Estimating stiffness and damping of a novel variable impedance actuator based on adjusting viscoelastic properties of thermoresponsive polycaprolactone in harmonic motions

Abstract

This paper presents a methodology to estimate the stiffness and damping of a novel variable impedance actuator designed to adjust its effective impedance by regulating the temperature of a thermoresponsive polymer, Polycaprolactone (PCL). The actuator’s PCL temperature is controlled using embedded flexible Peltier elements. However, due to the absence of internal temperature sensors, it is necessary to develop a reliable estimation method for effective stiffness and damping during harmonic motion. The proposed approach leverages experimental data, including trajectory measurements and phase delay analyses, to estimate these parameters under varying temperatures and frequencies. Experimental results demonstrate that stiffness and damping can be effectively modulated by altering the PCL temperature, with higher temperatures leading to decreased stiffness and damping due to reduced viscoelasticity. Additionally, it was observed that the system’s dynamic response is frequency-dependent, which presents a challenge for precise impedance regulation solely through temperature control. Despite this limitation, the proposed estimation methodology accurately captures the system’s viscoelastic behavior, offering valuable insights into the actuator’s performance and potential for applications requiring adaptive impedance control.

Introduction

As robotic systems are increasingly deployed in unstructured and dynamic environments, the likelihood of encountering direct physical Human-Robot Interaction (pHRI) is steadily growing. In such contexts, ensuring safe and reliable interaction is paramount, which requires robots not only to achieve accurate motion but also to dynamically adapt their physical behavior to external forces and unexpected disturbances. This adaptive behavior is commonly described in terms of impedance, a mechanical property that encapsulates resistance to motion and is defined by three key elements: stiffness, damping, and inertia¹. The ability to modulate impedance is critical for maintaining human safety while simultaneously enhancing the performance of the robot, particularly in tasks where compliant interaction and energy exchange are necessary.

A class of actuators known as Variable Impedance Actuators (VIAs) has emerged as a prominent solution for this challenge². VIAs are designed to modulate their mechanical impedance in real time by actively altering stiffness^3–5, damping characteristics^6,7, or even inertia⁸. By enabling this tunability, VIAs provide both safety and performance advantages in physical interaction scenarios, ranging from collaborative manipulation to assistive and wearable robotics. Among these categories, variable stiffness actuators (VSAs)⁹ have received the greatest attention within the robotics community. VSAs specifically target the modulation of stiffness in the output link, thereby offering direct control over one of the most influential parameters governing compliant behavior during interaction with uncertain or dynamic environments.

Other subsets of VIAs include variable damping actuators^7,10 and variable inertia actuators¹¹. Although these devices extend the spectrum of impedance modulation, their adoption has been comparatively limited. The primary reason for this reduced interest lies in their absence of energy-storing or energy-releasing components, unlike VSAs. As a result, they offer fewer benefits for reducing energy consumption in periodic or oscillatory motions^12–15. Nevertheless, damping regulation remains indispensable in situations where the system must prevent uncontrolled energy build-up for stability, such as in robotic knee joints during bipedal locomotion, where controlled dissipation is required for stability and safety.

For an actuator to be truly considered a VIA, it must provide control over at least two distinct aspects of its behavior: the trajectory of the link and the impedance characteristics of the interaction. This requirement implies the necessity of at least two independent mechanical power sources–typically realized in the form of flow and effort actuators¹⁶. For example, when both stiffness and damping are to be modulated simultaneously in addition to trajectory control, three independent actuation sources are required. Electric motors are commonly used in such architectures^17,18, but the addition of each motor inevitably increases both the complexity and physical size of the system.

Conventional VIA designs achieve impedance variation by embedding specialized mechanical mechanisms directly into the actuator structure¹⁹. While effective, this approach typically results in bulky and heavy actuators, with dimensions significantly larger than those of traditional rigid actuators or Series Elastic Actuators (SEAs)²⁰. The additional mass and complexity associated with these designs increase inertia, which can degrade safety margins in human-robot interaction scenarios^21–23. Consequently, despite the advantages offered by VIAs, SEAs are still often preferred, particularly in locomotion tasks^24–27 and manipulation systems^28–30, where compactness and reliability are critical. On the smaller scale, heating-based actuators using shape-memory alloys have been studied as low-power alternatives³¹, although their limited efficiency has constrained their applicability.

In response to these limitations, the present work introduces a thermo-active VIA design that eliminates the reliance on bulky mechanical mechanisms for impedance modulation. As illustrated in Fig. 1, the actuator maintains a compact and lightweight configuration by leveraging the thermal properties of Polycaprolactone (PCL), a thermoplastic polymer that undergoes a marked transition in viscoelastic properties when heated³². At lower temperatures, PCL exhibits rigid and stiff behavior, whereas at elevated temperatures it softens, thus allowing direct modulation of the actuator’s stiffness and damping properties. Flexible polyimide heaters, embedded within the actuator body, are employed to regulate the temperature of the PCL components, enabling active control of impedance through thermal excitation³³. This strategy represents a departure from traditional multi-motor VIA designs, as it achieves impedance tunability with minimal mechanical complexity, paving the way for compact, lightweight, and energy-efficient systems.

