Macro Fiber Composite-Actuated Soft Robotic Fish: A Gray Box Model-Predictive Motion Planning Strategy Under Limited Actuation

Abstract

This work experimentally investigates a model-predictive motion planning strategy to impose oscillatory and undulation movements in a macro fiber composite (MFC)-actuated robotic fish. Most of the results in this field exploit sinusoidal input signals at the resonance frequency, which reduces the device's maneuverability. Differently, this work uses body/caudal fin locomotion patterns as references in a motion planning strategy formulated as a model-based predictive control (MPC) scheme. This open-loop scheme requires the modeling of the device, which is accomplished by deriving a gray box state-space model using experimental modal data. This state-space model considers the electromechanical coupling of the actuators. Based on the references and the model, the MPC scheme derives the input signals for the MFC actuators. An experimental campaign is carried out to verify two references for mimicking the locomotion patterns of a fish under limited actuation. The experimental results confirm the motion planning scheme's capability to impose oscillatory and undulation movements.

Introduction

There is an ever-increasing demand for underwater inspections to assess environmental risks and extreme events. During these inspections, devices can acquire data that can enrich these analyses. Underwater robots can accomplish this task.^1–3 However, they should move in unstructured environments requiring maneuverability and energy efficiency. Bio-inspired underwater robots can reach these requirements. One of the advantages of animal mimicry is that natural selection has already determined efficiently biological mechanisms for locomotion patterns and anatomies in unstructured environments. Raj and Thakur⁴ identify two locomotion patterns/anatomies for fish-like robots: body/caudal fin (BCF) and median/paired fin based locomotion. BCF-based locomotion requires oscillatory and undulatory movements but can present poor maneuverability. Figure 1 illustrates how oscillatory and undulatory movements may impact the fish's biological anatomy according to Raj and Thakur.⁴

FIG. 1.

BCF-based locomotion regarding oscillatory and undulatory motion. BCF, body/caudal fin.

Sfakiotakis et al.⁵ detailed the movements of some aquatic animals; in particular, fishes' movements that present BCF-based locomotion patterns. One strategy to mimic animal dynamics emerged in the 2000s: the Soft Robots.^6,7 These devices are built using bendable or flexible materials,^8,9 which demand not only unusual actuation and control strategies^10,11 but also highly specialized manufacturing strategies.^12,13 Among the actuation strategies for fish-like robotic systems, the use of ionic polymer metal composites (IPMCs),^14,15 dielectric elastomer actuators,¹⁶ shape memory alloys (SMAs),^17,18 pneumatic actuators,^19,20 and magnetostrictive thin films²¹ have shown promising strategies.

An alternative for actuating flexible systems is the macro fiber composites (MFCs).²² These devices, developed by NASA Langley Research Center,^23,24 feature high flexibility, low density, and high force-imposing capability. Some works reported the use of these mechanisms^25–29 and demonstrate their applicability in flexible robot applications. Tan et al.³⁰ and Meng et al.³¹ modeled material and geometric nonlinearities in bioinspired devices using MFCs. Recently, Tan et al.³² presented experimental studies on a trout-inspired multifunctional robotic fish. This device is simultaneously an underwater swimmer and an energy harvester device.

Tan et al.,³² Lou et al.,³³ and Shahab and Ertuk³⁴ imposed oscillatory propulsion by resonant actuation of MFCs. In other words, most fish-like robots actuated by MFCs using BCF-locomotion explore sinusoidal input signals at resonant frequencies.^25,29,31 This approach might be optimal from the perspective of energy efficiency. In a different manner, Ming et al.²⁹ and Shao and Xu³⁵ have used triangular and quadratic signals. However, sinusoidal, triangular, and quadratic signals don't contribute to the device's maneuverability since no motion planning has been foreseen. Moreover, sinusoidal input signals in unevenly distributed MFC actuators' configurations would lead to undesired motion. Therefore, researchers should also propose motion planning strategies considering maneuverability and different MFC actuators' configurations.

Maneuverability is a critical feature regarding robotic inspection devices. In the case of fish-like robotics, position control and motion planning strategies have been proposed for tail-actuated robotic fishes. Both Cataño and Tan³⁶ and Zhang et al.³⁷ proposed nonlinear model predictive control approaches to the path following of a tail-actuated robotic. This methodology can cope with the nonlinear dynamics and actuation constraints while minimizing the control effort. Moreover, Zhang et al.³⁷ extended this approach using a two-stage orientation–velocity nonlinear model predictive controller for this application.

