Control of Yaw and Pitch Maneuvers of a Multilink Dolphin Robot

Abstract

Creatures from across the animal kingdom provide a multitude of design inspirations and principles to make robots more lifelike and practical.

6.1 Introduction

Creatures from across the animal kingdom provide a multitude of design inspirations and principles to make robots more lifelike and practical [1, 2]. Since the first fishlike robotic prototype, RoboTuna, was developed at the MIT [3], there have been a number of studies on bioinspired swimming robots owing to their potential in specific underwater applications. Specifically, the state of art in AUV technology hardly satisfies the increasing requirements of long range, high maneuverability , station-keeping, or energy saving. Learning from aquatic animals (e.g., fish and dolphins) and creating bioinspired robots will thereby shed light on new and innovative AUVs with satisfactory propulsive efficiency, speed, acceleration, maneuverability, flow control, and even stealth [4,5,6,7,8,9,10,11]. Hopefully, this endeavor is bridging the gap between the available technologies and the expected mission requirements.

Dolphins, as a typical kind of cetaceans, always surprise people with their amazing swimming skills. Their Froude (propulsive) efficiency is as high as 0.75–0.90 and maximum swimming speed is more than 11 m/s [12]. Not surprisingly, dolphins are capable of leaping up to several meters and performing beautiful acrobatic stunts. This fundamentally benefits from their unique propulsive configurations which are different from common fish. Dolphins have flattened and horizontal caudal fins (i.e., flukes) in conjunction with tails oscillating in the vertical plane, permitting highly flexible pitch maneuvers. On the other hand, fish laterally oscillate in the horizontal plane, achieving striking yaw turn maneuverability. With the complement of flexible flippers and spinal flexion, however, dolphins can also achieve good turn maneuverability in the horizontal plane. Dolphins can reportedly produce high-speed (561.6°/s), small-radius (0.20 BL) turn, which are not much worse than those of fish [13]. Although the pioneer work on evaluating swimming energetics of dolphins, also known as “Gray’s Paradox”, is proved to be flawed, it has greatly advanced our understanding of drag reduction, thrust production, and locomotion control [14, 15].

A general view holds that, for an aquatic animal, flexibility of the body and mobility of the control surfaces determine the performance level of achievable maneuverability [12]. In the past decades, Fish’s group has been very active in exploring the propulsion and maneuvering issues of dolphin swimming. For instance, Fish et al. examined the stabilization mechanism of dolphin swimming [16], qualitative and quantitative maneuverability [13, 17], and even aerial maneuvers of spinner dolphins [18]. However, the turn maneuvers in the vertical plane were rarely investigated.

From the perspective of bioinspired engineering, many issues have been discussed associated with dolphin-like propulsion and maneuvering over the last two decades, such as mechanical design of dorsoventral propulsion [19,20,21], locomotion control [22, 23], as well as roll and pitch maneuvers [24]. As an example, Nakashima et al. used the pitch movement to substitute the yaw maneuver with the help of the roll maneuver so as to realize 3D maneuverability [24]. As a matter of fact, the robot still moved in the horizontal plane. Although many researchers mentioned that a striking feature of dolphin swimming is to have highly agile pitch maneuvers via asymmetric oscillations of the tail, the main effects of the pitch maneuvers, individually or cooperatively, remain unexplored in the context of bio-robotics. Hence, developing a dolphin robot that realizes more agile yaw and pitch maneuvers will significantly contribute to a much enhanced understanding of such dolphin maneuvers.

The main purpose of this chapter is to develop more efficient and innovative yaw and pitch maneuvers for a multilink dolphin robot capable of 3D swimming, based on our previous work on the turning control [25, 26]. Towards this objective, we theoretically and experimentally perform maneuvering control as well as performance assessment. More specifically, for our dolphin robot with a yaw joint and multiple pitch joints, we laterally superpose directed biases to the yaw joint to control yaw motions, and employ asymmetric oscillations to pitch up and down in a wide range. Compared with the previous research on robotic dolphin swimming [19,20,21], a closed-loop pitch angle control strategy based on the intrinsic oscillatory property in dorsoventral propulsion is proposed. It basically involves two aspects, i.e., pitching to a desired angle and maintaining the acquired pitch angle while swimming forward. Another notable advantage is the online implementation of two acrobatic stunts: frontflip and backflip. Compared to the yaw and pitch performance of Nakashima’s dolphin robot in [24], the robot capable of 360° rotation presented here is far superior, revealing that the pitch maneuverability of dolphin robots can greatly be capitalized on. To the best of our knowledge, this is the first time that the acrobatic maneuvers of frontflip and backflip, as well as their combination, have been successfully performed on a physical dolphin robot .

In the remainder of the chapter, the overall design and control scheme of a free-swimming multilink dolphin robot are described in Sect. 6.2. The controls of yaw and pitch maneuvers are elaborated in Sects. 6.3 and 6.4, respectively. Some simulations and experiments are further provided in Sect. 6.5. Finally, discussion and concluding remarks are summarized in Sects. 6.6 and 6.7, respectively.

6.2 Overview of the Dolphin Robot

With the purpose of designing maneuverable and operable 3D motions, we attempt to fuse both mechanical structure and functional characteristics of the physical robot . In our previous work [22], the mechanical design and fabrication issues of a multilink dolphin-like robot equipped with artificial flippers were considered. In contrast, we introduce an internal moving slider as an alternative to the mechanical flippers in realizing up-and-down motion. Internal moving elements are not only protected from corrosion and biological fouling, but also supply effective actuation even in low-velocity circumstances where fins often lose their control authority [27]. The detailed mechanical configuration of the dolphin robot employed in this chapter is schematically shown in Fig. 6.1. The corresponding technical parameters of the developed robot are further listed in Table 6.1.