Fig. 1.

Prototype of the thermo-active Variable Impedance Actuator constructed from 3D-printed PETG components with embedded flexible polyimide heaters. A test weight of 0.91 kg is attached to the output link to demonstrate load-bearing capability.

The remainder of this paper is structured as follows. Section "Mechanical design and components of the proposed variable impedance actuator" introduces the mechanical design of the proposed thermo-active actuator and details its components. Section "Modeling the viscoelasticity of the proposed VIA" develops the modeling framework and estimation approach for characterizing viscoelastic properties of the actuator. Section “Experimental results” presents experimental validation of the proposed design through impedance modulation and trajectory tracking studies. Finally, Sect. "Discussion and conclusion" discusses the implications of the findings and outlines potential directions for future development of thermal-based variable impedance technologies.

Mechanical design and components of the proposed variable impedance actuator

The proposed Variable Impedance Actuator (VIA) integrates conventional electromechanical components with a thermo-responsive impedance modulation module. At its core, the actuator consists of a DC motor and gearbox assembly, which generate controlled motion and torque. This actuation system is placed in series with a custom thermo-active impedance module, thereby combining the precision of electromechanical control with tunable compliance and damping properties. The thermo-active module itself incorporates two principal elements: compression springs that establish a baseline elastic compliance and a Polycaprolactone (PCL)-based medium whose viscoelastic properties vary significantly with temperature. The output of this thermo-active module is coupled directly to the actuator’s link, allowing the impedance to be regulated in real time through thermal excitation. This section elaborates on both the conceptual mechanical design of the actuator and the fabrication process used in developing a functional prototype.

Mechanical design

The thermo-active impedance module represents the defining feature of the actuator’s design. As illustrated in Fig. 2, the module comprises an outer cylindrical ring firmly anchored to a fixed base. The internal cavity of this housing contains PCL material, which serves as the thermally tunable damping and stiffness element. To achieve controlled heating, three flexible polyimide heaters are radially mounted in slots distributed along the outer ring. These heaters are in direct thermal contact with the PCL and are positioned to allow partial exposure to the surrounding environment. This mounting configuration is intentional: by providing an accessible path for heat dissipation, the design mitigates the risk of thermal buildup that could otherwise compromise nearby electronic components or lead to premature failure of sensitive elements.

Fig. 2.

CAD model of the proposed thermo-active Variable Impedance Actuator, highlighting the integration of spring and PCL-based modules.

The transmission of motion from the motor to the output link is achieved via a spiked wheel housed inside the outer ring. The wheel is mechanically coupled to the actuator’s output link, while the housing is rigidly fixed to the motor shaft. The spikes along the wheel’s circumference penetrate into the PCL, thereby engaging with the thermoplastic medium during motion. As the wheel rotates, the spikes deform the PCL, producing a shear interaction between the rotating wheel and the thermally softened polymer. This interaction is central to the modulation of impedance: at low temperatures, the PCL remains stiff and resists deformation, while at elevated temperatures, the softened PCL allows greater compliance and damping.

The compliance of the system is further enhanced by the inclusion of compression springs, which connect each spike of the wheel to the fixed housing. When the wheel rotates due to motor input, the springs undergo deformation, producing a restoring elastic force that mimics the behavior of a Series Elastic Actuator (SEA). However, unlike conventional SEAs, the proposed design incorporates PCL to enable thermal modulation of impedance. By regulating the polymer’s viscoelastic properties through controlled heating, the actuator achieves a continuously tunable impedance response that combines the benefits of elastic compliance and temperature-dependent damping. This dual mechanism not only improves safety during physical interaction but also allows the actuator to adapt its behavior to different operational contexts. A prototype of the proposed actuator is presented in Fig. 1.

The overall mechanical design emphasizes compactness, low weight, and functional simplicity. Rather than relying on additional motors or complex gearing to achieve impedance variation, the actuator leverages the thermal characteristics of PCL, resulting in a design that is inherently more lightweight and energy-efficient. The following subsection discusses in greater detail the specific materials and components employed in fabricating the prototype.

Flexible heater design and operation

The stiffness modulation of the actuator is achieved through localized heating of the thermoresponsive polymer (PCL) using flexible polyimide heaters. Each heater is a Kapton^®-based thin-film element composed of etched resistive traces laminated onto a polyimide substrate. These heaters are lightweight, conformable, and capable of uniform heat distribution across curved surfaces. The nominal watt density of the selected heaters is approximately 5–10 W/in², corresponding to a total heating power of 5–15 W when driven at the rated supply voltage. Three identical heaters are arranged circumferentially around the PCL cavities to ensure even temperature distribution and minimize local thermal gradients.