Regarding fish-like robots using BCF-based locomotion, maneuverability is also critical. Barbosa et al.³⁸ numerically investigated the applicability of the model-based predictive control (MPC) for planning oscillatory and undulatory movements for a fish-like robot actuated by MFC bimorph actuators. This investigation proposed using a white-box model derived using the Euler–Bernoulli beam theory considering the electromechanical coupling of the actuators in an MPC scheme. The proposed MPC scheme was an open-loop approach, also denoted as a feedforward control strategy, that delivers the required inputs for the bimorph actuators for generating the imposed locomotion pattern. Despite its promising results, this proposal is highly subjected to modeling uncertainties, is only valid for evenly distributed MFC actuators' configurations, and has not been experimentally investigated.

In the present work, we experimentally investigate the use of MPC for motion planning for a fish-like robotic system. This strategy deals with the actuation limitation, unevenly distributed actuators, and the actual system dynamics. Therefore, the proposal in this work is the derivation of a state-space model using experimental modal data. We use a gray-box model in the MPC scheme to derive appropriate inputs for generating the required locomotion pattern. Since most of the results in this field exploit sinusoidal input signals at the resonance frequency, they cannot deal with actuation limitation and its uneven actuation distribution. It is important to highlight that this proposal is an open-loop strategy, and no feedback signal is used. Moreover, the designer can improve the device's maneuverability if it can perform not only sinusoidal motion but also asymmetrical and nonsinusoidal motion. Therefore, the main contribution of this work is to propose a gray-box model-based motion planning strategy for a soft fish-like robot that can perform different oscillatory and undulatory movements.

The prototype and the experimental campaign are described in Materials and Methods section. Theory/Calculation section presents the motion planning strategy, including the system modeling and the model predictive control strategy. Finally, experimental results are presented and discussed in Results and Discussion section, while conclusions are drawn in Conclusions section.

Materials and Methods

Details about the prototype and the experimental campaigns are given in this section. We carried out two experimental campaigns. In the first campaign, we acquired the first resonance frequencies and mode shapes of the structure, which are required for the fish-like robot's dynamic modeling. This information is used for deriving a gray-box modal state-space model.

The MPC exploits this model to derive the system's inputs for getting the desired motion. In this way, we experimentally investigated the actual prototype's kinematic behavior when imposing these inputs in the second campaign. The gray-box modal state-space model and the MPC-based motion planning strategy are described in the next section.

Prototype

The following conditions, which can improve the fish swimming efficiency, were considered in the prototype's design^39,40:

The speed wave propagated in the fish body is higher than the motion velocity;

The body's stiffness should decrease along the head-tail axis;

The presence of a tail improves the propulsion; and

The range of motion at the tail end should be the largest in the structure.

The prototype structure is a composite beam with a nonuniform cross-section to meet these conditions. The beam's core is made of carbon fiber due to its elastic properties. The MFC patches, used as actuators, are glued to this structure using the DP 460 Epoxy. The M8514-P1 (MFC patches) are allocated in the beam in the bimorph configuration.

The MFC patches, unlike ceramics, are flexible and can act on highly flexible arrangements. An MFC is composed of piezoelectric fibers, epoxy, and electrodes. Piezoelectric devices present different operation modes, denoted as $d_{i j}$ , where i indicates the direction of polarization and j the direction of the generated effect, and i and j can vary from 1 to 3. In the present configuration, direction 3 is in line with the z-direction (Fig. 2). However, since the $d_{33}$ mode provides higher actuation levels, this operation mode is exploited in this work.

FIG. 2.

(a) Illustration of the dimensions of the prototype structure. (b) Description of the MFC's locations, and (c) picture of the fish-like robotic system. MFC, macro fiber composite.

Figure 2a shows the dimensions of the prototype structure. The thickness of the structural elements and the MFC actuators is 3 mm. The distance between each actuator pair is 20 mm. A rectangular piece is added in this gap to attenuate a sudden cross-section change. Figure 2b depicts the description of the MFC's locations used in the modeling procedure.

