Hybrid Parameter Multiple Body Research Articles

A Neuro-Muscular Elasto-Dynamic Model of the Human Arm Part 1: Model Development

In this paper we develop an elasto-dynamic model of the human arm for use in neuro-muscular control and dynamic interaction studies. The motivation for this work is to present a case for developing and using non-quasistatic models of human musculo-skeletal biomechanics. The model is based on hybrid parameter multiple body system (HPMBS) variational projection principles. In this paper, we present an overview of the HPMBS variational principle applied to the full elasto-dynamic model of the arm. The generality of the model allows one to incorporate muscle effects as either loads transmitted through the tendon at points of origin and insertion or as an effective torque at a joint. Though the technique is suitable for detailed bone and joint modeling, we present in this initial effort only simple geometry with the bones discretized as Rayleigh beams with elongation, while allowing for large deflections. Simulations demonstrate the viability of the method for use in the companion paper and in future studies.

A Neuro-Muscular Elasto-Dynamic Model of the Human Arm Part 2: Musculotendon Dynamics and Related Stress Effects

In this paper we develop an elasto-dynamic model of the human arm that includes effects of neuro-muscular control upon elastic deformation in the limb. The elasto-dynamic model of the arm is based on hybrid parameter multiple body system variational projection principles presented in the companion paper. Though the technique is suitable for detailed bone and joint modeling, we present simulations for simplified geometry of the bones, discretized as Rayleigh beams with elongation, while allowing for large deflections. Motion of the upper extremity is simulated by incorporating muscle forces derived from a Hill-type model of musculotendon dynamics. The effects of muscle force are modeled in two ways. In one approach, an effective joint torque is calculated by multiplying the muscle force by a joint moment arm. A second approach models the muscle as acting along a straight line between the origin and insertion sites of the tendon. Simple arm motion is simulated by utilizing neural feedback and feedforward control. Simulations illustrate the combined effects of neural control strategies, models of muscle force inclusion, and elastic assumptions on joint trajectories and stress and strain development in the bone and tendon.

Modeling Loose Joints in Elastic Structures— Experimental Results and Validation

This article is the last of a set of four companion articles. In these articles, a hybrid parameter multiple body system (HPMBS) methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of a cantilever beam undergoing planar slewing motion. In this preliminary model, a “rigid” joint is utilized, which allows the members of the joint to be rigid, each attached to the elastic beam sections. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled with the idea of instantly applied nonholonomic constraints. This article addresses the experimental justification for the model, and our experimental apparatus is described herein. Simulation results and experimental results for the rigid joint configuration are compared and discussed. The theoretical model, the momentum transfer equations utilizing the concept of instantly applied nonholonomic constraints, and the numerical solution scheme are presented in the three companion articles. The motivation for this work is the need for low order but nonetheless accurate models of complicated nonlinear phenomena in structural systems.

Modeling Loose Joints in Elastic Structures— Simulation Algorithm and Results

This is the third paper in a four companion paper sequence. In these papers, a hybrid parameter multiple body system (HPMBS) methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of a cantilever beam undergoing planar slewing motion. In this preliminary model, a “rigid” joint is utilized, allowing the members of the joint to be rigid, each attached to an elastic beam section. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled with the idea of instantly applied nonholonomic constraints. In this paper, the contact and switching algorithms, the numerical solution scheme, and simulation results are presented. The theoretical model of the variable structure dynamic system, the momentum transfer equations and the experimental comparison are presented elsewhere. The motivation for this work is the need for accurate low order models of complicated nonlinear phenomena in structural systems.

Modeling Loose Joints in Elastic Structures— Momentum Transfer Model Development

This is the second article in a sequence of four companion articles. In these articles, a hybrid parameter multiple body system methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of an elastic cantilever beam undergoing planar slewing motion. In this preliminary model, a “rigid” joint is utilized, which allows the members of the joint to be rigid, each attached to one of the elastic beam sections. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled using the idea of instantly applied nonholonomic constraints. The momentum transfer equations will be developed herein. A review of the theoretical development is presented, followed by a step by step outline of the process. The novelty of the approach is the use of pseudo-speeds to realize the momentum transfer during the contact and impact events. The dynamical equations, numerical solution scheme, and experimental comparisons are presented in the companion articles. The motivation for this work is to produce low order models, based on first principles, of nonlinear structural phenomena, especially those associated with loose joints.