Based on the thermal energy estimate presented in Sect. "Components of the VIA prototype", and under the nominal heater power (5–15 W), the polymer temperature rises from within approximately 30–60 s. This heating duration was confirmed experimentally through thermocouple measurements placed on the actuator housing surface. The time constant is primarily determined by the PCL thermal mass, heater–polymer interface resistance, and convective heat loss to the surroundings.

Uniform heating is critical because the thermal conductivity of PCL is relatively low (). Insufficiently uniform temperature fields can lead to nonuniform stiffness within the actuator, local partial melting, or asymmetric deformation, which may bias the experimentally observed impedance characteristics. The circumferential heater configuration reduces these effects, but slight gradients are unavoidable near mechanical boundaries. To further improve homogeneity, thermal interface materials and reflective insulation were applied around the heater assembly.

Future iterations of the actuator will incorporate embedded miniature thermistors or thin-film RTDs within each PCL cavity, enabling closed-loop temperature feedback control. This addition will allow precise real-time modulation of polymer temperature, improve repeatability of stiffness control, and compensate for ambient variations during extended operation.

Components of the VIA prototype

The structural framework of the prototype, including the housing, output link, and base, was fabricated using additive manufacturing. Polyethylene terephthalate glycol (PETG) was selected as the filament material due to its advantageous combination of toughness, impact resistance, and thermal stability. Unlike polylactic acid (PLA), which is more brittle and prone to softening under moderate heating, PETG maintains its mechanical integrity at the elevated temperatures generated by the embedded heaters. This ensures reliable structural performance during prolonged operation of the thermo-active module.

The key functional element of the design is the Polycaprolactone (PCL) polymer. PCL is a biodegradable aliphatic polyester characterized by its low melting point (approximately C) and exceptionally low glass transition temperature (around C). These properties make PCL particularly well-suited for applications where temperature-dependent changes in stiffness and damping are desired. PCL is synthesized via ring-opening polymerization of cyclic ester monomers, a chain-growth polymerization technique that produces long polymer chains with high molecular weight³⁴. This method of synthesis, illustrated in Fig. 3a, imparts thermoplastic characteristics to PCL, allowing the material to undergo reversible softening and hardening with heating and cooling cycles³⁵.

Fig. 3.

Viscoelastic structure and behavior of PCL under varying temperatures and loads.

The viscoelastic nature of PCL arises from the interplay of its molecular-scale dynamics. Polymer chains can exhibit motion across multiple time scales, ranging from atomic vibrations to the diffusion of entire chain segments. Smaller segments respond rapidly to applied stress, while larger chains rearrange much more slowly, giving rise to the time-dependent deformation that characterizes viscoelastic materials. In the absence of external loading, the morphology of PCL is governed by internal factors such as covalent bonding, intermolecular interactions, segmental mobility, chain entanglement, and crystalline domains³⁶. At equilibrium, these factors collectively minimize the system’s energy, resulting in a disordered arrangement of polymer chains.

When subjected to external forces, however, the chains partially align along the direction of loading. Entanglement points act as anchors that resist this reorganization, pulling the chains back toward their original configuration once the force is removed³⁷. At small deformations, the response is predominantly elastic, and the material recovers its original morphology once unloaded, as illustrated in Fig. 3b. Under higher loads or prolonged stress, however, irreversible processes such as chain slippage or bond breakage can occur^38,39, leading to permanent deformation and the establishment of a new equilibrium state.

The thermo-responsive behavior of PCL is a direct consequence of its thermoplastic nature. At low temperatures, strong intermolecular interactions maintain rigidity, but as temperature increases, these interactions weaken, enabling the polymer chains to slide past one another more easily. This transition results in a substantial reduction in stiffness and an increase in damping capacity⁴⁰. Consequently, PCL is capable of shifting from a rigid, load-bearing state to a compliant, energy-dissipating state, making it a highly versatile material for impedance modulation in robotic actuators.

To experimentally quantify the thermal dependence of PCL impedance, static compression tests were performed on a PCL sponge under controlled temperature conditions. In these experiments, a weight of 4.5 kg was applied while the temperature was varied between and . The vertical displacement of the sponge was measured at each temperature and normalized by the initial thickness to compute strain, while the applied load was divided by the cross-sectional area to calculate stress. Representative results, shown in Fig. 3c, clearly demonstrate the temperature-dependent compliance of PCL, with the material exhibiting significantly greater deformation as the temperature increases. These findings validate the material’s suitability for thermo-active variable impedance applications.

Modeling the viscoelasticity of the proposed VIA

In order to fully characterize the impedance behavior of the proposed Variable Impedance Actuator (VIA), it is essential to capture the viscoelastic response of the Polycaprolactone (PCL) material that forms the thermo-active element of the design. The viscoelasticity of PCL is described in terms of its storage moduli, (tensile) and (shear), as well as its loss moduli, (tensile) and (shear), which are strongly dependent on temperature T. These parameters respectively represent the elastic energy stored and the viscous energy dissipated during mechanical loading. Fig. 4 illustrates the temperature-dependent behavior of the storage and loss moduli of PCL. A marked decrease is observed in both tensile storage and loss moduli as the material is heated beyond . At lower temperatures, below this threshold, the storage modulus is greater than the loss modulus ( and ), which indicates that the material behaves predominantly as an elastic solid. As the temperature increases, particularly above , the situation reverses, with the viscous contribution becoming dominant (). In this regime, PCL softens considerably and transitions into a more viscous-like material rather than retaining its initial elasticity. This strong temperature dependence of mechanical behavior provides a natural mechanism for implementing tunable impedance within the actuator design.

