Determination of Feasible Muscle Forces in Human Walking Using Static Optimization and Hill's Muscle Dynamics Constraints

Static optimization is a widely used technique for solving the redundant problem of muscle-force determination for given inverse dynamics. There are different cost functions, such as minimal metabolic energy, minimal forces, minimal fatigue, which lead to different results. One problem in this setting is that the resulting time histories of the ensuing equilibrium forces may not be feasible with respect to the muscle dynamics, i.e., the resulting force gradients at the muscle cannot be reproduced by the dynamics of the corresponding muscle. In this paper, a combined method is presented which takes into account the limitations induced by the muscle dynamics by applying static optimization techniques at each time step and prescribing minimal and maximal constraints for the forces by extrapolating the force values from previous steps using feasible muscle activation values. Thus, both the advantages of fast static optimization and the guarantee of feasible time histories of the muscle forces can be achieved. Moreover, joint forces turn out to be 20–40% higher in swing phase than those predicted with classical static optimization. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and identification of emotional aspects of locomotion

The studies of emotional facial expressions and emotional body language are currently receiving a lot of attention in the cognitive sciences. In this project, we study implicit bodily expression of emotions during standard motions, such as walking forwards.An underlying assumption of our work is that all human motion is optimal in some sense and that different emotions induce different objective functions, which result in different deformations of normal motion.We created a 3D rigid-body model of a human of which we use the forward dynamics simulation in an optimal control context. We performed two kinds of optimizations: (i) reconstruction of dynamic quantities, such as joint torques, of pre-recorded data of emotional walking motions and (ii) forward optimization that generates neutral and varied walking motions using different objective functions. Optimizations are performed with the software package MUSCOD-II, which uses a direct multiple-shooting discretization scheme. The results of this work form the foundation for further analysis of emotional motions using inverse optimal control methods.     

Interval oriented multi-section techniques for global optimization

This paper deals with two different optimization techniques to solve the bound-constrained nonlinear optimization problems based on division criteria of a prescribed search region, finite interval arithmetic and interval ranking in the context of a decision maker’s point of view. In the proposed techniques, two different division criteria are introduced where the accepted region is divided into several distinct subregions and in each subregion, the objective function is computed in the form of an interval using interval arithmetic and the subregion containing the best objective value is found by interval ranking. The process is continued until the interval width for each variable in the accepted subregion is negligible. In this way, the global optimal or close to global optimal values of decision variables and the objective function can easily be obtained in the form of an interval with negligible widths. Both the techniques are applied on several benchmark functions and are compared with the existing analytical and heuristic methods.     

Identification of Finite-Motion Parameters of Serial Chains via Measurement of 6 × 6 Stiffness Matrices

This paper describes a procedure for identifying geometric and stiffness parameters of a mechanical serial chain of know structure by measuring spatial 6×6 stiffness matrices at different positions. The method uses standard optimization routines to determine model parameters such that the model stiffness matrix features in the Frobenius norm the closest distance possible to the measured matrix. From this local identification, a rough model of parameters of finite-motion is created, from which new measuring positions are guessed. By applying this step repeatedly, a model for finite-displacement parameters can be obtained by a sequence of small force-displacement tests. The method is tested with a dummy device consisting of a revolute joint connecting two rigid links dressed with soft material to mimic for example muscle masses of a surrogate mechanism for the elbow joint of a human arm. Two robots grasping the upper and lower arm generate the motion while the force measurement is carried out by a six-axis force sensor. This makes the method potentially suitable for detecting anatomical parameters by in-vivo measurements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Criteria and dimension reduction of linear multiple criteria optimization problems

Two of the main approaches in multiple criteria optimization are optimization over the efficient set and utility function program. These are nonconvex optimization problems in which local optima can be different from global optima. Existing global optimization methods for solving such problems can only work well for problems of moderate dimensions. In this article, we propose some ways to reduce the number of criteria and the dimension of a linear multiple criteria optimization problem. By the concept of so-called representative and extreme criteria, which is motivated by the concept of redundant (or nonessential) objective functions of Gal and Leberling, we can reduce the number of criteria without altering the set of efficient solutions. Furthermore, by using linear independent criteria, the linear multiple criteria optimization problem under consideration can be transformed into an equivalent linear multiple criteria optimization problem in the space of linear independent criteria. This equivalence is understood in a sense that efficient solutions of each problem can be derived from efficient solutions of the other by some affine transformation. As a result, such criteria and dimension reduction techniques could help to increase the efficiency of existing algorithms and to develop new methods for handling global optimization problems arisen from multiple objective optimization.     

