Energy Harvesting for Robots with Adaptive Morphology

Abstract

Robots primarily made of soft and elastic materials have potential applications such as traveling in confined spaces due to their adaptive morphology. However, their energy efficiency is still subject to improvement. Although a possible approach to increase efficiency is by harvesting the energy used during their behavioral motion, it is not trivial to do so due to their complex dynamics. This work seeks to pioneer a study that exploits the tight coupling between a robot's adaptive morphology, control, and consequent behaviors to harvest energy and increase energy efficiency. It is hypothesized that since varying the robot's morphology may change the energy use that leads to contrasting behavior and efficiency, harvesting the robot's energy will need to be adapted to its morphology. To verify the hypothesis, we developed a shape-changing robot with an elastic structure that achieves locomotion via vibration controlled by a single motor, such that the complex dynamics of the robot can be characterized through its resonance frequencies. It will be shown that harvesting energy at opportune occasions is more important than maximizing the harvest capacity to increase energy efficiency. We will also show how the robot's shape affects energy use in locomotion and how energy harvesting will feedback additional energy that increases the magnitude and affects the robot's behavior. We conclude with an understanding of the role of the robot's morphology, that is, shape, in using the energy provided to the robot and how the understanding can be used to harvest the robot's energy to increase its efficiency.

Introduction

Soft robots are a rapidly evolving breed of machines that are more adaptable than traditional rigid robots due to their ability to deform.^1–5 This ability has previously been achieved through material compliance such as fluid-driven elastomers,^6,7 elastic cavities filled with granular matter,^8,9 as well as structural compliance like tensegrity structures^10,11 or deformable elastic beams.^12,13 These soft robots pose boundless potential applications, realizing themselves as grippers, assistive devices, and locomotion approaches.^14–18

The deformability and compliance in soft robot systems may lead to more complex dynamics but, at the same time, is also considered an opportunity to achieve intelligent behaviors.^3–5 The concept that intelligent behaviors such as control, computation, adaptation, or learning can be encoded in the robot's body instead of in the brain as its neural intelligence is known as physical intelligence.¹⁹ The physical intelligence concept is closely related to another relevant concept known as morphological computation: a concept that suggests that morphological properties, such as the shape and form of a body, as well as dynamic properties like compliance or resonance, play a crucial role in the emergence of intelligent behaviors.^20,21

Also, although both concepts are mentioned in the study of soft robots,^3–5,19 a strict definition of physical intelligence is related to but not the same as embodied intelligence.¹⁹ Physical intelligence focuses on the intelligence encoded in the body, whereas embodied intelligence focuses on the tight coupling between the robot's brain or control and body morphology (and the environment) that leads to intelligent behaviors.^19,22–27

Following this train of thought, adaptive morphology such as shape in robots is an emerging design principle that may lead to different behaviors and added functionality.^28–32 Shape adaptation has been investigated in soft robots based on origami, tensegrity structures, adhesive materials, as well as growing features,^33–38 whereas another active direction is stiffness variation.^39–43 In these examples, it can be seen that the study of adaptive morphology can focus more on demonstrating physical or embodied intelligence. For example, origami structures demonstrate physical intelligence by achieving multiple states, that is, flexible and rigid, by introducing a temperature-driven self-locking mechanism.³³

On the other hand, embodied intelligence is demonstrated by showing that a tight coupling among the “brain,” that is, control of applied magnetic field frequency, and the body of an origami robot with different shapes will lead to different locomotion speeds.³⁵ Overall, the examples show that robots with adaptive morphology (either passive, active, with local distributed mechanical or external concentrated electrical control) can be seen as an instance of physical intelligence, morphological computation, or embodied intelligence, given that these terms are sometimes loosely used interchangeably.

A recent review argued that energy efficiency and harvesting in robots made of soft and elastic materials are underexplored and that energy efficiency can potentially seek improvement through harvesting.⁴⁴ Energy harvesting can be defined as techniques to accumulate energy for enabling self-sufficient and energy-efficient mechatronic systems.^45,46 Energy harvesting methods are also used in self-powered microsystems, represented as inertial spring and mass systems.⁴⁷ In this work, energy harvesting refers to electrical energy, excluding methods that store potential energy using mechanical components such as clutches.⁴⁸

Similar to this work, in piezoelectric energy harvesting, ambient vibration causes structures to deform and results in mechanical stress and strain, leading to high output voltage but low current level.¹³ Energy harvesting in soft robots has been studied via the development of soft material-based energy storage and generators,⁴⁹ energy harvesting in standalone fluidic actuators,⁵⁰ and the design of a soft robotic mouth toward realizing energetically autonomous aquatic robots.⁵¹ Nonetheless, these works have not studied energy harvesting from the perspective of physical or embodied intelligence, that is, considering the relationship between harvested energy, resulting efficiency, and changing morphology and behaviors of the robots.

This work seeks to pioneer a study that exploits the tight coupling between a robot's adaptive morphology, control, and consequent behaviors to harvest energy and increase energy efficiency. It is hypothesized that energy harvesting will need to be adapted to a robot's morphology since the morphology may change the robot's energy usage and subsequent behavior and efficiency. The approach to verify this hypothesis focuses on a locomoting robot with structural compliance that relies on elastic beams to deform and vibrate, whose dynamics and behaviors can be characterized by resonance frequencies.