Fig. 6.1

Mechanical design of the multilink dolphin robot with yaw and pitch joints. a Concept design. b Robotic prototype

Table 6.1 Technical specification of the dolphin robot prototype

The built robot is composed of several components: (1) a moving slider primarily comprising a servomotor loaded by a large copper cylinder, corresponding to pitch joint J₄; (2) a rigid forebody, internally offering a fairly large space for housing the moving slider, control circuits, sensors, rechargeable batteries, and balancing weights; (3) a yaw unit, corresponding to a laterally oscillated joint J₀ responsible for turn maneuvers in the horizontal plane; and (4) a concatenated multilink mechanism connected by means of pitch joints J₁–J₃ for dorsoventral oscillations, followed by a slim peduncle made of polyvinyl chloride, and then a polyurethane fluke possessing a certain amount of elasticity. All links are connected in series with metal skeletons, covered by a compliant waterproof skin. Notice that the flippers and the dorsal fin in the present version solely serve the purpose of decoration and balancing. With such a multilink mechanism, the dolphin-like dorsoventral movements tracing a sinusoidal pathway can be achieved. According to the biological observations on dolphin swimming, Romanenko [28] suggested to use (6.1) to describe the periodic excursions of the body centerline in dorsoventral movements.

$$ h\left( {x_{n} ,t} \right) = h_{T} f\left( {x_{n} } \right)\sin \left( {2\pi ft} \right) $$

(6.1)

where h_T denotes the maximal vertical excursion of the fluke. x_n = x/BL represents the longitudinal coordinate measured from the rostrum, divided by the BL. f(x_n) is the polynomial expression of x_n, taking the form of $0.21 - 0.66x_{n} + 1.1x_{n}^{2} + 0.35x_{n}^{8}$. f is the tail beat frequency; and t is the time.

For the convenience of engineering applications, f(x_n) is simplified as $0.1 - 1.3x_{n} + 2.2x_{n}^{2}$ based on a polynomial fit of typical dolphin-like swimming features [22]. Furthermore, this sinusoidal motion is discretized both in time and distance. Specifically, the discretization in time is to transform a continuous movement into a series of position information instances at different times, whereas the discretization in distance is to approximately utilize folding lines instead of a continuous wave. By wirelessly modulating oscillatory frequency and amplitude, various speeds can be accomplished. Besides typical dorsoventral movements exhibited in the multijoint tail and the fluke, the dolphin robot can freely pitch and heave through adjusting the rotation angle of J₄ anteriorly and posteriorly, thus gaining the ability to move in 3D aquatic environments. In this case, the moving slider acting as a centroid adjustor is equivalent to a low-range pitch propulsor. In order to vary the pitch angle evidently, the moving slider should be mounted in the front of the head far from the robot’s centroid. Note that the moving slider remains its neutral position parallel to gravity direction except for pitching and diving motions. For more information on the detailed control and analysis of the moving slider, please refer to [29].

As a versatile aquatic animal, dolphins have become well known for performing multifarious 3D maneuvers. Since we focus our attention on turn maneuvers, further discussion of other maneuvers is beyond the scope of this chapter. In general, two types of turns can roughly be categorized: (1) turn in the horizontal plane, i.e., yaw turn, and (2) turn in the vertical plane, i.e., pitch turn. In the following sections, we will elaborate these two representative turn maneuvers and further integrate them into the physical robot .

6.3 Analysis and Control of Yaw Turn

6.3.1 Analysis of Yaw Turn

Natural dolphins execute yaw turn maneuvers in the horizontal plane by either lateral flexion of the anterior body or adduction and rotation of the flippers. For a dolphin robot , two methods are thereby available for the flexible yaw turns based on different propulsors. One is utilizing lateral flexion of the forebody with a specially designed yaw unit as fish do. In this fashion, dynamic biases can be added to the laterally moved yaw unit enabling the shift to the left or the right, while maintaining dorsoventral oscillations to produce thrust. Another idea is to employ asymmetric drives of the flippers to accomplish directed maneuvers. The difference of hydrodynamic forces between the bilateral flippers will generate a yawing moment, often accompanied by an anteriorly directed force on one side and another posteriorly directed force on the other side. Considering that the current miniature dolphin robot has no multifunctional mechanical flippers, we only apply the former method to generate yaw turns. It is worth noting that only one yaw joint is provided due to a tradeoff between the functionality and the structure complexity.

6.3.2 A Two-Segment Model for Yaw Turns

There is little discussion on how to achieve yaw turns in dolphin robots . To parameterize the yaw turn maneuvers in our robot , a simplified two-segment linked propulsive model is presented. As shown in Fig. 6.2, the whole robot is considered as a two-segment propulsive organism comprising a laterally deflected forebody and a dorsoventral multijoint tail. During the implementation of yaw turns, the forebody capable of sidewise movements by exerting variable biases serves as the rudder leading the heading, while the multijoint tail plays the role of the moving paddle.

Fig. 6.2

Illustration of a two-segment model for yaw turns

Suppose that θ₀ is the bias angle between the forebody and the tail, V is the relative velocity between the water and the moving robot , and F_t is the thrust resulted from dorsoventral oscillations of the tail. Meanwhile, the robot is assumed to be in a state of neutral buoyancy. That is, its center of mass coincides with the buoyancy in the Z-axis, indicating that the effect of the gravity can be neglected. From a perspective of classical mechanics, the behaved yaw turn is viewed as a combined action incorporating rotation and revolution, which is induced by the resultant forces primarily consisting of fluid drags and forward thrust. Furthermore, for a desired turn with given ω and turning radius (r′), a centripetal force (F_n) directed toward the turning center and a continuous turning moment (M_s) should be offered, i.e., having the following relationship:

$$ \left\{ {\begin{array}{*{20}l} {F_{n} = mV^{2} /r^{{\prime }} = m\omega^{2} r^{{\prime }} } \hfill \\ {M_{s} = m\dot{\omega }r^{{{\prime }2}} } \hfill \\ \end{array} } \right. $$

(6.2)

where m is the total mass of the whole robot .

Substituting F_n and M_s with detailed expressions containing fluid drag and forward thrust on the participated propulsive elements, we get an explicit yaw turn relation as (6.3) via some superposition and reduction operations.

$$ r^{{\prime }} = \frac{{mV^{2} }}{{\alpha V^{2} \sin \theta_{0} - F_{t} \sin \frac{{\theta_{0} }}{2}}} $$

(6.3)

where α is a factor closely relevant to the wetted areas of the forebody and the tail. For more details about the derivation process, please refer to [26, 30].

