Research and Publications - Department of Mechanical Engineering

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(1984-12-10) Sandy, John H.
Kinetic Model of a Collisional Admixture in Dusty Gas and its Application to Calculating Flow Past Bodies
(2000) Volkov, Alexey N.; Tsirkunov, Yury M.; University of Alabama Tuscaloosa
Using the methods of statistical physics, the basic kinetic equation describing the dynamics of a polydisperse admixture of solid particles in a dilute dusty-gas flow is derived. Particle rotation, inelastic collisions, and interaction with the carrier gas are taken into account. The basic kinetic equation is used to obtain a Boltzmann-type equation for the one-particle distribution function, for which the boundary conditions for the problem of dusty-gas flow past a body are formulated. On the basis of the kinetic model developed, using direct statistical modeling, the flow patterns and the fields of the dispersed-phase macroparameters in a uniform crosswise dusty-gas flow past a cylinder are obtained for various free-stream particle sizes and concentrations.
Numerical Modelling of the Magnus Force and the Aerodynamic Torque on a Spinning Sphere in Transitional Flow
(2007) Volkov, Alexey N.; University of Alabama Tuscaloosa
Three dimensional transitional flow over a spinning sphere is studied numerically by the direct simulation Monte Carlo method. The flow is assumed to be steady-state, gas molecules interact with each other as hard spheres and the speculardiffuse scattering model describes the interaction between molecules and the sphere surface. The translational and rotational velocities of the sphere is assumed to be perpendicular to each other. The drag coefficient, the Magnus force coefficient and the torque coefficient are found as functions of the Mach and Reynolds numbers and the dimensionless rotation parameter for subsonic and supersonic flows. Computational results are compared with the analytical solution for a spinning sphere in free molecular flow and with available semi-empirical data. The "critical" Knudsen number when the Magnus force is equal to zero is found as a function of the Mach number.
3D Numerical Modelling of a Rarefied Gas Flow in the Nearby Atmosphere around a Rotating Cometary Nucleus
(2007) Volkov, Alexey N.; Lukyanov, German A.; University of Alabama Tuscaloosa
A combined 3D model of a nearby atmosphere around an arbitrary rotating spherical cometary nucleus is developed. The model includes a 3D unsteady model of solar radiation absorption and heating of the nucleus material (water ice), its evaporation and condensation and a 3D quasi-stationary kinetic model of flow inside the near nucleus coma. This model can be used to predict the coma flow in conditions typical for rendezvous projects such as ESA project Rossetta. Calculations are carried out with the help of this model to reveal the influence of nucleus rotation on its temperature field and the flow field in the nearby atmosphere. It was found that the nucleus rotation influences significantly the nucleus temperature field and the coma flow. Vapor flow around a rotating nucleus is essentially three dimensional and differs qualitatively from the coma flow around a non-rotating nucleus.
Fluid/Kinetic Hybrid Simulation of Atmospheric Escape: Pluto
(2011) Tucker, Orenthal J.; Erwin, Justin T.; Johnson, Robert E.; Volkov, Alexey N.; Cassidy, Timothy A.; University of Alabama Tuscaloosa
A hybrid fluid/molecular kinetic model was developed to describe the escape of molecules from the gravitational well of a planet's atmosphere. This model was applied to a one dimensional, radial description of molecular escape from the atmosphere of Pluto and compared to purely fluid dynamic simulations of escape for two solar heating cases. The hybrid simulations show that the atmospheric temperature vs. altitude and the escape rates can differ significantly from those obtained using only a fluid description of the atmosphere.
