Dynamics of Non-ideal Topology Transitions in Multibody Mechanical Systems

Mechanical systems with time-varying topology appear frequently in various applications. In this paper, topology changes that can be modeled by means of bilateral impulsive constraints are analyzed. We present a concept to project kinematic and kinetic quantities to two mutually orthogonal subspaces of the tangent space of the mechanical system. This can be used to obtain decoupled formulations of the kinetic energy and the dynamic equations at topology transition. It will be shown that the configuration of the multibody system at topology change significantly influences the projection of non-ideal forces to both subspaces. Experimental analysis, using a dual-pantograph robotic prototype interacting with a stiff environment, is presented to illustrate the material. NOMENCLATURE A Jacobian associated with constrained motion. B Jacobian associated with tangential directions of contact. M Mass matrix of the system. Pa Projector associated with the space of admissible motion. Pc Projector associated with the space of constrained motion. q Generalized coordinates of the system. v Generalized velocities of the system. va Generalized velocities of the space of admissible motion. vc Generalized velocities of the space of constrained motion. ∗Address all correspondence to this author. f̄A Generalized applied impulses. f̄N Generalized non-ideal impulses. f̄R Generalized constraint impulses. T Kinetic energy of the system. Ta Kinetic energy of the space of admissible motion. Tc Kinetic energy of the space of constrained motion. INTRODUCTION Variable topology mechanical systems are present in various fields of applications such as robotics, biomechanics and mechanism science. The dynamic analysis of such systems depends on the time-varying nature of the connections between the elements of the system and the environment. This complicates the analysis because in most cases a different dynamic model must be developed for each constraint condition. Typical situations that occur in variable topology systems are the following: (1) The number of degrees of freedom of the system decreases via the development of certain connections. An example for this can be the grasping/capturing of a moving payload, which may also represent the interaction of two robotic mechanisms, or a human and a payload. The effect of mass capture on flexible multibody systems was studied in [1] and [2]. This group of problems includes two possibilities depending on whether the developed connections exist for a finite period of time or they represent an instantaneous situation. 1 Copyright c © 2009 by ASME (2) The constraint configuration is changing: some constraints are added and some become passive. But, the effective number of degrees of freedom may stay the same. An example for this situation can be found in the analysis of (active/passive) dynamic walking machines [3]. In those systems, the heel strike event represents a sudden change of topology where some constraints are imposed on the foot that makes contact, and other are released from the foot that leaves the ground [4, 5]. Discontinuous constraints have been a known concept in analytical mechanics [6–8]. As discussed earlier, two particular cases of such discontinuous constraint configurations can be the sudden removal and the sudden addition of constraints. The sudden removal of constraints alone does not instantaneously change the energy and momentum distribution of the system unless other impulsive forces (applied or constraint forces) are present. The sudden addition of constraints does cause instantaneous changes. Therefore, this is the truly critical event during the motion of variable topology systems. Such events can be characterized using “inert constraints” which are a class of impulsive constraints [6, 8]. This paper focuses on this event of sudden addition of constraints. We consider the general case of non-ideal development of constraints, i.e., we consider that impulsive forces can also be present along the tangential direction of the contact, e.g., due to friction or finite tangential stiffness of materials. It is not the aim of the work to model these impulses, but to understand their effect on the impulsive dynamics of topology transition. The dynamic analysis conducted in the paper is based on an analytical approach that allows a complete decoupling of the dynamic equations and the kinetic energy to two subspaces of the tangent space of the system, i.e., the spaces of constrained and admissible motions [9]. Based on this approach, it will be shown that the effects of non-ideal impulsive forces on the decoupled dynamic equations vary depending on the system configuration at topology transition. To illustrate this, detailed experimental analysis using a dual-pantograph robotic device is conducted. DYNAMICS MODELING Let us consider that the configuration of the system can be described by n generalized coordinates that are represented by n× 1 dimensional array q. The time derivatives of these generalized coordinates q̇ give a possible set of generalized velocities of the system. We will use a more general description for the velocities employing components collected in v, which can be interpreted as general linear combinations of the time derivatives of the generalized coordinates as v = Nq̇ and q̇ = N−1v, where N is an n×n transformation matrix that can depend on the generalized coordinates and time. We will consider that this parameterization represents a minimum set of generalized coordinates and velocities with respect to the continuous constraints imposed on the system. In this paper, we will primarily consider systems where the kinetic energy can be expressed as a quadratic function of the generalized velocities