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Human gait analysis is a complex problem in biomechanics because of highly nonlinear human motion equations, muscle dynamics, and foot-ground contact.
Despite a large number of studies in human gait analysis, predictive human gait simulation is still challenging researchers to increase the accuracy and computational efficiency for evaluative studies (e.g., model-based assistive device controllers, surgical intervention planning, athletic training, and prosthesis and orthosis design).
To assist researchers in this area, this review article classifies recent predictive simulation methods for human gait analysis according to three categories: (1) the human models used (i.e., skeletal, musculoskeletal and neuromusculoskeletal models), (2) problem formulation, and (3) simulation solvers.
Human dynamic models are classified based on whether muscle activation and/or contraction dynamics or joint torques (instead of muscle dynamics) are employed in the analysis. Different formulations use integration and/or differentiation or implicit-declaration of the dynamic equations. A variety of simulation solvers (i.e., semi- and fully-predictive simulation methods) are studied. Finally, the pros and cons of the different formulations and simulation solvers are discussed.

Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community. Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary. A number of potential application problems are also discussed. To offer the readers insights on the prominent developments in the field of bilevel optimization, we have performed an automated text-analysis of an extended list of papers published on bilevel optimization to date. This paper should motivate evolutionary computation researchers to pay more attention to this practical yet challenging area.

Abstract The conventional robotic assistive device was based on pre-programmed functions by the robot expert. This makes it difficult for stroke patients use it effectively due to difficulty of torque setting that is suitable for the user movement. Electromyography (EMG) signal measures the electrical signal of muscle contraction.The EMG-based robotics assistive technology would enable the stroke patients to control the robot movement according to the user's own strength of natural movement.This paper discusses the mapping of surface electromyography signals (sEMG) to torque for robotic rehabilitation. Particle swarm optimization (PSO) has been applied as a control algorithm for a number of selected mathematical models. sEMG signals were determined as input data to the mathematical model where parameters of the mathematical model were optimized using PSO. Hence, the good correlated estimated torque as output was obtained.

This paper reports on the use of a genetic algorithm-based technique, GABBA, to solve bi-level linear programming (BLLP) problems. GABBA is used to generate the leader's decision vector, and the follower's reaction is obtained from the solution to the linear program. GABBA is different from the usual genetic algorithms because we only use mutations, alleles of base-10 numbers, and a survival strategy that is suited to BLLP. Results show that while it takes more cpu time, GABBA gets closer to the global optimum than Bard's [1983] grid search techniques for problems of most sizes.

The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

Bilevel optimization problems are a class of challenging optimization
problems, which contain two levels of optimization tasks. In these problems,
the optimal solutions to the lower level problem become possible feasible
candidates to the upper level problem. Such a requirement makes the
optimization problem difficult to solve, and has kept the researchers busy
towards devising methodologies, which can efficiently handle the problem.
Despite the efforts, there hardly exists any effective methodology, which is
capable of handling a complex bilevel problem. In this paper, we introduce
bilevel evolutionary algorithm based on quadratic approximations (BLEAQ) of
optimal lower level variables with respect to the upper level variables. The
approach is capable of handling bilevel problems with different kinds of
complexities in relatively smaller number of function evaluations. Ideas from
classical optimization have been hybridized with evolutionary methods to
generate an efficient optimization algorithm for generic bilevel problems. The
efficacy of the algorithm has been shown on two sets of test problems. The
first set is a recently proposed SMD test set, which contains problems with
controllable complexities, and the second set contains standard test problems
collected from the literature. The proposed method has been evaluated against
two benchmarks, and the performance gain is observed to be significant.

A general optimization formulation for transition walking prediction using 3D skeletal model is presented. The formulation is based on a previously presented one-step walking formulation (Xiang et al., Int J Numer Methods Eng 79:667–695, 2009b). Two basic transitions are studied: walk-to-stand and slow-to-fast walk. The slow-to-fast transition is used to connect slow walk to fast walk by using a step-to-step transition formulation. In addition, the speed effects on the walk-to-stand motion are investigated. The joint torques and ground reaction forces (GRF) are recovered and analyzed from the simulation. For slow-to-fast walk transition, the predicted ground reaction forces in step transition is even larger than that of the fast walk. The model shows good correlation with the experimental data for the lower extremities except for the standing ankle profile. The optimal solution of transition simulation is obtained in a few minutes by using predictive dynamics method.