Fig. 4.

Storage and Loss (tensile and shear) Moduli of PCL at different temperatures.

The integration of this material into the actuator requires an appropriate viscoelastic model to represent its dynamic response. In our proposed VIA, the radial compression springs surrounding the spiked wheel are modeled as fixed elastic springs, while the PCL is treated as a variable viscoelastic element, consisting of a spring in parallel with a dashpot. The theoretical frameworks most commonly used for modeling such viscoelastic systems are the Maxwell model⁴¹ and the Kelvin-Voigt model⁴². Both frameworks represent the elastic component with a spring and the viscous component with a dashpot, but the difference lies in the way stress and strain are distributed between these elements.

In the Maxwell model, the spring and dashpot are connected in series, such that both experience the same stress while each undergoes independent strains. This allows the Maxwell model to effectively capture stress relaxation and recovery behavior, making it particularly well-suited for analyzing materials during unloading or after the removal of an applied load. On the other hand, the Kelvin-Voigt model places the spring and dashpot in parallel, where both elements experience the same strain while sustaining independent stresses. This configuration is more effective in describing creep phenomena, in which a material continues to deform gradually under constant stress, and it also captures the strong time-dependent strain response of viscoelastic materials. For engineering applications where elasticity remains a dominant feature and where time-dependent deformation under load is a primary consideration, the Kelvin-Voigt model has been widely adopted⁴³.

Given the requirements of our VIA, we adopt the Kelvin-Voigt representation for modeling PCL. The justification lies in the fact that the compression springs mounted radially around the wheel provide a strong elastic backbone to the system, while the temperature-sensitive PCL introduces a variable degree of viscous damping. When the system is operated at lower temperatures, the stiffness of PCL, in conjunction with the compression springs, ensures elastic-dominated performance, while at elevated temperatures, the viscous effect introduced by the dashpot element in the Kelvin-Voigt model more accurately captures the increased damping behavior. Thus, the model provides a robust framework for describing the smooth transition between elastic and viscous regimes observed in the material as the operating temperature varies.

The schematic of the complete VIA model is shown in Fig. 5. The DC motor drives the output link, which is mechanically coupled through the thermo-active impedance module. This module consists of the viscoelastic PCL element represented by a Kelvin-Voigt model, in parallel with the radially arranged compression springs, thereby forming the core of the variable impedance mechanism.

Fig. 5.

Schematic model of the proposed variable impedance module, located in series between the motor and the output link and powered through the flexible heaters.

System description and assumptions

The overall system can be conceptualized as a motor actuating an output link through a parallel combination of a viscoelastic material and a linear spring. The viscoelastic element is formed by the Polycaprolactone (PCL), whose storage modulus and loss modulus are explicitly temperature-dependent and are tuned by means of integrated heating devices. When the output link rotates, the spiked wheel compresses the springs and simultaneously deforms the PCL, thereby generating a combined elastic and viscous response. The elastic contribution arises from both the fixed compression springs and the temperature-dependent stiffness of PCL, while the viscous component originates exclusively from the internal damping properties of PCL.

The spring stiffness is denoted as , and the viscoelastic element is characterized geometrically by its cross-sectional area A and initial thickness . The motor and the output link are described by their respective masses and moments of inertia, denoted as and for the motor, and and for the output link. The trajectories of the motor and the output link are expressed as and , with their time derivatives representing angular velocities and accelerations. These quantities form the basis for evaluating the system’s dynamic response.

The complete set of physical parameters for the system is given in Table 1, which includes material properties, geometric dimensions, and thermal dependencies. The experimental data presented in Fig. 4 provide the functional relationship between temperature and the viscoelastic moduli of PCL, which are incorporated directly into the model. By minimizing external sources of damping, such as friction at the bearings on both the motor and the link side, we isolate the intrinsic damping of PCL, ensuring that the observed impedance modulation arises solely from the thermo-active behavior of the material.

Table 1.

System Parameters.

Parameter	Value
Individual spring stiffness	1.16 N/mm
Total spring stiffness	13.92 N/mm
Distance of springs from center axis	22.20mm
Average distance of PCL from center axis	20.75mm
Length of output link	140mm
Moment of inertia of output link
VIA weight	0.86kg
Weight attached to output link	0.91kg
Maximum angular deflection
Amount of PCL in each cavity	1.10g
Volume of PCL in each cavity	8800

The ultimate objective of this modeling framework is to predict and analyze the system’s impedance as a function of both frequency and temperature. By applying harmonic excitations to the motor input and recording the resulting trajectories of the output link, the system response can be evaluated across a wide range of operating conditions. This analysis allows us to demonstrate the tunability of the actuator impedance and to validate the proposed VIA design as a feasible solution for applications requiring adaptable compliance and damping.

