Controllability and observability in the problem of stabilizing steady motions of non-holonomic mechanical systems with cyclic coordinatest

The approach to the solution of stabilization problems for steady motions of holonomic mechanical systems [1, 2] based on linear control theory, combined with the theory of critical cases of stability theory, is used to solve the analogous problems for non-holonomic systems. It is assumed that the control forces may affect both cyclic and positional coordinates, where the number r of independent control inputs may be considerably less than the number n of degrees of freedom of the system, unlike in many other studies (see, e.g., [3–5]), in which as a rule r = n. Several effective new criteria of controllability and observability are formulated, based on reducing the problem to a problem of less dimension. Stability analysis is carried out for the trivial solution of the complete non-linear system, closed by a selected control. This analysis is a necessary step in solving the stabilization problem for steady motion of a non-holonomic system (unlike holonomic systems), since in most cases such a system is not completely controllable.     

Inverse dynamics of underactuated multibody systems

The present work deals with controlled mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The governing equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. The rotationless formulation of multibody dynamics will be considered [1]. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [2]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical integration of differential-algebraic equations with mixed holonomic and control constraints

The present work aims at the incorporation of control (or servo) constraints into finite–dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential–algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed–loop multibody systems, the present formulation accommodates the so–called rotationless formulation of multibody dynamics. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Right-hand solutions of the differential equations of dynamics for mechanical systems with sliding friction

As a further development of Painlevé's theory [1], the existence, continuability and uniqueness of righ-hand solutions of the differential equations of dynamics, and, under certain additional conditions, of the equations of motion of holonomic mechanical systems with sliding friction [2] are considered. In classical mechanics, acceleration is essentially defined as the right-hand derivative of velocity (see [3, 4]). Hence the most meaningful definition of the “solution of a differential equation” in problems of the dynamics of mechanical systems with sliding friction is that using the concept of right derivative [5].     

A numerical method for the servo constraint problem of underactuated mechanical systems

Servo constraints are used in inverse dynamics simulations of discrete mechanical systems, especially for trajectory tracking control problems [1], whose desired outputs are represented by state variables and treated as servo constraints [2]. Servo constraint problems can be classified into fully actuated and underactuated multibody systems, and the equations of motion take the form of differential algebraic equations (DAEs) including holonomic and servo constraints. For fully actuated systems, control inputs can be solved from the equations by model inversion, as the input distribution matrix is nonsingular and invertible. However, underactuated systems have more degrees of freedom than control inputs. The input distribution matrix is not invertible, and in contrast to passive constraints, the realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Therefore, it is challenging for the determination of control inputs which force the underactuated system to realize the partly specified motion. For differentially flat underactuated systems, the differentiation index of DAEs may exceed three. Hence we need to apply specific index reduction techniques, such as the projection approach applied in [3], [4], and [6]. The present work applies index reduction by minimal extension [5] to differentially flat underactuated crane systems and shows that the index can be reduced from five to three and even to one. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integral variational principles in Poincaré and Chetayev variables

A holonomic mechanical system with k degrees of freedom is considered, its state being characterized by n k defining coordinates, p < k Poincaré parameters [1] and k - p Chetayev parameters [2]. In these variables, generalized Routh equations are introduced and expressions are given for the integral variational principles of Hamilton-Ostrogradskii and Hamilton (the third form), as well as Hölder's principle and the Lagrange and Jacobi versions of the principle of least action.     

On the influence of gyroscopic forces on the stability of steady-state motion : PMM vol. 39, n 6, 1975, pp. 963–973

The question of the influence of gyroscopic forces on the stability of steady-state motion of a holonomic mechanical system when the forces depend upon the velocities of only the position coordinates was answered by the Kelvin-Chetaev theorems [1] on the influence of gyroscopic and dissipative forces on the stability of equilibrium. However, if the gyroscopic forces depend as well on the velocities of the ignorable coordinates, then their influence on the stability of steady-state motions can, as the two problems in [2] show, prove to be entirely different from the influence of gyroscopic forces depending only on the velocities of the position coordinates. In this paper we investigate the influence of gyroscopic forces depending linearly on the velocities of the generalized coordinates, including the ignorable ones, on the stability of the steady-state motion of a holonomic conservative system. We prove that when the gyroscopic forces applied with respect to the ignorable coordinates are given as total time derivatives of certain functions of the position coordinates, the gyroscopic forces can both stabilize as well as destabilize the steady-state motion. Under certain conditions, this influence is also preserved for the action of dissipative forces depending on the velocities of only the position coordinates. In the case of action of dissipative forces depending also on the velocities of the ignorable coordinates, we have indicated the stability and instability conditions of the steady-state motion. Examples are considered. In conclusion, we discuss the conditions under which the application of gyroscopic forces to the system is equivalent to adding terms depending linearly on the generalized velocities to the Lagrange function.     