As pointed out by Azuma,³⁹ the range of motion of a BCF fish must be larger in the tail. A reduction in the Young modulus or the moment of inertia of the beam across the head-to-tail direction can promote this kind of motion.³⁸ Therefore, a triangular piece is added to the prototype reducing the beam bending stiffness across the head-tail direction. Both the triangular and rectangular pieces are made of rigid PVC. These materials have been selected due to their elastic behavior. The Brascoved^® silicone was the glue used to embed them in the structure.

Figure 2c shows the prototype in a fixed-free condition. The experimental campaigns are carried out under this boundary condition. It is important to highlight that the designer can use the proposed motion planning scheme for other boundary conditions. Different conditions yield the mode shapes, which modify the modal state-space model required for the MPC-based motion planning strategy. Figure 3 shows the MFC actuator pairs and their connections. A DSpace 1103 is responsible for sending the input signals generated in a computer to the amplifiers (Model HVA 1500-1/50, single-channel; Smart Material Corporation) through Bayonet Neill Concelman cables. Safe high-voltage cables connect the MFCs and the amplifiers. There are two bimorph MFC actuator pairs: A-B and C-D (Fig. 3b).

FIG. 3.

(a) Illustration of the MFC B and MFC D and their connections. (b) Illustration of the MFCs and their connections, and (c) Input signal cabling.

Experimental campaign: resonance frequencies and mode shapes

The modeling procedure requires the assessment of the resonance frequencies and modal shapes of the prototype. In this way, we conducted an experimental campaign to assess the first three resonance frequencies and mode shapes. Details about this campaign, composed of two stages, can be found in Barbosa and Da Silva.⁴¹

The first stage comprised an impact test. A modal hammer from PCB PIEZOTRONICS^® (model 086E80) excited the structure, and a PCB transducer acquired this input signal excitation. A laser from Polytec^® (model CLV 700) was responsible for measuring the responses. The system was excited by an impact using the modal hammer in 20 different points.⁴¹ In this first stage, the MFC actuators were inactive. Therefore, these data were used to assess the resonance frequencies' range.

The mode shapes were acquired in this experimental campaign's second stage: an excitation test. The MFC actuator pairs were connected, as illustrated in Figure 3a and b. The following sinusoidal voltage input was furnished for the amplifiers (Fig. 3c): $V_{j} (t) = 1000 s i n (ω t + k_{j} π) + 500$ (1)

where j can be 1 or 2 (Fig. 3b), V is the input voltage in Volts, $ω$ is the exciting frequency in rad/s, and t is the time in seconds. The constant k_j is responsible for the delay in the MFC voltages, being equal to 1 for $j = 1$ and 0 for $j = 2$ . The input frequency $ω$ was varied in the frequency range acquired in the impact test to assess the resonance frequencies and mode shapes.

Experimental campaign: motion planning strategy

This experimental campaign slightly differs from the former (performed to assess the resonance frequencies and mode shapes). Unfortunately, the MFC-A has been permanently damaged. This issue would be irreparable for the strategies using sinusoidal signals^32–34,42 since it would cause asymmetric behavior.

Nevertheless, our proposal for deriving the actuators' inputs can deal with any number of actuators and reference points. This capability is essential since we can mimic the desired motion despite its shortcoming. Furthermore, our proposal extends the MPC-based motion planning strategy introduced by Barbosa et al.³⁸ since it considers actuation limitation and uneven actuation distribution.

A camera acquired the motion at a 30 FPS rate. Then, we processed the image to obtain the reference points' movements using the 4k-Shortcut software. Figure 4 shows a picture of the device during these experiments, where $w (x, t)$ is the device's motion in the z-direction.

FIG. 4.

Pictures of the fish-like robotic system (a) an upper view and (b) a front view.

Theory/Calculation

The authors adopt linear relationships in the vast majority of works presenting the modeling of piezoelectric materials. However, the electric fields in an MFC device may be nonlinear. Moreover, the MFC device has a periodic arrangement between the electrodes/epoxy/piezoelectric fibers. Due to this periodicity, Deraemaeker et al.⁴³ proposed the division of the MFC structure into the Representative Volume Elements (RVEs). As a result, the behavior of the RVEs can be linearized using the mixing rules technique.