The robot's vibration is driven by a single motor, whereas a commercial piezoelectric transducer is added for energy harvesting. The robot's shape can be changed on command to investigate the effects of robot morphology on its behavior and energy. With the motor control, the control of the robot's shape, and harvester operating conditions being centrally determined by the “brain” of the robot, this work seeks to clarify how the tight coupling between the “brain” and the robot's morphology (i.e., its embodied intelligence) can increase energy efficiency via harvesting.

The rest of the article is organized as follows. The Materials and Methods section details the experimental setup and procedures, introducing the robot and its mathematical model. The Experiment Result and Analysis section will then present the work results, followed by closing remarks and the future potential of this work.

Materials and Methods

The developed robot

The robot used in this work is shown in Figure 1a and will be referred to as the Inverted U-shape curved beam robot (IUCBR). This is due to its inverted “U” shape elastic beam structure made of two thin steel strips, forming its structural compliance and ability to change shape. These attributes facilitate travel in confined spaces, a common challenge in soft robot locomotion.^6,8 The relevant mechanical parameters are described in Table 1.

FIG. 1.

The IUCBR. (a) Side view of the robot. (b) Front view of the IUCBR showing the piezoelectric transducer. IUCBR, inverted U-shape curved beam robot.

Table 1.

The Mechanical Parameters and Their Values of Inverted U-Shaped Curved Beam Robot

No.	Parameter	Value	Unit
1	Height of the robot for 50°, 60°, and 70° configurations, respectively	0.136, 0.124, 0.108	m
2	Length of the robot	0.183	m
3	Length of rotating mass	0.047	m
4	Mass of robot body	0.034	kg
5	Mass of DC motor with mount	0.022	kg
6	Mass of the accelerometer	0.040	kg
7	Rotating mass	0.001	kg
8	Mass of IUCBR	0.127	kg

IUCBR, inverted U-shape curved beam robot.

The robot's locomotion is actuated by a DC motor attached to the rear leg of the IUCBR that rotates a mass on the arm connected to its output shaft. Vibrations from the motor are sampled by a Mide 509-PPA-1021 piezoelectric transducer placed on the other leg of the robot, as shown in Figure 1b. It has a capacitance value of 22nF and a full-scale voltage range of ±200 V. The robot's acceleration in three axes is measured with a Gulf Coast Data Concepts x200-4 USB impact accelerometer attached to the top of the body.

An Arduino UNO microcontroller with a 2 A motor driver controls the DC and servo motors via Pulse Width Modulation (PWM). The PWM signal, which dictates the DC motor input energy and servo motor angles, is varied using a graphical user interface (GUI) that sends the configurations to the microcontroller via serial communication. An oscilloscope and a set of multimeters are used to record values from energy harvesting.

The shape-changing feature of the robot is driven by an HD-1160A servo motor that extends the foot's distance, as shown in Figure 1a. The motor housing and the links connecting the feet to the servo motor output were three-dimensional printed in Polylactic Acid. For compliance, the links are loosely connected to the feet with zip-ties. These wide feet are made of similar steel strips to the body and fitted with rubber padding for the IUCBR's stability.

The relevant mathematical model

This section outlines the relevant model used to analyze the dynamics and behavior of the robot in relation to different morphologies from its shape-changing ability. As shown in Figure 2, two extreme shapes of the robot were evaluated, with the elastic beam raised to a more upright position or compressed to make the robot flatter.

FIG. 2.

Diagrams of the developed robot. (a) A schematic diagram of the IUCBR. (b, c) A simplified version of the schematic diagram of the IUCBR in the two of its configurations used for the purpose of mathematical modeling. (b) β is the angle between the robot body and the vertical where small values of β means that the robot is in a more upright position. (c) Therefore, (90° – β) is the angle between the robot leg and horizontal and large values of β means that the robot is in a flatter position.

The robot was decomposed to three-point masses in a two-linkage mass-spring system to simplify the model. The point masses do not include the curved beam mass and foot body mass, which are insignificant compared with the motors and joint housings. Figure 2 shows the notations used, where m_h and m_f are the masses at the lowest point of the hindfoot and front foot, respectively. The longitudinal stiffness and damping coefficients from the elastic curved beam body are represented by k_L and d_L, whereas k_β and d_β represent the torsional stiffness and damping coefficient of the body. d_L and d_β will not be used in this study for simplicity.

A concentration of the robot mass, comprising electronic components, is positioned at the center of the top of the robot body as mass, m. The legs of the robot that change in shape are labeled l, and their angle about the vertical axis at the top of the robot is β, or (90° − β) with the horizontal axis. Figure 2b and c shows the range of β with two configurations, a more upright standing configuration as β approaches 0° and a flatter laying configuration as β approaches 90°.

IUCBR in β ≈ 0° position

Referring to the two shape configurations shown in Figure 2b and c, the Lagrangian method is used to determine the torsional and longitudinal resonance frequencies that fundamentally rule the robot dynamics. Using the mass-spring system previously introduced with insignificant inertial effects on the masses, the kinetic and potential energies of the system (T and V respectively) are arranged as L = T – V. Using this Lagrangian L, the equation of motion of the robot will be derived with its Euler-Lagrange equation: $\frac{d}{d t} (\frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = 0,$ (1)

with conservative forces assumed to be zero and q representing the position vector of mass m in generalized coordinates. To complete Equation. (1), the kinetic energy, including the centripetal force generated by the rotating mass for the β ≈ 0° position, T is now:

