Pointing mechanism is widely used in aerospace field,and its pointing accuracy and stability have high requirements.The pointing mechanism will be affected by external interference when it works.In order to eliminate ...Pointing mechanism is widely used in aerospace field,and its pointing accuracy and stability have high requirements.The pointing mechanism will be affected by external interference when it works.In order to eliminate the impact of interference forces on the output accuracy of the mechanism,firstly,this paper proposes a design method for highprecision pointing mechanisms based on interference separation,aiming at the high-precision pointing requirements of pointing mechanisms.Based on the screw theory,a synthesis method for inner compensation mechanisms has been proposed.And a new type of double-layer parallel mechanism has been designed to compensate for interference forces.Then,the kinematics and dynamics of the mechanism are carried out.An evaluation index for compensating external interference forces is proposed.The interference compensation analysis is conducted for the pointing mechanism.The correctness of the proposed interference force compensation coefficient is verified.Finally,in order to find the optimal solution for the workspace and interference force compensation coefficient of the pointing mechanism,multi-objective optimization design of the structural parameters of the mechanism was carried out based on the particle swarm optimization algorithm.This provides a theoretical basis for the prototype design of the subsequent double-layer parallel mechanism.This double-layer parallel mechanism combines the advantages of large load-bearing capacity,large workspace,and high output accuracy.It can be better applied in the aerospace field where high-precision pointing and force interference compensation are integrated.展开更多
It is significant to develop a robot hand with high rigidity by a 6-DOF parallel manipulator（PM）.However,the existing6-DOF PMs include spherical joint which has less capability of pulling force bearing,less rotation...It is significant to develop a robot hand with high rigidity by a 6-DOF parallel manipulator（PM）.However,the existing6-DOF PMs include spherical joint which has less capability of pulling force bearing,less rotation range and lower precision under alternately heavy loads.A novel 6-DOF PM with three planar limbs and equipped with three fingers is proposed and its kinematics and statics are analyzed systematically.A 3-dimension simulation mechanism of the proposed manipulator is constructed and its structure characteristics is analyzed.The kinematics formulae for solving the displacement,velocity,acceleration of the platform,the active legs and the fingers are established.The statics formulae are derived for solving the active forces of PM and the finger mechanisms.An analytic example is given for solving the kinematics and statics of proposed manipulator and the analytic solved results are verified by the simulation mechanism.It is proved from the error analysis of analytic solutions and simulation solutions that the derived analytic formulae are correct and provide the theoretical and technical foundations for its manufacturing,control and application.展开更多
In order to solve the problem of weak stifness of the existing fully decoupled parallel mechanism, a new synthesis method of fully decoupled three translational (3T) parallel mechanisms (PMs) with closed-loop units an...In order to solve the problem of weak stifness of the existing fully decoupled parallel mechanism, a new synthesis method of fully decoupled three translational (3T) parallel mechanisms (PMs) with closed-loop units and high stifness is proposed based on screw theory. Firstly, a new criterion for the full decoupled of PMs is presented that the reciprocal product of the transmission wrench screw matrix and the output twist screw matrix of PMs is a diagonal matrix, and all elements on the main diagonal are nonzero constants. The forms of the transmission wrench screws are determined by the criterion. Secondly, the forms of the actuated and unactuated screws can be obtained according to their relationships with the transmission wrench screws. The basic decoupled limbs are generated by combination of the above actuated and unactuated screws. Finally, a closed-loop units construction method is investigated to apply the decoupled mechanisms in a better way on the high stifness occasion. The closed-loop units are constructed in the basic decoupled limbs to generate a high-stifness fully decoupled 3T PM. Kinematic and stifness analyses show that the Jacobian matrix is a diagonal matrix, and the stifness is obviously higher than that of the coupling mechanisms, which verifes the correctness of the proposed synthesis method. The mechanism synthesized by this method has a good application prospect in vehicle durability test platform.展开更多