Based on the RVEs' proposal, Barbosa et al.³⁸ described a general formulation for the time-variant moments of a fish-like structure and of the MFC actuator pairs. The authors have exploited the Euler–Bernoulli beam formation for deriving this formulation yielding a set of differential equations. Furthermore, the authors approximated the analytical white-box model by applying a modal transformation to this set of equations. However, this model, which can be rearranged to a modal state-space model,⁴⁴ is subjected to modeling uncertainties.

Therefore, in the present work, we propose the derivation of a gray-box model using modal data: the mode shapes, the natural frequencies, and the damping factors. These data have been acquired using the excitation text described in Experimental Campaign: Resonance Frequencies and Mode Shapes section. The gray-box modeling strategy improves the robustness of the proposal regarding modeling uncertainties. However, since the proposed MPC-based motion planning strategy is based on an open-loop scheme, it cannot deal directly with environmental disturbances. Despite this shortcoming, the motion of the robotic fish might still be undulatory and oscillatory, promoting swing capabilities for the device as numerically investigated by Barbosa et al.³⁸

This gray-box modal state-space model is as accurate as the greater the number of mode shapes/resonance frequencies used in the approximation.⁴⁴ The present work uses three mode shapes in the model's derivation. These ABCD matrices are derived considering two measurement points and two control inputs. The measurement points are given by:

where $y_{1} = w (0.5 L, t)$ denotes the motion at the z-direction of the measurement point located at $0.5 L$ , and where $y_{2} = w (, t)$ denotes the motion at the z-direction of the measurement point located at L (Fig. 4a), where $L = 332$ mm as shown in Figure 2. The control inputs are given by:

where V₁ is the input of the MFC C and V₂ is the input for the MFCs B and D. In the following subsection, this modeling procedure is summarized.

An MPC-based motion planning strategy exploits this modal state-space model by requiring reference inputs for the measurement points. This strategy is mathematically formulated as an optimization problem described in this section. This problem considers the required references, the measurement points, and the control inputs through the constraints' formulation.

Modal state-space model

Using the Euler–Bernoulli beam theory, the equations of motion can be derived, and the motion of a beam $w (x, t)$ can be calculated analytically.³⁸ This solution of this model can be approximated using the following modal transformation:

where are the mode shapes, $η_{r} (t)$ are the normal coordinate, and N is the number of normal modes used for this approximation. This approach yields the following approximated equation of motion: ${\ddot{η}}_{r} (t) + 2 ζ_{r} ω_{r} {\dot{η}}_{r} (t) + ω_{r}^{2} η_{r} (t) = φ_{r 1} V_{1} (t) + φ_{r 2} V_{2} (t),$ (5)

where $φ_{r 1}$ and $φ_{r 2}$ are the modal electromechanical couplings on the electrical configuration shown in Figure 3, $V_{1} (t)$ and $V_{1} (t)$ the input signals, $ω_{r}$ the natural frequencies, and $ζ_{r}$ the damping factors. These values can be experimentally acquired, as discussed in the next subsection.

The set of equations given by Equation (5) can be described in a modal state-space representation using the modal transformation [Equation (4)]. Considering $N = 3$ , the continuous-time modal state-space considering a bimorph actuation configuration is given by: $\begin{matrix} \dot{x} = A x + B u, \\ y = C x, \end{matrix}$ (6)

where the state variables ( $x \in ℛ^{12 \times 1}$ ) are: $x = \{\begin{matrix} ϕ_{1} (x = 0.5 L) η_{1} \\ ϕ_{1} (x = 0.5 L) {\dot{η}}_{1} \\ ϕ_{2} (x = 0.5 L) η_{2} \\ ϕ_{2} (x = 0.5 L) {\dot{η}}_{2} \\ ϕ_{3} (x = 0.5 L) η_{3} \\ ϕ_{3} (x = 0.5 L) {\dot{η}}_{3} \\ ϕ_{1} (x = L) η_{1} \\ ϕ_{1} (x = L) {\dot{η}}_{1} \\ ϕ_{2} (x = L) η_{2} \\ ϕ_{2} (x = L) {\dot{η}}_{2} \\ ϕ_{3} (x = L) η_{3} \\ ϕ_{3} (x = L) {\dot{η}}_{3} \end{matrix}\} = \{\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \\ x_{9} \\ x_{10} \\ x_{11} \\ x_{12} \end{matrix}\} .$ (7)