Modeling Loose Joints in Elastic Structures-Variable Structure Motion Model Development

In this article, a hybrid parameter multiple body system (HPMBS) methodology is utilized to model the frictional contact/impact of a loose bolted joint between two sections of a cantilever beam undergoing planar slewing motion. In this preliminary model, a "rigid" joint is utilized which allows the members of the joint to be rigid, each attached to the elastic beam sections. Frictional contact/impact is modeled at four contact points. The contact constraints and momentum transfer are modeled on a basis of instantly applied nonholonomic constraints. The theoretical model of the variable structure dynamic system is developed in this article, while the momentum transfer equations (utilizing the concept of instantly applied nonholonomic constraints), numerical solution scheme, and experimental comparisons are presented in companion articles. The underlying theme of this work is to produce low order models of complicated nonlinear structural dynamics, specifically, frictional contact and impact in joints with clearance, based on first principles.

Maxillo-facial surgery simulation with a simple patient specific material model

In this paper we develop an elasto-dynamic model of the human arm that includes effects of neuro-muscular control upon elastic deformation in the limb. The elasto-dynamic model of the arm is based on hybrid parameter multiple body system variational projection principles presented in the companion paper. Though the technique is suitable for detailed bone and joint modeling, we present simulations for simplified geometry of the bones, discretized as Rayleigh beams with elongation, while allowing for large deflections. Motion of the upper extremity is simulated by incorporating muscle forces derived from a Hill-type model of musculotendon dynamics. The effects of muscle force are modeled in two ways. In one approach, an effective joint torque is calculated by multiplying the muscle force by a joint moment arm. A second approach models the muscle as acting along a straight line between the origin and insertion sites of the tendon. Simple arm motion is simulated by utilizing neural feedback and feedforward control. Simulations illustrate the combined effects of neural control strategies, models of muscle force inclusion, and elastic assumptions on joint trajectories and stress and strain development in the bone and tendon.

On the efficacy of pseudo-coordinates—Part 1: moving interior constraints

In a recent publication, a Gibbs–Appell type projection methodology for modeling non-holonomic hybrid parameter multiple body systems (HPMBS) was presented. In this methodology, pseudo-coordinates and pseudo-speeds were introduced felicitously into the kinematic chain to allow rigorous inclusion of holonomic or non-holonomic constraints that occur among the hybrid parameter bodies; including intra-domain constraints applied within distributed parameter bodies in the system. In this paper, a HPMBS that exhibits intra-domain holonomic constraints and is, for comparison purposes, analytically tractable by hand via Hamilton's principle is studied. The purpose of this comparison and study is to evince the utility and effectiveness of the pseudo-variable based technique. This is accomplished by demonstrating the novel and judicious use of pseudo-coordinates and speeds in modeling HPMBS. It is shown, by allowing the natural description of the kinematic chain (open or closed) to be manifested with the natural pseudo-coordinates and speeds, that low-order models can accurately describe the dynamics of complicated HPMBS. Thus, the trend of ever-increasing degrees of freedom in the modeling of complicated flexible systems is also shown to be unnecessary, provided the proper natural finesse is applied.

