The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-di...The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-dimensional Hamiltonian operator with simple supported rectangular plate is proved to be invertible. Furthermore, by a certain compactness, we find that the spectrum of this operator consists only of isolated eigenvalues with finite geometric multiplicity, which will play a significant role in finding the analytical and numerical solution based on Hamiltonian system for a class of plate bending equations.展开更多
To implement restoration in a single motion blurred image,the PSF(Point Spread Function)is difficult to estimate and the image deconvolution is ill-posed as a result that a good recovery effect cannot be obtained.Cons...To implement restoration in a single motion blurred image,the PSF(Point Spread Function)is difficult to estimate and the image deconvolution is ill-posed as a result that a good recovery effect cannot be obtained.Considering that several different PSFs can get joint invertibility to make restoration wellposed,we proposed a motion-blurred image restoration method based on joint invertibility of PSFs by means of computational photography.Firstly,we designed a set of observation device which composed by multiple cameras with the same parameters to shoot the moving target in the same field of view continuously to obtain the target images with the same background.The target images have the same brightness,but different exposure time and different motion blur length.It is easy to estimate the blur PSFs of the target images make use of the sequence frames obtained by one camera.According to the motion blur superposition feature of the target and its background,the complete blurred target images can be extracted from the observed images respectively.Finally,for the same target images with different PSFs,the iterative restoration is solved by joint solution of multiple images in spatial domain.The experiments showed that the moving target observation device designed by this method had lower requirements on hardware conditions,and the observed images are more convenient to use joint-PSF solution for image restoration,and the restoration results maintained details well and had lower signal noise ratio(SNR).展开更多
From the point of mathematical mapping, this paper discusses the invertibility of nonlinear systems, and presents definition, equivalence theorem, sufficient and necessary conditions of strong invertibility for single...From the point of mathematical mapping, this paper discusses the invertibility of nonlinear systems, and presents definition, equivalence theorem, sufficient and necessary conditions of strong invertibility for single input single output systems. Then it analyzes the feasibility of inverse system control method, and yields some conclusions on feasibility.展开更多
In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we prop...In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.展开更多
Let 3V denote the set of all possible transfer function matrices of weakly invertible linear finite automata over a given finite field F. A classification and an enumeration on the infinite set are given.
We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various ...We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrum-preserving elementary operators. Several open problems are also mentioned.展开更多
In this paper,the so-called invertibility is introduced for rational univariate representations,and a characterization of the invertibility is given.It is shown that the rational univariate representations,obtained by...In this paper,the so-called invertibility is introduced for rational univariate representations,and a characterization of the invertibility is given.It is shown that the rational univariate representations,obtained by both Rouillier’s approach and Wu’s method,are invertible.Moreover,the ideal created by a given rational univariate representation is defined.Some results on invertible rational univariate representations and created ideals are established.Based on these results,a new approach is presented for decomposing the radical of a zero-dimensional polynomial ideal into an intersection of maximal ideals.展开更多
Ra, Rb transformations were successfully applied to establish invertibility theory for linear and quasi-linear finite automata over finite fields. In aprevious paper, the authors generalized R., Rb transformations to ...Ra, Rb transformations were successfully applied to establish invertibility theory for linear and quasi-linear finite automata over finite fields. In aprevious paper, the authors generalized R., Rb transformations to deal with nonlinear memory finite automata, and gave sufficient conditions for weak inverse andfor weakly invertible memory finite automata and inversion processes concerned;methods by transformation to generate a kind of nonlinear memory finite automatasatisfying one of these sufficient conditions were also given. This paper extends theconcepts, methods and results to general finite automata, in which states consist offinite input history, finite output history and finite 'inner state' history.展开更多