The state matrices are given by: $A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ - ω_{1}^{2} & - 2 ζ_{1} ω_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & - ω_{2}^{2} & - 2 ζ_{2} ω_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & - ω_{3}^{2} & - 2 ζ_{3} ω_{3} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & - ω_{1}^{2} & - 2 ζ_{1} ω_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - ω_{2}^{2} & - 2 ζ_{2} ω_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & - ω_{3}^{2} & - 2 ζ_{3} ω_{3} \end{matrix}],$ (8) $B = [\begin{matrix} 0 & 0 \\ ϕ_{1} (x = 0.5 L) φ_{11} & ϕ_{1} (x = 0.5 L) φ_{12} \\ 0 & 0 \\ ϕ_{2} (x = 0.5 L) φ_{21} & ϕ_{2} (x = 0.5 L) φ_{22} \\ 0 & 0 \\ ϕ_{3} (x = 0.5 L) φ_{31} & ϕ_{3} (x = 0.5 L) φ_{32} \\ 0 & 0 \\ ϕ_{1} (x = L) φ_{11} & ϕ_{1} (x = L) φ_{12} \\ 0 & 0 \\ ϕ_{2} (x = L) φ_{21} & ϕ_{2} (x = L) φ_{22} \\ 0 & 0 \\ ϕ_{3} (x = L) φ_{31} & ϕ_{3} (x = L) φ_{32} \end{matrix}],$ (9)

and $C = [\begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{matrix}] .$ (10)

The terms $φ_{r j}$ depend on the modal electromechanical couplings $θ_{r j}$ when the bimorph MFCs were connected in parallel.³⁴ The modal electromechanical couplings $θ_{r j}$ are given by the product of the electromechanical coupling $ϑ$ of the MFCs by the derivative of the mode shapes at the initial and final points where they are embedded^34,45: $θ_{r 1} = {ϑ \frac{d ϕ_{r} (x)}{d x}|}_{p_{1} (i n i c i a l)}^{p_{1} (f i n a l)}, a n d$ (11) $θ_{r 2} = {ϑ \frac{d ϕ_{r} (x)}{d x}|}_{p_{2} (i n i c i a l)}^{p_{2} (f i n a l)},$ (12)

where $p_{1} (i n i c i a l)$ , $p_{1} (f i n a l)$ , $p_{2} (i n i c i a l)$ , $p_{2} (f i n a l)$ indicate the initial and final points where the MFC actuator pairs are attached (Fig. 2b). The MFCs are asymmetrical concerning the operating voltage that may vary between −500 V and 1500 V.

The MFC actuators are electrically connected, as illustrated in Figure 3b. Considering these connections, the terms $φ_{r j}$ are given by: $φ_{r j} = f_{r} \frac{θ_{r 1} + θ_{r 2}}{2},$ (13)

where f_r are experimentally determined parameters added to modulate the motion range, r can vary from 1 to 3 according to the mode numbering, and j can vary from 1 to 2 according to the connected MFC actuator pairs. These f_r terms consider the modal electromechanical couplings and the proportionality constant of the mode shape, among other nonmodeled dynamics. In this way, $j = 1$ to MFCs A and C and $j = 2$ to MFCs B and D.

The MFC A was damaged during the experimental campaign, and its contribution should not be considered yielding: $φ_{r 1} = f_{r} \frac{0 θ_{r 1} + θ_{r 2}}{2} .$ (14)

In contrast, both MFCs B and D are working properly, yielding: $φ_{r 2} = f_{r} \frac{θ_{r 1} + θ_{r 2}}{2} .$ (15)

MPC-based motion planning

An MPC motion planning is exploited in this work.³⁸ This model-based strategy requires the discretization of the state-space model [Eq. (6)], which is carried out using the Tustin discretization method. A fixed sampling time $Δ t$ is used, and the discretization-related subindexes are omitted for simplicity.

Figure 5 illustrates the measurement points, the references, and the proposed MPC scheme for deriving the inputs for the MFC actuator pairs. This figure shows that the MPC scheme seeks to derive $u$ according to the model [Eq. (6)] and the references ( $y_{r e f}$ ) that are subjected to disturbances ( $d$ ). Moreover, this figure shows that the prediction errors, $e$ , are derived according to:

FIG. 5.

