Формальский А.М. | Lomonosov Moscow State University (original) (raw)
Papers by Формальский А.М.
Mechanisms and machine science, 2016
The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton fo... more The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton for human carrying a load. Human is modeled using a mathematical model of a planar bipedal five-link anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh and shin with point-foot. The exoskeleton is considered also as a five-link anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to biped (to human). If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip-, knee-, and ankle-joints coincide. We compare the walking of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; both models are with a load. A ballistic walking gait is designed in both cases. During the ballistic walking of the biped with exoskeleton the knee of the stance leg of the exoskeleton (and as a consequence of the biped) is locked. The locking can be realized in the knee of each leg of the exoskeleton by any mechanical brake device with no energy consumption. There are not any actuators in our exoskeleton. Therefore, we call it passive exoskeleton. The walking of the biped consists of alternating single- and double-support phases. In our study, the double-support phase is assumed as instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous stance leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone.
Mekhanika Tverdogo Tela, Mar 1, 1977
De Gruyter eBooks, Oct 29, 2015
Doklady Mathematics, Feb 1, 2010
Mechanics of Solids, Dec 1, 2018
Linear stationary systems with single-control (perturbing) action are considered. The impulse of ... more Linear stationary systems with single-control (perturbing) action are considered. The impulse of control actions is considered limited. Some properties of the boundaries of attainability domains are studied. It is shown that the boundary of the attainability domain can have flat regions, regions of ruled surfaces, edges, and conical angular points. An attainability domain is not strictly convex if there are straight edges and/or flat regions on the boundary. The behavior of the boundaries of the attainability domains with increasing time is studied. A third-order system with a threefold zero eigenvalue (triple integrator) is considered as an example. The structure of the attainability domain of this system is analytically investigated in three-dimensional space. An attainability domain is constructed numerically for some time values .
Journal of applied mathematics and mechanics, 2005
The forward motion of a monocycle is studied in terms of a mechanical model: a cylinder with an i... more The forward motion of a monocycle is studied in terms of a mechanical model: a cylinder with an inverted pendulum attached to it by a hinge. The rotation of the pendulum about the cylinder is implemented by an electric drive. The monocycle can be moved (rolled) over a surface with the help of the same drive. The control parameter is a voltage of bounded magnitude. A control law (with saturation), linear in the phase coordinates, is constructed, under which the pendulum is stabilized in the upper unstable equilibrium position, while the monocycle is maintained in position or moved. The domain of attraction of the desired steady state may be made the maximum possible (in the linear approximation). The domain of stability of the equilibrium state of the system is constructed in the plane of the following parameters: the total feedback gain and the delay. Delay appears in the control loop when there is an inductance in the rotor circuit of the motor.
Multibody System Dynamics, Nov 15, 2017
The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton fo... more The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton for human walking and carrying a load. Human is modeled using a planar bipedal anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh, shin, and foot (massless). The exoskeleton is considered also as an anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to the biped (to human)-the topology of the exoskeleton is the same as of the biped (human). Each of models of human and of exoskeleton has seven links and six joints. The hip-joint connects the trunk and two thighs of the two legs. If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip-, knee-, and ankle-joints coincide. We compare the walking gaits of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; for both cases the same load is transported. The problem is studied in the framework of ballistic walking model. During the ballistic walking of the biped with exoskeleton the knee of the support leg is locked, but the knee of the swing leg is unlocked. The locking and unlocking can be realized in the knees of the exoskeleton by any mechanical brake devices without energy consumption. There are not any actuators in the exoskeleton. Therefore, we call it passive exoskeleton. The walking of the biped consists of alternating single-and double-support phases. In our study, the double-support phase is assumed as instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous support leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone. For transportation of a load with mass 40 kg, the economy of the energy is approximately 28%, if the length of the step and its duration are equal to 0.5 m and 0.5 s respectively.