THERMALLY DRIVEN ATMOSPHERIC ESCAPE: TRANSITION FROM HYDRODYNAMIC TO JEANS ESCAPE
(IOP Publishing, 2011-02-16) Volkov, Alexey N.; Johnson, Robert E.; Tucker, Orenthal J.; Erwin, Justin T.; University of Virginia; New York University; University of Alabama Tuscaloosa
Thermally driven escape from planetary atmospheres changes in nature from an organized outflow (hydrodynamic escape) to escape on a molecule-by-molecule basis (Jeans escape) with increasing Jeans parameter, lambda, the ratio of the gravitational to thermal energy of the atmospheric molecules. This change is described here for the first time using the direct simulation Monte Carlo method. When heating is predominantly below the lower boundary of the simulation region, R-0, and well below the exobase of a single-component atmosphere, the nature of the escape process changes over a surprisingly narrow range of Jeans parameters, lambda(0), evaluated at R-0. For an atomic gas, the transition occurs over lambda(0) similar to 2-3, where the lower bound, lambda(0) similar to 2.1, corresponds to the upper limit for isentropic, supersonic outflow. For lambda(0) > 3 escape occurs on a molecule-by-molecule basis and we show that, contrary to earlier suggestions, for lambda(0) > similar to 6 the escape rate does not deviate significantly from the familiar Jeans rate. In a gas composed of diatomic molecules, the transition shifts to lambda(0) similar to 2.4-3.6 and at lambda(0) > similar to 4 the escape rate increases a few tens of percent over that for the monatomic gas. Scaling by the Jeans parameter and the Knudsen number, these results can be applied to thermally induced escape of the major species from solar and extrasolar planets.
Kinetic simulations of thermal escape from a single component atmosphere
(American Institute of Physics, 2011-06-06) Volkov, Alexey N.; Tucker, Orenthal J.; Erwin, Justin T.; Johnson, Robert E.; University of Virginia; University of Alabama Tuscaloosa
The one-dimensional steady-state expansion of a monatomic gas from a spherical source in a gravity field is studied by the direct simulation Monte Carlo method. Collisions between molecules are described by the hard sphere model, the distribution of gas molecules leaving the source surface is assumed to be Maxwellian, and no heat is directly deposited in the simulation region. The flow structure and the escape rate (number flux of molecules escaping the atmosphere) are analyzed for the source Jeans parameter lambda(0) (ratio of the gravitational energy to thermal energy of the molecules) and Knudsen number Kn(0) (ratio of the mean free path to the source radius) ranging from 0 to 15 and from 0.0001 to infinity, respectively. In the collisionless regime, flows are analyzed for lambda(0)=0-100 and analytical equations are obtained for asymptotic values of gas parameters that are found to be non-monotonic functions of lambda(0). For collisional flows, simulations predict the transition in the nature of atmospheric loss from escape on a molecule-by-molecules basis, often referred to as Jeans escape, to an organized outflow, often referred to as hydrodynamic escape. It is found that the structure of the flow and the escape rate exhibit drastic changes when lambda(0) varies over a narrow transition range 2-3. The lower limit of this range approximately corresponds to a critical Jeans parameter equal to 2.06, which is the upper limit for isentropic, supersonic outflow of a monatomic gas from a body in a gravity field. Subcritical, lambda(0)<= 2, flows are qualitatively similar to free outgassing in the absence of gravity, resulting in hypersonic terminal Mach numbers and escape rates that are independent of lambda(0) in the limit of small Knudsen numbers. Supercritical, lambda(0)>= 3, flows are controlled by thermal conduction and demonstrate qualitatively different trends. The ratio of the actual escape rate to the Jeans escape rate at the source surface is found to be a non-monotonic function of Kn(0) spanning the range from similar to 0.01 to similar to 2. At lambda(0)>= 6, the ratio of the actual escape rate to the Jeans escape rate at the exobase is found to be similar to 1.4-1.7. This is unlike the predictions of the slow hydrodynamic escape model, which is based on Parker's model for the solar wind and intended for the description of the atmospheric loss at lambda(0)>similar to 10. At lambda(0) < 6, the actual escape rate can be well approximated by a modified Jeans escape rate, which accounts for non-zero gas velocity. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3592253]
Effect of bending buckling of carbon nanotubes on thermal conductivity of carbon nanotube materials