Illustration of the MPC scheme for deriving the inputs for the MPC actuator pairs. MPC, model-based predictive control.

e = y_{r e f} - y - d .

(16)

In this work, the error $e$ can assume values bounded by the interval $[e_{l o w}, e_{h i g h}]$ . The desired references vary in time which may affect the actual error bound values. In this way, the actual error bound values are modified according to their previous values using the concept of dead bands³⁸: $e_{l o w} = d b_{l o w} - τ_{c} \frac{Δ e_{l o w}}{Δ t}, a n d$ (17) $e_{h i g h} = d b_{h i g h} - τ_{c} \frac{Δ e_{h i g h}}{Δ t},$ (18)

where $d b_{h i g h} = d b_{h i g h} 1 m$ and $d b_{l o w} = d b_{l o w} 1 m$ , $d b_{h i g h}$ and $d b_{l o w}$ define the dead bands, $1 m$ is a vector $[1, 1, \dots, 1] \in ℛ^{n_{m}}$ and n_m is the number of measurement points, and $τ_{c}$ is the time constant. The same low and high dead bands are used for the measurement points in this work. Moreover, the designer can define these values.

An optimization problem considering N_p samples in the Prediction Horizon is solved at each sample time. The decision variable of these optimization problems is the inputs $u$ , space-states $x$ , and errors $e$ for the subsequent N_c samples in the control horizon.

In this work, the objective function is composed of a weighted sum of values related to error values. In contrast, the constraints are related to the system's dynamics and actuators' limitation yielding: $\begin{matrix} {min}_{x, e, u} \sum_{k = 1}^{N_{p} - 1} \begin{matrix} w_{h i g h}^{T} (e (k) - e_{h i g h}) + \dots \\ w_{l o w}^{T} (e_{l o w} - e (k)) + w^{T} e (k) \end{matrix} \\ s u b j e c t t o \\ x_{k + 1} = A x_{k} + B u_{k}, \\ y_{k} = C x_{k}, \\ u_{l o w} \leq u_{k} \leq u_{h i g h}, \\ Δ u_{L} \leq Δ U, \\ Δ u_{U} \leq Δ U, \end{matrix}$ (19)

where $w = w 1 m$ , $w_{h i g h} = w_{h i g h} 1 m$ , $w_{l o w} = w_{l o w} 1 m$ are weighting vectors that penalize the error values. The input variations ( $Δ u_{U} = u_{k} - u_{k - 1}$ and $Δ u_{L} = u_{k - 1} - u_{k}$ ) are bounded by $Δ U = Δ U 1 i$ , where $1 i$ is a vector $[1, 1, \dots, 1] \in ℛ^{n_{i}}$ and n_i is the number of input signals. Moreover, the input values are also bounded by lower and the upper bounds given by $u_{l o w} = u_{l o w} 1 i$ and $u_{h i g h} = u_{h i g h} 1 i$ , respectively.

The MPC algorithm may be computationally expensive. Therefore, these algorithms usually exploit convex formulations. For this reason, we propose using linear objective and constraint functions. The convex optimization problem given by Equation (19) is solved using the Interior-Point Method⁴⁶ at every time step.

This proposal can derive the inputs for diverse MFC actuators' configurations, including unevenly distributed ones. Therefore, if the device presents a malfunctioning actuator, adequate inputs can be derived by solving the optimization problem given by Equation (19). The input values can be found rapidly since our proposal is a convex optimization problem. However, it is important to highlight that the device should be monitored by sensors capable of indicating this malfunctioning actuator. This monitoring is out of the scope of this work.

Results and Discussion

In this section, we describe the experimental results regarding two distinct references. Therefore, we first assessed the experimental parameters. Then, we experimentally evaluated the capability of the proposed MPC-based motion planning strategy to mimic oscillatory and undulatory movements.

Experimental parameters

The modeling strategy requires the assessment of the mode shapes, the natural frequency, and the damping factor. Experimental Campaign: Resonance Frequencies and Mode Shapes section describes the experimental campaign carried out to assess these experimental parameters. First, we processed the digital images obtained during sinusoidal excitation at resonance frequencies for deriving the mode shapes.⁴¹ We acquired the motion of several points of the fish-like structure. The measurement accuracy is ±2.5 mm at a maximum amplitude (for the first mode) in order of magnitude of 100 mm. The first three mode shapes and the experimental parameters are presented in Figure 6. The matrix $A$ is entirely determined by damping factors $ζ_{r}$ and the natural frequencies $ω_{r}$ , while the mode shapes are used in matrix $B$ . Since the modeling strategy requires function for describing these mode shapes, fourth-degree polynomials have been interpolated. The interpolated fourth-degree polynomials are:

FIG. 6.