Efficient integration of flexible multibody dynamics

The modelling of flexible multibody dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way. The discrete null space method developed in [0] provides an energy-momentum conserving integration scheme for the DAEs of motion of constrained mechanical systems. It relies on the elimination of the constraint forces from the discrete system along with a reparametrisation of the nodal unknowns. The resulting reduced scheme performs advantageously concerning different aspects: the constraints are fulfilled exactly, the condition number of the iteration matrix is independent of the time step and the dimension of the system is reduced to the minimal possible number saving computational costs. A six-body-linkage possessing a single degree of freedom is analysed as an example of a closed loop structure. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Whittaker方程对非完整力学系统的推广

1904年Whittaker利用能量积分将一个完整保守力学系统问题降阶为一个带有较少自由度系统问题.并得到了Whittaker方程[1].本文推导对于非完整力学系统的这类方程.并称之为广义Whittaker方程;然后把这些方程变换为Nielsen形式;最后举例说明新方程的应用.     

Stabilization of the motions of mechanical systems with variable masses

A holonomic mechanical system with variable masses and cyclic coordinates is considered. Such a system can have generalized steady motions in which the positional coordinates are constant and the cyclic velocities under the action of reactive forces vary according to a given law. Sufficient Routh-Rumyantsev-type conditions for the stability of such motions are determined. The problem of stabilizing a given translational-rotational motion of a symmetric satellite in which its centre of mass moves in a circular orbit and the satellite executes rotational motion about its axis of symmetry is solved.     

On the determination op forces op constraint reaction : PMM vol. 39, n 6, 1975, pp. 1129–1134

The equations of motion of mechanical systems with multipliers are reduced to the form enabling the separation of these equations into two groups, the first group describing the motions of the system, and the second group defining the multipliers. Each multiplier is determined independently of the remaining multipliers, and this makes it easy to assess the dynamic effect of each constraint on the system. On the basis of this approach, we study the following problems: determination of the constraint reactions [1], study of the motion of controlled systems with prescribed constraints [2, 3] and utilization of the method of nonholonomic mechanical systems in the case when the first integrals exist [4].     

A Prediction Model for State Observation and Model Predictive Control of Biped Robots

Biped walking robots present a class of mechanical systems with many different challenges such as nonlinear multi-body dynamics, a large number of degrees of freedom and unilateral contacts. The latter impose constraints for physically feasible motions and in stabilization methods as the robot can only interact due to pressure forces with the environment. This limitation can cause the system to fall under unknown disturbances such as pushing or uneven terrain. In order to face such problems, an accurate and fast model of the robot to observe the current state and predict the state evolution into the future has to be used. This work presents a nonlinear prediction model with two passive degrees of freedom (dof), point masses and compliant unilateral contacts. We show that the model is applicable for real-time model predictive optimization of the robot's motion. Experiments on the biped robot LOLA [1] underline the effectiveness of the proposed model to increase the system's long term stability under large unknown disturbances. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attraction for autonomous mechanical systems with sliding friction

Issues on attraction in autonomous mechanical systems with ideal holonomic bilateral constraints acted upon by potential gyroscopic dissipative forces and forces of sliding friction are considered. In particular, the semi-invariance of ω-limit sets and the conditions for the dichotomy of such systems are established. The investigation is based on the invariance principle using several Lyapunov functions, combining the methods of [1] with the La Salle invariance principle [2, 3] applied to autonomous systems with a discontinuous right-hand side.     

A comparison of three methods of constructing Lyapunov functions

Chetayev's effective method [1] for constructing Lyapunov functions in the form of a set of first integrals of the equations of perturbed motion has been widely used since the 1950s in Russia. In the 1980s the energy-Casimir method [2] was developed in the U.S.A. as well as the energy-momentum method [3], employed for Hamiltonian systems. A comparison of these methods for systems with a finite number of degrees of freedom has shown that the energy-Casimir method is a more complicated version of Chetayev's method, while the energy-momentum method is essentially the Routh-Lyapunov method [4,5], stated in modern geometrical language. Some examples are considered.     

Controllability and observability in the problem of stabilizing mechanical systems with cyclic coordinates

An approach based on linear control theory is used to solve the problem of stabilizing the steady motions of holonomic mechanical systems in which only cyclic coordinates are controllable [1–3]. The most general structure of forces acting on the system is considered and it is assumed that the constraints imposed are time-independent. The set of new criteria of controllability and observability based on the reduction of the problem under consideration is obtained. The reduction enables one to reduce the investigation of these problems to an analysis of a problem of less dimensions.     

On the existence of time-optimal control of mechanical manipulators

This paper is concerned with the problem of the existence and structure of time-optimal control for models derived from Lagrange equations of motion of mechanical systems involving links. The condition which ensures the existence of time-optimal control is demonstrated. The study conducted in this paper involves a highly nonlinear mathematical model of a two-degree-of-freedom mechanical system. However, the procedure and the results presented in this paper can be extended to mechanical systems with any finite number of degrees of freedom.The authors wish to thank Professor D. G. Hull and the reviewers for their most valuable comments and suggestions.     

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.     

Nuovi criteri per l'esistenza globale in futuro dei moti dei sistemi olonomi scleronomi

Summary By means of the comparison method and an appropriate choice of the Liapunov function, new criteria for the global existence in the future of the motions of several classes of holonomic scleronomic systems are obtained. The main advantage of these criteria respect to those provided in [1], [2]is that the potential energy of the mechanical system need not be bounded from below, as required in the above-mentioned papers, where the total energy of the system is chosen as Liapunov function.