We characterize the additive singularity preserving almost surjective maps on Mn （F）, the algebra of all n×n matrices over a field F with char F=0. We also describe additive invertibility preserving surjective ...We characterize the additive singularity preserving almost surjective maps on Mn （F）, the algebra of all n×n matrices over a field F with char F=0. We also describe additive invertibility preserving surjective maps on Mn （F） and give examples showing that all the assunlptions in these two theorems are indispensable.展开更多
The weak invertibility problem of finite automata (FAs) has received continuous considerations. The weak invertibility theory has applications in cryptography, both conventional cryptosystems and public key cryptosyst...The weak invertibility problem of finite automata (FAs) has received continuous considerations. The weak invertibility theory has applications in cryptography, both conventional cryptosystems and public key cryptosystems. Given a finite commutative ring (?) with identity, it is known, that the weak invertibility of a linear finite automaton(LFA) over (?) depends only on its transfer function matrix. Let H be the set of all possible transfer function matrices which weakly invertible LFAs over (?) can have. Our展开更多
We investigate some probabilistic properties of a new class of nonlinear time series models. A sufficient condition for the existence of a unique causal, strictly and weakly stationary solution is derived. To understa...We investigate some probabilistic properties of a new class of nonlinear time series models. A sufficient condition for the existence of a unique causal, strictly and weakly stationary solution is derived. To understand the proposed models better, we further discuss the moment structure and obtain some Yule-Walker difference equations for the second and third order cumulants, which can also be used for identification purpose. A sufficient condition for invertibility is also provided.展开更多
Complementary inverter is the basic unit for logic circuits,but the inverters based on full oxide thin-film transistors(TFTs)are still very limited.The next challenge is to realize complementary inverters using homoge...Complementary inverter is the basic unit for logic circuits,but the inverters based on full oxide thin-film transistors(TFTs)are still very limited.The next challenge is to realize complementary inverters using homogeneous oxide semiconduc-tors.Herein,we propose the design of complementary inverter based on full ZnO TFTs.Li-N dual-doped ZnO(ZnO:(Li,N))acts as the p-type channel and Al-doped ZnO(ZnO:Al)serves as the n-type channel for fabrication of TFTs,and then the complemen-tary inverter is produced with p-and n-type ZnO TFTs.The homogeneous ZnO-based complementary inverter has typical volt-age transfer characteristics with the voltage gain of 13.34 at the supply voltage of 40 V.This work may open the door for the development of oxide complementary inverters for logic circuits.展开更多
With the development of high-frequency and highvoltagetraction machines(TM)incorporating hairpin windings(HW)and SiC inverters for electric vehicles(EV),both theinterturn voltage stress and temperature within HW are r...With the development of high-frequency and highvoltagetraction machines(TM)incorporating hairpin windings(HW)and SiC inverters for electric vehicles(EV),both theinterturn voltage stress and temperature within HW are rising,increasing the risk of partial discharge(PD),and presentingsignificant challenges to insulation safety.Therefore,this paperaddresses this issue and proposes potential solutions.Firstly,thepaper examines an 8-pole,48-slot,6-layer HW TM to highlightthe unique characteristics of this winding structure,and explainsthe uneven distribution of interturn voltage stress andtemperature.Subsequently,a high-frequency equivalent circuitmodel of the HW TM prototype is developed.The error ofsimulation and experiment is only 5.7%,which proves theaccuracy of the model.Then,an improved HW scheme isproposed to lower the maximum voltage stress by 29.3%.Furthermore,the temperature distribution of HW TM isanalyzed to facilitate a detailed examination of the impact oftemperature on insulation PD.Finally,the partial dischargeinception voltage(PDIV)of interturn insulation,consideringtemperature effects,is calculated and verified throughexperiment.The paper proposes a reliability-oriented designmethod and process for HW TM.It demonstrates that thereliability-oriented design can achieve PD-free performance inthe design stage of HW.展开更多
In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the...In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the Drazin spectrum and the Aluthge transform of an operator M ∈ B(H), respectively.展开更多
The technology of electric propulsion aircraft(EPA)represents an important direction and an advanced stage in the development of aviation electrification.It is a key pathway for green development in aviation industry ...The technology of electric propulsion aircraft(EPA)represents an important direction and an advanced stage in the development of aviation electrification.It is a key pathway for green development in aviation industry and can significantly enhance the energy efficiency of aircraft propulsion system.Electric motor is the most critical electromechanical energy conversion component in an aircraft electric propulsion system(EPS).High-performance electric motors,power electronic converters and EPS control form the foundation of the EPA.This paper provides an overview of the characteristics of electric motors for EPA,analyzes the inverter topologies of EPSs,and reviews ongoing EPA projects.The article highlights the latest advancements in three types of motors:superconducting motors(SCMs),permanent magnet synchronous motors(PMSMs),and induction motors(IMs).It summarizes the control system architectures of current EPA initiatives and,building on this foundation,proposes future research directions for EPSs.These include cutting-edge areas such as high-performance motors and advanced manufacturing technologies,Ga N-or Si C-based inverter integration and innovation,electric propulsion control systems,and optimization of wiring systems.展开更多