Potential energy component V is similarly decomposed into gravitational energy and elastic energy from the springs: $V = m g l c o s β + 2 * \frac{1}{2} {(k_{L} l)}^{2} + 2 * \frac{1}{2} {(k_{β} β)}^{2}$ (3)

Since the position vector is made of two components l and β, Equation (2) is used to form the equations of motion for both generalized coordinates where l is assumed to be constant when solving for β and vice versa. $\ddot{β} + \frac{2 k_{β} - m g l}{m l^{2}} β = 0$ (4)

Defining the torsional frequency as ω_β, the generalized equation of motion can be written as: $\ddot{β} + ω_{β}^{2} β = 0$ (5)

By comparing Equations (4) and (5), we obtain the torsional resonance frequency as: $ω_{β \approx 0^{\circ}} = \sqrt{\frac{2 k_{β} - m g l}{m l^{2}}}$ (6)

In the case where β is constant and Equation (2) is applied for l, the homogeneous part of the Euler-Lagrange equation is formed as stated earlier. Similarly using small-angle approximation, cosβ ≈ 1 since β ≈ 0. $m \ddot{l} + 2 k_{L} l = 0 \Rightarrow \ddot{l} + \frac{2 k_{L}}{m} l = 0$ (7)

The longitudinal frequency, defined as ω_L, can then form the equation of motion as follows: $\ddot{l} + ω_{L}^{2} l = 0$ (8)

Therefore, by comparing Equations (7) and (8), we obtain the longitudinal frequency, ω_L, as: $ω_{L} = \sqrt{\frac{2 k_{L}}{m}}$ (9)

IUCBR in β ≈ 90° position

For the β ≈ 90° configuration shown in Figure 2b, which is also equivalent to (90° – β), measured anti-clockwise from the x-axis: $\ddot{β} + \frac{m g l + 2 k_{β}}{m l^{2}} β = 0$ (10)

On the other hand, the general form of the equation of motion with a defined torsional frequency ω_90°, is as follows: $\ddot{β} + ω_{β}^{2} β = 0$ (11)

By comparing Equations (10) and (11), we obtain the torsional resonance frequency as: $ω_{β \approx 9 0^{\circ}} = \sqrt{\frac{2 k_{β} + m g l}{m l^{2}}}$ (12)

Applying Equation (2) again for constant ω_β _{≈ 90°} to find the homogeneous part of the Euler-Lagrange equation where β ≈ 90°. $m \ddot{l} + 2 k_{L} l = 0 \Rightarrow \ddot{l} + \frac{2 k_{L}}{m} l = 0$ (13)

The general form of the equation of motion involving the longitudinal frequency, ω_L, is as follows: $\ddot{l} + ω_{L}^{2} l = 0$ (14)

Therefore, by comparing Equations (14) and (15), the longitudinal resonance frequency is of the form $ω_{L} = \sqrt{\frac{2 k_{L}}{m}}$ (15)

The model shows that locomotion behavior reacts to the change of robot morphology, as will be emphasized through experimental results. With the assumption that torsional frequency is larger than the longitudinal frequency, it can be seen from Equations (6), (9), (12), and (15) that the two frequencies will be closer in value to each other when the robot is in β ≈ 0° configuration. This relationship is the opposite when the torsional frequency is smaller than the longitudinal frequency.

The longitudinal frequency enacts a “sliding” lateral motion along the x-axis, whereas the torsional frequency induces a vertical “hopping” behavior along the z-axis. These lead to a possible behavior where the robot hops diagonally forward when the two frequencies are similar in value. In cases where the two frequencies are highly distinct, having a bias where the induced frequency is closer to one of the two resonant frequencies would produce more distinct behaviors of either purely “sliding” or “hopping” in its place.

Therefore, though not the focus of this work, the tendency for sliding behavior from β ≈ 90° configurations benefit travel in confined spaces. The β ≈ 0° configuration, where the two resonance frequencies are close, may be helpful in open terrain as the behavior allows more tendency to hop diagonally forward.

To verify the model, similar to the procedure explained in our previous work,¹² we determine the torsional and longitudinal resonance frequencies of the robot by fixing it on a wooden table and gradually increase the voltage supplied to the DC motor that actuates the rotating mass while observing the vibration patterns of the robot in a more upright β = 50° configuration and flatter β = 70° configuration. The result is shown in Table 2, which confirms that in a β = 50° configuration, the frequencies are further apart.

Table 2.

Torsional and Longitudinal Frequencies for Inverted U-Shape Curved Beam Robot in Two Configurations Along with the Corresponding Spring Constants Derived by the Robot Model

Robot configuration	Torsional frequency, $ω_{T}$ (Hz)	Longitudinal frequency, $ω_{L}$ (Hz)
50°	23.386	19.315
70°	25.148	11.611

In contrast, in a β = 70° configuration, the torsional and longitudinal frequencies have closer values. The experimental result shown in Table 2 and the derived theoretical framework support each other and will help in explaining the energy efficiency in the latter part of the article.

It must also be kept in mind that a robot's behavior will also depend on the environment, that is, the ground surface, as a behavior is a result of a tight coupling interaction among control input, body morphology, and the environment.^3,23 At this stage of the study, we perform the experiments on a fixed surface, that is, a wooden table. In addition, in our previous work,^12,52 we also investigated other parameters such as sizes and different value of the rotating mass. Although a thorough investigation of all these other factors is outside the scope of the article, we will discuss both the effect of the environment and other parameters in the Discussion on Other Parameters that May Affect Robot Behavior section and Supplementary Data.