Bilevel programming problems are often found in practice. In this paper, we handle one such bilevel application problem from the domain of environmental economics. The problem is a Stakelberg game with multiple objectives at the upper level, and a single objective at the lower level. The leader in this case is the regulating authority, and it tries to maximize its total tax revenue over multiple periods while trying to minimize the environmental damages caused by a mining company. The follower is the mining company whose sole objective is to maximize its total profit over multiple periods under the limitations set by the leader. The solution to the model contains the optimal taxation and extraction decisions to be made by the players in each of the time periods. We construct a simplistic model for the Stackelberg game and provide an analytical solution to the problem. Thereafter, the model is extended to incorporate realism and is solved using a bilevel evolutionary algorithm capable of handling multiple objectives.

Unilateral below-knee amputees develop abnormal gait characteristics that include bilateral asymmetries and an elevated metabolic cost relative to non-amputees. In addition, long-term prosthesis use has been linked to an increased prevalence of joint pain and osteoarthritis in the intact leg knee. To improve amputee mobility, prosthetic feet that utilize elastic energy storage and return (ESAR) have been designed, which perform important biomechanical functions such as providing body support and forward propulsion. However, the prescription of appropriate design characteristics (e.g., stiffness) is not well-defined since its influence on foot function and important in vivo biomechanical quantities such as metabolic cost and joint loading remain unclear. The design of feet that improve these quantities could provide considerable advancements in amputee care. Therefore, the purpose of this study was to couple design optimization with dynamic simulations of amputee walking to identify the optimal foot stiffness that minimizes metabolic cost and intact knee joint loading. A musculoskeletal model and distributed stiffness ESAR prosthetic foot model were developed to generate muscle-actuated forward dynamics simulations of amputee walking. Dynamic optimization was used to solve for the optimal muscle excitation patterns and foot stiffness profile that produced simulations that tracked experimental amputee walking data while minimizing metabolic cost and intact leg internal knee contact forces. Muscle and foot function were evaluated by calculating their contributions to the important walking subtasks of body support, forward propulsion and leg swing. The analyses showed that altering a nominal prosthetic foot stiffness distribution by stiffening the toe and mid-foot while making the ankle and heel less stiff improved ESAR foot performance by offloading the intact knee during early to mid-stance of the intact leg and reducing metabolic cost. The optimal design also provided moderate braking and body support during the first half of residual leg stance, while increasing the prosthesis contributions to forward propulsion and body support during the second half of residual leg stance. Future work will be directed at experimentally validating these results, which have important implications for future designs of prosthetic feet that could significantly improve amputee care.

We propose a computational procedure for inferring the cost functions that, according to the Principle of Optimality, underlie experimentally observed motor strategies. In the current use of optimization-based mathematical models of neuro-musculoskeletal systems, the cost functions are not known a-priori, since they can not be directly observed or measured on the real bio-system. Consequently, cost functions need to be hypothesized for any given motor task of interest, based on insight into the physical processes that govern the problem.This work tries to overcome the need to hypothesize the cost functions, extracting this non-directly observable information from experimental data. Optimality criteria of observed motor tasks are here indirectly derived using: (a) a mathematical model of the bio-system; and (b) a parametric mathematical model of the possible cost functions, i.e. a search space constructed in such a way as to presumably contain the unknown function that was used by the bio-system in the given motor task of interest. The cost function that best matches the experimental data is identified within the search space by solving a nested optimization problem. This problem can be recast as a non-linear programming problem and therefore solved using standard techniques.The methodology is here formulated for both static and dynamic problems, and then tested on representative examples.

The purpose of this paper is to present inverse optimal control as a promising approach to transfer biological motions to
robots. Inverse optimal control helps (a) to understand and identify the underlying optimality criteria of biological motions
based on measurements, and (b) to establish optimal control models that can be used to control robot motion. The aim of inverse
optimal control problems is to determine—for a given dynamic process and an observed solution—the optimization criterion that
has produced the solution. Inverse optimal control problems are difficult from a mathematical point of view, since they require
to solve a parameter identification problem inside an optimal control problem. We propose a pragmatic new bilevel approach
to solve inverse optimal control problems which rests on two pillars: an efficient direct multiple shooting technique to handle
optimal control problems, and a state-of-the art derivative free trust region optimization technique to guarantee a match
between optimal control problem solution and measurements. In this paper, we apply inverse optimal control to establish a
model of human overall locomotion path generation to given target positions and orientations, based on newly collected motion
capture data. It is shown how the optimal control model can be implemented on the humanoid robot HRP-2 and thus enable it
to autonomously generate natural locomotion paths.