On the historical development of human walking dynamics

Human walking is an interdisciplinary research topic. It started in the Ancient World with early observations and questions in philosophy, it was treated in the Middle Ages with experiments and data collection by physiologists, and in the 20th century models were designed, equations of motion were generated and simulations by multibody dynamics approaches were performed. More recently parameter optimization was used to overcome the problem of muscle overactuation and inverse dynamics methods were introduced. In the first part of the paper the early developments and mechanism models are described while in the second part parameter optimization is applied to gait disorder simulations as an example of recent research results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

复合材料破坏准则的唯象理论

本文研究了复合材料的二阶型和三阶型张量多项式破坏准则的几种主要理论。提出了用非线性优化方法来确定准则中的影响系数,并将优化结果与几种典型的复合材料实验数据及各种理论的计算结果进行了讨论和比较,结果表明系数优化的方法是有效的。     

A probabilistic description of footbridges vibration serviceability

Vibration serviceability of pedestrian bridges is considered. Imperfections in the human walking force are described as changes in walking factors. Probability distributions of walking frequencies, step length, and amplitude of walking force are taken into account. The imperfections can influence the level of footbridge vibration response. The Monte Carlo method is applied. The generated realizations of the loads formulate a set of initial data for numerical calculations. The probability distribution of maximal vertical acceleration is obtained. The results are presented as a probability that the required level of vibration will not be exceeded. A reliability describing the footbridges serviceability is also estimated. The probabilistic based approach can be used for future optimalizations of footbridge design codes due to the dynamic force loads. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sensitivity Analysis of Human Leg Metabolical Costs

The human walking is characterized by skeletal dynamics and muscle excitation patterns minimizing the metabolical energy. This criterion is applied to assess the performance of lower limb prosthetic devices, and to evaluate therapies for patients presenting gait disorders. It is desirable, therefore, to dispose models of the human normal and pathological gaits capable of estimating the metabolical energy expenditure. For the swing phase of normal and pathological gaits a musculoskeletal model of the lower limb is presented to estimate metabolical energy expenditure. The mechanical model has three degrees of freedom and is actuated by eight Hill-type muscle units, and the model for the metabolical costs is adopted from literature. In this paper a combination of inverse and direct dynamics is used, and a sensitivity analysis of the dynamical behavior and the corresponding metabolical costs estimations with respect to parametrized neural excitations is performed. The leg motions are based on experiments in a gait analysis laboratory. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotation-Symmetric Referencebodys For Energy Efficient Flow Around Bodies

Topology optimization is used to optimize problems of flow around bodies and problems of guided flow. Within the context of research, optimization criteria are developed to increase the energy efficiency of these problems [1] [2] [3] [4] [5]. In order to evaluate the new criteria in respect to the increasing of energy efficiency, reference bodies for different Reynolds numbers in combination with given design space limitations are needed. Therefore, an optimal body at Reynolds number against 0 was analytically determined by Bourot [6]. At higher Reynolds numbers, in the range of laminar and turbulent flows, no analytical solution is known. Accordingly, reference bodies are calculated by CFD calculations at three technical relevant Reynolds numbers (1.000, 32.000, 100.000) in combination with parameter optimization. The cross section of the bodies is described by a parameterized model. To get the optimal body, a parameter optimization based on a “brute force”; algorithm is used to optimize with regard to the friction loss and pressure loss in order to minimize the total loss (c_d-value). The result is an optimal parameter constellation, depending on the Reynolds number. Within the results, it is possible to develop the optimal geometries. The identified characteristics of the flow field around these bodies are used as base for new optimization criteria. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forward dynamics simulation of human body under tilting perturbations

Human body uses different strategies to maintain its stability and these strategies vary from fixed-foot strategies to strategies which foot is moved in order to increase the support base. Tilting movement of foot is one type of the perturbations usually is exposed to human body. In the presence of such perturbations human body must employ appropriate reactions to prevent threats like falling. But it is not clear that how human body maintains its stability by central nervous system (CNS). At present study it is tried that by presenting a musculoskeletal model of human lower extremity with four links, three degrees of freedom (DOF) and eight skeletal muscles, the level of muscle activations causes the maintenance of stability, be investigated. Using forward dynamics solution, leads to a more general problem, rather than inverse dynamics. Hence, forward dynamics solution by forward optimization has been used for solving this highly nonlinear problem. To this end, first the system’s equations of motion has been derived using lagrangian dynamics. Eight Hill-type muscles as actuators of the system were modeled. Because determination of muscle forces considering their number is an undetermined problem, optimization of an appropriate goal function should be practiced. For optimization problem, the characteristics of genetic algorithms as a method based on direct search, and the direct collocation method, has been profited. Also by considering requirements of problem, some constraints such as conservation of model stability are entered into optimization procedure. Finally to investigate validation of model, the results from optimization and experimental data are compared and good agreements are obtained.     

Analysis and assessment of foot posture in children with flat feet

The paper presents the methods for foot assessment such as: X-ray, graphic contour testing, ground reaction force, and pressure distribution. Some of this methods can be used just in static. The platforms with sensors are very good tools in assessment foot posture during human walking. The evaluation was carried on 20 typical subjects and 60 children with foot deformities. GRF parameters, compared to clinical radiographic measurement demonstrated generally a good correlation. The analysis of fround reaction force and pressure distribution represent a strong tool to validate clinical tests and can be used for screening, diagnosis and treatment choice in clinical practice. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modular and efficient approach to computational modeling and sensitivity analysis of robot and human motion dynamics

