaits occur at different velocities, e.g., as in horse gaits.) The timing of the arm coordination patterns, corresponding to each of the gaits described next, are summarized in table 3, which indicates the phase of each arm, as a fraction of the stride period T_s, with respect to arm L1.

G1 gait: The most straightforward arm coordination pattern is when all eight arms perform the sculling motion in synchrony. The arm trajectories are shown in figure 4(a). This gait can be related to the arm-swimming mode observed in live octopuses.

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G2 gait: This gait is produced by the synchronized sculling movement of pairs of diagonally opposite arms with a phase difference of T_s/4 between adjacent pairs of arms. The trajectories of the four pairs of arms are shown in figure 4(b). Gait G2 results in a smooth forward movement of the robot.

G3 gait: This gait is produced by the synchronized sculling movement of pairs of adjacent arms, in addition to a phase difference of T_s/4 between adjacent pairs of arms. This arm activation pattern gives rise to a relatively slow and wobbling movement of the robot.

G4 gait: This gait is produced by the synchronized sculling movement of two sets of four arms, one set containing arms L1, L3, R2, R4 and the other, arms L2, L4, R1, R3. There is a phase difference of T_s/2 between the two sets of arms. The trajectories of the two sets of arms are shown in figure 4(c).

In the preceding gaits, the action of the arms is equally distributed over the whole duration of the stride period T_s. Numerous variations of the preceding gaits are possible by altering the phase between the arms so that their action is no longer equally distributed over T_s.

The present study focuses on gaits G1, G2, and G4, as these were found to be the most effective for straight-line propulsion. Indicative simulation results, demonstrating forward propulsion by these three gaits, for the compliant-arms system, are shown in figure 4, for ω = 50°/s, β = 5, A = 15° and ψ = 40°. In addition, the snapshots in figure 5 provide an illustration of the arms' bending motion over one sculling period for gait G1. The lower plot of figure 5 shows the orientation angle for the base and the tip of the arm during one sculling period. It can be seen that the orientation of the base segment follows the prescribed angular trajectory ${{\varphi }_{1}}(t)$ of the active joint, while the orientation of the tip exhibits oscillations superimposed on this profile. These oscillations arise from the flexibility of the arm.

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To aid in the visualization of the proposed sculling-based gaits, a series of animations of the eight-arm mechanism, swimming under gaits G1, G2, and G4, are provided in the accompanying videos of the supplementary material.

The results in figure 4 indicate that, although the average steady-state speed V is approximately the same for the three gaits, the different patterns for the coordination of the arms' sculling motion have a significant impact on the profile of the instantaneous velocity v_b(t). In gait G1, v_b(t) exhibits quite pronounced peaks that coincide with the occurrence of the power stroke by the arms' synchronized motions. On the other hand, the phasing of the arms' power strokes in G2 results in thrust generation being more evenly distributed over the duration of T_s. The variations of v_b(t), which occur with a period equal to T_s/4, are considerably reduced, resulting in a smoother overall forward motion of the system. Correspondingly, the phasing of gait G4 yields a velocity profile with characteristics that fall in between those of the other two gaits. These results are, for the most part, similar to those obtained in our prior study for propulsion with rigid arms [30]. One notable differentiating feature of the response obtained with compliant arms is the high frequency ripples in the instantaneous velocity v_b(t) of the system, which are associated with the flexibility of the unactuated joints.

In order to assess the effect of the various kinematic parameters of the sculling profile on the propulsive velocity attained by the compliant-arms swimmer, a series of parametric simulation studies were performed. The parameter space, considered for these simulations, was specified on the basis of the capabilities of the actuators employed in the developed robotic prototype (see section 5.1) to facilitate comparison with experimental results. The results in figure 6(a) show the average steady-state speed V attained by the compliant-arms system in gait G1, when varying the recovery stroke velocity ω and the velocity ratio β, while retaining the sculling amplitude and offset constant ($A={{25}^{{}^\circ }}$ and ψ = 40°, respectively). As expected, V increases with both β and ω, while forward propulsion is not possible when β = 1. In figure 6(b), the attained speeds from these simulation runs are shown against βω. It can be seen that, at least for the investigated parameter space, the swimming speed is effectively dependent on this quantity, which corresponds to the arms' angular velocity during the power stroke.