Uneven power distribution,transient voltage,and frequency deviations are observed in the photovoltaic storage hybrid inverter during the switching between grid-connected and island modes.In response to these issues,th...Uneven power distribution,transient voltage,and frequency deviations are observed in the photovoltaic storage hybrid inverter during the switching between grid-connected and island modes.In response to these issues,this paper proposes a grid-connected/island switching control strategy for photovoltaic storage hybrid inverters based on the modified chimpanzee optimization algorithm.The proposed strategy incorporates coupling compensation and power differentiation elements based on the traditional droop control.Then,it combines the angular frequency and voltage amplitude adjustments provided by the phase-locked loop-free pre-synchronization control strategy.Precise pre-synchronization is achieved by regulating the virtual current to zero and aligning the photovoltaic storage hybrid inverter with the grid voltage.Additionally,two novel operators,learning and emotional behaviors are introduced to enhance the optimization precision of the chimpanzee algorithm.These operators ensure high-precision and high-reliability optimization of the droop control parameters for photovoltaic storage hybrid inverters.A Simulink model was constructed for simulation analysis,which validated the optimized control strategy’s ability to evenly distribute power under load transients.This strategy effectively mitigated transient voltage and current surges during mode transitions.Consequently,seamless and efficient switching between gridconnected and island modes was achieved for the photovoltaic storage hybrid inverter.The enhanced energy utilization efficiency,in turn,offers robust technical support for grid stability.展开更多
This paper gives a necessary condition for a kind of weakly invertible, invertible, weak inverse or inverse finite automata by linear RaRb transformation sequence. For such finite automata the existence of terminating...This paper gives a necessary condition for a kind of weakly invertible, invertible, weak inverse or inverse finite automata by linear RaRb transformation sequence. For such finite automata the existence of terminating RaRb transformation sequence is also established.展开更多
Let p be an analytic polynomial on the unit disk.We obtain a necessary and sufficient condition for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when all coefficients of p are real numb...Let p be an analytic polynomial on the unit disk.We obtain a necessary and sufficient condition for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when all coefficients of p are real numbers.Furthermore,we establish several necessary and sufficient,easy-to-check conditions for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when some coefficients of p are complex numbers.展开更多
1Introduction To date,in model-based gait-planning methods,the dynamics of the center of mass(COM)of bipedal robots have been analyzed by establishing their linear inverted pendulum model(LIPM)or extended forms(Owaki ...1Introduction To date,in model-based gait-planning methods,the dynamics of the center of mass(COM)of bipedal robots have been analyzed by establishing their linear inverted pendulum model(LIPM)or extended forms(Owaki et al.,2010;Englsberger et al.,2015;Xie et al.,2020).With regard to model-based gait-generation methods for uphill and downhill terrain,Kuo(2007)simulated human gait using an inverted pendulum,which provided a circular trajectory for the COM rather than a horizontal trajectory.He found that a horizontal COM trajectory consumed more muscle energy.Massah et al.(2012)utilized a 3D LIPM and the concept of zero moment point(ZMP).They developed a trajectory planner using the semi-elliptical motion equations of an NAO humanoid robot and simulated walking on various sloped terrains using the Webots platform.展开更多
基金supported by National Natural Science Foundation of China under Grant No.10562002Natural Science Foundation of Inner Mongolia under Grant Nos.200508010103 and 200711020106
文摘The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-dimensional Hamiltonian operator with simple supported rectangular plate is proved to be invertible. Furthermore, by a certain compactness, we find that the spectrum of this operator consists only of isolated eigenvalues with finite geometric multiplicity, which will play a significant role in finding the analytical and numerical solution based on Hamiltonian system for a class of plate bending equations.
