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.55)The scalar à plays the role of a switch that enables/disables the parametricvariation.In several application (e.g.telemanipulation) when a certain setof physical parameters is reached, it has to be kept constant despite of anyinternal energy flows.Thus, when it is necessary to change the parameters, Ãis set equal to 1 and when a set of parameters has to e kept constant à is sett 0.Remark 5.21.Notice that when à = 0 the Dirac structures given in Eq.(5.54)and Eq.(5.53) are the same as the one given in Eq.(5.51) meaning that whenà = 0 the extended port-Hamiltonian system behaves as a fixed parametersport-Hamiltonian system.Remark 5.22.Some extra dissipating element can be added to dissipate energystored in the parametric part of the state.In fact if we connect to the para-metric power ports a dissipative element, the energy absorption leads to adecrease of the parametric part of the state.This strategy can be used todecrease parametric states independently of the energy flowing along the ex-tended Dirac structure. 194 5 Transparency in Port-Hamiltonian Based Telemanipulation5.5.2 Parameters Associated to Energy Dissipation andInterconnectionParameters relative to energy dissipation play the role of modulating energyabsorption.Changing values of these parameters is safe since different va-lues would imply only a different rate of absorption.There is only one thingto be aware of: as reported in Sec.1.4, the power flow towards dissipativeelements must always be positive and, therefore, we cannot take a set of pa-rameters which makes this power flow negative.If this happened dissipativeelements would be transformed in power injecting elements which, therefore,would destroy passivity of the port-Hamiltonian system.This constraint inthe choice of dissipative parameters is not very restrictive.In fact, in severalapplications (e.g.telemanipulation), port-Hamiltonian controllers are used toreproduce their physical equivalent and modulating physical parameter hasthe aim to obtain the same kind of behavior but characterized by differentphysical properties.Transforming, by a certain choice of parameters, dissipa-tion in energy production we obtain a totally different kind of behavior whichis always an undesired feature.Parameters relative to the interconnection modulation can be freely chan-ged since they modulate transformations which are lossless independently oftheir modulation constant.5.5.3 SimulationsThe aim of this section is to provide some simulations in order to validatethe obtained results.Consider the system shown in Fig.5.7, where the hu-man operator interacts with a device, modeled as a mass m, controlled by anport-Hamiltonian impedance controller equivalent to the mechanical parallelof a linear spring, characterized by a stiffness k, and of a damper, with dam-ping coefficient b.In this simplified scheme, we assume that the value of thedamping coefficient b is fixed, while the stiffness k can be adapted in order tochange the sensation rendered to the human operator.Suppose that we want to change the value of the stiffness of the impedancecontroller and to drive it to the value kref.The energy storage function of thecontroller is:1H(k, x) = kx2 (5.56)2where the rest length of the spring is supposed to be zero.The power portexpressing the elongation:"H, -‹ =(kx,-‹ ) =(ex, fx) (5.57)"xThe following parametric port describes the power flow related to the stiffnessvariation: 5.5 A Passivity Preserving Tuning of Port-Hamiltonian Systems 195krefxlx kMbIPCFig.5.7.A 1-DOF variable parameters port-Hamiltonian impedance controller"H x2Ù Ù, -k = , -k = ek, fk) (5.58)"k 2Notice that the effort ek is always positive and the sign of the power exchangedthrough the parametric port is determined only by the flow fk.Finally, theimpedance felt by the user is described by the external power port (e, f), andthe goal of the stiffness adaptation is to compute an external effort e as closeas possible to that generated by a real spring of stiffness kref:f = ‹ , e = krefx (5.59)We choose to deviate part of the energy that is flowing towards the elon-gation port to the parametric port in order to passively change the stiffnessof the spring; therefore, we extend the Dirac structure of the fixed parametersport-Hamiltonian system in the way reported in Eq.