Viscoelastic properties of PCL

The PCL element’s viscoelastic behavior is described using the temperature-dependent storage and loss moduli, and , which are functions of temperature . The effective dynamic modulus of PCL, , is expressed as:

1

The dynamic stiffness of the PCL element, , is related to as:

2

The real component, , represents the elastic stiffness, while the imaginary component, , represents damping.

Equation of motion

The motor applies a torque to drive the output link, and the viscoelastic-spring assembly transmits this torque while opposing motion through damping and stiffness. The total torque acting on the output link is given by:

3

where is the moment of inertia of the output link, is the viscous damping coefficient, is the angular displacement, and and are the angular acceleration and velocity, respectively.

The torque transmitted through the viscoelastic element and spring is a combination of elastic and damping forces:

4

where is the relative angular velocity, and and denote the real and imaginary parts of , respectively.

The torque balance for the motor is:

5

where and are the moment of inertia and viscous damping coefficient of the motor, respectively.

Combining Eq. 3, Eq. 4, and Eq. 5, the coupled equations of motion for the system are:

6

Frequency and temperature dependency

To investigate the frequency and temperature dependency, the motor is commanded to follow a sinusoidal trajectory:

7

where is the amplitude and is the angular frequency of the motor’s trajectory.

The output link trajectory, , is obtained by solving the coupled equations of motion (Eq. 6) using the temperature-dependent moduli and to determine the corresponding dynamic stiffness and damping. we know that the output link trajectory follows:

8

where is the phase delay. The phase delay and magnitude of the output link are key indicators of the system’s dynamic response.

Decay and phase delay

The Decay is defined as the difference between the motor trajectory amplitude and the output link trajectory amplitude :

9

To express , we note that the system’s response is influenced by the total effective stiffness and damping :

10

11

The amplitude ratio between the link and motor is determined from the frequency response of the system. The governing equation for is:

12

Substituting into the Decay equation:

13

and the phase delay is:

14

The Decay and phase delay depends the following:

• Frequency (): Higher frequencies increase the inertial term , leading to greater Decay and larger phase delays.

• Temperature (): Changes in and directly affect and , modifying and , influencing both the phase delay and Decay.

• Material Properties: The storage and loss moduli and dictate the viscoelastic response.

For a perfectly rigid system, and , leading to .

The comparison between and at various temperatures and frequencies provides insights into the viscoelastic behavior of PCL, the role of temperature modulation, and the system’s dynamic performance.

Estimating effective stiffness and damping by analyzing the actuator’s performance

To characterize the dynamic behavior of the system, it is essential to quantify the effective stiffness and damping , which collectively describe how the actuator-link system resists motion and dissipates energy. These parameters are not intrinsic material constants alone, but rather encapsulate the combined effects of the viscoelastic properties of PCL, the mechanical compliance of the spring assembly, and the influence of the motor-link inertia.

The estimation of and is performed by analyzing the output link trajectory in response to a harmonic motor excitation. The measured Decay reflects the relative amplitude reduction between the motor input and the link response, while the phase delay captures the temporal lag caused by damping and viscoelastic effects. Both quantities are frequency- and temperature-dependent, and their careful measurement allows the identification of the system’s effective dynamic parameters.

In practice, and as will be seen in the experimental section, and are extracted from the steady–state portions of the recorded trajectories in Fig. 6 (0.5 and 1.0 Hz). The normalized attenuation curves in Fig. 7 correspond to the amplitude ratio used in (12), providing the direct experimental inputs for estimating at each temperature.

Fig. 6.

Following a sinusoidal trajectory for the output link at different temperatures versus expected trajectory.

Fig. 7.

Decay at room (), intermediate (), and high () temperatures, normalized to room temperature between 0.5Hz (left) and 1.0Hz (right) trajectories.

The effective stiffness can be interpreted as the static-equivalent spring constant that would produce the same amplitude ratio between input and output under harmonic excitation. By using the experimentally measured amplitude ratio from Eq. 12, is determined iteratively through the relationship:

15

Similarly, the effective damping quantifies the rate at which energy is dissipated in the system. The measured phase lag contains information about both the viscous and viscoelastic contributions to damping, allowing to be extracted using:

16

Because the stiffness and damping are interdependent in the equations above, an iterative computation is generally required. Starting with an initial estimate for one parameter, the other is calculated, and the process is repeated until convergence is achieved. This iterative approach ensures that both parameters consistently satisfy the experimentally observed amplitude ratio and phase delay.

The resulting values of and provide direct insight into the system’s dynamic characteristics. High values of indicate a relatively rigid system with minimal deformation under applied torque, while higher values reflect greater energy dissipation and stronger attenuation of oscillatory behavior. These effective parameters also enable the prediction of system response to arbitrary inputs, the design of control strategies, and the assessment of performance under varying environmental conditions such as changes in temperature that influence the viscoelastic properties of PCL.