Energy efficiency

The effects of energy harvesting on energy efficiency are observed by observing the Cost of Transport (CoT) of the robot with and without the energy harvesting module: $C o T = \frac{P}{m g v} \Rightarrow C o T = \frac{V I t}{m g d}$ (16)

Where P is the power consumed to move the system of mass m with velocity v under normal gravitational acceleration g, with known values of input voltage V, current I, and travel distance d under time t replacing velocity v, the equation can be re-written as shown in Equation (16).

Power coefficients

The robot dynamics are captured with the accelerometer placed on top of the robot. The x-axis represents the forward direction, and the z-axis captures the motion in the vertical plane. The y-axis data have insignificant variation due to the broad feet of the IUCBR. $M = ||(P_{x}, P_{z})|| = \sqrt{P_{x}^{2} + P_{z}^{2}}; γ = (\frac{P_{z}}{P_{x}})$ (17)

To observe the robot's dynamics and behaviors, with reference to Figure 2a and Equation (17), we find the resultant magnitude and phase angle of P_x and P_z, which are power coefficients obtained by using Fast Fourier Transforms on accelerometer measurements in x and z direction. Here, M is the magnitude of power coefficients, and angle γ is measured from the ground.

Energy harvesting module

The energy harvesting circuit uses a Mide 509-PPA-1021 piezoelectric transducer. This piezoelectric strip is attached tightly to the frontal apex of the curved beam robot body, parallel to the position of the vibration motor on the hind apex. The main assumption here is that there is no slip and all vibrations experienced by the robot body can be picked up by the piezoelectric strip. The aim of such a harvesting setup is to capture accelerations and vibrations that do not lead to robot propulsion, that is, accelerations in the z-axis.

With the focus on how robot behavior interacts with energy harvester (EH) operation and vice versa, there is less concern on capturing the small interactions with the piezoelectric strip that affect performance such as efficiency loses from mechanical and electrical coupling. As will be shown later in The Relationship Among Robot Behavior, Energy Harvesting, and Energy Efficiency section, the results are validated by observing the trends in robot behavior that align with the trends proposed by our robot model, which does not explicitly take motor and piezoelectric strip placement as well as electrical circuit modeling into account. Therefore, modeling of the earlier components in the mechatronic system was left out of the scope of this work but may be needed for optimization of robot performance in future works.

A non-controlled full-wave rectifier helps harness the sinusoidal energy generated by transducing vibrational energy. It comprises a diode bridge (D₁ – D₄), resistor R, and capacitor C and is a part of the energy harvesting module shown in Figure 3b, whereas the complete circuit is shown in Figure 3c. The piezo element in the circuit is shown as the driving voltage source, and the output of the circuit is across the R component.

FIG. 3.

Experiment setup and notations (a) The power coefficients along x- and z-axes, and the corresponding magnitude and angle. (b) The block diagram of experimental procedure. Here, the energy harvesting module is realized using a bridge rectifier with resistor R and capacitor C. (c) A schematic diagram of full-wave rectifier with diode circuit as energy storing circuit.

A time constant τ = RC dictates the duration of the charge and discharge cycle of the capacitor with respect to the values of the resistor R and the capacitor C. If the value of R is kept constant, a higher capacitor value will increase the time taken until the load can be powered whereas a lower value will decrease τ. In this experiment, the value of R is kept constant at 100 Ω whereas capacitors of three different values were used: 0.47, 4.7, and 47 μF to determine the best capacitance value for higher energy efficiency.

Experiment procedure

The block diagram in Figure 3b shows how different experiment elements are connected. The GUI helps monitor the input to the DC and servo motors. The input voltage control unit actuates the DC and servo motors based on the configuration sent from the GUI on the computer. As a baseline, the robot is run without the energy harvesting module for three different shapes, and the CoT is measured. We conducted 10 trials of the robot running on a wooden table for 10 s with set input voltages supplied to the DC motor, measuring the distance traveled. At each voltage V, we record the supplied current I and time t required to calculate the energy E supplied to the robot using E = V × I × t.

The energy harvesting module is then integrated for similar runs, and its impact on the CoT is observed by a comparison with results before energy harvesting. The capacitor in the energy harvesting module is varied to study its effect on the CoT.

Experiment Result and Analysis

This section presents the experimental results and relevant analysis. The goal of the experiments is to explore the relationship among the harvested energy, resulting efficiency, and the robot's behavior and morphology. Figure 4 shows the IUCBR in motion for the β = 60° configuration on the experiment bench. The straight marking lines with a spacing of 10 cm served as a reference for measuring the distance.

FIG. 4.

Images of the IUCBR in motion: The IUCBR in β = 60° configuration on an experimenting table with four straight lines drawn at spacing of 10 cm each with the value shown at the top right shows the time taken to reach that point from the starting point x = 0 at t = 0. The two rows differ in the input energy supplied to the IUCBR, which are 0.029 J (V = 1.1 V, I = 0.026 A) and 0.078 J (V = 1.7 V, I = 0.046 A) before and after the energy harvesting module is used, respectively. (a–c) and (d–f) show the different points reached by the robot at the same time but different supplied energy.

Figure 5 illustrates the results of CoT against the input energy to the robot. Since the harvested energy nonlinearly affects the input energy of the robot (the x-axis), the right column (Fig. 5c, f) presents plots that have x-axis intervals re-ordered by subtracting the measured harvested energy from the total input energy to the robot, and the corresponding CoT is also affected in this way. This is done so that Figure 5c and f can be compared with Figure 5a and d to analyze the effects of the EH on the CoT. Additional details on this re-ordering process can be found in the Supplementary Data.