Therefore, parametric relationships of r′, V (or ω), and θ₀ according to (6.3) in diverse yaw turns can theoretically be evaluated. To this end, the presented yaw turn control is further integrated into a 3D dynamic model developed for dolphin-like swimming within the framework of multibody dynamics [30]. Thus simulation results about various yaw turns, as will be addressed in Sect. 6.4, can be achieved in the MATLAB/Simulink graphical development environment, which offers a qualitative estimation. We remark that the proposed two-segment model assumes that the robot has a special yaw unit, facilitating tests in comparison with fish robots . A flaw with this model is its inability to predict the yaw turn with active flippers.

6.4 Analysis and Control of Pitch Turn

6.4.1 Analysis of Pitch Turn

As mentioned previously, the dolphin fluke that is firmly attached to the posterior skeleton is flattened and horizontal. Because of the particular orientation of the fluke, dolphins are competent in effecting pitch turns in the vertical plane. The dolphin robot , in the context of biorobotics, is expected to easily execute maneuverable pitch turns by asymmetric oscillations of the tail about the longitudinal axis of the body. For the multilink tail mimicking the common propulsive curve of live dolphins, symmetric oscillations will cause straight swimming, while asymmetric oscillations (e.g., adding some biases to some of the moving links) will trigger up-and-down motions in the vertical plane. Notice that the moving tail in the pitch turn not only offers the thrust, but also produces a nonzero time-averaged moment leading to a change in the vertical heading.

Inspired by fast yet maneuverable C-start escape response (hereafter referred to as “C-type turn”) in Chap. 3, we make an effort to replicate this mechanism to achieve wide-range pitch maneuvers in dolphin-like swimming. In general, C-type turn involves two stages: stage 1 (the bending phase) and stage 2 (the unbending phase), distinguished by different functions and body shapes [31]. The bending phase means the formation of a C-shaped configuration by bending all fore-and-aft pitch joints simultaneously to one side of the longitudinal body axis, whereas the unbending phase indicates the recovery of the moving body from being curved to being straight. In principle, the bending phase is relatively short, having a positive effect on increasing the turning angle; while the relatively long unbending phase provides a negative effect, leading to a back turn of the head. Therefore, the recoil of the turning angle in the unbending phase should be minimized so as to maximize the turning angle. However, little research has been conducted on the design and optimization of the unbending phase. In order to restrain the possible recoils, a natural strategy is to slow down the unbending. Following this strategy, the dolphin robot will meet with another difficulty in overcoming its own gravity. If the dolphin moves slowly in a pitch maneuver, the gravity will force it back to the horizontal plane and to sink or even turn over sideways. But this is not the case for a fish maneuvering in the horizontal plane. So a novel approach to carry on the unbending phase should be sought for a maximum pitch performance.

6.4.2 Design of the Unbending Phase

From the geometric viewpoint, the task of the moving pitch links in the unbending phase is to recover from the C-shaped configuration to a straight form. Consider a multilink dolphin robot , where L_i (i = 0, 1, 2, 3) denotes the i-th link, l_i indicates the length of the i-th link, and J_i stands for the ith joint. With the pitch-up direction as positive, the pitch angle of the robot head is indicated by θ, whereas the rotation angle θ_i with a subscript is defined for joint J_i. Since the origin of the reference frame J₁-xz is fixed at the first pitch joint J₁, the movements of all links can be divided into two directions, longitudinal (x-axis) and dorsoventral (z-axis). According to classical mechanics, the caudal movement will thrust the robot body, whereas the dorsoventral oscillation will produce an unwanted nose-down pitching moment to reduce the acquired pitch-up angle. Moreover, for any moving rigid body through a fluid, if its speed is constant, the drag from the fluid is directly correlated with its wetted area. To minimize the undesirable pitching moment, it is feasible to make sure that the dorsoventral movements contain as few wetted areas as possible. Evidently, the smallest wetted area for an elliptical cylinder link is its cross-section, implying that each link should ideally move along its longitudinal axis. Unfortunately, this is unfeasible in that all links are serially connected. So a practical way is to maintain the present position, changing as little as possible.

As shown in Fig. 6.3a, J₁ is the most unadjustable joint, which can only rotate back to x-axis. During this process, L₂ should keep as close to its previous position as possible. That is, the end point of L₂, i.e., J₃ should stay on the original position of L₂. As shown in Fig. 6.3h, let $J_{2}^{{\prime }}$ be the updated position of J₂ after a short time interval Δt. To keep $L_{2}^{{\prime }}$ as close to as its previous position, the updated end point $J_{3}^{{\prime }}$ of $L_{2}^{{\prime }}$ should remain at L₂ geometrically, satisfying

Fig. 6.3

Illustration of the unbending phase. a J₁ is the active joint. b J₁ has reached the straight form. c–f J₂ acts as the active joint. g J₂ has reached the straight form. h Diagram of calculation of $\theta_{i}^{{\prime }}$. Notice that the dotted lines denote the previous positions of the links, while the solid lines represent the new positions after a short time interval Δt. In particular, the last link in (f) cannot keep close to its own previous position, since the last joint has arrived at its maximum bending limit

$$ \frac{{x_{3}^{{\prime }} - x_{2} }}{{x_{2} - x_{3} }} = \frac{{z_{3}^{{\prime }} - z_{2} }}{{z_{2} - z_{3} }} $$

(6.4)

where, (x₂, z₂) and (x₃, z₃) are the coordinates of J₂ and J₃, and $\left( {x_{3}^{{\prime }} ,z_{3}^{{\prime }} } \right)$ is the coordinate of $J_{3}^{{\prime }}$. Meanwhile, $J_{2}^{{\prime }}$ and $J_{3}^{{\prime }}$ are the two end points of L₂ whose length is l₂, so

$$ \left( {x_{3}^{{\prime }} - x_{2}^{{\prime }} } \right)^{2} + \left( {z_{3}^{{\prime }} - z_{2}^{{\prime }} } \right)^{2} = l_{2}^{2} $$