In this paper a new class library for the computation of the forward multi-body-system (MBS) dynamics of robots and biomechanical models of human motion is presented. By the developed modular modeling approach the library can be flexibly extended by specific modeling elements like joints with specific geometry or different muscle models and thus can be applied efficiently for a number of dynamic simulation and optimization problems. The library not only provides several methods for solving the forward dynamics problem (like articulated body or composite rigid body algorithms) which can transparently be exchanged. Moreover, the numerical solution of optimal control problems, like in the forward dynamics optimization of human motion, is significantly facilitated by the computation of the sensitivity matrix with respect to the control variables. Examples are given to demonstrate the efficiency of the approach. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tracking analysis for general linearly coupled dynamical systems

Tracking analysis problem is studied for general linearly coupled dynamical systems in this paper. One challenging and essential question for this issue is that: At least how many nodes should be informed about the objective tracking signal? This paper is devoted to answer this question. Two dynamical network models are considered. The first one, each individual has its own dynamics and simultaneously influenced by its neighbors’ information. The dynamics of itself could be stable, periodic, semi-periodic, and chaotic. The second one, each individual update its state just according to the error states different from its communicated neighbors. The main contribution of this paper is that the minimum number of controllers is designed to force the state of each agent to the desired objective by fully utilizing the structure of the network. The convergence rate can also be estimated. The topology of the underlying network can be directed and hierarchical. Some simple criteria are given to judge whether the tracking control can be successful. In addition, numerical examples are given to show the validity of the analytical results.     

Particle swarm optimization with age-group topology for multimodal functions and data clustering

This paper proposes particle swarm optimization with age-group topology (PSOAG), a novel age-based particle swarm optimization (PSO). In this work, we present a new concept of age to measure the search ability of each particle in local area. To keep population diversity during searching, we separate particles to different age-groups by their age and particles in each age-group can only select the ones in younger groups or their own groups as their neighbourhoods. To allow search escape from local optima, the aging particles are regularly replaced by new and randomly generated ones. In addition, we design an age-group based parameter setting method, where particles in different age-groups have different parameters, to accelerate convergence. This algorithm is applied to nonlinear function optimization and data clustering problems for performance evaluation. In comparison against several PSO variants and other EAs, we find that the proposed algorithm provides significantly better performances on both the function optimization problems and the data clustering tasks.     

Axiomatizations of the Euclidean compromise solution

A multicriteria optimization problem is one of choosing an alternative that optimizes several—possibly conflicting—objective functions simultaneously. The utopia point of a multicriteria optimization problem is the vector that specifies for each objective function the most favorable feasible value. The Euclidean compromise solution in multicriteria optimization is a solution that selects from a feasible set the alternative such that its vector of criteria values has minimal Euclidean distance to the utopia point. This paper provides several axiomatic characterizations of the Euclidean compromise solution that are based on consistency properties.     

Optimization of the number of components in the mixed model using multi-criteria decision-making

The distributions of empirical data are often complex. Such complexity cannot be sufficiently addressed by the individual theoretical statistical distribution function. Furthermore, the selection of the distribution function becomes more complicated when the empirical data present a multi-peak feature. In such a case, the multiple testing criteria and the mixed model must be considered during the selection of an appropriate distribution function. Aiming at this vague challenge, the present paper proposes a novel method for establishing a mixed model that can describe accurately the distribution characteristics of empirical data. Apart from combining the Akike and Bayesian information criteria to define the feasible solutions of the mixed model, this study also utilizes the root mean squared deviation, coefficient of determination, Kolmogorov–Smirnov test statistic, average deviation in cumulative distribution function, and average deviation in probability distribution function as the testing criteria. In addition, a non-linear programming is used to find the weighting factors of each criterion. The multi-criteria decision-making technology is adopted to comprehensively and objectively integrate these testing criteria into a synthetic indicator. Finally, an optimization algorithm is proposed to determine the optimal number of components in the mixed model. The illustrated results of the simulated data and measured signals confirm that this approach can estimate precisely the number of components as well as establish a highly accurate mixed model.     

Investigation of optimal bipedal walking gaits subject to different energy-based objective functions

Optimal bipedal walking gaits subject to different energy-based objective functions are investigated using a simple planar rigid body model of a bipedal robot with upper body, thighs and shanks. The robot's segments are connected by revolute joints actuated by electric motors. The actuators' torques are generated by a trajectory tracking controller to produce periodic walking gaits. A numerical optimization routine is used to find optimal reference trajectories for average speeds in the range of 0.3 – 2.3 m/s to investigate the influence of different objective functions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reverse Convex Programming Approach in the Space of Extreme Criteria for Optimization over Efficient Sets

The problem of minimizing a convex function over the difference of two convex sets is called ‘reverse convex program’. This is a typical problem in global optimization, in which local optima are in general different from global optima. Another typical example in global optimization is the optimization problem over the efficient set of a multiple criteria programming problem. In this article, we investigate some special cases of optimization problems over the efficient set, which can be transformed equivalently into reverse convex programs in the space of so-called extreme criteria of multiple criteria programming problems under consideration. A suitable algorithm of branch and bound type is then established for globally solving resulting problems. Preliminary computational results with the proposed algorithm are reported.