Dynamics of Tensegrity Systems: Compact Forms

Analytical Mechanics. Foundations of Engineering Mechanics Series

Sets of atoms collectively behaving as rigid bodies are often used in molecular dynamics to model entire molecules or parts thereof. This is a coarse-graining strategy that eliminates degrees of freedom and supposedly admits larger time steps without abandoning the atomistic character of a model. In this paper, we rely on a particular factorization of the rotation matrix to simplify the mechanical formulation of systems containing rigid bodies. We then propose a new derivation for the exact solution of torque-free rotations, which are employed as part of a symplectic numerical integration scheme for rigid-body dynamics. We also review methods for calculating pressure in systems of rigid bodies with pairwise-additive potentials and periodic boundary conditions. Finally, simulations of liquid phases, with special focus on water, are employed to analyze the numerical aspects of the proposed methodology. Our results show that energy drift is avoided for time step sizes up to 5 fs, but only if a proper smoothing is applied to the interatomic potentials. Despite this, the effects of discretization errors are relevant, even for smaller time steps. These errors induce, for instance, a systematic failure of the expected equipartition of kinetic energy between translational and rotational degrees of freedom.

A method for dynamic analysis of large-scale constrained system of mixed rigid and flexible bodies with intermittent motion is presented. The system equations of motion are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. Equations of motion are computer generated and integrated forward in time using an explicit-implicit integration algorithm. Points in time at which sudden events of the intermittent behavior occur are monitored by an event predictor which controls the integration algorithm and forces a solution for the system impulse-momentum relation at those points. Solutions of impulse-momentum relations define the jump discontinuities in the composite velocity vector as well as the generalized impulses of the reaction forces at different joints of the mechanical system.

In the present study, the equations of motion for planar mechanical systems that consist of interconnected rigid bodies with common types of kinematic joint are derived on the basis of particle dynamics and a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then, for the resulting equivalent system of particles, the concepts of linear and angular momenta are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For an open-loop system, the equations of motion are generated recursively along the open chains. For a closed-loop system, the system is transformed to open chains by cutting suitable kinematic joints with the addition of cut-joint kinematic constraints. Examples of multibranch closed-loop systems are chosen to demonstrate the generality and simplicity of the proposed method.

Suitable for both senior-level and first-year graduate courses, this fully revised edition provides a unique and systematic treatment of engineering dynamics that covers Newton–Euler and Lagrangian approaches. New to this edition are: two completely revised chapters on the constraints on, and potential energies for, rigid bodies, and the dynamics of systems of particles and rigid bodies; clearer discussion on coordinate singularities and their relation to mass matrices and configuration manifolds; additional discussion of contravariant basis vectors and dual Euler basis vectors, as well as related works in robotics; improved coverage of navigation equations; inclusion of a 350-page solutions manual for instructors, available online; a fully updated reference list. Numerous structured examples, discussion of various applications, and exercises covering a wide range of topics are included throughout, and source code for exercises, and simulations of systems are available online.

A new strategy for full pose and velocity consensus control of multi‐agent rigid body systems in the presence of communication delays is presented. Specifically, consensus protocols are proposed for a system of N heterogeneous rigid bodies on the Banach manifold associated with the tangent bundle , under a fixed and undirected communication topology, where the attitudes of the rigid bodies are described in terms of rotation matrices. The stability argument is strengthened from that used in prior studies by using an extension of Morse–Lyapunov–Krasovskii approach, and sufficient conditions are derived to achieve almost global asymptotic stability of the consensus subspace. Finally, illustrative examples are given to demonstrate the proposed method where both homogenous and heterogeneous communication delays are considered.