Izvestiâ Akademii nauk SSSR. Tehničeskaâ kibernetika, 2020
Moscow University Mechanics Bulletin, Mar 1, 2020
An example of a linear completely contrallable time-invariant system is considered to study the c... more An example of a linear completely contrallable time-invariant system is considered to study the change of its attainability region after the transition to a reduced system with a decreased order. The original system is a triple integrator. After reduction, a new system (a double integrator) whose control belongs to the set of piecewise continuous bounded functions is considered. The attainability set of the reduced system and the projection of the attainability set of the original system onto the coordinate plane of the reduced system are compared.
De Gruyter eBooks, Oct 29, 2015
De Gruyter eBooks, Oct 29, 2015
De Gruyter eBooks, Oct 29, 2015
Journal of applied mathematics and mechanics, 1997
Abstract A uniform circular membrane, the edges of which are clamped to a large, absolutely rigid... more Abstract A uniform circular membrane, the edges of which are clamped to a large, absolutely rigid ring, is considered. In its undeformed state, the membrane lies in the same plane as the ring. One-dimensional displacements of the ring and membrane in a direction perpendicular to that plane are investigated. The problem of stabilizing the state of the system by means of an external control force operating in that direction on the ring is studied. Stabilizing control is realized by linear feedback with respect to the displacement and velocity of the ring, and the integral of the displacement and deformations of the membrane. Allowance is made for a delay in the control loop. The stability of the control process is considered, and the regions of asymptotic stability of the desired position of equilibrium of the membrane in the space of the feedback coefficients are found.
Doklady Mathematics, 2006
De Gruyter eBooks, Oct 28, 2015
This monograph is dedicated to the designing motion control for objects that may have desired wor... more This monograph is dedicated to the designing motion control for objects that may have desired working regimes unstable without additional guidance. For example, an object of such kind is the recently invented vehicle "Segway" that, having a passenger on board, is an unstable inverted pendulum mounted on a platform with a wheel pair. There exists also a single-wheel "Segway" (one-wheel scooter) called "Solowheel". This single-wheel scooter is even less stable than the two-wheel "Segway". There are many more unstable devices. Most links of a two-legged anthropomorphic walker (like the "links" of its prototype, the human being) are unstable inverted pendulums. Flying vehicles are unstable if the center of pressure is located in front of the center of mass. If the center of pressure is "far" behind the mass center, the flying vehicle has a large static stability margin and can not be agile. In order to increase its agility, it is necessary to reduce the stability margin and, for some special purposes (like military aircraft), even "enter" the instability domain. The problems of stabilizing a desired operating mode also arise in relation with design of an exoskeleton and of different unstable systems that have magnetic or electrostatic support. If the desired operating regime is stable without control, i.e., the object of control is naturally stable [105], then the purpose of the control system is to improve the quality of transient processes. These transients may occur when the object deviates from the desired mode due to external disturbances. But if the desired operating regime is unstable without control, i.e., the object is naturally unstable, the primary goal of the control system becomes stabilization of this regime. In this case, it is virtually impossible to implement the desired operating regime without any control, and the matters of transient quality fade into insignificance. The problems of designing a control law for an unstable object and stabilizing a desired operating regime are concerned with certain difficulties. Actually, control resources are in some way or another limited in any real system. Hence an unstable object cannot be translated to a desired operating regime from any state. In other words, the set of states from which the object can be driven to the desired operation mode for a set of control resource limitations occupies an area of the phase space. This set is referred to as "controllability domain". The domain of attraction of the desired operating regime, appearing when some specific control law is designed, for example, as a feedback, belongs to the controllability domain; mostly, it occupies only a part of this domain. The domain of attraction usually means a set of initial states from which the system can be moved asymptotically to the desired regime by means of the control law (this definition is used further in this book; some authors refer to it as "basin of attraction"). If the domain of attraction is small with respect to practically possible disturbances of the object motion, then the desired operating regime is practically unrealizable. This means that the control resources are insufficient, or the control law is not efficient. Thus, if limitations are imposed on control resources, the problem of con
Journal of applied mathematics and mechanics, 1971
Mechanics of Solids, Oct 1, 2008
We study the plane motion of a double pendulum with fixed suspension point. The pendulum is contr... more We study the plane motion of a double pendulum with fixed suspension point. The pendulum is controlled by a single moment applied to the internal hinge between the links. The moment is assumed to be bounded in absolute value. We construct a feedback control law bringing the pendulum from the position in which both links hang vertically downwards into the unstable upper position in which both links are inverted. The same feedback ensures the asymptotic stability of the pendulum in the upper equilibrium position. Since the pendulum can be brought to the lower equilibrium position from any initial states, it follows that the constructed control law ensures the global stability of the inverted pendulum.