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The second series of simulations was focused on the effect of the amplitude A and the offset ψ of the sculling motion, retaining constant the other two parameters with ω = 50°/s and β = 5. The results, which are summarized in figure 7, indicate that A has limited influence on the attained speed of the system, particularly for gait G1, while the optimal value for ψ is near 30°. It is noted that the variance of V with respect to ψ, for the compliant arms, is reduced in comparison to what was observed for propulsion with rigid arms in [30].

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Changes in the overall heading direction of the system can be generated by a variety of asymmetric motions of the robot's arms that result in a range of 'turning gaits'. Multiple such gaits could be implemented by appropriate arm coordination in any three-dimensional direction. It is noted that not all eight arms need to move for the generation of a turning gait; however, non-moving arms were found to facilitate the motion by being held at their maximum angular position.

To systematically study turning on the eight-arm mechanism presented, and facilitate its implementation on the robotic prototype (see section 5), here, we focus our investigation on planar turning gaits using sculling-only movements of the compliant arms. Three such gaits, for turns in the xy plane (with reference to figure 2(a)), are presented next:

GR1 gait: The first turning gait is generated by the synchronized sculling movement of three adjacent arms on the same side of the system, e.g., {L2, L3, L4}, while the remaining five arms are all extended outward, at their maximum angular position, i.e., at (ψ + A). The trajectory obtained with this gait, as well as the temporal variation of the arms' angle and of the turning rate of the main body, is presented in figures 8(a) and (b).

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GR2 gait: In this gait, turning is produced by the synchronized sculling motion of three opposite pairs of lateral arms, namely {R1, L4}, {R2, L3}, and {R3, L2}. The sets of arms {R1, R2, R3} and {L4, L3, L2} rotate sideways in synchrony. The two remaining arms, {L1} and {R4}, are extended outward at the maximum angular position (ψ + A). Simulations illustrating this gait are shown in figures 8(c) and (d).

GR3 gait: The third gait is achieved by the synchronized sculling movement of three opposite pairs of lateral arms, {R1, L4}, {R2, L3}, and {R3, L2}. The three adjacent arms {R1, R2, R3} open and close in synchrony with the opposite set of arms {L4, L3, L2}, while performing the sculling motion at half the amplitude (at the same T_s). Arms {L1} and {R4} are extended outward at the maximum angular position (ψ + A). Indicative results for gait GR3 are shown in figures 8(e) and (f).

The computational framework of section 3 was employed to investigate the characteristics of these turning gaits, through a series of parametric studies. In all of the simulations shown here, the recovery stroke velocity was set at ω = 50°/ s, with a velocity ratio β = 5. Indicative results for the three gaits, obtained with A = 25° and ψ = 30°, are provided in figure 8. The trajectory plots in this figure indicate that gaits GR1 and GR2 essentially implement in-place rotation, with however, different characteristics, particularly with regard to the orientation of the system. On the other hand, gait GR3 appears to be more appropriate for combining heading direction changes with an overall translational motion (see also figure 10). Furthermore, the plots with the temporal evolution of the head segment yaw velocity indicate that the latter exhibits a higher average value, as well as larger variance during each sculling period, for gait GR2.

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The stride angle θ_s, representing the average angle by which the heading direction changes during each sculling period, is provided in figure 9. The results show that, for all three of the investigated turning gaits with compliant arms, θ_s increases with the sculling amplitude A, while it exhibits little dependence on the sculling offset. It can also be seen that, in line with the observations arising from figure 8, the higher stride angles are obtained with GR2 and GR1, while GR3 is the least effective in incurring rapid changes to the system's heading direction.