The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process.However,the traditional representations cannot achieve this because of the absence of coordinate invar...The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process.However,the traditional representations cannot achieve this because of the absence of coordinate invariance.Therefore,the coordinate invariant method is an important research issue.First,the rigid-body acceleration,the time derivative of the twist,is proved to be a screw,and its physical meaning is explained.Based on the twist and the rigid-body acceleration,the acceleration of the end-effector is expressed as a linear-bilinear form,and the kinematics Hessian matrix of the manipulator（represented by Lie bracket）is deduced.Further,Newton-Euler＇s equation is rewritten as a linear-bilinear form,from which the dynamics Hessian matrix of a rigid body is obtained.The formulae and the dynamics Hessian matrix are proved to be coordinate invariant.Referring to the principle of virtual work,the dynamics Hessian matrix of the parallel manipulator is gotten and the detailed dynamic model is derived.An index of dynamical coupling based on dynamics Hessian matrix is presented.In the end,a foldable parallel manipulator is taken as an example to validate the deduced kinematics and dynamics formulae.The screw theory based method can simplify the kinematics and dynamics of a manipulator,also the corresponding dynamics Hessian matrix can be used to evaluate the dynamical coupling of a manipulator.展开更多
The determination of virtual constraints is always one of the key and difficult problems in traditional mobility calculation. To make mobility calculation simple, considering avoiding virtual constraints, some new for...The determination of virtual constraints is always one of the key and difficult problems in traditional mobility calculation. To make mobility calculation simple, considering avoiding virtual constraints, some new formulae have been presented, however these formulae can hardly intuitively reflect general link group＇s restrictions on output member and its influences on independence of output parameters, which is premise to the judgment of the properties of mobility. Towards the problem to reveal the intrinsic relationship between the degree of freedom（DOF） of a mechanism, the link group, and the dimension of output parameters, also to avoid determination of virtual constraint, based on the new concepts of the ＂DOF of general link group＂ and ＂node parameters＂, a new formula in the calculation of the mobility of mechanisms is presented that is expressed with DOFs of the general link groups and rank of motion parameters of base point of the output link. It is named GOM（mobility of groups and output parameter） formula. On the basis of new concepts of＂effective parameters＂ and ＂invalid parameters＂, a rule is put forward for solving the DOF of mechanisms with invalid parameters by GOM formula, that is, the base point parameters are the subset of effective parameters of link group. Thereafter, several examples are enumerated and the results coincide with the prototype data, which proves the validity of the proposed formula. Meanwhile, it is obtained that the necessary and sufficient condition for the judgment of output parameters independence is that each of the DOF of the link group is not less than zero. The proposed formula which is simple in calculation provides theoretical basis for the judgment of independence of output parameters and provides references for type synthesis of novel parallel mechanisms with independence requirements of their output parameters.展开更多
A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grübler-Kutzbach criterion for mobility has be...A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grübler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grübler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grübler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. Firstly, the mobility research background and the Modified Grübler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grübler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grübler-Kutzbach criterion is demonstrated based on screw theory. The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grübler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grübler-Kutzbach criterion.展开更多
Although it is common to eliminate the singularity of parallel mechanism by adding the branched chain with actuation redundancy, there is no theory and method for the configuration synthesis of the branched chain with...Although it is common to eliminate the singularity of parallel mechanism by adding the branched chain with actuation redundancy, there is no theory and method for the configuration synthesis of the branched chain with actuation redundancy in parallel mechanism. Branched chains with actuation redundancy are synthesized for eliminating interior singularity of 3-translational and 1-rotational（3T1R） parallel mechanisms. Guided by the discriminance method of hybrid screw group according to Grassmann line geometry, all the possibilities are listed for the occurrence of interior singularities in 3T1R parallel mechanism. Based on the linear relevance of screw system and the principles of eliminating parallel mechanism singularity with actuation redundancy, different types of branched chains with actuation redundancy are synthesized systematically to indicate the layout and the number of the branched chainsinterior with actuation redundancy. A general method is proposed for the configuration synthesis of the branched chains with actuation redundancy of the redundant parallel mechanism, and it builds a solid foundation for the subsequent performance optimization of the redundant actuation parallel mechanism.展开更多