(6.5)

By solving (6.4) and (6.5), the new coordinate of J₂ is determined. Repeating this adjustment down to the last joint of the robot , we can obtain the new positions of all pitch joints. More generally, the following equations hold

$$ \left\{ {\begin{array}{*{20}l} {\frac{{x_{i + 1}^{{\prime }} - x_{i} }}{{x_{i} - x_{i + 1} }} = \frac{{z_{i + 1}^{{\prime }} - z_{i} }}{{z_{i} - z_{z + 1} }}} \\ {\left( {x_{i + 1}^{{\prime }} - x_{i}^{{\prime }} } \right)^{2} + \left( {z_{i + 1}^{{\prime }} - z_{i}^{{\prime }} } \right)^{2} = l_{i}^{2} } \\ \end{array} } \right. $$

(6.6)

where (x_i, z_i) and $\left( {x_{i}^{{\prime }} ,z_{i}^{{\prime }} } \right)$ (i = 2, 3) are the coordinates of J_i before and after Δt, respectively.

Further let $\theta_{i}^{{\prime }}$ (see Fig. 6.3h) be the rotation angle of J_i after Δt. It easily follows

$$ \theta_{i}^{{\prime }} = \arccos \frac{{\overrightarrow {{J_{i}^{{\prime }} J_{{i{ + 1}}}^{{\prime }} }} \cdot \overrightarrow {{J_{i - 1}^{{\prime }} J_{i}^{{\prime }} }} }}{{\left| {\overrightarrow {{J_{i}^{{\prime }} J_{{i{ + 1}}}^{{\prime }} }} } \right|\left| {\overrightarrow {{J_{i - 1}^{{\prime }} J_{i}^{{\prime }} }} } \right|}},\quad i = 2,3. $$

(6.7)

Consequently, all the updated joint angles can be determined by means of the iterative calculation of (6.6) and (6.7).

In practice, once the unbending phase is initiated, J₁ rotates at its full speed and the other joints all follow their previous joints. Hence, J₁ is referred to as the active joint, the others as the passive joints. After J₁ returns to the straight form, J₂ will be the next active joint. This suggests that, like a propagating wave, the posterior joints will become the active joint alternately down to the last joint. An overall unbending phase is illustrated in Fig. 6.3a–g. Notice that if the calculated joint angle exceeds a preset maximum bending limit (e.g., a case shown in Fig. 6.3f), this limit will temporarily be used as the control input. We also remark that this new trajectory tracking strategy for the unbending phase is not specifically designed for pitch turns in dolphins. With proper modifications, this tracking strategy can be extended to fast turns in fish, further speeding up the whole turning maneuvers.

6.4.3 Maintaining the Pitch Angle

In addition to pitching to a desired angle through fast turns, maintaining the acquired pitch angle while swimming forward is another issue to achieve a relatively precise control of the pitch angle. Due to intrinsic properties of asymmetric oscillations of the tail and the fluke in pitch turns, it is very difficult to seek an accurate quantitative relationship between the oscillation of the joints and the acquired pitch angle θ. Indeed, like its biological counterpart, the robot head exhibits vertical recoil motions due to the large thrust generated by the fluke [16]. As a result, an instant θ does actually not indicate the desired pitch angle, more exactly, nor the “average” pitch angle. Note that the angle between the horizontal plane and the actual trajectory of the robot can further be categorized into the pitch-down related diving angle and the pitch-up related surfacing angle. Unlike the previous open-loop yaw turn control, with the aid of the feedback of θ measured by the onboard gyroscope, we propose a closed-loop strategy to make θ fall in the range of [θ_Ref − θ_A, θ_Ref + θ_A], where θ_Ref indicates the desired reference pitch angle and θ_A denotes the nominal recoil amplitude of the dolphin. Figure 6.4 shows a flowchart illustrating how to maintain a given pitch angle. If the measured θ reaches the lower bound (i.e., θ_Ref − θ_A), the dolphin robot will pitch up. Likewise, if the measured θ peaks the upper bound (i.e., θ_Ref + θ_A), the robot will pitch down. As for the low-level control implementation, simple PID algorithms are employed to regulate the rotation rate of the active joint in both bending phase and normal swimming so as to achieve the desired pitch angle [32].

Fig. 6.4

A flowchart describing how to maintain a preset pitch angle

In the interest of simplicity, we give an example of the implementation of straight swimming. As was discussed earlier, the presented unbending phase is of assistance in reducing the dorsoventral force while enhancing the thrust. If the tail’s oscillations are made up of two unbending phases in opposite directions, the robot will certainly swim straight. Even so, the accompanying recoil motions exhibited in the robot head (i.e., θ_A) always exist. Practically, θ_A is served as a threshold to inversely trigger pitch motions.

Consider a half period oscillation in Fig. 6.5. When θ − θ_Ref > θ_A will begin to rotate ventral-ward. Meanwhile, all the other pitch joints will passively adjust their joint angles. Once J₁ has reached its swim-limit, the posterior joints will successively take the role of active joint to further bend ventral-ward. Notice that the swim-limit is not the maximum oscillation limit of the joint, but a virtual limit which is smaller than the maximum oscillation limit. Furthermore, if this swim-limit of each joint increases from head to tail, the swimming robot will display a body wave with an increasing amplitude of oscillation.

Fig. 6.5

Illustration of a half period oscillation. In this half period, only J₁ acts as the active joint. In (c), J₃ cannot keep on its original trajectory, because J₃ has reached its own swim-limit. Notice that the dotted lines mean the previous positions of the links, while the solid ones show the new position after a short time interval Δt

On the other hand, when θ − θ_Ref < −θ_A, J₁ will retake the role of the active joint and resume an inverse rotation. In case of all the joints attaining their swim-limits, a bending phase with the same joint rotation rate may be inserted to compensate for a gap between θ and −θ_A. It is worthwhile to note that, instead of an arbitrary constant, θ_A is dependent on the defined swim-limits. A preferable θ_A tends to make only the anterior half joints have an opportunity to become the active joints, thereby replicating a full period of the sinusoid body wave .