In addition, figure 10 shows the effect of the sculling offset angle ψ on the trajectory of the system for the three investigated gaits. The results indicate that, for gait GR1, the overall trajectory of the system is smoother for smaller ψ values, while the opposite is true for gait GR2. Moreover, the turning radius of the trajectory attained with gait GR3 can be seen to be approximately constant for $\psi \geqslant 40{}^\circ$.

As a final comment, we note that the turning performance for all three gaits presented here is influenced considerably also by the hydrodynamic characteristics of the main body, as determined through the drag force coefficients in (4).

For the purposes of this work, a novel compliant-body robotic prototype (figures 11, 12) was developed, based on an octopus-inspired design. The prototype incorporates eight compliant arms, mounted to the rear of a compliant platform, which encloses the battery, electronics, and buoyancy elements (figure 12(a)). All parts are made of PU (PMC-746 urethane rubber) after being cast in purpose-designed molds (figures 12(b)–(d)).

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The arms are conical, with length of 200 mm, base radius of 10 mm, and tip radius of 1 mm, and include 38 cylindrical sucker-like protrusions along their length. The latter are arranged in a staggered configuration, covering about one-third of the arm's surface area [49, 51]. Each arm is driven by a dedicated waterproof micro-servomotor (HS-5086WP, Hitech, USA) that allows a 110° span of rotation. The arms are arranged symmetrically at the rear side of the platform (inserted in custom-made compartments, figure 12(d)), such that diametrically opposite arms move in the same plane and with the protrusions side facing inward. The platform is circular with a diameter of 16 cm, while the octopus-like mantle is a half ellipsoid with major and minor axes of 16 cm and 8 cm, respectively. The mantle encloses an empty cavity that can be filled with water to adjust the buoyancy of the robot (figure 12(a)). The overall weight of the submerged prototype with the buoyancy cavity filled with water is 2.68 kg. The compliant parts of the robot were cast in molds (figures 12(b) and (c)) that were custom designed in SolidWorks^® and fabricated in ABSplus material, using a three-dimensional printer (Dimension Elite, Statasys, USA). A mixture of liquid urethane was poured in the molds, at room temperature, avoiding air entrapment, and was left overnight to cure. The material's specific gravity is 1.035 g/cc, while its dynamic viscosity is 5 kg m⁻¹s⁻¹; the tensile strength is 4.826 N mm⁻², with a Young's modulus E of 3.605 N mm⁻². Evidence from analytical models of stiffness in uniform cylindrical beams shows that the stiffness increases fast, in a non-linear manner, as a function of the radius of the beam. This is consistent with the variation of the stiffness from base to tip of the arm that we employed in the simulation studies.

The robotic prototype is energetically autonomous, being powered by an on-board Li-Po battery that allows for about 1 hour of continuous operation. It is also fully untethered and tested underwater in an experimental water tank of length 200 cm, width 70 cm, and height 60 cm. An 8-bit microcontroller platform (Arduino pro mega) is used to program the arm trajectories for implementation of the various swimming gaits, and to collect data regarding the electrical power consumption of the servos. Communication between the microcontroller and a local PC is achieved wirelessly through a dedicated RF link (RFM22B-S2), operating at 433 MHz ISM.

The position and orientation of the robotic prototype inside the water tank are estimated by computer vision methods with a high-definition camera (at 25 fps). A checkerboard marker of known size (figure 11(a)) is placed on the robot during the experiments. The pose of the marker is estimated in each camera frame by calculating the homography between the camera's image plane and the marker's plane, up to a scale factor.

In addition, a custom test-rig, shown in figure 13, has been set up to measure the forces generated by the arm-swimming gaits. A high-precision digital force gauge (Alluris FMI-210A5) is mounted on top of a dexion frame and placed in the water tank such that the dynamometer lies above the water's surface. The generated force from the submerged robot is transmitted to the dynamometer via a rigid stainless steel shaft, acting as a lever. The lower end of the lever is attached to the platform, while the upper end pushes against the force gauge. A spherical bearing is placed at the lever's fulcrum, positioned at the point where the ratio of power transmission is half. All orange parts shown in figure 13 have been fabricated in ABSplus material. This setup allows measurement of forces generated by the arm motions while the robot is restrained from moving. The measurement data are acquired from the force gauge via a custom interface running under LabView in the host PC.