The lower-mobility parallel mechanism has been widely used in the engineering field due to its numerous excellent characteristics.However,little work has been devoted to the actuator selection and placement that best ...The lower-mobility parallel mechanism has been widely used in the engineering field due to its numerous excellent characteristics.However,little work has been devoted to the actuator selection and placement that best satisfy the system's functional requirements during concept design.In this study,a unified approach for synthesizing the actuation spaces of both rigid and flexure parallel mechanisms has been presented,and all possible combinations of inputs could be obtained,laying a theoretical foundation for the subsequent optimization of inputs.According to the linear independence of actuation space and constraint space of the lower-mobility parallel mechanism,a general expression of actuation spaces in the format of screw systems is deduced,a unified synthesis process for the lower-mobility parallel mechanism is derived,and the efficiency of the method is validated with two selective examples based on screw theory.This study presents a theoretical framework for the input selection problems of parallel mechanisms,aiming to help designers select and place actuators in a correct and even optimal way after the configuration design.展开更多
Current research of automatic transmission(AT)mainly focuses on the improvement of driving performance,and configuration innovation is one of the main research directions.However,finding new configurations of ATs is o...Current research of automatic transmission(AT)mainly focuses on the improvement of driving performance,and configuration innovation is one of the main research directions.However,finding new configurations of ATs is one of the main limitations of configuration innovation.In the present study,epicyclic gear trains(EGTs)are applied to investigate mechanisms of 9-speed ATs.Then four kinematic configurations are proposed for automatic transitions.In order to evaluate the performance of proposed mechanisms,the lever analogy method is applied to conduct kinematic and mechanical analyses.The power flow analysis is conducted,and then transmission efficiencies are calculated based on the torque method.The comparative analysis between the proposed and existing mechanisms is carried out where obtained results show that proposed mechanisms have reasonable performance and can be used in ATs.The prototype of an AT is manufactured and the speed test is conducted,which proves the accuracy of analysis and the feasibility of proposed mechanisms.展开更多
In today’s society,parallel manipulators(PMs)are widely used in industrial production,aerospace,and other fields,where their forward kinematic analyses often serve as the foundation for various tasks,such as design,c...In today’s society,parallel manipulators(PMs)are widely used in industrial production,aerospace,and other fields,where their forward kinematic analyses often serve as the foundation for various tasks,such as design,calibration,and control.In the past few decades,this issue has seemingly been repeatedly addressed using various numerical methods,intelligent algorithms,and algebraic tools.While it is undeniable that solving the equations is easier with current technology,the problem of“how to formulate solvable equations”is often overlooked.This analysis issue typically involves establishing non-linear,multi-parameter,high-dimensional,and strong-coupled mathematical equations,which,from a geometric perspective,is also considered a process of solving a spatial polyhedron.When considering the temporal dimension of motion between two isomorphic polytopes,based on calculus theory,it has been found that this non-linear problem can be transformed into the superposition of multiple iteratively linear equations.Consequently,we propose an original method for the forward kinematic analysis of PMs,namely the finite-step-integration(FSI)method.In this study,the mathematical principles and modeling methods of the FSI method are elucidated,and the modeling and programming processes of the FSI method are demonstrated using general 6-UPS and 3-UPS/S manipulators as examples.Through the analysis of its unique algebraic structure,the methods for singularity determination and motion tracking characteristic analysis are investigated.This method addresses the long-standing challenges in the forward kinematic modeling of PMs,which is applicable for design,calibration,and control,while also offering novel insights for modeling and solving certain non-linear engineering problems.展开更多
基金Supported by the National Natural Science Foundation of China(52275032)the Natural Science Foundation of Hebei Province(E2022203077)+1 种基金the Hebei Provincial Science and Technology Plan(22371801D)the Hebei Provincial Science and Technology Research and Development Program-Central Guidance for Local Science and Technology Development Fund(246Z1818G).