Lower limb amputees, on average, experience reduced mobility and higher metabolic rates than non-amputees. It may be possible to improve their mobility and metabolic rate with an optimized robotic prosthesis. However, we still need to determine what specific parameters we should optimize. Here, we use large-scale trajectory optimization on a simulated transtibial amputee with a robotic prosthesis to obtain metabolic energyminimizing walking gaits with multiple prosthesis controllers. Using this simulation, we were able to obtain trends in the energetics and kinematics for various levels of prosthesis work. We find that the net metabolic rate has a roughly quadratic relationship with the net prosthesis work rate. This simulation predicts that metabolic rate could be reduced below that of a non-amputee, although such gaits are highly asymmetric and not seen in experiments with amputees. Walking simulations with left-right symmetry in kinematics or ground reaction forces have higher metabolic rates than asymmetric gaits, suggesting a potential reason for asymmetries in amputee walking. Our findings suggest that a computational framework such as the one presented here could augment the experimental approaches to prosthesis design iterations, although quantitatively accurate prediction of experiments from simulation remains an open problem.

We propose a new method to estimate muscle activity in a straightforward manner with high accuracy and relatively small computational costs by using the external input of the joint angle and its first to fourth derivatives with respect to time. The method solves the inverse dynamics problem of the skeletal system, the forward dynamics problem of the muscular system, and the load-sharing problem of muscles as a static optimization of neural excitation signals. The external input including the higher-order derivatives is required for a calculation of constraints imposed on the load-sharing problem. The feasibility of the method is demonstrated by the simulation of a simple musculoskeletal model with a single joint. Moreover, the influences of the muscular dynamics, and the higher-order derivatives on the estimation of the muscle activity are demonstrated, showing the results when the time constant of the activation dynamics are very small, and the third and fourth derivatives of the external input are ignored, respectively. It is concluded that the method has a potential to improve estimation accuracy of muscle activity.

Optimality in footfall pattern use is often studied in relation to running performance and injury risk. The typical variables assessed (metabolic cost, impact force) represent only two of many potential variables runners might want to minimize situationally. Here we used optimal control theory to predict optimal model-based running mechanics with 44 different cost functions. We tallied the frequency of different footfall patterns, then examined which patterns minimized which types of cost functions. When the model wore shoes, rearfoot striking (RFS) was predicted by 57% of the cost functions and was consistently optimal for functions related to whole-body energy expenditure and peak joint contact forces. No other footfall pattern was predicted by more than 25% of the functions. Non-RFS patterns tended to be optimal for functions that gave equal weight to all muscles, avoiding localized muscle fatigue. Non-RFS patterns were also predicted when minimizing average joint contact forces. Similar predictions were seen when the model ran barefoot, where RFS was optimal for 55% of the functions. The results suggest that RFS is the most versatile footfall pattern (optimal for the greatest number of goals), and may explain why RFS is the most common pattern in recreational shod runners. We argue that natural non-RFS runners are not necessarily behaving "sub-optimally", but rather may be optimizing their gaits on factors not tested here (e.g. comfort, which is difficult to quantify). In addition, switching from RFS to non-RFS may reduce the joint load accumulated during a run if speed and step length are maintained.
Copyright © 2015 Elsevier Ltd. All rights reserved.

When the leader's objective function of a nonlinear bilevel programming problem is nondifferentiable and the follower's problem of it is nonconvex, the existing algorithms cannot solve the problem. In this paper, a new effective evolutionary algorithm is proposed for this class of nonlinear bilevel programming problems. First, based on the leader's objective function, a new fitness function is proposed that can be easily used to evaluate the quality of different types of potential solutions. Then, based on Latin squares, an efficient crossover operator is constructed that has the ability of local search. Furthermore, a new mutation operator is designed by using some good search directions so that the offspring can approach a global optimal solution quickly. To solve the follower's problem efficiently, we apply some efficient deterministic optimization algorithms in the MATLAB Toolbox to search for its solutions. The asymptotically global convergence of the algorithm is proved. Numerical experiments on 25 test problems show that the proposed algorithm has a better performance than the compared algorithms on most of the test problems and is effective and efficient.

In this paper, we give necessary optimality conditions for the nonlinear bilevel programming problem. Furthermore, at each feasible point, we show that the steepest descent direction is obtained by solving a quadratic bilevel programming problem. We give indication that this direction can be used to develop a descent algorithm for the nonlinear bilevel problem.

A bilevel program is a mathematical program involving functions defined implicitly as solutions to another mathematical program. We discuss a method for extracting derivative information on the implicit function, which is especially efficient when the lower-level problem has simple bounds on the variables and/or many inactive constraints. Computational experience on problems with up to 230 variables and 30 constraints is presented.