(American Institute of Physics, 2012-03-01) Volkov, Alexey N.; Shiga, Takuma; Nicholson, David; Shiomi, Junichiro; Zhigilei, Leonid V.; University of Tokyo; University of Virginia; University of Alabama Tuscaloosa
The effect of bending buckling of carbon nanotubes (CNTs) on thermal conductivity of CNT materials is investigated in atomistic and mesoscopic simulations. Nonequilibrium molecular dynamics simulations of the thermal conductance through an individual buckling kink in a (10,10) single-walled CNT reveal a strong dependence (close to inverse proportionality) of the thermal conductance of the buckling kink on the buckling angle. The value of the buckling kink conductance divided by the cross-sectional area of the CNT ranges from 40 to 10 GWm(-2) K-1 as the buckling angle changes from 20 to 110 degrees. The predictions of the atomistic simulations are used for parameterization of a mesoscopic model that enables calculations of thermal conductivity of films composed of thousands of CNTs arranged into continuous networks of bundles. The results of mesoscopic simulations demonstrate that the conductivity of CNT films is sensitive to the angular dependence of the buckling kink conductance and the length of the individual CNTs. For a film composed of 1 mu m-long CNTs, the values of the in-plane film conductivity predicted with a constant conductance of 20 GWm(-2) K-1 and the angular-dependent conductance obtained in atomistic simulations are about 40 and 20% lower than the conductivity predicted for the same film with zero thermal resistance of the buckling kinks, respectively. The weaker impact of the angular-dependent buckling kink conductance on the effective conductivity of the film is explained by the presence of a large fraction of kinks that have small buckling angles and correspondingly large values of conductance. The results of the simulations suggest that the finite conductance of the buckling kinks has a moderate, but non-negligible, effect on thermal conductivity of materials composed of short CNTs with length up to 1 mu m. The contribution of the buckling kink thermal resistance becomes stronger for materials composed of longer CNTs and/or characterized by higher density of buckling kinks. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3687943]
Heat conduction in carbon nanotube materials: Strong effect of intrinsic thermal conductivity of carbon nanotubes
(American Institute of Physics, 2012-07-24) Volkov, Alexey N.; Zhigilei, Leonid V.; University of Virginia; University of Alabama Tuscaloosa
Computational study of thermal conductivity of interconnected networks of bundles in carbon nanotube (CNT) films reveals a strong effect of the finite thermal conductivity k(T) of individual nanotubes on the conductivity k of the CNT materials. The physical origin of this effect is explained in a theoretical analysis of systems composed of straight randomly dispersed CNTs. An analytical equation for quantitative description of the effect of finite k(T) on the value of k is obtained and adopted for continuous networks of bundles characteristic of CNT films and buckypaper. Contrary to the common assumption of the dominant effect of the contact conductance, the contribution of the finite k(T) is found to control the value of k at material densities and CNT lengths typical for real materials. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737903]
Modelling and experimental study of fatigue of powder metal steel (FC-0205)
(Taylor & Francis, 2013) Allison, P. G.; Hammi, Y.; Jordon, J. B.; Horstemeyer, M. F.; United States Department of Defense; United States Army; U.S. Army Corps of Engineers; U.S. Army Engineer Research & Development Center (ERDC); Mississippi State University; University of Alabama Tuscaloosa
The microstructure sensitive multistage fatigue model captured the fatigue life of a powder metal FC-0205 steel alloy. Uniaxial strain controlled fatigue data and microstructure information from sets of high and low porosity specimens calibrated the model. Strain-life behaviour depicted that above the plastic strain limit of 0.002 mm mm(-1) in the low cycle fatigue regime, where ubiquitous plasticity occurred, the different porosity levels gave distinct, visibly different results. However, specimens tested below the plastic limit in the high cycle fatigue regime, where failure was dominated by local cyclic microplasticity, showed unclear fatigue lives at different porosity levels. Fractography using scanning electron microscopy showed no clear presence of striations; however, asserted striations in powder metal specimens were similar to geometrical features observed on fracture surfaces of monotonically loaded specimens. The experimental and microstructure data calibrated a fatigue model that allowed for satisfactory prediction of the varying porosity specimen strain-life curves.