The first three mode shapes, natural frequencies, and damping factors.

\begin{matrix} ϕ_{1} (x) = & - 7.18 x^{4} + 5.35 x^{3} - 0.93 x^{2} + \dots \\ \dots + 0.14 x - 0.00033, \end{matrix}

(20)

\begin{matrix} ϕ_{2} (x) = & - 13.79 x^{4} + 6.62 x^{3} - 1.03 x^{2} + \dots \\ \dots + 0.07 x + 0.00037, a n d \end{matrix}

(21)

\begin{matrix} ϕ_{3} (x) = & - 2.64 x^{4} + 4.18 x^{3} - 1.34 x^{2} + \dots \\ \dots + 0.10 x - 0.00033 . \end{matrix}

(22)

where $0 \leq x \leq L = 332 m m$ .

The values of the modal electromechanical couplings [Eqs. (11) and (12)] are $ϑ = 6.891 0^{- 6} N m ∕ V$ for MFCs M8514-P1.^34,45 These values are found in slightly different conditions than the ones in this work. However, as described in Modal State-Space Model section, the modulation terms, f_r, consider the electromechanical couplings, the proportionality constant required for the mode shape, and other phenomena. The values f_r used in the matrix $B$ were obtained by exciting the prototype at the resonance frequencies ( $ω_{1}$ , $ω_{2}$ , and $ω_{3}$ in Fig. 6). Using the electrical connection depicted in Figure 3b, the input signals were V₁ and V₂, for: $V_{1} = 1000 s i n (ω_{r} t) + 500,$ (23)

and $V_{2} = 1000 s i n (ω_{r} t + π) + 500,$ (24)

where $r = 1 \dots 3$ .

We manually adjusted the values of f₁, f₂, and f₃ so that the outcome of the actual prototype (movements acquired by the camera and image processing) and the model matched. These values are shown in Table 1. In addition, the parameters used in the MPC-based motion planning scheme described in Equation (19) are also given in Table 1.

Table 1.

The Parameters of the Model-Based Predictive Control-Based Motion Planning Scheme

Parameter	Value	Description
f ₁	1.225 × 10⁶ 1/kg	Modulation factor
f ₂	4.500 × 10⁷ 1/kg	Modulation factor
f ₃	2.500 × 10⁷ 1/kg	Modulation factor
n_i	2	Number of input signals
n_m	2	Number of meas. points
$Δ t$	0.001 s	Sampling time
N_p	80 samples	Prediction horizon
N_c	5 samples	Control horizon
$τ_{c}$	$1 \times 1 0^{- 5}$ s	Time constant
$u_{l o w}$	−500 V	Lower bound
$u_{h i g h}$	1500 V	Upper bound
$Δ U$	50 V	Bound
$d b_{l o w}$	$- 5 \times 1 0^{- 4}$ s	Lower bound of dead band
$d b_{h i g h}$	$5 \times 1 0^{- 4}$ s	Upper bound of dead band
$w_{l o w}$	20	Weighting factor
$w_{h i g h}$	20	Weighting factor
w	1	Weighting factor

The experimental parameters mitigate the uncertainties arising from the simplifying hypotheses, whether related to electromechanical couplings or the Euler–Bernoulli modeling. Another consideration that must be made concerns the values of the amplitudes of the functions that describe the vibrating modes.

Reference 1

First, we imposed a sinusoidal motion at the first resonance frequency to the tail and no motion to the middle of the body. This reference attends to one of the conditions to improve the fish swimming efficiency: the range of motion at the tail end should be the largest in the structure.³⁹ In this way, the references in the MPC-based motion planning scheme [Eq. (16)] are described as: $r_{1} (x = L) = 0.08 s i n (ω_{1} t) a n d r_{1} (x = 0.5 L) = 0,$ (25)

where the references are in meters, and t is the time given in seconds.