文摘Pointing mechanism is widely used in aerospace field,and its pointing accuracy and stability have high requirements.The pointing mechanism will be affected by external interference when it works.In order to eliminate the impact of interference forces on the output accuracy of the mechanism,firstly,this paper proposes a design method for highprecision pointing mechanisms based on interference separation,aiming at the high-precision pointing requirements of pointing mechanisms.Based on the screw theory,a synthesis method for inner compensation mechanisms has been proposed.And a new type of double-layer parallel mechanism has been designed to compensate for interference forces.Then,the kinematics and dynamics of the mechanism are carried out.An evaluation index for compensating external interference forces is proposed.The interference compensation analysis is conducted for the pointing mechanism.The correctness of the proposed interference force compensation coefficient is verified.Finally,in order to find the optimal solution for the workspace and interference force compensation coefficient of the pointing mechanism,multi-objective optimization design of the structural parameters of the mechanism was carried out based on the particle swarm optimization algorithm.This provides a theoretical basis for the prototype design of the subsequent double-layer parallel mechanism.This double-layer parallel mechanism combines the advantages of large load-bearing capacity,large workspace,and high output accuracy.It can be better applied in the aerospace field where high-precision pointing and force interference compensation are integrated.
基金Supported by National Natural Science Foundation of China(Grant No.51175447)Key Planned Project of Hebei Province,China(Grant No.11962127D)
文摘It is significant to develop a robot hand with high rigidity by a 6-DOF parallel manipulator（PM）.However,the existing6-DOF PMs include spherical joint which has less capability of pulling force bearing,less rotation range and lower precision under alternately heavy loads.A novel 6-DOF PM with three planar limbs and equipped with three fingers is proposed and its kinematics and statics are analyzed systematically.A 3-dimension simulation mechanism of the proposed manipulator is constructed and its structure characteristics is analyzed.The kinematics formulae for solving the displacement,velocity,acceleration of the platform,the active legs and the fingers are established.The statics formulae are derived for solving the active forces of PM and the finger mechanisms.An analytic example is given for solving the kinematics and statics of proposed manipulator and the analytic solved results are verified by the simulation mechanism.It is proved from the error analysis of analytic solutions and simulation solutions that the derived analytic formulae are correct and provide the theoretical and technical foundations for its manufacturing,control and application.
基金Supported by National Natural Science Foundation of China(Grant No.52275032)Key Project of Hebei Provincial Natural Science Foundation of China(Grant No.E2022203077)Hebei Provincial Key Research and Development Plan of China(Grant No.202230808010057).
文摘In order to solve the problem of weak stifness of the existing fully decoupled parallel mechanism, a new synthesis method of fully decoupled three translational (3T) parallel mechanisms (PMs) with closed-loop units and high stifness is proposed based on screw theory. Firstly, a new criterion for the full decoupled of PMs is presented that the reciprocal product of the transmission wrench screw matrix and the output twist screw matrix of PMs is a diagonal matrix, and all elements on the main diagonal are nonzero constants. The forms of the transmission wrench screws are determined by the criterion. Secondly, the forms of the actuated and unactuated screws can be obtained according to their relationships with the transmission wrench screws. The basic decoupled limbs are generated by combination of the above actuated and unactuated screws. Finally, a closed-loop units construction method is investigated to apply the decoupled mechanisms in a better way on the high stifness occasion. The closed-loop units are constructed in the basic decoupled limbs to generate a high-stifness fully decoupled 3T PM. Kinematic and stifness analyses show that the Jacobian matrix is a diagonal matrix, and the stifness is obviously higher than that of the coupling mechanisms, which verifes the correctness of the proposed synthesis method. The mechanism synthesized by this method has a good application prospect in vehicle durability test platform.