THERMAL ESCAPE IN THE HYDRODYNAMIC REGIME: RECONSIDERATION OF PARKER's ISENTROPIC THEORY BASED ON RESULTS OF KINETIC SIMULATIONS
(IOP Publishing, 2013-03-10) Volkov, Alexey N.; Johnson, Robert E.; University of Virginia; New York University; University of Alabama Tuscaloosa
The one-dimensional steady-state problem of thermal escape from a single-component atmosphere of mon- and diatomic gases is studied in the hydrodynamic (blow-off) regime using the direct simulation Monte Carlo method for an evaporative-type condition at the lower boundary. The simulations are performed for various depths into an atmosphere, indicated by a Knudsen number, Kn(0), equal to the ratio of the mean free path of molecules to the radial position of the source surface, ranging from 10 to 10(-5), and for the range of the source Jeans parameter, lambda(0), equal to the ratio of gravitational and thermal energies, specific to blow-off. The results of kinetic simulations are compared with the isentropic model (IM) and the Navier-Stokes model. It is shown that the IM can be simplified if formulated in terms of the local Mach number and Jeans parameter. The simulations predict that at Kn(0) < similar to 10(-3) the flow includes a near-surface non-equilibrium Knudsen layer, a zone where the flow can be well approximated by the IM, and a rarefied far field. The corresponding IM solutions, however, only approach Parker's critical solution as lambda(0) approaches the upper limit for blow-off. The IM alone is not capable for predicting the flow and requires boundary conditions at the top of the Knudsen layer. For small Kn(0), the scaled escape rate and energy loss rate are found to be independent of lambda(0). The simulation results can be scaled to any single-component atmosphere exhibiting blow-off if the external heating above the lower boundary is negligible, in particular, to sublimation-driven atmospheres of Kuiper belt objects.
Atomistic simulations, mesoscopic modeling, and theoretical analysis of thermal conductivity of bundles composed of carbon nanotubes
(American Institute of Physics, 2013-09-10) Volkov, Alexey N.; Salaway, Richard N.; Zhigilei, Leonid V.; University of Virginia; University of Alabama Tuscaloosa
The propensity of carbon nanotubes (CNTs) to self-organize into continuous networks of bundles has direct implications for thermal transport properties of CNT network materials and defines the importance of clear understanding of the mechanisms and scaling laws governing the heat transfer within the primary building blocks of the network structures-close-packed bundles of CNTs. A comprehensive study of the thermal conductivity of CNT bundles is performed with a combination of non-equilibrium molecular dynamics (MD) simulations of heat transfer between adjacent CNTs and the intrinsic conductivity of CNTs in a bundle with a theoretical analysis that reveals the connections between the structure and thermal transport properties of CNT bundles. The results of MD simulations of heat transfer in CNT bundles consisting of up to 7 CNTs suggest that, contrary to the widespread notion of strongly reduced conductivity of CNTs in bundles, van der Waals interactions between defect-free well-aligned CNTs in a bundle have negligible effect on the intrinsic conductivity of the CNTs. The simulations of inter-tube heat conduction performed for partially overlapping parallel CNTs indicate that the conductance through the overlap region is proportional to the length of the overlap for CNTs and CNT-CNT overlaps longer than several tens of nm. Based on the predictions of the MD simulations, a mesoscopic-level model is developed and applied for theoretical analysis and numerical modeling of heat transfer in bundles consisting of CNTs with infinitely large and finite intrinsic thermal conductivities. The general scaling laws predicting the quadratic dependence of the bundle conductivity on the length of individual CNTs in the case when the thermal transport is controlled by the inter-tube conductance and the independence of the CNT length in another limiting case when the intrinsic conductivity of CNTs plays the dominant role are derived. An application of the scaling laws to bundles of single-walled (10,10) CNTs reveals that the transition from inter-tube-conductance-dominated to intrinsic-conductivity-dominated thermal transport in CNT bundles occurs in a practically important range of CNT length from similar to 20 nm to similar to 4 mu m. (C) 2013 AIP Publishing LLC.
MOLECULAR-KINETIC SIMULATIONS OF ESCAPE FROM THE EX-PLANET AND EXOPLANETS: CRITERION FOR TRANSONIC FLOW
(IOP Publishing, 2013-12-06) Johnson, Robert E.; Volkov, Alexey N.; Erwin, Justin T.; University of Virginia; New York University; University of Alabama Tuscaloosa
The equations of gas dynamics are extensively used to describe atmospheric loss from solar system bodies and exoplanets even though the boundary conditions at infinity are not uniquely defined. Using molecular-kinetic simulations that correctly treat the transition from the continuum to the rarefied region, we confirm that the energy-limited escape approximation is valid when adiabatic expansion is the dominant cooling process. However, this does not imply that the outflow goes sonic. Rather large escape rates and concomitant adiabatic cooling can produce atmospheres with subsonic flow that are highly extended. Since this affects the heating rate of the upper atmosphere and the interaction with external fields and plasmas, we give a criterion for estimating when the outflow goes transonic in the continuum region. This is applied to early terrestrial atmospheres, exoplanet atmospheres, and the atmosphere of the ex-planet, Pluto, all of which have large escape rates.