FIG. 5.

Results of energy efficiency experiments. (a) CoT plotted against Energy (J) for different shapes of IUCBR. (b) Shows the values recorded with energy harvesting module and (c) is obtained after subtracting and reordering the input energy. The inset figure shows the average value for each shape. The second row shows the plots of robot CoT, with energy harvesting module having different values of capacitor used, namely (d) 0.47 μF, (e) 4.7 μF, (f) 47 μF. Please note that (b) and (f) are two identical figures. (g–i) Show the CoT plot with respect to input energy for β = 50°, 60°, and 70° robot configurations respectively, as an inverse of (d–f). CoT, cost of transport.

The effect of energy harvesting on energy efficiency

The first result related to energy efficiency is shown in Figure 5a, where the CoT is plotted against IUCBR input energy without an EH. The minimum recorded CoT values for the 50° (red), 60° (blue), and 70° (black) robot shapes are 0.727, 0.546, and 0.708 at 1.63, 0.89, and 1.01 J, respectively. This is reflected in the inset figure showing the average CoT of each shape, where the 60° shape (blue bar) shows the lowest average CoT.

The energy harvesting module with a capacitor value of 47 μF is then integrated to potentially increase energy efficiency. The results are shown in Figure 5b. Here, the harvested energy is fed back to the DC motor, in addition to the input energy from the power supply. Thus, the overall input energy shifts a bit, affecting the robot's behavior accordingly as will be explained in the following sub-section.

In Figure 5c, we plot the CoT obtained with the energy harvesting module against the actual input energy applied to the DC motor, neglecting the harvested energy, and therefore the CoT-energy pairs are ordered like Figure 5a. The trend shown in Figure 5c is similar to Figure 5a and numerically more energy efficient, with some differences in trends at lower input energies for 60° and 70° configurations. These are the result of the interactions among the energy harvesting, the tendency for these configurations to “slide,” and the resulting interactions with the environment (i.e., friction), as will be explained further in the next sections.

Compared with Figure 5a, the average CoT is lower for all shapes with a similar trend. Notably, the average CoT before and after the EH for 50°, 60°, and 70° configuration are statistically significantly different with values 0.958, 0.941, and 1.095 and 0.591, 0.577, and 0.777, respectively.

Given the same piezoelectric transducer and resistor R in the energy harvesting circuit, a large factor affecting the results is the capacity C. Therefore, we evaluated capacitors with different values to determine the best capacitance for the time constant τ = RC, to increase harvested energy. Determining the optimal time constant for the complex and changing behaviors of the robot may prove interesting as higher amounts of harvested energy may not equate to higher energy efficiency.

Figure 5d–f shows the CoT-energy plot for different IUCBR configurations with EH using 0.47, 4.7, and 47 μF as capacitor values, respectively. The inset figures show that larger amounts of harvested energy do not necessarily lead to higher energy efficiency. For example, for the 50° configuration, the average CoT for C values 0.47, 4.7 μF (Fig. 5d, e, respectively) are lower than when the EH module used C of 47 μF shown in Figure 5f. In detail, the average CoT for 50°, 60°, and 70° configuration where the capacitor values equal to 0.47, 4.7, and 47 μF are 0.456, 0.674, 0.866; 0.463, 0.534, 0.928; and 0.591, 0.577, 0.777, respectively.

We also plot the above set of figures grouped based on the robot shape for different capacitor values, as shown in Figure 5g–I, that help us to compare the results from a different perspective. It is shown that for 50° configuration in Figure 5g, the 0.47 μF results in the lowest CoT, whereas for the other shapes, the best CoT is shared between the 4.7 and 47 μF capacitor values.

Concretely, the average CoT for capacitor values of 0.47, 4.7, and 47 μF where the shape is in the 50°, 60°, and 70° configuration are 0.456, 0.463, 0.591; 0.674, 0.534, 0.577; and 0.866, 0.928, 0.777 respectively. From the last two paragraphs, the best capacitance values for 50°, 60°, and 70° configurations are 0.47, 4.7, and 47 μF, respectively.

The effect of energy harvesting on the robot's behavior

Figure 6 shows plots of parameters to analyze the behavior of the robot. The first is energy E supplied to the robot. The other is the power coefficients of the robot's vibration, P_x and P_z, in x and z direction of the robot's motion, respectively.

FIG. 6.

Plots of robot behavior (power coefficients) with respect to input energy. The first, second, and third columns of the figures show plots for different shapes for the normal run, with the EH and after subtracting and reordering the input energy, respectively. (a), (d), and (g) show the power coefficients in x and z direction for different configurations plotted against input energy whereas (j) shows the ratio of P_x and P_z as the consolidated plot for three configurations. (b), (e), (h), and (k) show the same set of figures after the EH is used whereas (c), (f), (i), and (l) show them with the harvested energy subtracted from the input energy and reordered. EH, energy harvester.

Figure 6a–c shows the plot of these parameters when the robot is in β = 50° configuration, whereas Figure 6d–f and Figure 6g–i show the plot when the robot is in 60° and 70° configurations, respectively. Finally, the bottom row, that is, Figure 6j–l, shows the consolidated variation for the three configurations that plots the ratio between P_x and P_z over the entire range of input energy.