基金Supported by National Natural Science Foundation of China(Grant Nos.51375420,51105322)
文摘The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process.However,the traditional representations cannot achieve this because of the absence of coordinate invariance.Therefore,the coordinate invariant method is an important research issue.First,the rigid-body acceleration,the time derivative of the twist,is proved to be a screw,and its physical meaning is explained.Based on the twist and the rigid-body acceleration,the acceleration of the end-effector is expressed as a linear-bilinear form,and the kinematics Hessian matrix of the manipulator（represented by Lie bracket）is deduced.Further,Newton-Euler＇s equation is rewritten as a linear-bilinear form,from which the dynamics Hessian matrix of a rigid body is obtained.The formulae and the dynamics Hessian matrix are proved to be coordinate invariant.Referring to the principle of virtual work,the dynamics Hessian matrix of the parallel manipulator is gotten and the detailed dynamic model is derived.An index of dynamical coupling based on dynamics Hessian matrix is presented.In the end,a foldable parallel manipulator is taken as an example to validate the deduced kinematics and dynamics formulae.The screw theory based method can simplify the kinematics and dynamics of a manipulator,also the corresponding dynamics Hessian matrix can be used to evaluate the dynamical coupling of a manipulator.
基金supported by National Natural Science Foundation of China(Grant Nos.51275438,51005195)Hebei Provincial Natural Science Foundation of(Grant No.E2011203214)Development Program of Qinhuangdao City,China(Grant No.201101A069)
文摘The determination of virtual constraints is always one of the key and difficult problems in traditional mobility calculation. To make mobility calculation simple, considering avoiding virtual constraints, some new formulae have been presented, however these formulae can hardly intuitively reflect general link group＇s restrictions on output member and its influences on independence of output parameters, which is premise to the judgment of the properties of mobility. Towards the problem to reveal the intrinsic relationship between the degree of freedom（DOF） of a mechanism, the link group, and the dimension of output parameters, also to avoid determination of virtual constraint, based on the new concepts of the ＂DOF of general link group＂ and ＂node parameters＂, a new formula in the calculation of the mobility of mechanisms is presented that is expressed with DOFs of the general link groups and rank of motion parameters of base point of the output link. It is named GOM（mobility of groups and output parameter） formula. On the basis of new concepts of＂effective parameters＂ and ＂invalid parameters＂, a rule is put forward for solving the DOF of mechanisms with invalid parameters by GOM formula, that is, the base point parameters are the subset of effective parameters of link group. Thereafter, several examples are enumerated and the results coincide with the prototype data, which proves the validity of the proposed formula. Meanwhile, it is obtained that the necessary and sufficient condition for the judgment of output parameters independence is that each of the DOF of the link group is not less than zero. The proposed formula which is simple in calculation provides theoretical basis for the judgment of independence of output parameters and provides references for type synthesis of novel parallel mechanisms with independence requirements of their output parameters.
基金supported by National Natural Science Foundation of China(Grant Nos. 51175446,50875227)the Science Supporting Plan of Education Department of Hebei Province,China(Grant No. 2008150)
文摘A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grübler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grübler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grübler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. Firstly, the mobility research background and the Modified Grübler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grübler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grübler-Kutzbach criterion is demonstrated based on screw theory. The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grübler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grübler-Kutzbach criterion.
基金Supported by Research Fund for the Doctoral Program of Higher Education,China(Grant No.20131333110008)
文摘Although it is common to eliminate the singularity of parallel mechanism by adding the branched chain with actuation redundancy, there is no theory and method for the configuration synthesis of the branched chain with actuation redundancy in parallel mechanism. Branched chains with actuation redundancy are synthesized for eliminating interior singularity of 3-translational and 1-rotational（3T1R） parallel mechanisms. Guided by the discriminance method of hybrid screw group according to Grassmann line geometry, all the possibilities are listed for the occurrence of interior singularities in 3T1R parallel mechanism. Based on the linear relevance of screw system and the principles of eliminating parallel mechanism singularity with actuation redundancy, different types of branched chains with actuation redundancy are synthesized systematically to indicate the layout and the number of the branched chainsinterior with actuation redundancy. A general method is proposed for the configuration synthesis of the branched chains with actuation redundancy of the redundant parallel mechanism, and it builds a solid foundation for the subsequent performance optimization of the redundant actuation parallel mechanism.