Journal of applied mathematics and mechanics, 1996
Abstract The problem of the fastest motion (bending) of a plane two-link mechanism with a load on... more Abstract The problem of the fastest motion (bending) of a plane two-link mechanism with a load on the end from a specified initial configuration to a specified final configuration is considered. The moduli of the controlling moments at the fixed hinge and the hinge between the links are bounded. It is shown that motions for which the links remain folded over the whole time interval are optimal. A control which satisfies the Pontryagin maximum principle is constructed numerically for the case when the links are not superposed in the boundary configurations. In the case of this control, the second link oscillates around the first during the bending of the two-link mechanism without being superposed on it in any time interval.
Automation and Remote Control, Dec 1, 2003
Control systems described by linear difference equations are studied, assuming that control actio... more Control systems described by linear difference equations are studied, assuming that control actions are bounded both in magnitude (geometrically) and norm (integrally). The states in the phase space to which a system can be transferred from the origin and the states from which a system can be transferred to the origin are studied. Examples are given.
Journal of applied mathematics and mechanics, May 1, 2006
The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendul... more The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.
Mechanisms and machine science, 2016
The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton fo... more The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton for human carrying a load. Human is modeled using a mathematical model of a planar bipedal five-link anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh and shin with point-foot. The exoskeleton is considered also as a five-link anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to biped (to human). If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip-, knee-, and ankle-joints coincide. We compare the walking of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; both models are with a load. A ballistic walking gait is designed in both cases. During the ballistic walking of the biped with exoskeleton the knee of the stance leg of the exoskeleton (and as a consequence of the biped) is locked. The locking can be realized in the knee of each leg of the exoskeleton by any mechanical brake device with no energy consumption. There are not any actuators in our exoskeleton. Therefore, we call it passive exoskeleton. The walking of the biped consists of alternating single- and double-support phases. In our study, the double-support phase is assumed as instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous stance leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone.
Mekhanika Tverdogo Tela, Mar 1, 1977
De Gruyter eBooks, Oct 29, 2015
Doklady Mathematics, Feb 1, 2010
Mechanics of Solids, Dec 1, 2018
Linear stationary systems with single-control (perturbing) action are considered. The impulse of ... more Linear stationary systems with single-control (perturbing) action are considered. The impulse of control actions is considered limited. Some properties of the boundaries of attainability domains are studied. It is shown that the boundary of the attainability domain can have flat regions, regions of ruled surfaces, edges, and conical angular points. An attainability domain is not strictly convex if there are straight edges and/or flat regions on the boundary. The behavior of the boundaries of the attainability domains with increasing time is studied. A third-order system with a threefold zero eigenvalue (triple integrator) is considered as an example. The structure of the attainability domain of this system is analytically investigated in three-dimensional space. An attainability domain is constructed numerically for some time values .
Journal of applied mathematics and mechanics, 2005
The forward motion of a monocycle is studied in terms of a mechanical model: a cylinder with an i... more The forward motion of a monocycle is studied in terms of a mechanical model: a cylinder with an inverted pendulum attached to it by a hinge. The rotation of the pendulum about the cylinder is implemented by an electric drive. The monocycle can be moved (rolled) over a surface with the help of the same drive. The control parameter is a voltage of bounded magnitude. A control law (with saturation), linear in the phase coordinates, is constructed, under which the pendulum is stabilized in the upper unstable equilibrium position, while the monocycle is maintained in position or moved. The domain of attraction of the desired steady state may be made the maximum possible (in the linear approximation). The domain of stability of the equilibrium state of the system is constructed in the plane of the following parameters: the total feedback gain and the delay. Delay appears in the control loop when there is an inductance in the rotor circuit of the motor.