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Here, we present experimental results from a series of tests to assess the propulsion performance of the robotic prototype, equipped with the compliant PU arms, under the various arm-swimming sculling gaits G1, G2, and G4 presented in section 3. The conducted tests confirmed the generation of forward propulsion for all three of the investigated gaits. Figure 14 displays snapshots from one test run, with the prototype swimming using gait G1. Indicative results for the displacement and the velocity profile of the robot swimming in gaits G1, G2, and G4, using the same sculling parameters, are shown in figure 15. As indicated in these graphs, the qualitative characteristics of the velocity profiles, associated with each gait, were found to be consistent with those identified in the simulation studies (see section 4.2). We also note that a 'twitching' of the compliant arms during their recovery stroke, which was particularly evident for G1, was visually observed during these tests. This is consistent with the high frequency ripple in the instantaneous velocity profiles in the simulation results (see figure 4). The fact that a similar ripple does not appear so prominently in the experimental data shown in figure 15 is attributed to the vision-based method employed to track the motion of the prototype.

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The data shown in figure 16 summarize the average speed V attained by the robot in gaits G1, G2, and G4, for different A and ψ values, when ω = 50°/s, and β = 5. For comparison purposes, the experimental data (denoted by solid lines with circular markers) are shown along with the simulation results (denoted by dashed lines), already presented in figure 7. The experimentally obtained values for V are, for all three gaits, in the range of approximately 70−100 mm s⁻¹, exhibiting moderate dependence on the sculling amplitude A, particularly for gaits G2 and G4. It can be seen that the simulation studies are quite effective in predicting the velocity of the system for gait G2, while more significant discrepancies appear for G4 and, even more so, for G1. It is noted that the latter two gaits are associated with a more discontinuous motion of the system (see figure 15). This suggests that the observed discrepancies are due to inherent limitations of the simple drag model employed for the fluid interactions of the system, which manifest themselves to a larger extent during unsteady phenomena.

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It is also of interest to note that, unlike the simulations, the experimental results indicate a slight shift of the optimal ψ value with A. Motivated by this observation, figure 17 plots the speed of the system against ψ − A (rather than ψ). The combined variable ψ − A provides a clearer picture of the system's behavior. It can be seen to effectively render the speed independent of the sculling amplitude, with the velocity being maximized when ψ − A is near zero. This suggests that, to optimize the forward speed for given β and ω values, the sculling offset should, in practice, be specified to be equal to the sculling amplitude.

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An additional set of measurements was performed, with the robot swimming in gait G1, whereby the sculling amplitude and offset were held constant ($A={{25}^{{}^\circ }}$ and ψ = 30°), while varying the velocity ratio β and the recovery stroke velocity ω. The forward speeds attained by the robot during these tests, for three different ω values, are summarized, along with corresponding simulation results, in figure 18. It can be seen that the model underestimates the robot's velocity by about 10%, a discrepancy that, as explained previously, could be attributed to the unsteady characteristics of the system's motion in gait G1. Nevertheless, it should be pointed out that the experimental results confirm the simulations' prediction that the main control variable, upon which the forward swimming velocity depends, is the arms' power stroke angular velocity βω (see figure 18(b)).

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5.2.1. Cost of Transport (CoT)

The propulsive efficiency of the developed prototype is examined here using the non-dimensional cost of transport (CoT) metric, calculated as

$$\begin{eqnarray}&&CoT={{P}_{in}}/(Vmg),\end{eqnarray} \tag{ 7 }$$

where P_in denotes the average electrical power consumed by the servos driving the arms during a sculling period, m is the mass of the swimming platform with the eight compliant arms, g is the acceleration of gravity, while V is the average steady-state swimming velocity.