Figures 7a and b show these references and model and actual prototype outcomes. Despite the references' symmetrical nature, the prototype's motion is asymmetrical. This behavior is a result of the prototype limitations. For example, the actuation is bounded, and the MFC A was damaged. Moreover, the model was able to predict this asymmetrical motion. As shown in Figure 7, the error associated with the point at $x = 0.5 L$ is greater than the error related to the point at $x = L$ . The designer can modify this condition by choosing different weighting factors in the optimization problem [Eq. (19)] according to the importance of tracking error in each measurement point. In this work, the same weights are selected for both measurement points.

FIG. 7.

Reference 1, experimental and model outputs: (a) $x = L$ and (b) $x = 0.5 L$ .

Figure 8 depicts the input signals derived by the MPC-based motion planning scheme. As can be seen, these input signals have periodic behavior but cannot be described by a sum of a couple of sinusoidal functions. Most of the references in the field exploit sinusoidal functions at resonance frequencies to excite such devices. The proposed motion planning strategy is versatile and can deal with different imposed motions.

FIG. 8.

Input signals for the Reference 1: (a) V₁ and (b) V₂.

Reference 2

Second, we also imposed a sinusoidal motion at the first resonance frequency to the tail and no motion to the middle of the body. However, a smaller motion amplitude is required. Figure 8 shows that both input signals have reached the bounds. In this way, evaluating a situation that might not violate these bounds is important to fully address the proposal's capability. In this way, the references in the MPC-based motion planning scheme [Eq. (16)] are described as: $r_{2} (x = L) = 0.04 s i n (ω_{1} t) a n d r_{2} (x = 0.5 L) = 0,$ (26)

where the references are in meters, and t is the time given in seconds.

Regarding this second symmetrical reference, Figure 9a and b shows that the model and actual prototype outcomes also present a symmetrical nature. As depicted in Figure 10, the higher bound (1500 V) does not impose any actuation limitation. This fact contributes to the capability of the proposed MPC-motion planning strategy to impose the required motion. Figure 10 also indicates that the maximum required value of V₁ is higher than that required by V₂. Again, this difference is due to the damaged MFC.

FIG. 9.

Reference 2, experimental and model outputs: (a) $x = L$ and (b) $x = 0.5 L$ .

FIG. 10.

Input signals for the Reference 2: (a) V₁ and (b) V₂.

Finally, we can also observe in Figure 9 that the fish-like robot presents movements in x = 0.5L. This fact, also perceived for Reference 1, indicates a structural limitation. In summary, it is a fact that the system cannot follow any references due to actuation and structural constraints.

Conclusions

This work experimentally evaluates an MPC-based motion planning scheme for fish-like soft robots actuated by MFC actuators. Despite exploiting sinusoidal signals at the resonance frequency as most works in the field, the proposal may impose locomotion patterns, improving the device's maneuverability.

The motion planning scheme is based on optimization problems, subject to the system's dynamics and actuation limitations, for minimizing the difference between the predicted motions of the device (model-based strategy) and the imposed references. Using experimental data, the authors modeled the system's dynamics as a gray-box modal state-space model. Based on the references (required locomotion pattern) and the modal state-space model, the MPC scheme derives the input signals for the MFC actuators considering two references. During the experiments, a short circuit damaged an MFC patch. Therefore, these input signals consider this limitation.

The first reference imposed a motion to the structure that required input signals higher than the bounds. Due to this limitation, the actual motion could not fully follow the reference yielding asymmetrical behavior. This asymmetry could be verified not only by the experiments but also by the proposed model. In contrast, the second reference imposed smaller-amplitude movements, requiring lower input signals. Under these conditions, the fish-like robot system's actual motion was similar to the imposed reference.

The experimental results confirm the capability of the proposed motion planning scheme for the fish-like robotic system immersed in air. Furthermore, these results corroborate the maneuverability of such devices since diverse motion profiles can be imposed.

Footnotes

Author Disclosure Statement

No competing financial interests exist. The coauthors have seen and agree with the contents of the article. We certify that the submission is original work and is not under review at any other publication.

Funding Information

This research is supported by FAPESP 2022/07119-2 and FAPESP 2018/15894-0. Moreover, A.B.S. and M.M.d.S. are grateful for their research grants, CAPES 88887 498358/2020-00 and CNPq 303884/2021-5, respectively.

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