The multi-level linear programs of Candler, Norton and Townsley are a simple class of sequenced-move games, in which players are restricted in their moves only by common linear constraints, and each seeks to optimize a fixed linear criterion function in his/her own continuous variables and those of other players. All data of the game and earlier moves are known to a player when he/she is to move. The one-player case is just linear programming.We show that questions concerning only the value of these games exhibit complexity which goes up all levels of the polynomial hierarchy and appears to increase with the number of players.For three players, the games allow reduction of the
2 and
2 levels of the hierarchy. These levels essentially include computations done with branch-and-bound, in which one is given an oracle which can instantaneously solve NP-complete problems (e.g., integer linear programs). More generally, games with (p + 1) players allow reductions of
p
and
p
in the hierarchy.An easy corollary of these results is that value questions for two-player (bi-level) games of this type is NP-hard.

Low back pain is an extremely common problem that most people experience at some point in their life. While substantial heterogeneity exists among low back pain epidemiological studies limiting the ability to compare and pool data, estimates of the 1 year incidence of a first-ever episode of low back pain range between 6.3% and 15.4%, while estimates of the 1 year incidence of any episode of low back pain range between 1.5% and 36%. In health facility- or clinic-based studies, episode remission at 1 year ranges from 54% to 90%; however, most studies do not indicate whether the episode was continuous between the baseline and follow-up time point(s). Most people who experience activity-limiting low back pain go on to have recurrent episodes. Estimates of recurrence at 1 year range from 24% to 80%. Given the variation in definitions of remission and recurrence, further population-based research is needed to assess the daily patterns of low back pain episodes over 1 year and longer. There is substantial information on low back pain prevalence and estimates of the point prevalence range from 1.0% to 58.1% (mean: 18.1%; median: 15.0%), and 1 year prevalence from 0.8% to 82.5% (mean: 38.1%; median: 37.4%). Due to the heterogeneity of the data, mean estimates need to be interpreted with caution. Many environmental and personal factors influence the onset and course of low back pain. Studies have found the incidence of low back pain is highest in the third decade, and overall prevalence increases with age until the 60-65 year age group and then gradually declines. Other commonly reported risk factors include low educational status, stress, anxiety, depression, job dissatisfaction, low levels of social support in the workplace and whole-body vibration. Low back pain has an enormous impact on individuals, families, communities, governments and businesses throughout the world. The Global Burden of Disease 2005 Study (GBD 2005) is currently making estimates of the global burden of low back pain in relation to impairment and activity limitation. Results will be available in 2011. Further research is needed to help us understand more about the broader outcomes and impacts from low back pain.

A three-dimensional, neuromusculoskeletal model of the body was combined with dynamic optimization theory to simulate normal walking on level ground. The body was modeled as a 23 degree-of-freedom mechanical linkage, actuated by 54 muscles. The dynamic optimization problem was to calculate the muscle excitation histories, muscle forces, and limb motions subject to minimum metabolic energy expenditure per unit distance traveled. Muscle metabolic energy was calculated by slimming five terms: the basal or resting heat, activation heat, maintenance heat, shortening heat, and the mechanical work done by all the muscles in the model. The gait cycle was assumed to be symmetric; that is, the muscle excitations for the right and left legs and the initial and terminal states in the model were assumed to be equal. Importantly, a tracking problem was not solved. Rather only a set of terminal constraints was placed on the states of the model to enforce repeatability of the gait cycle. Quantitative comparisons of the model predictions with patterns of body-segmental displacements, ground-reaction forces, and muscle activations obtained from experiment show that the simulation reproduces the salient features of normal gait. The simulation results suggest that minimum metabolic energy per unit distance traveled is a valid measure of walking performance.

A numerical technique for solving nonlinear optimal control
problems is introduced. The state and control variables are expanded in
the Chebyshev series, and an algorithm is provided for approximating the
system dynamics, boundary conditions, and performance index. Application
of this method results in the transformation of differential and
integral expressions into systems of algebraic or transcendental
expressions in the Chebyshev coefficients. The optimum condition is
obtained by applying the method of constrained extremum. For
linear-quadratic optimal control problems, the state and control
variables are determined by solving a set of linear equations in the
Chebyshev coefficients. Applicability is illustrated with the
minimum-time and maximum-radius orbit transfer problems

Handbook of human motion

- P B Müller
- D S I Wolf
- P D . G.-P. Brueggemann
- Z Deng
- D A Mcintosh
- D F Miller

Introduction to IPOPT: a tutorial for downloading, installing, and using IPOPT

- waechter a