基金funding of Natural Science Foundation of Shandong Province(ZR2013F0025),www.sdnsf.gov.cn.
文摘To implement restoration in a single motion blurred image,the PSF(Point Spread Function)is difficult to estimate and the image deconvolution is ill-posed as a result that a good recovery effect cannot be obtained.Considering that several different PSFs can get joint invertibility to make restoration wellposed,we proposed a motion-blurred image restoration method based on joint invertibility of PSFs by means of computational photography.Firstly,we designed a set of observation device which composed by multiple cameras with the same parameters to shoot the moving target in the same field of view continuously to obtain the target images with the same background.The target images have the same brightness,but different exposure time and different motion blur length.It is easy to estimate the blur PSFs of the target images make use of the sequence frames obtained by one camera.According to the motion blur superposition feature of the target and its background,the complete blurred target images can be extracted from the observed images respectively.Finally,for the same target images with different PSFs,the iterative restoration is solved by joint solution of multiple images in spatial domain.The experiments showed that the moving target observation device designed by this method had lower requirements on hardware conditions,and the observed images are more convenient to use joint-PSF solution for image restoration,and the restoration results maintained details well and had lower signal noise ratio(SNR).
文摘From the point of mathematical mapping, this paper discusses the invertibility of nonlinear systems, and presents definition, equivalence theorem, sufficient and necessary conditions of strong invertibility for single input single output systems. Then it analyzes the feasibility of inverse system control method, and yields some conclusions on feasibility.
文摘In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.
基金supported by the National Natural Science Foundation of China.
文摘Let 3V denote the set of all possible transfer function matrices of weakly invertible linear finite automata over a given finite field F. A classification and an enumeration on the infinite set are given.
文摘We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrum-preserving elementary operators. Several open problems are also mentioned.
基金the National Natural Science Foundation of China under Grant No.12161057。
文摘In this paper,the so-called invertibility is introduced for rational univariate representations,and a characterization of the invertibility is given.It is shown that the rational univariate representations,obtained by both Rouillier’s approach and Wu’s method,are invertible.Moreover,the ideal created by a given rational univariate representation is defined.Some results on invertible rational univariate representations and created ideals are established.Based on these results,a new approach is presented for decomposing the radical of a zero-dimensional polynomial ideal into an intersection of maximal ideals.
文摘Ra, Rb transformations were successfully applied to establish invertibility theory for linear and quasi-linear finite automata over finite fields. In aprevious paper, the authors generalized R., Rb transformations to deal with nonlinear memory finite automata, and gave sufficient conditions for weak inverse andfor weakly invertible memory finite automata and inversion processes concerned;methods by transformation to generate a kind of nonlinear memory finite automatasatisfying one of these sufficient conditions were also given. This paper extends theconcepts, methods and results to general finite automata, in which states consist offinite input history, finite output history and finite 'inner state' history.
基金supported in part by a grant from the Ministry of Science of Slovenia
文摘We characterize the additive singularity preserving almost surjective maps on Mn （F）, the algebra of all n×n matrices over a field F with char F=0. We also describe additive invertibility preserving surjective maps on Mn （F） and give examples showing that all the assunlptions in these two theorems are indispensable.
