An improved inverse dynamics formulation for estimation of external and internal loads during human sagittal plane movements

Planar musculoskeletal models are common in the inverse dynamics analysis of human movements such as walking, running and jumping. The continued interest in such models is justified by their simplicity and computational efficiency. Related to a human planar model, a unified formulation for both the flying and support phases of the sagittal plane movements is developed. The actuation involves muscle forces in the lower limbs and the resultant muscle torques in the other body joints. The dynamic equations, introduced in absolute coordinates of the segments, are converted into useful compact forms using the projective technique. The solution to a determinate inverse dynamics problem allows for the explicit determination of the external reactions (presumed to vanish during the flying phases) and the resultant muscle torques in all the model joints. The indeterminate inverse dynamics problem is then focused on the assessment of muscle forces and joint reaction forces selectively in the supporting lower limb. Numerical results of the inverse dynamics simulation of sample sagittal plane movements are reported to illustrate the validity and effectiveness of the improved formulation.     

Internal models for motor control and trajectory planning

A number of internal model concepts are now widespread in neuroscience and cognitive science. These concepts are supported by behavioral, neurophysiological, and imaging data; furthermore, these models have had their structures and functions revealed by such data. In particular, a specific theory on inverse dynamics model learning is directly supported by unit recordings from cerebellar Purkinje cells. Multiple paired forward inverse models describing how diverse objects and environments can be controlled and learned separately have recently been proposed. The 'minimum variance model' is another major recent advance in the computational theory of motor control. This model integrates two furiously disputed approaches on trajectory planning, strongly suggesting that both kinematic and dynamic internal models are utilized in movement planning and control.     

Dynamic analysis of load carriage biomechanics during level walking

This paper describes an investigation into the biomechanical effects of load carriage dynamics on human locomotion performance. A whole body, inverse dynamics gait model has been developed which uses only kinematic input data to define the gait cycle. To provide input data, three-dimensional gait measurements have been conducted to capture whole body motion while carrying a backpack. A nonlinear suspension model is employed to describe the backpack dynamics. The model parameters for a particular backpack system can be identified using a dynamic load carriage test-rig. Biomechanical assessments have been conducted based on combined gait and pack simulations. It was found that the backpack suspension stiffness and damping have little effect on human locomotion energetics. However, decreasing suspension stiffness offers important biomechanical advantages. The peak values of vertical pack force, acting on the trunk, and lower limb joint loads are all moderated. This would reduce shoulder strap pressures and the risk of injury when heavy loads are carried.     

Prediction of ground reaction forces and moments during various activities of daily living

Inverse dynamics based simulations on musculoskeletal models is a commonly used method for the analysis of human movement. Due to inaccuracies in the kinematic and force plate data, and a mismatch between the model and the subject, the equations of motion are violated when solving the inverse dynamics problem. As a result, dynamic inconsistency will exist and lead to residual forces and moments. In this study, we present and evaluate a computational method to perform inverse dynamics-based simulations without force plates, which both improves the dynamic consistency as well as removes the model?s dependency on measured external forces. Using the equations of motion and a scaled musculoskeletal model, the ground reaction forces and moments (GRF&Ms) are derived from three-dimensional full-body motion. The method entails a dynamic contact model and optimization techniques to solve the indeterminacy problem during a double contact phase and, in contrast to previously proposed techniques, does not require training or empirical data. The method was applied to nine healthy subjects performing several Activities of Daily Living (ADLs) and evaluated with simultaneously measured force plate data. Except for the transverse ground reaction moment, no significant differences (P>0.05) were found between the mean predicted and measured GRF&Ms for almost all ADLs. The mean residual forces and moments, however, were significantly reduced (P>0.05) in almost all ADLs using our method compared to conventional inverse dynamic simulations. Hence, the proposed method may be used instead of raw force plate data in human movement analysis using inverse dynamics.     

A 3-D dynamic model of human finger for studying free movements.