This chapter describes the kinematics of point masses and rigid bodies when non-inertial coordinate systems (frames) are used to describe their motion. We obtain relative velocity and acceleration expressions for moving frames and then apply those expressions to find the velocities and accelerations of constrained systems of rigid bodies at specific instances of time, similar to what is done in many elementary dynamics texts. However, we also show that it is possible to determine the kinematics of a constrained system of rigid bodies more completely as a function of time. In some cases this can be done analytically but in general the solution must be done numerically since the positional constraints are normally nonlinear. To solve the positional constraints we use the Newton-Raphson method. Kinematics is treated in this chapter both by the traditional vector approach and by an equivalent matrix-vector method that is more readily suited to dealing with complex systems. The matrix-vector approach for planar problems is covered in Sects. 4.4 and 4.5 while more general three-dimensional problems are treated in Sect. 4.6 and those that follow. Three-dimensional rotations are described in terms of both Euler angles and Euler parameters as these are the most commonly used generalized rotational coordinates. Some classical examples of the dynamics effects seen in rotating coordinate systems, such as the Foucault pendulum, are also given.

Stabilization of a reentry vehicle (RV) by a partial spin-up of it is considered for the case of uncontrolled descent into the atmosphere. In this case, the vehicle is a composite construction consisting of two rigid bodies, a return capsule and a stabilizing block, which is put in rotation. A model is developed for the spatial motion of the reentry vehicle considered as a system of coaxial rigid bodies rotating about a common axis of symmetry. The free motion is studied, and the stability of steady-state regimes is analyzed. The spatial motion of the system is considered for the case of a small asymmetry due to displacement of the axes of dynamic symmetry of the bodies with respect to the spin axis, and approximate solutions for the motion parameters of the free system are found.

Determination of Christoffel symbols of the first kind for a class of rigid body systems by topological considerations

From d’Alembert’s principle exact nonlinear differential equations of motion are derived for systems which are composed of an arbitrary number of rigid bodies with an arbitrary interconnection structure and with arbitrary ideal constraints in hinges between bodies (the term hinge is used for any kind of connection between two bodies). The paper represents a generalization of earlier investigations by Fischer [1], Hooker [2, 3], Margoulis [2], Roberson [4, 5], Boland/Samin/Willems [6], Lilov [7] and Wittenburg [4, 7, 8]. A comprehensive textbook on the subject was written by Wittenburg [9].

In the present study, the equations of motion for generalized planar linkages that consist of a system of rigid bodies with all common types of kinematic joints are derived using a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then for the resulting equivalent system of particles, the concepts of linear and angular momentums are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For the open loop case, the equations of motion are generated recursively along the open chains. For the closed loop case, the system is transformed to open loops by cutting suitable kinematic joints and introducing cut-joints kinematic constraints. An example of a multi-branch closed-loop system is chosen to demonstrate the generality and simplicity of the proposed method.

During the last decade, tensegrity systems have been the focus of numerous investigations exploring the possibility of adopting them for planetary landing and exploration applications. Early approaches mainly focused on locomotion aspects related to tensegrity systems, where mobility was achieved by actuating the cable members of the system. Later efforts focused on understanding energy storage mechanisms of tensegrity systems undergoing landing events. More precisely, it was shown that under highly dynamic events, buckling of individual members of a tensegrity structure does not necessarily imply structural failure, suggesting that efficient structural design of planetary landers could be achieved by allowing its compression members to buckle. In this work, we combine both aspects of previous research on tensegrity structures, showing a possible lattice-like structural configuration able to withstand impact events, store pre-impact kinetic energy, and utilize a part of that energy for the locomotion process. Our work shows the feasibility of this proposed approach via both experimental and computational means.

The self-equilibrium problem of the Class-Theta tetrahedral tensegrity module

Any articulated system of rigid bodies defines a statically equivalent serial chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. An SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed, followed by the experimental validation of SESC modeling. Second, the problem of generating an SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.

The paper deals with dynamic analysis of a wind power generator as a large flexible structure with high speed rotating machines and considerable masses. The dynamic model is considered as a multibody system of rigid and flexible bodies. Nonstationary and transitional processes caused because of eccentricity of the high speed rotating machines, as well as, of the propeller vibrations are simulated and analyzed. Analytical method is applied for dynamic simulation. The results are verified by numerical procedures. Example of wind power generator with three propellers is presented.