By the way, if there is no feedback of the pitch angle, it is possible to produce a traveling body wave and swim forward merely by transferring the active joint from the first joint to the middle one. This is also an empirical method to determine θ_A by experiments. Since aquatic perturbations always exist, a significant flaw is that the robot is unable to swim straight.

6.5 Results and Discussions

6.5.1 Experimental Setup

In order to evaluate the presented methods to achieve yaw and pitch turns, robotic prototypes and an experimental testbed have been developed. In particular, another multilink dolphin robot similar to the prototype shown in Fig. 6.1b has been specifically modified to perform wide-range pitch maneuvers. The primary differences lie in the absence of the moving slider and the rearrangement of static balancing weights. Currently a “trial and error” method is employed to adjust the mass and position of balancing weights so that the COM of the robot closes to and lies beneath its CB at rest. Notice that an additional purpose of removing the slider is to experimentally verify the potential of pitch turns in the vertical plane. Systematic tests on the turning performance and control methods have been conducted in two swim tanks: the larger one is approximately 7 m × 3.5 m × 1.2 m (length × width × height) and the other is about 1.68 m × 0.87 m × 0.8 m with a transparent wall permitting close inspection. Commands either from the human operator or from the interactive control console are sent to the dolphin robot via a wireless connection able to penetrate water down to a depth of 1.2 m. The dolphin robot receives the upper commands and thereby changes its swimming speed and direction. At the same time, an overhead vision measuring system is applied to provide an effective estimation of the planar motion performance. Moreover, a dual-axis LPY5150AL gyroscope fixed in the robot’s head measures angular velocity along the yaw and pitch axes with a full scale of ±1500°/s.

6.5.2 Testing of Yaw Turns

To ensure a stable turning performance and facilitate the comparison, a canonical circular motion is chosen as the test condition. In tests, the dolphin robot was required to uninterruptedly perform circular yaw turns, in which maneuverability related parameters were then assessed. As an illustrative example, Fig. 6.6 shows a yaw turn sequence, in which the robot completed a circular motion with a period of approximately 13 s. Actually, a combination of varying circular motions will effect complicated motions such as loop-the-loop and figure-of-eight motions. As an extension, Fig. 6.7 shows a comparison of simulated and experimental data for various circular maneuvers. Simulation results according to (6.3) in connection with the dynamic swimming model indicate that the turning rate is inversely related to the turning radius within the overall range of bias angles. More specifically, the turning rate increases but the turning radius declines with the increase of biases from 15° to 60°. However, the measured values are somewhat different from the simulated ones: the measured turning radii chiefly tally with the calculated ones, but the measured turning rates are lower than the simulated data. This deviation may be due to other unmodelled factors, such as wave drag, mechanical friction loss and elasticity, which participate in and contribute to aquatic locomotion directly or indirectly. In contrast to circular motions performed by the multilink fish robots , the achievable turning rate of the dolphin robot with the same bias angle (about 32°/s with a bias angle of 60°) is considerably inferior to that of the fish robot which peaked 52°/s [25].

Fig. 6.6

Snapshot sequence of a circular yaw turn near the water surface

Fig. 6.7

Comparison of the relationship of bias angle, turning radius, and turning rate in circular motions with an oscillatory frequency of 1 Hz. Notice that “S” and “E” shown in parentheses stand for “simulated data” and “experimental data”, respectively

Considering that 3D swimming can be viewed as a composition of yaw maneuvers and up-and-down motions, we tested a hybrid maneuver involving diving, yaw turns, and surfacing. In this case, the tail plus the fluke acted as the planar propulsor, whereas the moving slider served as the pitching propulsor. Figure 6.8 illustrates a snapshot sequence of a 3D combined motion lasting 53 s. By and large, three stages can be shown: (1) By the aid of an anteriorly tilted slider, the dolphin robot with zero speed dived continuously till touching ground during the 0–13 s; (2) A series of circular motions, i.e., loop-the-loop behaviors, were performed by imposing a directed bias angle on the yaw joint within the 14–51 s interval; and (3) The robot thereafter rapidly rose up to the surface of the water with a posteriorly tilted slider. A careful inspection revealed that the robot took approximately 12 s to dive down to about 0.6 m and merely 3 s to return to the water surface. This indicated that surfacing was much faster and more efficient than diving for the moving slider induced pitch motions. Although the moving slider is a relatively simple means to regulate the vertical movement, it mostly goes into effect in low-speed circumstances. Additionally, due to many constraints on available internal space and mechatronic devices, the acquired maximum pitching by the slider is much smaller than 90°, only 60° for our robot . It is apparent that the robot with the moving slider is fundamentally unable to vertically move up and down, much less execute a 360° rotation in the vertical plane.

Fig. 6.8

Snapshot sequence of a combined case of 3D motion including diving, yaw turns, and surfacing

6.5.3 Testing of Wide-Range Pitch Turns

Besides the small-angle pitching led by the moving slider, other wide-range pitch maneuvers were tested. Specifically, an acrobatic flip comprising a sequence of body movements accompanied by one or more rotations was tested on the multilink dolphin robot . Although acrobatic flip is generally performed in gymnastics, dance, tricking, free running, and various special activities, these acrobats have no application to dolphin robots yet. In this work, we try to create two hybrid acrobatic stunts, frontflip and backflip, in the context of robotics. For our dolphin robot , there are three basic elements to the flips: diving, surfacing, and maintaining the pitch angle. The parameters relevant to the dolphin flip s are further listed in Table 6.2, where the swim-limits and maximum oscillation limits are all relative values. It should also be remarked that the maximum oscillation limits are considerably constrained by the stiffness of the outer skin and by the control objective. The controlled joint angle is then computed by means of the pitch control method developed in Sect. 6.4.