Multibody System Dynamics, Nov 15, 2017
The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton fo... more The paper aim is to show theoretically the feasibility and efficiency of a passive exoskeleton for human walking and carrying a load. Human is modeled using a planar bipedal anthropomorphic mechanism. This mechanism consists of a trunk and two identical legs; each leg consists of a thigh, shin, and foot (massless). The exoskeleton is considered also as an anthropomorphic mechanism. The shape and the degrees of freedom of the exoskeleton are identical to the biped (to human)-the topology of the exoskeleton is the same as of the biped (human). Each of models of human and of exoskeleton has seven links and six joints. The hip-joint connects the trunk and two thighs of the two legs. If the biped is equipped with an exoskeleton, then the links of this exoskeleton are attached to the corresponding links of the biped and the corresponding hip-, knee-, and ankle-joints coincide. We compare the walking gaits of a biped alone (without exoskeleton) and of a biped equipped with exoskeleton; for both cases the same load is transported. The problem is studied in the framework of ballistic walking model. During the ballistic walking of the biped with exoskeleton the knee of the support leg is locked, but the knee of the swing leg is unlocked. The locking and unlocking can be realized in the knees of the exoskeleton by any mechanical brake devices without energy consumption. There are not any actuators in the exoskeleton. Therefore, we call it passive exoskeleton. The walking of the biped consists of alternating single-and double-support phases. In our study, the double-support phase is assumed as instantaneous. At the instant of this phase, the knee of the previous swing leg is locked and the knee of the previous support leg is unlocked. Numerical results show that during the load transport the human with the exoskeleton spends less energy than human alone. For transportation of a load with mass 40 kg, the economy of the energy is approximately 28%, if the length of the step and its duration are equal to 0.5 m and 0.5 s respectively.
Izvestiâ Akademii nauk SSSR. Tehničeskaâ kibernetika, 2020
Moscow University Mechanics Bulletin, Mar 1, 2020
An example of a linear completely contrallable time-invariant system is considered to study the c... more An example of a linear completely contrallable time-invariant system is considered to study the change of its attainability region after the transition to a reduced system with a decreased order. The original system is a triple integrator. After reduction, a new system (a double integrator) whose control belongs to the set of piecewise continuous bounded functions is considered. The attainability set of the reduced system and the projection of the attainability set of the original system onto the coordinate plane of the reduced system are compared.
De Gruyter eBooks, Oct 29, 2015
De Gruyter eBooks, Oct 29, 2015
De Gruyter eBooks, Oct 29, 2015
Journal of applied mathematics and mechanics, 1997
Abstract A uniform circular membrane, the edges of which are clamped to a large, absolutely rigid... more Abstract A uniform circular membrane, the edges of which are clamped to a large, absolutely rigid ring, is considered. In its undeformed state, the membrane lies in the same plane as the ring. One-dimensional displacements of the ring and membrane in a direction perpendicular to that plane are investigated. The problem of stabilizing the state of the system by means of an external control force operating in that direction on the ring is studied. Stabilizing control is realized by linear feedback with respect to the displacement and velocity of the ring, and the integral of the displacement and deformations of the membrane. Allowance is made for a delay in the control loop. The stability of the control process is considered, and the regions of asymptotic stability of the desired position of equilibrium of the membrane in the space of the feedback coefficients are found.