A summary of the thus estimated CoT values, derived from the parametric studies for the three gaits (see figure 16) is provided in figure 19. It can be seen that, for all three gaits, the CoT generally lies in the range between 1.5 and 2.0, exhibiting limited variance with respect to the sculling amplitude A. Moreover, figure 20 shows the variance of the CoT with respect to β and ω, derived from the measurement set in figure 18, involving different β and ω values. The results indicate that the efficiency of propulsion increases considerably with the velocity ratio β.

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5.2.2. Effect of number of active arms

An additional set of measurements was performed with the eight-arm swimmer to investigate the propulsion characteristics of the system with respect to the number of arms participating in propulsion. The conducted tests involved swimming in gait G1 with varying ψ values, while employing only two or four arms, with A = 25°, ω = 50°/s, and β = 5. The inactive arms were held at an angle of 0° (this requires some energy consumption).

The results of these measurements, compared against those obtained for swimming with all eight arms, in terms of the attained speed V, the overall power consumption P_in, and the CoT, are provided in figure 21. In can be seen that, for fewer arms participating in propulsion, the CoT becomes essentially independent of the sculling offset angle. The mean CoT, averaged over the range of ψ values, is approximately 0.98 for two arms, 1.07 for four arms, and 1.63 for eight arms.

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The observed rise in the cost of transport as the number of arms increases is consistent with other results for the CoT in multi-appendage aquatic robots [53]. It can be attributed to the fact that, as the number of arms is increased, the average swimming velocity also increases. In turn, this entails a larger hydrodynamic resistance during the arms' recovery stroke, which places higher demands in the servomotors driving the arms. Since this resistance scales quadratically with velocity, the increase in the power requirement P_in, is more substantial than the increase in velocity attained by employing more arms for propulsion (see figures 21(a) and (b)).

It is noted that, since figures 16 and 19 indicate that the average attained velocity and the cost of transport are approximately the same for eight-arm swimming in the three investigated forward gaits (assuming a given set of the arms' sculling motion parameters), we expect that the preceding findings regarding the variance of the CoT with respect to the number of arms also apply to gaits G2 and G4.

5.2.3. Propulsive forces

Indicative plots for the axial force measurements obtained with the eight-arm system are shown in figure 22. It can be seen that, although the average generated force is approximately the same for all three gaits, the instantaneous force profiles exhibit marked differences. Similar to the free-swimming velocity profiles (see figures 4 and 15), these can be attributed to the different phasing of the arms' power stroke in the three gaits. More specifically, the generated force under gait G1, where all eight arms move in synchrony, exhibits a single large positive peak during each sculling period. In G2 there are two distinct such peaks, of a lesser amplitude in comparison to G1, related to the fact that the arms move in two sets of four. Finally, the phasing of the arms' power strokes in G2 results in thrust generation being evenly distributed over the duration of the sculling period. In addition, figure 23, shows the average magnitude of the positive force peaks, recorded for G1, as a function of ψ + A (i.e., the angle of the arms when the power stroke is initiated). As would be expected, higher peak forces are attained along the axial direction with increasing ψ + A values.

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Experimental results obtained with the eight-arm robotic prototype are presented for the turning gaits GR1, GR2, and GR3, prescribed as in the computational studies. Figure 24 displays snapshots from an indicative case for gait GR1 (A = 25°, ψ = 25°, ω = 50°/s and β = 5). Initially, the robot is parallel to the long side of the tank and then it performs a clockwise turn of approximately 70°, achieved over six sculling periods (corresponding to 10.5 s).

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Estimates for the average stride angle of the robotic swimmer, obtained from the analysis of the vision system data, are provided for the three gaits in figure 25, as a function of the sculling amplitude and sculling offset. Along with the experimental data (denoted by solid lines with markers), the plots also show the results from corresponding simulation runs (denoted by dashed lines). Although there are quantitative differences, the experimental data indicate that the model captures effectively a number of characteristics related to each of the studied gaits. In particular, it can be seen that the stride angles for gaits GR1 and GR2 are similar and larger than the ones for gait GR3. Moreover, the experimental data also confirm the predictions of the simulations, regarding the reduced dependency of the stride angle on the sculling amplitude for GR3, compared to the other two gaits.