基金Supported by National Natural Science Foundation of China(Grant No.51775475).
文摘The lower-mobility parallel mechanism has been widely used in the engineering field due to its numerous excellent characteristics.However,little work has been devoted to the actuator selection and placement that best satisfy the system's functional requirements during concept design.In this study,a unified approach for synthesizing the actuation spaces of both rigid and flexure parallel mechanisms has been presented,and all possible combinations of inputs could be obtained,laying a theoretical foundation for the subsequent optimization of inputs.According to the linear independence of actuation space and constraint space of the lower-mobility parallel mechanism,a general expression of actuation spaces in the format of screw systems is deduced,a unified synthesis process for the lower-mobility parallel mechanism is derived,and the efficiency of the method is validated with two selective examples based on screw theory.This study presents a theoretical framework for the input selection problems of parallel mechanisms,aiming to help designers select and place actuators in a correct and even optimal way after the configuration design.
基金Supported by National Natural Science Foundation of China(Grant Nos.51975544,51675495).
文摘Current research of automatic transmission(AT)mainly focuses on the improvement of driving performance,and configuration innovation is one of the main research directions.However,finding new configurations of ATs is one of the main limitations of configuration innovation.In the present study,epicyclic gear trains(EGTs)are applied to investigate mechanisms of 9-speed ATs.Then four kinematic configurations are proposed for automatic transitions.In order to evaluate the performance of proposed mechanisms,the lever analogy method is applied to conduct kinematic and mechanical analyses.The power flow analysis is conducted,and then transmission efficiencies are calculated based on the torque method.The comparative analysis between the proposed and existing mechanisms is carried out where obtained results show that proposed mechanisms have reasonable performance and can be used in ATs.The prototype of an AT is manufactured and the speed test is conducted,which proves the accuracy of analysis and the feasibility of proposed mechanisms.
基金supported by the National Natural Science Foundation of China(Grant Nos.52105035 and 62203094)the Special Central Funds for Guiding Local Scientific and Technological Development,China(Grant No.236Z1801G)+2 种基金the Higher Education Youth Top Talent Project of Hebei Province,China(Grant No.BJK2024042)the Natural Science Foundation of Hebei Province,China(Grant Nos.E2021203109 and F2023501021)the Graduate Student Innovation Capability Training and Support Project of Hebei Province,China(Grant No.CXZZBS2024053).
文摘In today’s society,parallel manipulators(PMs)are widely used in industrial production,aerospace,and other fields,where their forward kinematic analyses often serve as the foundation for various tasks,such as design,calibration,and control.In the past few decades,this issue has seemingly been repeatedly addressed using various numerical methods,intelligent algorithms,and algebraic tools.While it is undeniable that solving the equations is easier with current technology,the problem of“how to formulate solvable equations”is often overlooked.This analysis issue typically involves establishing non-linear,multi-parameter,high-dimensional,and strong-coupled mathematical equations,which,from a geometric perspective,is also considered a process of solving a spatial polyhedron.When considering the temporal dimension of motion between two isomorphic polytopes,based on calculus theory,it has been found that this non-linear problem can be transformed into the superposition of multiple iteratively linear equations.Consequently,we propose an original method for the forward kinematic analysis of PMs,namely the finite-step-integration(FSI)method.In this study,the mathematical principles and modeling methods of the FSI method are elucidated,and the modeling and programming processes of the FSI method are demonstrated using general 6-UPS and 3-UPS/S manipulators as examples.Through the analysis of its unique algebraic structure,the methods for singularity determination and motion tracking characteristic analysis are investigated.This method addresses the long-standing challenges in the forward kinematic modeling of PMs,which is applicable for design,calibration,and control,while also offering novel insights for modeling and solving certain non-linear engineering problems.