Figure 6 can also be seen from a different perspective, where the first column (Fig. 6a, d, g) shows the results before using the EH and the second (Fig. 6b, e, h) and third columns (Fig. 6c, f, i) show the same plot with EH using 47 μF as the capacitor, and after the reordering process.

Observing different rows in Figure 6, when angle β is low, P_x and P_z are quite close to each other, whereas in the opposite case, P_x tends to be larger than P_z. The average difference between P_x and P_z for 50°, 60°, and 70° configurations are 0.828 ± 0.598, 3.081 ± 1.205, and 2.615 ± 1.548, respectively, before EH is used and becomes 0.931 ± 0.495, 3.287 ± 1.991, and 3.546 ± 1.446, respectively, with the harvester. Also, the difference of these values between the 50° configuration and the other two is statistically significant, whereas it is not the case for both cases between 60° and 70° configurations.

In summary, the average behavior of the IUCBR for each shape remains the same while the magnitude is increased. More specifically, for the 50° configuration, P_x is not dominant over P_z regardless of increased magnitude from the EH. On the other hand, for flatter positions such as 60° and 70°, P_x is still dominant on P_z after the harvester is used. The additional increase in the magnitude of P_x and P_z is shown by the bars stacked on top of them. Here, the same colors of red, blue, and black are used for 50°, 60°, and 70° respectively, and varied shades of these colors are used to denote the P_x and P_z components.

The ratios of power coefficients, P_x/P_z, without the use of the EH module (Fig. 6j), after the use (Fig. 6k), and after reordering for comparison purposes (Fig. 6l) are also presented. These plots also show additional data points at higher input energies compared with the bar charts as it was observed that the behavior of the robot becomes less predictable at those energy ranges, that is, entering the chaotic region.¹² By observing Figure 6j and l, we confirm that after neglecting the additional energy harvested, the trend of behavior shown by P_x/P_z will be similar to when the robot does not use an EH. Further, by observing Figure 6j and k, the additional energy may non-linearly change the overall order of the behaviors concerning the original input energy.

To visually represent how robot behavior is affected by the input energy, P_x and P_z are plotted into M and γ and mapped to a polar plot. Figure 7a shows M plotted against energy E given to the robot for the three different shapes of the robot without the EH module. Figure 7b shows a similar figure for γ. These lead to Figure 7c, a polar plot that characterizes the robot's behavior through a graphical representation of the effect of magnitude M and angle γ on the robot's motion.

FIG. 7.

The magnitude, angle, and phase plot of the power coefficients of IUCBR over the entire energy input range for β = 50°, 60°, and 70° configurations before EH is used. (a, b) shows the line plot of magnitude and angle, respectively, whereas (c) is analogous to the polar plot, which graphically points the resultant magnitude, M, acting in a direction γ as experienced by IUCBR that shows its motion vector.

When angle β is low, the arrows shown in the polar plot tend to point diagonally upward due to comparable values between P_x and P_z. On the other hand, when angle β is high, the arrows tend to point forward as P_x is significantly larger than P_z, as predicted by the model in the Materials and Methods section.

Figure 8 shows the polar plot representation of the robot's behavior. To visually clarify how the behavior may change, we apply a threshold of 25° to angle γ, which is set as half the maximum angle value. The robot's behavior is described as tending to hop if γ is larger than the threshold and a tendency to slide otherwise. It can be seen in Figure 8a that without energy harvesting, low input energy causes a tendency to slide, and high input energy causes a tendency to hop. However, additional harvester energy changes each behavior and the overall order with respect to input energy, as shown by Figure 8b.

FIG. 8.

Polar plots of vectorized robot behavior. The polar plot shown in Figure 7c can be used to indicate different behaviors of the robot by applying a threshold value to the angle γ that acts on the robot. As a result, the tendency to slide is shown by dashed red lines whereas the tendency to diagonally hop forward is shown by solid green lines. The robot in different configuration of β = 50°, 60°, and 70° under normal run and with EH is shown in (a–c), respectively. (a) This polar plot shows the IUCBR without EH. (b) With EH in place, the polar plot is updated with the order of behaviors may change due to the non-linearity of the additional energy. (c) Shows the reordered version of polar plot with EH such that behaviors are comparable with (a) for each input energy. Darker shade of color implies that CoT has improved from the previous value.

After subtracting additional energy and re-ordering, the trend of the behaviors shown in Figure 8c is similar to Figure 8a. Further, although each behavior may change with respect to input energy, the average behavior for each shape does not change, as shown by the polar plot at the right-end of Figure 8a–c.

The average values of γ are 38.323°, 25.314°, and 23.720° for 50°, 60°, and 70° configuration respectively without EH and 39.610°, 28.106°, and 27.586° with the EH module showing less tendency to hop and more tendency to slide as β is higher. We confirm no statistically significant difference between γ before and after the harvester is used, that is, the average behavior for each shape remains the same.

On the other hand, the magnitude M before and after the EH is used for 50°, 60°, and 70° configuration are 5.168, 6.429, 4.708, and 6.979, 8.425, 6.759, respectively. The difference is statistically significant at the 0.05 level, showing that magnitude has considerably increased.

To illustrate these findings with an example of a robot run, Figure 9 shows two scenarios of robot runs with the highest energy efficiency (lowest CoT) identified through our tests. The left configuration of Figure 9c represents the best robot setup, running in open terrain, whereas the right configuration represents the robot running with highest efficiency in a height constrained terrain. Figure 9b shows a filtered motion plot of a robot run in the x- and z-axes, allowing us to see how the 60° configuration has more perturbations in the accelerometer caused by propensity for hopping motions as identified earlier and seen in the polar plots in Figure 9a.