(5.53).This leads to:Ùf = ‹ and k = Ã(t)f (5.60)where Ã(t) is the modulation law that has to be computed in order to achievethe desired behavior on the external port.Notice that the stiffness variationÙk is proportional to the external flow f and it takes place only when Ã(t) =0.This choice satisfies the first equation of Eq.(5.59) and straightforwardcomputations show that the elongation perceived externally is equal to x.By expressing the Dirac structure in kernel form, we obtain the followingequations:ëø öø ëø öø ëø öø ëø öø101 -‹ 00 0 exÙíø01Ãøø íø-køø íø00 0øø íøekøø+ = 0 (5.61)000 f 1Ã-1 eFirst of all, consider the situation when the user is extracting power from theexternal port, namely P = e, f 0 and adiffeomorphismh : U × n ’! À-1(U)such that, for each m " U, the correspondence x ’! h(m, x) defines an iso-morphism between the vector space n and the vector space À-1(m).The pair(U, h) is called a local coordinate system for ¾.The vector space À-1(m) iscalled fiber over m for any m "M.For a very complete treatment of vector bundles see [204].A.2 TensorsTensors are the generalization of the concepts of vector and matrix and, infact, once a coordinate frame has been defined, they can be represented asmultidimensional matrices.For a complete treatment see, for example, [185].Definition A.19 (Multilinear map).Consider n+1 vector spaces V1,.,Vn, W.Amultilinear map is a map:L : V1 ×· · · ×Vn ’!W (A.13) A.3 Lie Groups and Rigid Motions 207such that, for each i =1,., n, the maps:Li(v1,., vi-1, vi+1,., vn) : Vi ’!W vi ’! L(v1,., vn) (A.14)are linear for each v1,., vi-1, vi+1,., vn.The set of multilinear operators defined on Vi,.,Vn, W is indicated asLn(V1,., Vn; W)and it can be proven that it is a vector space.Consider a vector space V and denote its dual space, namely the vector spaceformed by the linear operators on V, by V".Definition A.20 (Tensors).Given a vector space V, the set of tensors oforder p + q of type (p, q), namely p contravariant and q covariant, is definedby:Tqp(V) =Lp+q(V,., V, V",.V"; ) (A.15)q times p timesNotice that T01(V) =V and that T10(V) =V".It is possible to define tensors over a differentiable manifold.In Sec.A [ Pobierz caÅ‚ość w formacie PDF ]
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.55)The scalar à plays the role of a switch that enables/disables the parametricvariation.In several application (e.g.telemanipulation) when a certain setof physical parameters is reached, it has to be kept constant despite of anyinternal energy flows.Thus, when it is necessary to change the parameters, Ãis set equal to 1 and when a set of parameters has to e kept constant à is sett 0.Remark 5.21.Notice that when à = 0 the Dirac structures given in Eq.(5.54)and Eq.(5.53) are the same as the one given in Eq.(5.51) meaning that whenà = 0 the extended port-Hamiltonian system behaves as a fixed parametersport-Hamiltonian system.Remark 5.22.Some extra dissipating element can be added to dissipate energystored in the parametric part of the state.In fact if we connect to the para-metric power ports a dissipative element, the energy absorption leads to adecrease of the parametric part of the state.This strategy can be used todecrease parametric states independently of the energy flowing along the ex-tended Dirac structure. 194 5 Transparency in Port-Hamiltonian Based Telemanipulation5.5.2 Parameters Associated to Energy Dissipation andInterconnectionParameters relative to energy dissipation play the role of modulating energyabsorption.Changing values of these parameters is safe since different va-lues would imply only a different rate of absorption.There is only one thingto be aware of: as reported in Sec.1.4, the power flow towards dissipativeelements must always be positive and, therefore, we cannot take a set of pa-rameters which makes this power flow negative.If this happened dissipativeelements would be transformed in power injecting elements which, therefore,would destroy passivity of the port-Hamiltonian system.This constraint inthe choice of dissipative parameters is not very restrictive.In fact, in severalapplications (e.g.telemanipulation), port-Hamiltonian controllers are used toreproduce their physical equivalent and modulating physical parameter hasthe aim to obtain the same kind of behavior but characterized by differentphysical properties.Transforming, by a certain choice of parameters, dissipa-tion