Table 6.2 Parameters relevant to the dolphin flips

(1) Frontflip: First, with the fine-tuned pitch algorithms, the robot sharply pitches up (i.e., surfacing) to 45° followed by pitching down (i.e., diving) to around −45°. Then, the robot will bend all pitch joints ventral-ward simultaneously until it exceeds −90° in an instant (referred to as a frontflip). When the robot body is tucked, the relative positions of the COM and the CB will be inverted. Due to the remaining rotational speed and the gravity, the robot will retain the curved body and tumble forward until its head is upward. Finally, another pitch-down procedure will be deployed to adjust the pitch angle back to the horizontal plane. Accompanied the successful implementation of a dolphin frontflip depicted in Fig. 6.9, we summarize the detailed control steps as follows:

Fig. 6.9

Snapshot sequence of a successful dolphin frontflip

• (1) All the pitch joints oscillate upward, effecting a level-up of the robot head, and then gradually stretch down based on the proposed pitch algorithm. After several oscillation cycles, the robot swims away from the bottom of the tank and gets to the middle of the water (please see Fig. 6.9a–c).

• (2) The robot suddenly jerks down and employs a pitch-down method, so that it can obtain a pitch angle of −45° (see Fig. 6.9d).

• (3) All the pitch joints slightly bend dorsally, enlarging the swing distance, in order to maximize the pitching moment in the next action (see Fig. 6.9e).

• (4) The robot bends all pitch joints ventrally with the maximum speed (see Fig. 6.9f), resulting in a tuck position rolling forward. Once the pitch angle is over −90°, the COM will shift to the front of the CB (see Fig. 6.9g). The whole tuck position will be dominated by the model discussed later.

• (5) When the robot’s frontflip angle rate approaches zero, the COM will be instead located beneath the CB. At this moment, the robot stretches its tail to start an updated pitch-down maneuver. Hence, the relative positions of the COM and the CB will resume the normal state (see Fig. 6.9i, j).

• (6) After several repetitions of the pitch-down maneuvers, the robot will eventually return to a horizontal state (see Fig. 6.9k, l).

Figure 6.10 further plots the pitch angle of the robot head collected from the gyroscope during the execution of the frontflip. In the following part, we will provide a motion estimation of the tuck position while performing a frontflip.

Fig. 6.10

Comparison of simulated and actual pitch angle in the frontflip

In simplicity, the equivalent lengths are defined in accordance with the dimension of the physical robot : L_a = 0.3 m, L_p = 0.16 m, and L_c = 0.07 m, where L_a, L_p, and L_c sequently denote the equivalent lengths for the anterior immobile part, the dorsoventrally oscillatory part, and the fluke chord. According to a statistical analysis of eight experiment videos, the tuck position lasted t_tuck ≈ 0.666 ± 0.122 s (mean ± SD) on average and the fluke underwent a rotation of 180° + 45° + 30° = 255°, where 45° denotes a preparatory 45° pitching up for the purpose of hastening the forward flip. Notice also that a transient overshoot of about 30° was usually observed due to the rotational inertia. Particularly, the whole tuck process is considered as symmetric and the maximum rotation rate is estimated as 540 ± 33°/s. Considering that the acceleration is continuous and the force (or current) do not change abruptly, and to cover the overshoot-related slowdown phase, we use a cubic polynomial to kinematically fit the rotation angle of the fluke:

$$ \beta_{\text{front}} = 1726.4t^{3} - 1724.7t^{2} + 255,\quad t \in \left[ {0,0.666} \right] $$

(6.8)

where β_front is expressed in degrees. Given β_front and θ in the earth-fixed inertial reference frame, it follows that ψ = β_front − θ, where ψ represents the rotation angle of the multijoint tail. Figure 6.11 visually plots the rotation angle and rate of the fluke in the tuck position. As can be observed, the rotation angle of the fluke varied smoothly. Of course, other types of polynomials may serve to numerically fit the tuck position of dolphin flip .

Fig. 6.11

Estimate of the rotation angle and rate of the fluke in the tuck position. Note that the black lines stand for the estimated rotation angles, while the blue lines for the estimated rotation rates

On the other hand, a simplified dynamic model is formed to describe this tuck position. Since only a fraction of the posterior of the robot actively participates in the dorsoventral oscillations, whose mass is relatively small compared to that of the robot , it is reasonable that the moment of inertia of the whole robot is considered as constant. In practice, the tucked dolphin body is regarded as a cylinder whose moment of inertia can be calculated as $J_{D} = \frac{1}{12}mL_{\text{flip}}^{2}$, where L_flip denotes the equivalent length of the tucked robot . With the understanding that a clockwise direction from the lateral perspective is positive, for a cross-sectional area of the anterior immobile part shown in Fig. 6.9g, the resistance experienced is calculated as

$$ df = \frac{1}{2}\rho \cdot \left( {W_{a} \cdot dr} \right) \cdot C_{c} \cdot \left( {r \cdot \dot{\theta }} \right)^{2} $$

(6.9)

where ρ is the density of the water, W_a is the equivalent (average) dorso-ventral-ward projection width of the anterior immobile part, C_c is the profile drag coefficient of a cylinder, and r is the distance between the cross section to the current COM. Hence, the resistance moment experienced is derived as

$$ M_{f} = \int\limits_{0}^{{L_{a} }} {df \cdot r} = \frac{1}{8}\rho W_{a} C_{a} L_{a}^{4} \dot{\theta }^{2} . $$

(6.10)

In the same manner, the thrust moment is evaluated as

$$ \begin{aligned} M_{F} & = \, \int\limits_{0}^{{L_{p} }} {\frac{1}{2}\rho W_{p} C_{p} r^{2} \dot{\psi }^{2} \cdot rdr + } \int\limits_{{L_{p} }}^{{L_{p} + L_{c} }} {\frac{1}{2}\rho W_{c} C_{F} r^{2} \dot{\psi }^{2} \cdot rdr} \\ & { = }\frac{1}{8}\rho \dot{\psi }^{2} \left\{ {W_{p} C_{c} L_{p}^{4} + W_{c} C_{F} \left[ {\left( {L_{p} + L_{c} } \right)^{4} - L_{p}^{4} } \right]} \right\} \\ \end{aligned} $$

(6.11)

where W_p and W_c are the equivalent widths of the dorsoventrally oscillatory part and the fluke, respectively; and C_F is the profile drag coefficient of a real square flat plate perpendicular to the flow.