文摘The weak invertibility problem of finite automata (FAs) has received continuous considerations. The weak invertibility theory has applications in cryptography, both conventional cryptosystems and public key cryptosystems. Given a finite commutative ring (?) with identity, it is known, that the weak invertibility of a linear finite automaton(LFA) over (?) depends only on its transfer function matrix. Let H be the set of all possible transfer function matrices which weakly invertible LFAs over (?) can have. Our
基金supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2010121005)supported by the Scientific Research and Development Funds for Youth of Fujian University of Technology of China (Grant No. GY-Z09081)
文摘We investigate some probabilistic properties of a new class of nonlinear time series models. A sufficient condition for the existence of a unique causal, strictly and weakly stationary solution is derived. To understand the proposed models better, we further discuss the moment structure and obtain some Yule-Walker difference equations for the second and third order cumulants, which can also be used for identification purpose. A sufficient condition for invertibility is also provided.
基金supported by Zhejiang Provincial Natural Science Foundation of China(No.LZ24E020001).
文摘Complementary inverter is the basic unit for logic circuits,but the inverters based on full oxide thin-film transistors(TFTs)are still very limited.The next challenge is to realize complementary inverters using homogeneous oxide semiconduc-tors.Herein,we propose the design of complementary inverter based on full ZnO TFTs.Li-N dual-doped ZnO(ZnO:(Li,N))acts as the p-type channel and Al-doped ZnO(ZnO:Al)serves as the n-type channel for fabrication of TFTs,and then the complemen-tary inverter is produced with p-and n-type ZnO TFTs.The homogeneous ZnO-based complementary inverter has typical volt-age transfer characteristics with the voltage gain of 13.34 at the supply voltage of 40 V.This work may open the door for the development of oxide complementary inverters for logic circuits.
基金supported by the Project of National Natural Science Foundation of China under Grant 52407060 and 52422704supported by Liaoning Province science and technology plan doctoral project under Grant 2023-BSBA-255.
文摘With the development of high-frequency and highvoltagetraction machines(TM)incorporating hairpin windings(HW)and SiC inverters for electric vehicles(EV),both theinterturn voltage stress and temperature within HW are rising,increasing the risk of partial discharge(PD),and presentingsignificant challenges to insulation safety.Therefore,this paperaddresses this issue and proposes potential solutions.Firstly,thepaper examines an 8-pole,48-slot,6-layer HW TM to highlightthe unique characteristics of this winding structure,and explainsthe uneven distribution of interturn voltage stress andtemperature.Subsequently,a high-frequency equivalent circuitmodel of the HW TM prototype is developed.The error ofsimulation and experiment is only 5.7%,which proves theaccuracy of the model.Then,an improved HW scheme isproposed to lower the maximum voltage stress by 29.3%.Furthermore,the temperature distribution of HW TM isanalyzed to facilitate a detailed examination of the impact oftemperature on insulation PD.Finally,the partial dischargeinception voltage(PDIV)of interturn insulation,consideringtemperature effects,is calculated and verified throughexperiment.The paper proposes a reliability-oriented designmethod and process for HW TM.It demonstrates that thereliability-oriented design can achieve PD-free performance inthe design stage of HW.
基金the National Natural Science Foundation of China (No.10571113)
文摘In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the Drazin spectrum and the Aluthge transform of an operator M ∈ B(H), respectively.
基金supported by the National Nature Science Foundation of China(Grant No.52302507)。
文摘The technology of electric propulsion aircraft(EPA)represents an important direction and an advanced stage in the development of aviation electrification.It is a key pathway for green development in aviation industry and can significantly enhance the energy efficiency of aircraft propulsion system.Electric motor is the most critical electromechanical energy conversion component in an aircraft electric propulsion system(EPS).High-performance electric motors,power electronic converters and EPS control form the foundation of the EPA.This paper provides an overview of the characteristics of electric motors for EPA,analyzes the inverter topologies of EPSs,and reviews ongoing EPA projects.The article highlights the latest advancements in three types of motors:superconducting motors(SCMs),permanent magnet synchronous motors(PMSMs),and induction motors(IMs).It summarizes the control system architectures of current EPA initiatives and,building on this foundation,proposes future research directions for EPSs.These include cutting-edge areas such as high-performance motors and advanced manufacturing technologies,Ga N-or Si C-based inverter integration and innovation,electric propulsion control systems,and optimization of wiring systems.