Characterization Of Multi-layered Fish Scales (Atractosteus spatula) Using Nanoindentation, X-ray CT, FTIR, and SEM
(MyJove Corporation, 2014) Allison, Paul G.; Rodriguez, Rogie I.; Moser, Robert D.; Williams, Brett A.; Poda, Aimee R.; Seiter, Jennifer M.; Lafferty, Brandon J.; Kennedy, Alan J.; Chandler, Mei Q.; United States Department of Defense; United States Army; U.S. Army Corps of Engineers; U.S. Army Engineer Research & Development Center (ERDC); University of Alabama Tuscaloosa
The hierarchical architecture of protective biological materials such as mineralized fish scales, gastropod shells, ram's horn, antlers, and turtle shells provides unique design principles with potentials for guiding the design of protective materials and systems in the future. Understanding the structure-property relationships for these material systems at the microscale and nanoscale where failure initiates is essential. Currently, experimental techniques such as nanoindentation, X-ray CT, and SEM provide researchers with a way to correlate the mechanical behavior with hierarchical microstructures of these material systems. However, a well-defined standard procedure for specimen preparation of mineralized biomaterials is not currently available. In this study, the methods for probing spatially correlated chemical, structural, and mechanical properties of the multilayered scale of A. spatula using nanoindentation, FTIR, SEM, with energy-dispersive X-ray (EDX) microanalysis, and X-ray CT are presented.
A Study of Flexible Energy-Saving Joint for Biped Robots Considering Sagittal Plane Motion
(Springer International Publishing Switzerland, 2015-08) Zhang, Qiang; Teng, Lin; Wang, Yang; Xie, Tao; Xiao, Xiaohui
A flexible ankle joint for biped walking robots is proposed to investigate the influence of joint stiffness on motor’s peak torque and energy consumption of the sagittal plane motion during the single support phase. Firstly, an improved model of the inverted pendulum is established, which is the theoretical foundation of the flexible ankle joint. Then the analysis of the analytic method of flexible joint is presented based on the improved model of the inverted pendulum. Finally, dynamic simulations of the flexible joint are performed to examine the correctness of analytic method. The results show that the flexible joint can reduce the joint motor’s peak torque and energy consumption. Furthermore, there is an optimal joint stiffness of the flexible system, which can minimum peak torque with reduction of 45.99% and energy consumption with reduction of 51.65%.
VOLATILE LOSS AND CLASSIFICATION OF KUIPER BELT OBJECTS
(IOP Publishing, 2015-08-10) Johnson, R. E.; Oza, A.; Young, L. A.; Volkov, A. N.; Schmidt, C.; University of Virginia; New York University; Centre National de la Recherche Scientifique (CNRS); UDICE-French Research Universities; Sorbonne Universite; Universite Paris Cite; Universite Paris Saclay; Southwest Research Institute; University of Alabama Tuscaloosa
Observations indicate that some of the largest Kuiper Belt Objects (KBOs) have retained volatiles in the gas phase (e.g., Pluto), while others have surface volatiles that might support a seasonal atmosphere (e.g., Eris). Since the presence of an atmosphere can affect their reflectance spectra and thermal balance, Schaller & Brown examined the role of volatile escape driven by solar heating of the surface. Guided by recent simulations, we estimate the loss of primordial N-2 for several large KBOs, accounting for escape driven by UV/EUV heating of the upper atmosphere as well as by solar heating of the surface. For the latter we present new simulations and for the former we scale recent detailed simulations of escape from Pluto using the energy limited escape model validated recently by molecular kinetic simulations. Unlike what has been assumed to date, we show that unless the N-2 atmosphere is thin (< similar to 10(18) N-2 cm(-2)) and/or the radius small (< similar to 200-300 km), escape is primarily driven by the UV/EUV radiation absorbed in the upper atmosphere. This affects the discussion of the relationship between atmospheric loss and the observed surface properties for a number of the KBOs examined. Our long-term goal is to connect detailed atmospheric loss simulations with a model for volatile transport for individual KBOs.