As mentioned previously, an overall tuck position started at t₀ (corresponding to 7.1 s) and continued for a time of t_tuck. Afterwards, owing to the inertia along with the induced moment by the difference between the COM and the CB, the dolphin continued to tumble forward. Therefore, the whole frontflip maneuver was dynamically governed by the following equations of motion:

$$ J_{D} \ddot{\theta } = M_{F}^{{\prime }} + M_{f}^{{\prime }} + M_{G} $$

(6.12)

$$ M_{F}^{{\prime }} = \left\{ {\begin{array}{*{20}l} { - \frac{1}{8}\rho \dot{\psi }^{2} \left\{ {W_{p} C_{c} L_{p}^{4} + W_{c} C_{F} \left[ {\left( {L_{p} + L_{c} } \right)^{4} - L_{p}^{4} } \right]} \right\},} \hfill & {t \in \left[ {t_{0} ,t_{0} + t_{\text{tuck}} } \right]} \hfill \\ {0,} \hfill & {t > t_{0} + t_{\text{tuck}} } \hfill \\ \end{array} } \right. $$

(6.13)

$$ M_{f}^{{\prime }} = \left\{ {\begin{array}{*{20}l} { - \frac{1}{8}{\text{sign}}\left( {\dot{\theta }} \right)\dot{\theta }^{2} \rho W_{a} C_{a} L_{a}^{4} ,} \hfill & {t \in \left[ {t_{0} ,t_{0} + t_{\text{tuck}} } \right]} \hfill \\ { - \frac{1}{16}{\text{sign}}\left( {\dot{\theta }} \right)\dot{\theta }^{2} \rho \left( {W_{a} C_{a} L_{a}^{4} - 2W_{c} C_{F} L_{p}^{4} } \right),} \hfill & {t > t_{0} + t_{\text{tuck}} } \hfill \\ \end{array} } \right. $$

(6.14)

$$ M_{G} = \left\{ {\begin{array}{*{20}l} {mgd_{0} t\cos \left( {\dot{\theta }} \right)/t_{\text{tuck}} ,} \hfill & {t \in \left[ {t_{0} ,t_{0} + t_{\text{tuck}} } \right]} \hfill \\ {mgd_{0} \cos \left( {\dot{\theta }} \right),} \hfill & {t > t_{0} + t_{\text{tuck}} } \hfill \\ \end{array} } \right. $$

(6.15)

where $M_{F}^{{\prime }}$ and $M_{f}^{{\prime }}$ represent the thrust and resistance moments on the robot ; $M_{G}$ indicates the moment induced by the difference between the COM and the CB; and d₀ denotes the maximum distance between the COM and the CB.

For the convenience of simplicity, the relative angular speed between the dolphin head and the water flow is assumed to be $2\dot{\theta }$ since $\dot{\theta }$ is measured in the earth-fixed reference frame. Note that the induced water flow results from the drastic flap of the fluke at the end of the tuck position. To precisely determine this relative angular speed, extensive strict hydrodynamic tests are needed. As a comparison, the simulated pitch angle according to (6.13) is plotted in Fig. 6.10, with the used parameters and initial conditions listed in Table 6.3. It was found that the simulated pitch angle was largely consistent with the simulated one while in the tuck position. More specifically, first, an average pitch angle rate up to 208°/s was obtained within the 7.1–7.7 s. Second, there existed a critical point in both simulated and experimental data at about 7.7 s, corresponding to an abrupt change in the pitch angle rate dropping from 208 to 82°/s within the 7.7–8.8 s. It implied that the fluke providing the thrust was shifting to the drag pattern. Third, the slope of the simulated data was much steeper than that of the experimental data in the late phase of the tuck position, showing a slower rotation rate of the fluke results. This may be because the tucked dolphin body cannot be simply regarded as a circular cylinder whose drag coefficient will be increased notably. Last, an interesting phenomenon appeared that, after 8.8 s, the robot entered into a damped oscillatory mode of motion till the COM lay almost directly beneath the CB, if no active control worked. At that time, the waterproof outer skin was actually compressed to some extent so that the buoyancy experienced decreased and the robot went on downward. It also partly demonstrated the effectiveness of the active control in accomplishing controllable maneuvers.

Table 6.3 Parameter values used for the tuck position simulation

(2) Backflip: In relation to the dolphin frontflip, the backflip is harder to implement because the gravity predominantly serves as a resisting force in the foremost 270° rotation. Therefore, the robot has to pitch up 270° only by its powered propulsion. Especially, the robot will undergo an upside-down motion state. To restrain possible sideward roll, the COM should be altered to lie beneath the CB. For our robot , this can be achieved by bending J₁ ventral-ward sharply. With the aid of a snapshot sequence of a dolphin backflip shown in Fig. 6.12, the control procedure is elaborated as follows:

Fig. 6.12

Snapshot sequence of a successful dolphin backflip

• (1) Like the preparation of the frontflip, the robot first pitches up into the middle of the water (see Fig. 6.12a–e). A major difference lies in that the robot continues pitching up so as to locate itself in an upright position.

• (2) J₁ drastically bends 50° ventral-ward to make the COM get across the CB (see Fig. 6.12f–h). The robot thereby maintains an upside-down posture.

• (3) The robot ceaselessly pitches up till it attains a total pitch angle of more than 220°. Then, it extends all the pitch joints, preparing for the last action of the backflip (see Fig. 6.12i).

• (4) Swinging all the pitch joints as swiftly as possible (see Fig. 6.12j–m), the robot rolls backward and ultimately completes a backflip (see Fig. 6.12n, o).