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It should be noted that certain shortcomings of our experimental setup (e.g., boundary effects due to the limited width of the water tank, and difficulties in the acquisition of the visual data due to the limited field of view of the underwater camera) have a more significant impact when studying turning maneuvers, compared to forward swimming, and may have contributed to the discrepancies between the experimental data and the simulation results observed in figure 25.

By sequentially activating forward and turning gaits, it is possible for the system to move along complex paths. One such example is provided in figure 26, in which the robot describes a right-angle path by initially employing gait G1 to move forward, then switching to gait GR2 to perform a turning maneuver, and finally reverting to G1 for the last part of the motion. Similar combinations of the forward and turning gaits presented in the previous sections can be employed by the robotic swimmer, providing a range of combined trajectories. The ability to combine various straight-line and turning maneuvers, provides the system with the capacity to navigate along complex paths. This highlights the potential of implementing reactive behaviors, e.g., collision avoidance, pipe following, etc, that are important for underwater applications.

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Backward propulsion can also be obtained by modifying the timing of the sculling motion so that the power stroke occurs when the arms are extended toward the mantle, while the recovery stroke is performed during closing of the arms. This motion can be employed both in simulation and experimentally, as shown in the snapshots of figure 27. Here, the swimmer swims backward (i.e., arms first), by reversing the power and recovery strokes in the sculling profile. The ability of a robotic agent to perform both forward and backward propulsion, with similar efficiency, may have particular importance to inspection, and search and rescue applications.

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To highlight the potential of employing the arms both for propulsion and for manipulation, figure 28 provides a series of snapshots from an experiment in which two of the robot's arms (specifically, the diametrically opposite pair of arms L1 and R4) are employed in grasping an object (here, a plastic ball), while the other six are allocated to the propulsion of the system, implementing the G1 forward swimming gait. Depending on the overall dimensions and weight of the object, additional arms could be allocated to grasping. The compliance of the arms is of key significance for such schemes, as it allows for a safe and shape-adaptable handling of objects.

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It can be seen that our experimental results largely support the predictions of the lumped-element models proposed, which, although simplified, appear to be in good agreement with the robotic experiments, capturing adequately the behavior of the system in the operating envelope considered. Various discrepancies, between the experimental results and the simulation ones, may be due to inaccuracies in the specification of the modeling parameters, limitations of the fluid drag model used, imperfect buoyancy compensation, inherent limitations and performance variability of the servomotors used in the prototype, as well as limitations in the computer vision estimation methods used.

Toward expanding our array of computational tools for the analysis of the system, we are currently considering employing CFD methodologies for the numerical study of the time-dependent flow around the full robotic system, comprising the mantle and eight sculling arms, under the various proposed gaits. Such an approach would provide better insights regarding the coupling between the system and the surrounding fluid, the effect of which is not captured by the simple fluid drag models used in the lumped-element simulation tools. To this end, we present here indicative results from two preliminary CFD analyses, involving two different simplifications of the system [49, 51, 52]. The first one considers a single-arm self-propulsion system performing a two-stroke sculling motion (figures 29 and 30), that demonstrates the generation of forward thrust. The unsteady flow solution was obtained utilizing high-accuracy numerical methodologies (details in [51]). The sculling arm follows the motion profile described in section 4.1, while rotating as a straight unit around its base and sliding freely along the x-axis. A comprehensive analysis of the observed vortical structures is provided in [51]. The generated hydrodynamic force along the x-axis is shown for this case in figure 30(a), along with the attained velocity ${{v}_{b}}(t)$ and displacement ${{x}_{b}}(t)$ over the duration of 5 cycles (figure 30(b)). These results exhibit similar trends to those demonstrated by the computational and experimental results of sections 4 and 5, respectively.

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In addition to the above, we considered a CFD study of the entire system of the mantle and non-moving arms, translating at a steady speed within quiescent fluid (figure 31). The figures reveal a complex flow disturbance in the surrounding region for both forward and lateral motion (equivalently, for both flow directed toward the mantle tip and the lateral side) and demonstrate the effect of the mantle on the motion.

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