The purpose of this work is to develop a 3D inverse dynamic model of the human finger for estimating the muscular forces involved during free finger movements. A review of the existing 3D models of the fingers is presented, and an alternative one is proposed. The validity of the model has been proved by means of two simulations: free flexion-extension motion of all joints, and free metacarpophalangeal (MCP) adduction motion. The simulation shows the need for a dynamic model including inertial effects when studying fast movements and the relevance of modelling passive forces generated by the structures studying free movements, such as the force exerted by the muscles when they are stretched and the passive action of the ligaments over the MCP joint in order to reproduce the muscular force pattern during the simulation of the free MCP abduction-adduction movements.     

A Lagrange-based generalised formulation for the equations of motion of simple walking models

Simple 2D models of walking often approximate the human body to multi-link dynamic systems, where body segments are represented by rigid links connected by frictionless hinge joints. Performing forward dynamics on the equations of motion (EOM) of these systems can be used to simulate their movement. However, deriving these equations can be time consuming. Using Lagrangian mechanics, a generalised formulation for the EOM of n-link open-loop chains is derived. This can be used for single support walking models. This has an advantage over Newton–Euler mechanics in that it is independent of coordinate system and prior knowledge of the ground reaction force (GRF) is not required. Alternative strategies, such as optimisation algorithms, can be used to estimate joint activation and simulate motion. The application of Lagrange multipliers, to enforce motion constraints, is used to adapt this general formulation for application to closed-loop chains. This can be used for double support walking models. Finally, inverse dynamics are used to calculate the GRF for these general n-link chains. The necessary constraint forces to maintain a closed-loop chain, calculated from the Lagrange multipliers, are one solution to the indeterminate problem of GRF distribution in double support models. An example of this method’s application is given, whereby an optimiser estimates the joint moments by tracking kinematic data.     

Virtual trajectory and stiffness ellipse during multijoint arm movement predicted by neural inverse models

We predict the virtual trajectories and stiffness ellipses during multijoint arm movements by computer simulations. A two-link manipulator with four single-joint muscles and two double-joint muscles is used as a model of the human arm. Physical parameters of the model are derived from several experimental data. Among them, special emphasis is put on low values of the dynamic hand stiffness recently measured during single joint and multijoint movements. The feedback-error-learning scheme to acquire the inverse dynamics model and the inverse statics model is utilized for this prediction. The virtual trajectories are much more complex than the actual trajectories. This indicates that planning the virtual trajectory is as difficult as solving the inverse dynamics problem for medium and fast movements, and simply falsifies the advocated computational advantage of the virtual trajectory control hypothesis. Thus, we conclude that learning inverse models is essential even in the virtual trajectory control framework. Finally, we propose a new computational model to learn the complicated shape of the virtual trajectories by integrating the virtual trajectory control and the feedback-error-learning scheme.     

Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors

In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.     

The inverse dynamics problem of neuromuscular control

The myoskeletal inverse dynamics problem and the myocybernetic control inverse problem were investigated with respect to their ill-posedness. The first problem consists of finding from observed experimental motion and reaction force data the resultant muscle moments that generated the observed motion, while the second aims at finding the corresponding neural controls. It is shown that both problems belong to the class of incorrectly posed (ill-posed) problems that, by definition, do not possess unique solutions. To illustrate this point, results of a forward dynamics simulation of a comprehensive neuromusculoskeletal model of the human body are presented. These results demonstrate that fairly chaotic neural control perturbations have very little influence on the resulting motion trajectory, at least in the present example. While a regularization procedure may be applied to solve successfully the myoskeletal inverse dynamics problem, the myocybernetic control inverse problem is unsolvable. The latter fact has the important implication that, based on the somatosensory inputs it receives, the pars intermedia in the cerebellum is not able to control individual motor unit stimulation rates and recruitment patterns but only whole muscles by means of a single compound signal. The latter signal is identified as the “common drive.” Presumably at the spinal level, special neural circuits are used to decompose the common drive signal into motor unit recruitment patterns and stimulation rates that are specific for a given mode of contraction and probably obey certain optimality principles. Received: 26 February 1999 / Accepted in revised form: 11 June 1999     

Are natural coordinates a useful tool in modeling planar biomechanical linkages?