On the efficacy of pseudo-coordinates—Part 2: moving boundary constraints

In this paper an argument is presented in favor of utilizing felicitous or natural coordinates in the model formulation of complex hybrid parameter multiple body mechanical systems (HPMBS). Specifically for this paper, HPMBS that consist of continuua that are subjected to spatially and temporally varying non-holonomic boundary conditions. This is the second paper of a two part series of papers that is presented to clarify the novelty and usefulness of a recently developed Gibbs–Appell type projection based HPMBS modeling tool. The purpose of the paper is to show that with the novel use of pseudo-coordinates and speeds (as defined by the author) it is completely natural to provide minimal configuration space dimensionality yet still retain rigorous analytical formulation tractability. Presented in this work, as a demonstrative arguing point, is the development of the hybrid parameter motion equations for a rolling flexible-disk material cutting device. This device consists of a circular flexible continuum (the cutter) along with the requisite mounting rigid hub and handle. This non-holonomically constrained device is modeled executing spatial motion constrained to the plane via moving constraints applied to the boundary of the planar continuum. Also included in this work are numerical results bolstering the claims made herein. These numerical results demonstrate that the methodology elucidated provides low-order models suitable for modeling complicated devices. These low-order models are in contrast to the current modeling trend of ever-increasing degrees of freedom.

Systematic closed form modeling of hybrid parameter multiple body systems

Presented in this paper is a systematic approach to modeling non-holonomic hybrid parameter multiple body systems. The continuum bodies are represented with the postulates usually associated to the non-linear theories, the Timoshenko (like) beam theories, the higher order plate and shell theories, and the rational theories (e.g. rods) with intrinsic rotary inertia properties. The methodology is an extension of previous work. It is founded in variational principles, but uses vector algebra to eliminate tedium. The variational nature of the methodology allows rigorous equation formulation providing not only the complete non-linear hybrid differential equations, but also the boundary conditions. The methodology is formulated satisfying general non-holonomic constraints; it produces a minimal realization. The spatial dimensions of the continuua are not restricted and the inter-body connections are completely general. To demonstrate the application of the technique, a two-link elastic pendulum or manipulator is modeled. The algorithmic modeling steps are demonstrated. Numerical simulations are presented.

MODELLING, SIMULATION AND EXPERIMENTAL VERIFICATION OF CONTACT/IMPACT DYNAMICS IN FLEXIBLE MULTI-BODY SYSTEMS

Contact/impact in flexible multibody systems that undergo pre-contact free motion, contact/impact and post-impact constraint motion is modelled. The shortcomings of using coefficients of restitution, penalty parameters and Lagrange multipliers are overcome by using a methodology for modelling non-holonomic hybrid parameter multiple body systems (HPMBS). The minimum set of equations required for the systems were developed using the methodology. All the dynamics of the manipulator (elaborated in the section below) are modelled with very few degrees of freedom. The model was experimentally validated via a planar two-link flexible manipulator that undergoes the complete motion regime. The comparisons show that the model predicts the complete dynamics of the manipulator accurately and the transition between the different modes of motion is seamless.

Contact/Impact in Hybrid Parameter Multiple Body Mechanical Systems—Extensions for Higher-Order Continuum Models

In recent work the author presented a systematic formulation of hybrid parameter multiple body mechanical systems (HPMBS) undergoing contact/impact motion. The method rigorously models all motion regimes of hybrid multiple body systems (i.e., free motion, contact/impact motion, and constrained motion), utilizing minimal sets of hybrid differential equations; Lagrange multipliers are not required. The contact/impact regime was modeled via the idea of instantaneously applied nonholonomic constraints. The technique previously presented did not include the possibility of continuum assumptions along the lines of Timoshenko beams, higher order plate theories, or rational theories considering intrinsic spin-inertia. In this technical brief, the above-mentioned method is extended to include the higher-order continuum assumptions which eliminates the continuum shortfalls from the previous work. The main contributions of this work include: 1) the previous work is rigorously extended, and 2) the fact that coefficients of restitution are not required for modeling the momentum exchange between motion regimes of HPMBS. The field and boundary equations provide the needed extra equations that are used to supply post-collision pointwise relationships for the generalized velocities and velocity fields.