FIG. 9.

Illustration of the robot in action with most efficient configurations for two scenarios. (a) Shows the motion vectors at 2 s time intervals as polar plots. (b) Shows the displacement of the robot's center of mass as a motion plot, the top axis shows the plot with respect to time in seconds. (c) Shows the robot with a more upright position on the left before entering a tunnel with a flatter position on the right. The two text boxes show the parameters used in each configuration and the resulting CoT. The change in configuration was done manually, and the data recorded by the accelerometer were stitched together. The footage of the robot moving is also shown in the Supplementary Video S1.

The robot shape and harvesting setup were then changed manually before entering the tunnel setup, resulting in the drop in z-axis displacement in the motion plot by 1 cm when the accelerometer reading was stitched together. The β = 70° configuration fit into the tunnel and had more sliding motion for more stable locomotion, but it may be slower and less efficient.

The relationship among robot behavior, energy harvesting, and energy efficiency

To relate the observations in The Effect of Energy Harvesting on Energy Efficiency and The Effect of Energy Harvesting on the Robot's Behavior sections, the effect of the robot behavior (characterized by power vectors introduced in The Effect of Energy Harvesting on the Robot's Behavior section) on the energy harvested and how it improves the energy efficiency that is, CoT are compiled in Figure 10. These figures present a biharmonic interpolated surface plot from the experiment data points in The Effect of Energy Harvesting on Energy Efficiency and The Effect of Energy Harvesting on the Robot's Behavior sections to estimate the trends of EH performance and change of CoT after energy harvesting with respect to robot behavior.

FIG. 10.

Interpolated surface plots (warmer colored regions indicate higher magnitudes, alternatively, regions containing the maximum point would be higher magnitude regions) to illustrate trends in EH performance and subsequent CoT across different robot behaviors for each experiment configuration. (a) Energy harvested for β = 50° configuration, (b) difference in CoT before and after energy harvesting for β = 50°configuration. (c) Energy harvested for β = 60° configuration, (d) difference in CoT before and after energy harvesting for β = 60°configuration. (e) Energy harvested for β = 70° configuration, (f) difference in CoT before and after energy harvesting for β = 70°configuration. Note that higher difference in CoT is better, and the harvester has a more positive impact on robot energy efficiency for the given robot configuration and behavior.

The left column (Fig. 10a, c, e) shows the surface plots of P_x and P_z against the energy harvested, and the right column (Fig. 10b, d, f) shows the difference in CoT values (CoT before harvester subtracted CoT after harvester, higher difference is better) for the three robot configurations. The figures show how the energy harvested and CoT changes with respect to acceleration in x- and z-axes that represents the robot's behaviors. These plots help to investigate the EH characteristics and its impact on the improvement of the CoT.

Several initial observations include higher energy harvested along the z-axis when the robot does not exhibit large x-axis acceleration for forward motion. More specifically, it can be seen from the left column figures (Fig. 10a, c, e) that the harvester can capture more of the excess vibrations from the acceleration along the z-axis with generally less work done to transfer vibrational energy to propulsion.

As for the change in CoT values shown in the right column figures (Fig. 10b, d, f), the surfaces show that CoT improvement (larger reduction is better) is concentrated in areas as expected by the explanation in The Relevant Mathematical Model section and results in The Effect of Energy Harvesting on Energy Efficiency and The Effect of Energy Harvesting on the Robot's Behavior sections.

As predicted by the theoretical model and results in previous sections, when β is low, the robot has more tendency to hop diagonally forward as the two resonance frequencies are closer to each other. The result is a high improvement of the CoT that is distributed more equally along the two axes. The opposite gradually occurs when β is higher. In the case of β = 70°, the two resonance frequencies, that is, longitudinal and torsional, are clearly separated as shown in Table 2. This difference causes a change in the distribution of how the harvested energy improves robot CoT, as seen by a more split trend in Figure 10f.

Although there is improvement of the CoT at high x-axis accelerations, Figure 10f shows that the EH had more of an impact on CoT at low x-axis accelerations. Figure 6i helps explaining this phenomenon. In Figure 6i, it is shown that as input energy increases, the longitudinal frequency of the robot diverges further away from its potent resonance frequency identified in Table 2 and causes a drop in magnitude of P_x and brings the values of P_x and P_z much closer together.

These points constitute the lower left quadrant of Figure 10f, with a robot behavior that trends toward hopping forward. The energy harvested is, therefore, similarly distributed more evenly along the two axes, and more z-axis “hopping” can be harvested into improving CoT. However, since the phenomenon is limited to that range of higher input energy with low forward robot motion due to low magnitudes of Px, the harvested energy improves the CoT in a limited x-axis range. As explained in The Effect of Energy Harvesting on the Robot's Behavior section, overall, the β = 70° configuration still tends to exhibit “sliding” motions that can have harsher interaction with the environment and therefore the CoT does not improve as much as the other configurations.

The results can also be related with those in The Effect of Energy Harvesting on Energy Efficiency section that shows that the best capacitance value, in terms of lowest CoT, for 50°, 60°, and 70° configurations are 0.47, 4.7, and 47 μF, respectively. A time constant τ = RC dictates the duration of the charge and discharge cycle of the energy harvesting circuit. Since R is kept constant, higher capacitor values will increase the time taken until the load can be powered.