In the same fashion as the frontflip, we conducted backflip simulation and experimental comparison. Specifically, eight backflip experiments were used to numerically estimate the tuck position. The whole tuck process lasted t_tuck ≈ 0.575 ± 0.118 s, in which the fluke underwent a rotation of approximately 150° + 0° + 30° = 180°, where 0° means the removal of the preparatory pitching up and 30° indicates the accompanying overshoot. The obtained fitting equation to describe the rotation angle of the fluke is given below.

$$ \beta_{\text{back}} = 1893.6t^{3} + 1633.2t^{2} + 180,\quad t \in \left[ {0,0.575} \right] $$

(6.16)

According to the plot shown in Fig. 6.11, the fluke rotation angle of the backflip is smaller than that of the frontflip, which is caused by lack of both the preparatory 45° pitching up in the backflip and the fore-and-aft symmetry of the COM of the robot . Figure 6.13 further plots a comparison between simulation and experimental results for the pitch angle in the backflip corresponding to Fig. 6.12. It was shown that in general a good agreement was achieved to model the stuck position in the backflip. Specifically, the simulated tuck process initiated at about 8.1 s. Interestingly, if active control was removed, the robot would eventually enter into a damped oscillatory mode with a pitch angle of 450°, i.e., the rostrum of the robot stood upward. In reality, the robot only experienced a total pitch angle of approximately 360° from start to end due to the existence of supervised control. Extensive tests also confirmed that a 50° ventral-ward bending of J₁ adequately triggered and settled the upside-down swim.

Fig. 6.13

Comparison of simulated and actual pitch angle in the backflip

(3) Complex pitch maneuvers: Furthermore, other complicated pitch maneuvers were designed and tested. For instance, basically, the frontflip/backflip can be duplicated or mixed to produce more amazing stunts. More interestingly, the frontflip/backflip can further be combined with yaw turns to achieve a finely tuned path planning. As a simple illustrative case, Fig. 6.14 shows a snapshot sequence of the mix of a frontflip and a backflip, with the corresponding pitch angle depicted in Fig. 6.15. When the robot almost completed the frontflip at about 7.0 s, it suddenly switched to the backflip procedure by fully capitalizing the remaining kinetic energy and the inertia. A careful inspection also revealed that the waveform of the pitch angle shown in Fig. 6.15 closely resembled the separate frontflip (see Fig. 6.10) and backflip (see Fig. 6.13) processes, demonstrating a seamless combination.

Fig. 6.14

Snapshot sequence of a frontflip followed by a backflip

Fig. 6.15

Plot of the measured pitch angle in the frontflip plus backflip

6.6 Discussion

Since biologically underlying mechanisms of efficient and maneuverable dolphin swimming are hardly rigorously tested in a live specimen, the dolphin robot becomes an effective testbed permitting repeatable and free-swimming tests. However, concerning the achievable maneuverability, as summarized in Table 6.4, the developed dolphin robot is no match for dolphins in vivo. Note that the turn data listed in Table 6.4 are related to the cases of powered turns. In fact, unpowered turns of dolphins have smaller minimum radii than powered turns. The minimum radii were reported to range from 0.11 to 0.17 BL [17]. These small radius turns displayed by the dolphin often correspond to the biological features of an extremely flexible body and mobile flippers. The relatively simple mechatronic structure and control strategies may be the main reason for the inability of the dolphin robot to overtake its biological counterpart. Specifically, for lack of additional agile control surfaces, the current robot only relies on the feedback of the gyroscope to adjust the pitch joints. For instance, J₁ has to be utilized to make the COM beneath the CB in the backflip, resulting in a relatively small pitch moment in the upside-down phase. If a pair of flippers were imported, J₁ would be relieved to contribute more to the subsequent pitch maneuvers, thus hastening the process of the backflip. The successful implementation of the dolphin frontflip, the backflip, as well as their combination is a good demonstration. Meanwhile, this is fully consistent with the known biological observations [33]. We also remark that these stunts did not always succeed in the experiments. One of the main reasons was that the water depth was slightly more than 1 BL, making the robot hardly position itself exactly in the mid-depth of the swim tank. When a larger swim tank is available, more complex wide-range stunts are expected to be performed and the success rate of the stunts will hence be enhanced.

Table 6.4 Performance comparison of real dolphins (Tursiops truncatus) with the dolphin robot

Another issue to mention is pitch angle-based closed-loop control strategies in generating various pitch motions. On the one hand, to achieve a combined pitch turn (e.g., turning-while-diving), we do not need to decouple the pitch maneuver and the forward swimming, but finely control the unbending phase and maintain the pitch angle. On the other hand, other than importing feedback terms to alter the turning parameters or directly imposing a fixed bias to deflect the body, we utilize a half-period-oscillation-based trajectory tracking method to replicate a traveling wave. In this framework, using the pitch angle of the robot head as feedback, the robot is able to perform generalized straight swimming. Undeniably, a limitation of the pitch angle-based closed-loop control is that the full elimination of vertical recoil motions is impossible due to the intrinsic oscillatory property in dorsoventral propulsion.

In addition, as recently addressed by Lauder, 3D structure and control of propulsive surfaces in fish swimming have not received sufficient attention over the past years [34]. As a response, developing multiple propulsive surfaces and exploring 3D dolphin swimming will be a new research direction. More complicated maneuvers can be created as combinations of the basic ones and will make for a much better show.

6.7 Concluding Remarks

In this chapter, we have developed the active yaw and pitch control methods to maneuver a multilink dolphin robot without a pair of mechanical flippers. Taking into account that the robot mechanically consists of a yaw joint and a series of pitch joints, we impose directed biases upon the yaw joint to deflect the dolphin body so as to execute yaw turns in the horizontal plane. At the same time, a closed-loop pitch strategy is proposed to perform pitch turns in the vertical plane, principally involving pitching to a desired angle and maintaining the acquired pitch angle while swimming forward. Furthermore, the control approaches to implement the wide-range dolphin frontflip and backflip are presented and successfully tested. Both simulation and experimental results on various turns have primarily verified the effectiveness of the formulated control methods. It is revealed that the dolphin robot achieves better performance in pitch turns than it does in yaw turns.

The future work will focus on thoroughly investigating the 3D combined turns and the on-site path planning. Improving our mechanical design and adding more propulsive surfaces and sensors to endow the dolphin robot with enhanced locomotion and maneuverability, of course, is another ongoing endeavor.