Therefore, analyzing and as functions of excitation frequency and temperature reveals trends that are critical for understanding the viscoelastic behavior. For example, stiffness typically decreases and damping increases with temperature, reflecting the softening and increased energy dissipation of the PCL. Similarly, at higher frequencies, inertial effects become more pronounced, modifying the apparent effective stiffness and phase lag. These analyses provide a comprehensive framework to evaluate and predict the actuator’s dynamic performance in real-world conditions.

Thermal response time estimation

:

To evaluate the characteristic thermal response of the PCL medium, the energy required to raise its temperature from ambient conditions to the softening regime was estimated from material properties and heater power. Each cavity contains approximately 1.10 g of PCL, yielding a total polymer mass of about 3.3 g across the three cavities. Using a specific heat capacity of and a temperature increase from to (), the required sensible heat is approximately Including the latent heat of fusion (), the total energy input may reach if partial melting occurs. Given the effective power of the embedded flexible heaters (5–15 W delivered to the polymer), the corresponding heating time is on the order of tens of seconds to a few minutes, depending on boundary heat losses and heater contact. This theoretical estimate agrees with observed heating behavior during experiments. Such response times indicate that stiffness modulation can be achieved in quasi-static or low-frequency tasks, whereas real-time high-frequency impedance control would require enhanced thermal management.

It should be noted that the present formulation employs a linear Kelvin–Voigt approximation, which is valid within the moderate temperature range where deformation remains small and PCL behaves quasi-linearly. Near its melting point (around ), however, the polymer exhibits pronounced nonlinear viscoelastic and rate-dependent behavior. Future work will incorporate more advanced constitutive frameworks–such as generalized Maxwell models with Prony-series representation–to capture these nonlinear effects and temperature–frequency coupling.

Experimental results

Performance of our proposed VIA

In order to examine the capability of our proposed VIA in regulating the overall impedance of the output link, the motor was set to follow harmonic trajectories, as mentioned before. The actual position of the output link was then measured using the dedicated encoder as shown in Fig. 1. This experiment was done at different temperatures and frequencies. The PCL temperature was set at different levels from room temperature () to using the flexible heaters. The goal was to show how the temperature can affect the time-dependent impedance of PCL.

The motor shaft was programmed to execute a sinusoidal trajectory with an amplitude of and two distinct frequencies: , where was set to 0.5 Hz and 1 Hz. The temperature of the PCL material was controlled using flexible heaters to evaluate its viscoelastic behavior at varying thermal conditions. A rotary encoder (model E40S6-2500-3-T-24 from Autonics) with a resolution of 2500 pulses per revolution was employed to precisely measure the trajectory of the output link.

The compliant nature of the output link, characterized by its combined elastic and viscous behavior, inherently causes Decay and phase delay relative to the motor’s input trajectory. This Decay increases proportionally with the compliance of the output link, which is influenced by both the material properties of the PCL and the temperature. Furthermore, as the system can be modeled as a second-order mass-spring-damper system, a phase delay between the motor trajectory (input) and the output link trajectory (output) is expected. Both Decay and phase delay are highly dependent on the frequency of the motor’s sinusoidal excitation and the temperature-dependent viscoelastic properties of the PCL.

Figure 6 illustrates the output link’s trajectories under two temperature conditions, (room temperature) and , at the two specified frequencies of 0.5 Hz and 1 Hz. Initially, the output link exhibited irregular transient behavior as it transitioned from its stationary state to a dynamic steady-state response. In steady-state, the output link followed a sinusoidal trajectory with similar frequency to the motor input but displayed varying magnitudes and phase delays.

Across all experiments, Decay was observed in the output link’s trajectory, with its magnitude directly correlated to the temperature of the PCL. At higher temperatures, such as , a 30% increase in Decay was noted compared to room temperature (). This phenomenon is attributed to the reduction in the viscoelasticity of PCL at elevated temperatures, which decreases the material’s resistance to deformation. Despite these significant changes in Decay, the phase delay between the motor and the output link remained minimal across the temperature range.

However, it is important to note that the small phase delay does not imply that the effective damping, , is zero. In fact, the effective stiffness, , is considerably larger than the inertia of the output link multiplied by the square of the angular frequency, i.e., . As mentioned earlier, the excitation frequencies, ranging from 0.5 Hz to 1 Hz, are well below the natural frequency of the link, meaning that the system operates in a regime where effective damping still plays a crucial role in the dynamic behavior.

By analyzing the data extracted from the experiments, including the calculation of Decay and phase delay, and utilizing the methodology described in Sect. "Estimating effective stiffness and damping by analyzing the actuator’s performance", the effective damping and stiffness of the system have been estimated. The elastic stiffness, , is derived based on the Young’s modulus , which is temperature-dependent, the cross-sectional area , and the initial length . On the other hand, the effective damping, , is determined from the temperature-dependent damping modulus , the same cross-sectional area, and the initial length of the material.