Figure 10 shows that the energy can be mainly harvested from the excess vibrations from the acceleration along the z-axis, as the robot has to use some of the energy to propel forward. Since the 70° configuration has an overall tendency to purely slide forward, that is, less acceleration along the z-axis, it is reasonable that to harvest a meaningful amount of energy, it is better to have a larger time constant. On the other hand, the other configurations have more tendency to diagonally hop forward, which is a mixture of forward and, energy harvestable, vertical motion.

In this case, smaller time constants that lead to quicker bursts of energy are shown to be beneficial. It must be noticed that the balance between waiting for a significant amount of energy to be harvested and quickly having a burst of energy needs to be kept. For example, the best capacitance value for 60° configuration is 4.7 μF, which is the mid-value between 0.47 and 47 μF.

Discussion on other parameters that may affect robot behavior

Although the focus of this article is the change of the robot's morphology, it must be kept in mind that a robot's behavior depends on the whole interaction among its control, body morphology, and environments.^3,23 In this context, we show the ground interaction model that we used and verified in our previous work⁵² in the Supplementary Data. The vertical ground interaction forces are approximated by nonlinear spring-damper interactions, and the horizontal forces are calculated by a sliding stiction model.

The vertical and horizontal ground reaction forces depend on several parameters, such as the friction and damping coefficients, which will depend on the environment. In the experimental results of this work, the effect of the ground interaction forces is encapsulated in the power coefficients recorded by the accelerometer (P_x and P_z along the x and z direction respectively) although it is not explicitly modeled.

For instance, if the surface is not sufficiently hard and the vertical reaction force is low, it may lower down the value of the power coefficients along the z direction (P_z) even if the robot is in an upright position, that is, has a low value of β. While at this stage, we use a wooden surface where the effect of different resonance frequencies to the motion was shown to be significant.^12,52,53 Future work could see experiments on other types of surfaces.

Also, in this work we only explore shape-changing ability enabled by the elastic body of the robot. However, as investigated in our previous work,^12,52,53 other parameters such as different rotating masses will also affect the robot's behavior. For example, we have shown that increasing value of the rotating mass would increase the centripetal force that drives the robot's motion^12,52 as also explained further in the Supplementary Data.

Lastly, the focus of this work is the energy efficiency of the robot. Sometimes, it is interesting to focus on other variables such as the velocity of the robot. As shown by the model of the effect of the rotating mass in the Supplementary Data, increasing the energy input to the robot will increase the angular velocity of the mass, which will increase the centripetal force that drives the robot's motion and eventually increase the robot's velocity. In the Supplementary Data, we also show the plot of the robot's velocity before and after harvesting for each of the robot's configuration. However, as explained by Equation (16), increased value of velocity does not necessarily increase the efficiency of the robot as it depends on how much energy needed to achieve it.

Conclusion

Despite the potential of robots made of soft and elastic materials, they still require improvements in energy efficiency. Although an approach of improvement is by harvesting energy from behavioral motion, it is not straightforward due to the complex dynamics of the robot. This work showed that increasing energy efficiency via harvesting requires exploitation of the relationship between harvested energy and robot morphology.

We present the following findings through a structurally compliant shape-changing robot whose behavior is characterized by its resonance frequencies. First, harvesting energy at suitable occasions is more important than maximizing the amount. This additional energy, which is nonlinear to the input energy of the robot due to complex dynamics, dictates behavioral changes that affect efficiency. Second, the robot's morphology dictates the usage of harvested energy and subsequent robot behavior.

We have shown analytically and experimentally how different shapes lead to different uses of the additional energy in terms of the power coefficients that form the motion vector of the robot, leading to different behaviors. To maximize the energy efficiency, it is essential to increase the magnitude of the behavior and to have a suitable behavior with the additional energy.

Here, if the additional energy causes the robot to hop vertically on the same spot, the efficiency will reasonably be reduced as the robot does not move forward despite being provided with additional energy. Oppositely, when the horizontal motion of the robot becomes too dominant, it may not be an efficient behavior either due to possible resistance from friction. We have also shown that there is a best capacitance value for each shape, or, from another perspective, the best shape for each capacitance value.

In terms of applications, the work shows how the developed robot is able to deform while moving forward under confined spaces, such as in search and rescue scenarios while minimizing the required power to change its shape and maximizing the energy efficiency. Nevertheless, the study is in a fundamental stage, and it still requires a lot more work to bring it to the application stage.

Future plans include integrating variable capacitors in the harvester to tune additional energy supplied to the robot; this could then be autonomous based on the robot's changing morphology. Such approaches may also be varied with different stiffness or environments. We are also interested in applying the investigated concept in other types of robots, such as those made of viscoelastic materials that lead to larger structural damping.

Another interesting direction will be to focus on enabling local adaptive rules coded into the robot structure that can support regulating the leg angles without applying an external central electrical control unit, hence focusing more on the physical intelligence of the robot.

Footnotes

Authors' Contributions

S.A.K. developed the robot, performed and analyzed the experiments, and partially wrote the manuscript. L.Y.L. created the multimedia file, performed and analyzed the experiments, and partially wrote the manuscript. F.I. supervised the project. S.G.N conceived, supervised, and funded the project.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by the FRGS Grant (Project no: FRGS/1/2017/ICT02/MUSM/03/3) provided by the Ministry of Higher Education, Malaysia and the School of Engineering Seed Funding (2016), Monash University Malaysia.

Supplementary Material

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Supplementary Material

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