in energy production we obtain a totally different kind of behavior whichis always an undesired feature.Parameters relative to the interconnection modulation can be freely chan-ged since they modulate transformations which are lossless independently oftheir modulation constant.5.5.3 SimulationsThe aim of this section is to provide some simulations in order to validatethe obtained results.Consider the system shown in Fig.5.7, where the hu-man operator interacts with a device, modeled as a mass m, controlled by anport-Hamiltonian impedance controller equivalent to the mechanical parallelof a linear spring, characterized by a stiffness k, and of a damper, with dam-ping coefficient b.In this simplified scheme, we assume that the value of thedamping coefficient b is fixed, while the stiffness k can be adapted in order tochange the sensation rendered to the human operator.Suppose that we want to change the value of the stiffness of the impedancecontroller and to drive it to the value kref.The energy storage function of thecontroller is:1H(k, x) = kx2 (5.56)2where the rest length of the spring is supposed to be zero.The power portexpressing the elongation:"H, -‹ =(kx,-‹ ) =(ex, fx) (5.57)"xThe following parametric port describes the power flow related to the stiffnessvariation: 5.5 A Passivity Preserving Tuning of Port-Hamiltonian Systems 195krefxlx kMbIPCFig.5.7.A 1-DOF variable parameters port-Hamiltonian impedance controller"H x2Ù Ù, -k = , -k = ek, fk) (5.58)"k 2Notice that the effort ek is always positive and the sign of the power exchangedthrough the parametric port is determined only by the flow fk.Finally, theimpedance felt by the user is described by the external power port (e, f), andthe goal of the stiffness adaptation is to compute an external effort e as closeas possible to that generated by a real spring of stiffness kref:f = ‹ , e = krefx (5.59)We choose to deviate part of the energy that is flowing towards the elon-gation port to the parametric port in order to passively change the stiffnessof the spring; therefore, we extend the Dirac structure of the fixed parametersport-Hamiltonian system in the way reported in Eq.(5.53).This leads to:Ùf = ‹ and k = Ã(t)f (5.60)where Ã(t) is the modulation law that has to be computed in order to achievethe desired behavior on the external port.Notice that the stiffness variationÙk is proportional to the external flow f and it takes place only when Ã(t) =0.This choice satisfies the first equation of Eq.(5.59) and straightforwardcomputations show that the elongation perceived externally is equal to x.By expressing the Dirac structure in kernel form, we obtain the followingequations:ëø öø ëø öø ëø öø ëø öø101 -‹ 00 0 exÙíø01Ãøø íø-køø íø00 0øø íøekøø+ = 0 (5.61)000 f 1Ã-1 eFirst of all, consider the situation when the user is extracting power from theexternal port, namely P = e, f 0 and adiffeomorphismh : U × n ’! À-1(U)such that, for each m " U, the correspondence x ’! h(m, x) defines an iso-morphism between the vector space n and the vector space À-1(m).The pair(U, h) is called a local coordinate system for ¾.The vector space À-1(m) iscalled fiber over m for any m "M.For a very complete treatment of vector bundles see [204].A.2 TensorsTensors are the generalization of the concepts of vector and matrix and, infact, once a coordinate frame has been defined, they can be represented asmultidimensional matrices.For a complete treatment see, for example, [185].Definition A.19 (Multilinear map).Consider n+1 vector spaces V1,.,Vn, W.Amultilinear map is a map:L : V1 ×· · · ×Vn ’!W (A.13) A.3 Lie Groups and Rigid Motions 207such that, for each i =1,., n, the maps:Li(v1,., vi-1, vi+1,., vn) : Vi ’!W vi ’! L(v1,., vn) (A.14)are linear for each v1,., vi-1, vi+1,., vn.The set of multilinear operators defined on Vi,.,Vn, W is indicated asLn(V1,., Vn; W)and it can be proven that it is a vector space.Consider a vector space V and denote its dual space, namely the vector spaceformed by the linear operators on V, by V".Definition A.20 (Tensors).Given a vector space V, the set of tensors oforder p + q of type (p, q), namely p contravariant and q covariant, is definedby:Tqp(V) =Lp+q(V,., V, V",.V"; ) (A.15)q times p timesNotice that T01(V) =V and that T10(V) =V".It is possible to define tensors over a differentiable manifold.In Sec.A [ Pobierz caÅ‚ość w formacie PDF ]