The paper analyzes the use of natural coordinates in modeling and simulation of musculoskeletal models of the human body. A biomechanical model of the lower extremity of the human body was constructed for the analysis. It consists of three anatomical segments described by eight natural coordinates located at the joints. The developed model was applied to solve three classic dynamics problems of human motion: inverse dynamics, direct dynamics, and static optimization. The analysis covers the raising of a leg together with; the time characteristics of the resultant net torques at the basic joints of the leg (inverse dynamics), the time histories of natural coordinates and their velocities (direct dynamics) as well as the time-varying muscle force patterns (static optimization). In order to check the numerical efficiency of modeling in the natural coordinates' environment the results were compared with the ones received through generalized coordinates. Some conclusions drawn from this comparison and final remarks referring to the biomechanical identification of the analyzed motor task were included in the paper.     

Wide-field motion integration in fly VS cells: insights from an inverse approach

Fly lobula plate tangential cells are known to perform wide-field motion integration. It is assumed that the shape of these neurons, and in particular the shape of the subclass of VS cells, is responsible for this type of computation. We employed an inverse approach to investigate the morphology-function relationship underlying wide-field motion integration in VS cells. In the inverse approach detailed, model neurons are optimized to perform a predefined computation: here, wide-field motion integration. We embedded the model neurons to be optimized in a biologically plausible model of fly motion detection to provide realistic inputs, and subsequently optimized model neuron with and without active conductances (g(Na), g(K), g(K(Na))) along their dendrites to perform this computation. We found that both passive and active optimized model neurons perform well as wide-field motion integrators. In addition, all optimized morphologies share the same blueprint as real VS cells. In addition, we also found a recurring blueprint for the distribution of g(K) and g(Na) in the active models. Moreover, we demonstrate how this morphology and distribution of conductances contribute to wide-field motion integration. As such, by using the inverse approach we can predict the still unknown distribution of g(K) and g(Na) and their role in motion integration in VS cells.     

Cell-Specific Cardiac Electrophysiology Models

The traditional cardiac model-building paradigm involves constructing a composite model using data collected from many cells. Equations are derived for each relevant cellular component (e.g., ion channel, exchanger) independently. After the equations for all components are combined to form the composite model, a subset of parameters is tuned, often arbitrarily and by hand, until the model output matches a target objective, such as an action potential. Unfortunately, such models often fail to accurately simulate behavior that is dynamically dissimilar (e.g., arrhythmia) to the simple target objective to which the model was fit. In this study, we develop a new approach in which data are collected via a series of complex electrophysiology protocols from single cardiac myocytes and then used to tune model parameters via a parallel fitting method known as a genetic algorithm (GA). The dynamical complexity of the electrophysiological data, which can only be fit by an automated method such as a GA, leads to more accurately parameterized models that can simulate rich cardiac dynamics. The feasibility of the method is first validated computationally, after which it is used to develop models of isolated guinea pig ventricular myocytes that simulate the electrophysiological dynamics significantly better than does a standard guinea pig model. In addition to improving model fidelity generally, this approach can be used to generate a cell-specific model. By so doing, the approach may be useful in applications ranging from studying the implications of cell-to-cell variability to the prediction of intersubject differences in response to pharmacological treatment.     

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.     

Flexible multibody simulation approach in the analysis of tibial strain during walking

Strains within the bone tissue play a major role in bone (re)modeling. These small strains can be assessed using experimental strain gage measurements, which are challenging and invasive. Further, the strain measurements are, in practise, limited to certain regions of superficial bones only, such as the anterior surface of the tibia. In this study, tibial strains occurring during walking were estimated using a numerical approach based on flexible multibody dynamics. In the introduced approach, a lower body musculoskeletal model was developed by employing motion capture data obtained from walking at a constant velocity. The motion capture data were used in inverse dynamics simulation to teach the muscles in the model to replicate the motion in forward dynamics simulation. The maximum and minimum tibial principal strains predicted by the model were 490 and -588 microstrain, respectively, which are in line with literature values from in vivo measurements. In conclusion, the non-invasive flexible multibody simulation approach may be used as a surrogate for experimental bone strain measurements and thus be of use in detailed strain estimations of bones in different applications.     