基金received funding from the Postgraduate Research&Practice Innovation Program of Jiangsu Province(SJCX23_1633)2023 University Student Innovation and Entrepreneurship Training Program(202311463009Z)+1 种基金Changzhou Science and Technology Support Project(CE20235045)Open Project of Jiangsu Key Laboratory of Power Transmission&Distribution Equipment Technology(2021JSSPD12).
文摘Uneven power distribution,transient voltage,and frequency deviations are observed in the photovoltaic storage hybrid inverter during the switching between grid-connected and island modes.In response to these issues,this paper proposes a grid-connected/island switching control strategy for photovoltaic storage hybrid inverters based on the modified chimpanzee optimization algorithm.The proposed strategy incorporates coupling compensation and power differentiation elements based on the traditional droop control.Then,it combines the angular frequency and voltage amplitude adjustments provided by the phase-locked loop-free pre-synchronization control strategy.Precise pre-synchronization is achieved by regulating the virtual current to zero and aligning the photovoltaic storage hybrid inverter with the grid voltage.Additionally,two novel operators,learning and emotional behaviors are introduced to enhance the optimization precision of the chimpanzee algorithm.These operators ensure high-precision and high-reliability optimization of the droop control parameters for photovoltaic storage hybrid inverters.A Simulink model was constructed for simulation analysis,which validated the optimized control strategy’s ability to evenly distribute power under load transients.This strategy effectively mitigated transient voltage and current surges during mode transitions.Consequently,seamless and efficient switching between gridconnected and island modes was achieved for the photovoltaic storage hybrid inverter.The enhanced energy utilization efficiency,in turn,offers robust technical support for grid stability.
基金Project supported by the National Natural Science Foundation of China.
文摘This paper gives a necessary condition for a kind of weakly invertible, invertible, weak inverse or inverse finite automata by linear RaRb transformation sequence. For such finite automata the existence of terminating RaRb transformation sequence is also established.
基金supported by the Yunnan Natural Science Foundation(Grant No.201601YA00004)supported by National Natural Science Foundation of China(Grant No.11701052)+1 种基金Chongqing Natural Science Foundation(Grant No.cstc2017jcyjAX0373)the Fundamental Research Funds for the Central Universities(Grant Nos.106112016CDJRC000080 and 106112017CDJXY100007)。
文摘Let p be an analytic polynomial on the unit disk.We obtain a necessary and sufficient condition for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when all coefficients of p are real numbers.Furthermore,we establish several necessary and sufficient,easy-to-check conditions for Toeplitz operators with the symbol z+p to be invertible on the Bergman space when some coefficients of p are complex numbers.
基金supported by the National Natural Science Foundation of China(No.12332023)the Zhejiang Provincial Natural Science Foundation of China(No.LY23E050010).
文摘1Introduction To date,in model-based gait-planning methods,the dynamics of the center of mass(COM)of bipedal robots have been analyzed by establishing their linear inverted pendulum model(LIPM)or extended forms(Owaki et al.,2010;Englsberger et al.,2015;Xie et al.,2020).With regard to model-based gait-generation methods for uphill and downhill terrain,Kuo(2007)simulated human gait using an inverted pendulum,which provided a circular trajectory for the COM rather than a horizontal trajectory.He found that a horizontal COM trajectory consumed more muscle energy.Massah et al.(2012)utilized a 3D LIPM and the concept of zero moment point(ZMP).They developed a trajectory planner using the semi-elliptical motion equations of an NAO humanoid robot and simulated walking on various sloped terrains using the Webots platform.