The resultant storage modulus (related to the system’s stiffness) and loss modulus (related to the damping behavior) are then compared with experimental data, as shown in Fig. 4, at various temperatures as summarized in Table 2.

Table 2.

Estimated vs. Experimentally Obtained Values of Storage and Loss Moduli for Polycaprolactone.

T [° C]	[kPa]	[kPa]	[kPa]	[kPa]
22	1000+/−24	985.34+/−21	1000+/−20	834.45+/−12
40	0.9+/−0.02	0.92+/−0.03	1800+/−103	1820+/−25
50	0.25+/−0.03	0.4+/−0.02	1+/−0.02	1.23+/−0.04
60	0.02+/−0.001	0.18+/−0.02	0.1+/−0.02	0.12+/−0.01

Figure 7 further quantifies the relationship between Decay and temperature by normalizing Decay values to those measured at . The data reveal that at the lower frequency of 0.5 Hz, the effect of temperature on the overall impedance of the actuator is particularly pronounced. At , the Decay was nearly five times greater than at room temperature, demonstrating the system’s sensitivity to thermal modulation. At the higher frequency of 1 Hz, the temperature effect was less pronounced but still observable, indicating that the system’s impedance regulation capability extends to higher operating frequencies. These results highlight the actuator’s ability to modulate its impedance by adjusting the temperature of the PCL material (The datasets generated and/or analysed during the current study are available in https://github.com/ARMLAB2025/PCL2).

Energy efficiency and weight comparison with existing VIAs

In order to better contextualize the performance of the proposed thermally-tunable VIA using Polycaprolactone (PCL), we compare its mass, energy demands, and efficiency with several representative variable impedance/variable stiffness actuators (VIAs/VSA) from the literature, including AwAS, AwAS-II, MACCEPA, and VSA-II. Table 3 summarizes the key numbers; following that we discuss implications for energy efficiency.

Table 3.

Comparison of representative VIAs and the proposed PCL-VIA in terms of weight, stiffness mechanism, and energy efficiency.

Actuator	Mass (kg)	Energy (J)	Description
AwAS5	1.8	350	Lever-arm VIA with pre-compressed spring.
AwAS-II15	1.1	580	Rotary-spring system with improved energy storage.
MACCEPA 2.02	2.0	300	Variable lever geometry with antagonistic springs.
VSA-II4	2	700	Agonistic-Antagonistic design of springs and actuators
Proposed PCL-VIA	0.86	250–500	Thermal softening of PCL via integrated heaters.

The proposed PCL-VIA has a weight of approximately 0.86 kg (actuator body including PCL polymer and heater components). By contrast, AwAS weighs 1.8 kg, AwAS-II 1.1 kg, and MACCEPA-based designs often weigh in the range of 1.5–2.5 kg depending on configuration and torque capacity. Thus, our device is 20–50% lighter than many existing VIAs with similar torque output or stiffness modulation capability. The reduced mass is advantageous in terms of reduced inertia, better responsiveness, and lower power required for moving the actuator as part of a robotic system.

Energy Efficiency - Energy for Modulation: The proposed PCL-VIA requires only modest amounts of thermal energy for stiffness modulation. Based on the mass of PCL () and specific heat (), raising temperature from 22 °C to 60 °C ( K) requires on the order of 250 J sensible heat. If melting or phase transitions occur, extra latent heat (70 Jg^–1) may be needed, increasing the total to . Heating times at realistic heater power ( net to the polymer) are thus on the order of 25–50 s. Because stiffness changes are intermittent, the average power for many tasks is low.

Comparison with Mechanically Tuned VIAs: VIAs such as AwAS, AwAS-II, and MACCEPA modulate stiffness via mechanical means (lever arms, spring pre-compression, spring recruitment). The energy cost in such systems mainly comes from the motors adjusting the mechanism, possible friction, actuator inefficiencies, and losses in gearing. Some works (e.g., in MACCEPA prostheses) show that inclusion of compliance and mechanical energy storage can reduce motor power draw during cyclic or gait tasks significantly.

Trade-offs and Implications While the proposed PCL-VIA offers advantages in reduced weight and low average energy required for stiffness changes, there are trade-offs:

Speed of modulation is slower due to thermal time constants; mechanical systems (AwAS, MACCEPA etc.) generally adjust stiffness faster, especially for rapid or continuous modulation.

Duty cycle/frequency of stiffness changes matters: if frequent, high- cycling occurs, thermal inefficiencies and heat losses may dominate. Model validity and mechanical efficiency: Mechanical systems often have more consistent behavior (linear or designed stiffness curves), whereas thermal systems may show non-linear, hysteretic behavior, and thermal losses (radiation, conduction) reduce net efficiency.