Numerical simulation of cardiovascular dynamics with healthy and diseased heart valves

This paper presents a new concentrated parameter model for cardiovascular dynamics that includes an innovative model of heart valve dynamics, which is embedded in the overall model of the four chambers of the heart and the systemic and pulmonary circulation loops. The heart chambers are described with a variable elastance model, and the systemic and pulmonary loops are described with modified Windkessel models. In modelling the heart valve dynamics, the various factors that influence the valve motion are examined, and the governing differential equation for valve motion is derived. The heart valve model includes the influence of the blood pressure effect, the friction effect from the tissue, and from blood motion. As improvement from previous works, the contribution of the blood vortex effect in the vicinity of the valve leaflets to valve motion is specially considered. The proposed model is then used in simulation of healthy and certain pathological conditions such as mitral valve stenosis and aortic regurgitation. The predicted results agree well with results illustrated in cardiology textbooks.     

A 3-D model of tumor progression based on complex automata driven by particle dynamics

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.     

Inverse modeling of dog airway and respiratory system impedances

Mechanical parameters of the respiratory system are often estimated from respiratory impedances using lumped-element inverse models. One such six-element model is composed of an airway branch [with a resistance (Raw) and inertance (Iaw)] separated from a tissue branch [with a resistance (Rt), inertance (It), and compliance (Ct)] by a shunt compliance representing alveolar gas compression (Cg). Even though the airways are known to have frequency-dependent resistance and inertance, these inverse models have been composed of linear frequency-independent elements. In this study we investigated the use of inverse models where the airway branch was represented by a frequency-independent Raw and Iaw, a Raw that is linearly related to frequency and an Iaw that is independent of frequency, and a system of identical parallel tubes the impedance of which was computed from the tube radius and length. These inverse models were used to analyze airway and respiratory impedances between 2 and 1,024 Hz that were predicted from an anatomically detailed forward model. The forward model represented the airways by an asymmetrically branched network with a terminal impedance representative of known Cg, Rt, It, and Ct. For respiratory impedances between 2 and 128 Hz, all models fit the data reasonably well, and reasonably accurate estimates of Cg, Rt, It, and Ct were extracted from these data. For data above 200 Hz, however, only the multiple-tube model accurately fitted respiratory impedances (Zrs). This model fitted the Zrs data best when composed of 27 tubes, each having a radius of 0.148 cm and a length of 16.5 cm.     

Computational nature of human adaptive control during learning of reaching movements in force fields

Learning to make reaching movements in force fields was used as a paradigm to explore the system architecture of the biological adaptive controller. We compared the performance of a number of candidate control systems that acted on a model of the neuromuscular system of the human arm and asked how well the dynamics of the candidate system compared with the movement characteristics of 16 subjects. We found that control via a supra-spinal system that utilized an adaptive inverse model resulted in dynamics that were similar to that observed in our subjects, but lacked essential characteristics. These characteristics pointed to a different architecture where descending commands were influenced by an adaptive forward model. However, we found that control via a forward model alone also resulted in dynamics that did not match the behavior of the human arm. We considered a third control architecture where a forward model was used in conjunction with an inverse model and found that the resulting dynamics were remarkably similar to that observed in the experimental data. The essential property of this control architecture was that it predicted a complex pattern of near-discontinuities in hand trajectory in the novel force field. A nearly identical pattern was observed in our subjects, suggesting that generation of descending motor commands was likely through a control system architecture that included both adaptive forward and inverse models. We found that as subjects learned to make reaching movements, adaptation rates for the forward and inverse models could be independently estimated and the resulting changes in performance of subjects from movement to movement could be accurately accounted for. Results suggested that the adaptation of the forward model played a dominant role in the motor learning of subjects. After a period of consolidation, the rates of adaptation in the internal models were significantly larger than those observed before the memory had consolidated. This suggested that consolidation of motor memory coincided with freeing of certain computational resources for subsequent learning. Received: 01 January 1998 / Accepted in revised form: 26 January 1999