ON MODELLING VARIABLE STRUCTURE DYNAMICS OF HYBRID PARAMETER MULTIPLE BODY SYSTEMS

Recently a methodology for closed-form modelling of non-holonomic hybrid parameter multiple body systems (HPMBS) was presented. The tool provides a means to model non-holonomic HPMBS with sets of minimal closed-form hybrid differential equations; these equations are minimal in the sense that there are no Lagrange multiplier equations required. Also provided by the method is a means to generate closed-form models of the momentum change imparted on a HPMBS when the system undergoes impulsive loads or configuration space changes. In this paper, these techniques will be merged to form an algorithm for modelling HPMBS under the influence of changes in the configuration space due to the variable presence of, or lack of, constraints. It should be noted that restitution coefficients are not needed to describe momentum transferred within the system when the configuration changes. Included in this work is the explicit minimal closed-form model of a flexible two-link planar manipulator interacting with a work-piece and the numerical simulation of this system under various conditions. The process of modelling this system will elucidate the algorithm which is applicable to higher order systems.

SYMBOLIC EQUATION PROCESSING UTILIZING VECTOR/DYAD NOTATION

In previous work the author has presented a methodology for obtaining closed form equations of motion for general non-holonomic hybrid parameter multiple body systems (HPMBS). It is well known that generating closed form symbolic equations of motion for HPMBS of few degrees of freedom rapidly degenerates to an algebraic soup unless computer tools are utilized. Methods dependent on Lagrange multipliers are virtually unable to provide the constant free (minimal) symbolic equations for systems of moderate to high order due to the complication of eliminating the multipliers. In this paper, the intent is to provide the details, and example use of, a tool, based on commercial symbolic manipulation software, developed to assist the symbolic processing of the equations of motion as previously minimally formulated. The tool presented herein is novel in the way the vector notation of the formulation scheme is retained and the way that frame information is carried in the calculations. The symbolic tool provided in this paper allows the modelling methodology to be a practical tool for generating closed form equations of motion for extremely complicated HPMBS.

Contact/Impact in Hybrid Parameter Multiple Body Mechanical Systems

The multiple motion regime (free/constrained) dynamics of hybrid parameter multiple body (HPMB) systems is addressed. Impact response has characteristically been analyzed using impulse-momentum techniques. Unfortunately, the classical methods for modeling complex HPMB systems are energy based and have proven ineffectual at modeling impact. The problems are exacerbated by the problematic nature of time varying constraint conditions. This paper outlines the reformulation of a recently developed HPMB system modeling methodology into an impulse-momentum formulation, which systematically handles the constraints and impact. The starting point for this reformulation is a variational calculus based methodology. The variational roots of the methodology allows rigorous equation formulation which includes the complete nonlinear hybrid differential equations and boundary conditions. Because the methodology presented in this paper is formulated in the constraint-free subspace of the configuration space, both holonomic and nonholonomic constraints are automatically satisfied. As a result, the constraint-addition/deletion algorithms are not needed. Generalized forces of constraint can be directly calculated via the methodology, so the condition for switching from one motion regime to another is readily determined. The resulting equations provides a means to determine after impact velocities (and velocity fields for distributed bodies) which provide the after collision initial conditions. Finally the paper demonstrates, via example, how to apply the methodology to contact/impact in robotic manipulators and structural systems.

Modeling Hybrid Parameter Multiple Body systems: A different approach

In this paper, the Hybrid Parameter Multiple Body (HPMB) system modeling problem is addressed. Presented, is a systematic methodology for deriving equations of motion for these highly complex systems. The methodology is founded in variational principles, but uses vector algebra to eliminate tedium. The variational nature of the methodology allows rigorous equation formulation providing not only the complete non-linear, hybrid, differential equations, but also the boundary conditions. One novelty of the method lies in the fact that its variational nature is transparent to the user. This enables it to be effectively used by researchers having a working knowledge of Lagrange's method. The methodology is formulated in the system's constraint-free sub-space of the configuration space, thus all constraints (holonomic and nonholonomic) are automatically satisfied. Since Lagrange multipliers are not needed, the method produces a minimal realization. The spatial dimensions of the continuum bodies are not restricted and the inter-body connections are completely general.