Therefore, in tasks or applications where stiffness changes are needed relatively infrequently, at low frequency, and where weight/inertia is a critical concern, the proposed PCL-VIA provides a compelling trade-off: significantly reduced mass and modest energy cost for useful impedance modulation. For applications requiring rapid stiffness switching or high-frequency dynamic adjustments, mechanical VIAs may still have the upper hand, or hybrid approaches (mechanical + thermal) could be promising.

Discussion and conclusion

This study presents a novel methodology to estimate the stiffness and damping of a Variable Impedance Actuator (VIA) that regulates its effective impedance by controlling the temperature of a thermoresponsive polymer, Polycaprolactone (PCL). The actuator employs embedded flexible Peltier elements to modulate the temperature of the PCL, which in turn alters its viscoelastic properties. By analyzing experimental data, including trajectory measurements and phase delay calculations, the proposed methodology provides an effective means of characterizing the actuator’s stiffness and damping across varying thermal and frequency conditions.

The experimental results demonstrate that the effective stiffness and damping of the VIA are highly temperature-dependent (Fig. 4). As the temperature increases, the stiffness decreases due to the reduction in Young’s modulus , and the damping decreases as a result of reduced viscoelasticity . These changes directly influence the actuator’s dynamic response, with higher temperatures leading to increased Decay and reduced resistance to deformation. The observed Decay, particularly pronounced at lower frequencies (e.g., 0.5 Hz), underscores the material’s sensitivity to thermal modulation and highlights the importance of thermal control in achieving desired impedance characteristics.

There is a discrepancy in Table 2 between the estimated and experimental E₁ values at (0.02 vs. 0.18). The discrepancy likely arises from three main factors:

1. The estimation algorithm becomes numerically ill-conditioned at very low stiffness, making it highly sensitive to small measurement noise in phase and amplitude.

2. Near , PCL approaches its melting transition and exhibits nonlinear viscoelastic behavior not captured by the linear Kelvin–Voigt model (Fig. 5).

3. Residual temperature non-uniformity or geometry change in the softened polymer alters the effective cross-section used in modulus conversion.

Despite significant changes in Decay (Fig. 7), the phase delay between the motor input and the output link remained minimal across the tested temperature range. This phenomenon can be attributed to the relatively low excitation frequencies used in the experiments, which are well below the natural frequency of the system (Fig. 6). As a result, the output link maintained near-synchronous motion with the motor input. However, the small phase delay does not imply an absence of effective damping. The results indicate that the effective stiffness, , is significantly larger than the inertia term , suggesting that damping still plays a crucial role in the system’s overall dynamic behavior. Also, it should be mentioned that the impedance modulation was validated only for low-frequency motions (0.5–1 Hz), and that extension to higher frequencies will be the subject of future work.

The achievable impedance modulation rate in the present actuator is fundamentally constrained by the thermal dynamics of the PCL medium. Based on the measured and estimated heating rates (5–15 W heater input, 250–500 J energy demand), the time required to reach the target temperature range () lies within 30–90 s. Although adequate for low-frequency or quasi-static modulation, this timescale limits rapid real-time stiffness switching. To improve responsiveness, future designs may incorporate higher watt-density heaters, optimized thermal paths, or active cooling strategies (for instance, thermoelectric or forced-air cooling). These improvements would enable closed-loop impedance control at higher frequencies while preserving the compact, lightweight advantages of the thermally tunable concept.

The proposed estimation methodology effectively captures the actuator’s viscoelastic behavior, as validated by the close agreement between experimentally obtained and estimated storage and loss moduli. This capability is crucial for applications where precise impedance regulation is required, particularly in physical human-robot interaction (pHRI) and soft robotics. However, the frequency-dependence of the system’s response presents a challenge for achieving consistent impedance control solely through temperature modulation. This limitation points to the need for supplementary control strategies, such as feedback mechanisms, to enhance the system’s adaptability in dynamic environments. Also, the PCL polymer remained mechanically stable over the experimental cycles conducted, but systematic thermal and mechanical fatigue tests will be performed in future work to quantify lifetime and possible aging effects.

In conclusion, while the presented actuator demonstrates a proof-of-concept for thermally tunable impedance control. Further optimization, long-term testing, and faster thermal management will be required before deployment in practical robotic systems. The findings of this study highlight the potential of thermoresponsive polymers like PCL for adaptive impedance control in robotic systems. The proposed estimation framework provides a reliable approach for characterizing stiffness and damping without requiring internal temperature sensors, which simplifies the actuator design. The demonstrated ability to modulate impedance through temperature control, combined with the high accuracy of the estimation method, positions this VIA as a promising candidate for applications requiring dynamic and tunable mechanical properties. Future work will focus on integrating advanced control strategies and expanding the operational frequency range to further enhance the system’s performance and applicability.

Author contributions

T.E. did the design and experiments, D. J. did the experiments and prepared the figures, T.E., D.J., and A.J. wrote the manuscript text.

Funding

This work was funded by National Science Foundation NSF under Grant Number 2213263, and by the National Institute of Health (NIH) through grant T32GM136501.

Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its supplementary information files. Other data are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-30236-6.