A CRITERION FOR THE VALIDITY OF PARKER'S MODEL IN THERMAL ESCAPE PROBLEMS FOR PLANETARY ATMOSPHERES
(IOP Publishing, 2015-10-01) Volkov, A. N.; University of Alabama Tuscaloosa
Mass escape rate of mon- and diatomic gases from a planetary atmosphere is studied based on Parker's model for a broad range of surface conditions. The escape rate is found to follow two asymptotic regimes, namely, high-and low-density regimes, with a short intermediate regime between them. Equations for the escape rate in every asymptotic regime are found theoretically. A comparison of the obtained escape rates with results of recent kinetic simulations shows that Parker's model satisfactorily predicts escape rates only in the high-density regime. Based on this finding, a criterion of applicability of Parker's model for the calculation of the mass escape rate is established.
Compliant Joint for Bipod Robot Considering Energy Consumption Optimization
(Springer, 2015-11) Zhang, Qiang; Xiao, Xiaohui; Wang, Yang; You, Penghui; Xie, Tao
Abstract: In order to optimize the energy consumption of biped robot while walking, the compliant joint for biped walking robots is proposed to investigate the influence of ankle joint and knee joint stiffness on motor torque and energy consumption of the sagittal plane motion during the single support phase. Firstly, an improved model of the five-link biped robot is established, which is the theoretical foundation of the compliant joint. Then, with the method of gait planning based on natural Zero Moment Point (ZMP) trajectory, the robot’s center of mass (COM) track is obtained by setting reference of ZMP trajectory and the gait on a rigid path is acquired by interpolation. Finally, both the Lagrange equations analytic method and dynamic simulations are performed to analyze the influences of compliant joint stiffness on motor torque and energy consumption based on the improved model of the five-link biped robot. The results show that the compliant joint can reduce the joint motor torque and energy consumption effectively. Furthermore, there is an optimal stiffness of the compliant ankle joint and knee joint respectively, which can minimum the motor energy consumption with reduction of 89.87% and 90.11% in analytic method, as well as 88.66% and 81.23% in dynamic simulations.
HVAC control loop performance assessment: A critical review (1587-RP)
(Taylor & Francis, 2016) O'Neill, Zheng; Li, Yanfei; Williams, Keith; University of Alabama Tuscaloosa
This article presents a comprehensive review of control loop performance assessments in the context of building HVAC controls. Few studies are available for assessing HVAC control loop performance using a single control quality factor. A control quality factor should be an objective and quantitative metric with simple-to-interpret criteria and should only use data available from the actual control system, such as the control output. The authors systematically reviewed 34 indices and the associated methods of evaluating control loop performance and cataloged the drawbacks and merits of the different indices. Most of these performance assessment indices are currently used in process control industry applications. There were 14 of the 34 indices selected for further review, due to their particular suitability for implementation in HVAC control loop performance assessment. Finally, the selected 14 indices are implemented for assessments of three regulatory control loops with proportional-integral controllers: a heating coil outlet air temperature control loop and variable air volume room air temperature control loop using simulated data from a dynamic Modelica model, and variable air volume room air temperature control loop in a heating mode from real field data. Based on the review and preliminary results, the Normalized Harris Index and exponentially weighted moving averages based index are proposed as potential candidates for control quality factor, and further investigation of the use of them in HVAC control loop performance assessment is recommended.
Power Efficiency-Based Stiffness Optimization of a Compliant Actuator for Underactuated Bipedal Robot
(Springer International Publishing Switzerland, 2016-08-03) Zhang, Qiang; Xiao, Xiaohui; Guo, Zhao
Introducing compliant actuation to robotic joints can obtain better disturbance rejection performance and higher power efficiency than conventional stiff actuated systems. In this paper, inspired by human joints, a novel compliant actuator applied to underactuated bipedal robot is proposed. After modeling the stiffness of the compliant actuator, this paper gives the configuration of the bipedal robot actuated by compliant actuators. Compared with the elastic structure of MABEL, the compliant element of our robot is simplified. Based on the dynamics of the compliant actuator-driven bipedal robot, a feedback linearization controller is presented to implement position control of the compliant actuator for power efficiency analysis and stiffness optimization. Co-simulations of MATLAB and ADAMS are performed under the defined control trajectory by altering actuator stiffness. The simulation results indicate that, compared with the actuator maintaining very high stiffness like a rigid actuator, the power efficiency of the compliant actuator is improved, and the stiffness optimized to 375 N•m/rad can reach the highest power efficiency.

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