Doklady Mathematics, 2006
De Gruyter eBooks, Oct 28, 2015
This monograph is dedicated to the designing motion control for objects that may have desired wor... more This monograph is dedicated to the designing motion control for objects that may have desired working regimes unstable without additional guidance. For example, an object of such kind is the recently invented vehicle "Segway" that, having a passenger on board, is an unstable inverted pendulum mounted on a platform with a wheel pair. There exists also a single-wheel "Segway" (one-wheel scooter) called "Solowheel". This single-wheel scooter is even less stable than the two-wheel "Segway". There are many more unstable devices. Most links of a two-legged anthropomorphic walker (like the "links" of its prototype, the human being) are unstable inverted pendulums. Flying vehicles are unstable if the center of pressure is located in front of the center of mass. If the center of pressure is "far" behind the mass center, the flying vehicle has a large static stability margin and can not be agile. In order to increase its agility, it is necessary to reduce the stability margin and, for some special purposes (like military aircraft), even "enter" the instability domain. The problems of stabilizing a desired operating mode also arise in relation with design of an exoskeleton and of different unstable systems that have magnetic or electrostatic support. If the desired operating regime is stable without control, i.e., the object of control is naturally stable [105], then the purpose of the control system is to improve the quality of transient processes. These transients may occur when the object deviates from the desired mode due to external disturbances. But if the desired operating regime is unstable without control, i.e., the object is naturally unstable, the primary goal of the control system becomes stabilization of this regime. In this case, it is virtually impossible to implement the desired operating regime without any control, and the matters of transient quality fade into insignificance. The problems of designing a control law for an unstable object and stabilizing a desired operating regime are concerned with certain difficulties. Actually, control resources are in some way or another limited in any real system. Hence an unstable object cannot be translated to a desired operating regime from any state. In other words, the set of states from which the object can be driven to the desired operation mode for a set of control resource limitations occupies an area of the phase space. This set is referred to as "controllability domain". The domain of attraction of the desired operating regime, appearing when some specific control law is designed, for example, as a feedback, belongs to the controllability domain; mostly, it occupies only a part of this domain. The domain of attraction usually means a set of initial states from which the system can be moved asymptotically to the desired regime by means of the control law (this definition is used further in this book; some authors refer to it as "basin of attraction"). If the domain of attraction is small with respect to practically possible disturbances of the object motion, then the desired operating regime is practically unrealizable. This means that the control resources are insufficient, or the control law is not efficient. Thus, if limitations are imposed on control resources, the problem of con
Journal of applied mathematics and mechanics, 1971
Mechanics of Solids, Oct 1, 2008
We study the plane motion of a double pendulum with fixed suspension point. The pendulum is contr... more We study the plane motion of a double pendulum with fixed suspension point. The pendulum is controlled by a single moment applied to the internal hinge between the links. The moment is assumed to be bounded in absolute value. We construct a feedback control law bringing the pendulum from the position in which both links hang vertically downwards into the unstable upper position in which both links are inverted. The same feedback ensures the asymptotic stability of the pendulum in the upper equilibrium position. Since the pendulum can be brought to the lower equilibrium position from any initial states, it follows that the constructed control law ensures the global stability of the inverted pendulum.
Journal of applied mathematics and mechanics, 1996
Abstract The problem of the fastest motion (bending) of a plane two-link mechanism with a load on... more Abstract The problem of the fastest motion (bending) of a plane two-link mechanism with a load on the end from a specified initial configuration to a specified final configuration is considered. The moduli of the controlling moments at the fixed hinge and the hinge between the links are bounded. It is shown that motions for which the links remain folded over the whole time interval are optimal. A control which satisfies the Pontryagin maximum principle is constructed numerically for the case when the links are not superposed in the boundary configurations. In the case of this control, the second link oscillates around the first during the bending of the two-link mechanism without being superposed on it in any time interval.
Automation and Remote Control, Dec 1, 2003
Control systems described by linear difference equations are studied, assuming that control actio... more Control systems described by linear difference equations are studied, assuming that control actions are bounded both in magnitude (geometrically) and norm (integrally). The states in the phase space to which a system can be transferred from the origin and the states from which a system can be transferred to the origin are studied. Examples are given.
Journal of applied mathematics and mechanics, May 1, 2006
The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendul... more The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.