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Codependent No More. Rebel Flag. Wreck This Journal. Shop by Category. Format see all. Publication Year see all. Topic see all. Special Attributes see all. Dust Jacket. Subjects see all. Language see all. Guaranteed Delivery see all. No Preference. Condition see all. Brand New. Like New. Very Good. Not Specified. Please provide a valid price range. Buying Format see all. All Listings. Best Offer. Forward simulations based on the optimal controls and initial states closely matched the DC results at Fig. The optimal muscle activation patterns were consonant with the requirements of the simulated task and the minimum activation objective function.
Activations were uniformly low in muscles that generate exclusively extension moments Figs. The results for biarticular muscles were more variable. The rectus femoris Fig. When the rectus femoris activity ceased Fig.
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The activity in the hamstrings Fig. The dotted blue lines are the initial guess used at the 25 node density. The solid black lines are the optimal results for the node density. The dashed red lines overlying the solid black lines are the results of a forward dynamics simulation Forward Sim. This approach effectively combines the high-level programming, design and control capabilities of MATLAB with the musculoskeletal modeling, simulation and analysis tools provided by OpenSim.
Within this framework, we used the direct collocation technique to solve two predictive problems; a periodic motion problem using a simple musculoskeletal model and a discrete motion problem using a more realistic model of the human lower limb. The simple model optimal control problem could also be solved using the fmincon solver, but the computation times were too long to be of general use. Our intent is that this framework will facilitate the application of predictive biomechanical simulation to solving clinically-relevant human movement problems. However, IPOPT places more demands on the user, which can translate into considerable up-front costs.
To use fmincon for the types of problems described here, the user needs to provide functions that return the value of the objective function and the values of the equality constraints. IPOPT has similar requirements, but also obligates the user to provide functions that return the gradient of the objective function with respect to the unknowns, the constraints Jacobian matrix, and the sparsity pattern of the constraints Jacobian. The fmincon algorithm will automatically calculate finite difference approximations for any derivatives that are not provided by the user; however, that is not the case for IPOPT.
The process of determining the sparsity structure of the constraints Jacobian can itself be a time-consuming and error-prone task for problems with thousands of unknowns and constraint. However, that task need only be performed once for a particular model and movement problem, and the benefits can be substantial compare times in Figs.
The performance of both fmincon and IPOPT can benefit from analytical gradients and Jacobians if provided by the user, though IPOPT should still hold a considerable performance advantage due to the use of sparse linear algebra. Unfortunately, it is not always possible to obtain analytical expressions for the required derivatives when interfacing with OpenSim, which is a potential limitation of the approach presented here. The objective function used in this work was an explicit function of the model states; therefore, it would be possible to derive an analytical expression for the gradient of the objective function with respect to the unknown parameters.
However, the same is not true for the constraints Jacobian, which is where most of the time is spent in the optimization algorithms. An advantage of having the full symbolic equations is that analytical gradient vectors and Jacobian matrices can readily be determined, which should speed up the most time-consuming part of solving the NLP. There is a trade-off though, as the approach used by OpenSim has the advantage of greatly facilitating model development and analysis, while relieving the user from many lower-level details such as deriving symbolic equations of motions.
For the cases studied here, optimal results were obtained using IPOPT in times ranging from 15 s to 2 hr, depending on the node density and model complexity. These two factors should at least partially offset, suggesting that this comparison may be reasonable as a first approximation. While the actual time required to generate walking simulations using IPOPT with the present framework will need to be determined, even if it requires several hours it will be highly competitive with traditional shooting methods e.
Tracking problems involving large-scale, three-dimensional musculoskeletal models can be solved in a few hours or less using computed muscle control Saul et al. Despite being slower, DC may still be preferred over computed muscle control for some tracking problems due to the flexibility it affords, such as in defining the cost function, or in placing arbitrary constraints on the solution. Convergence with the fmincon algorithm from the MATLAB Optimization Toolbox was too slow to be of much practical value, even for the simple model optimal control problem.
This was due almost entirely to the inability of fmincon to make use of the known sparsity pattern of the constraints Jacobian. We informally compared the impact on performance of requiring IPOPT to use a dense Jacobian and found that it was only marginally faster than fmincon, rather than being over times faster when the sparse Jacobian was used.
Given the large number of independent elements in the constraints Jacobian, performance could be dramatically increased given enough compute nodes. However, even without any performance enhancements, fmincon is still useful for development work as it is easier to use than IPOPT. We found that problems could be more easily tested and debugged using fmincon, before switching to IPOPT to gain the performance advantage. A key aspect of the DC approach is deciding on the minimally acceptable grid density. For the lower limb discrete movement task, the node solution was nearly indistinguishable from the node solution, suggesting that the node density could be used for future studies.
However, this could only be determined by first solving the node case. The cumulative computation time for obtaining the node solution, including the grid refinement process, was over 3 hours. However, once that process was complete, related optimization problems, such as different final postures or different movement times, could be solved at the node density in about 20—30 min each. Indeed, one of the strengths of DC is in rapidly solving several closely related optimization problems, once an initial problem has been solved e.
While we used fixed grid spacing in this work, it is possible to optimize the spacing used in the grid refinement process based on estimates of the discretization error at each grid density, which may confer additional performance benefits Betts, That approach, and the one presented here, are indeed complimentary and simply suited to different purposes. The traditional approach has been to use a low-dimensional e. These other approaches could likely also be implemented using OpenSim and MATLAB and would be subject to many of the same strengths and weaknesses described herein.
The example code provided with this article may prove to be a useful starting point for researchers implementing these other approaches via OpenSim and MATLAB. This should facilitate the use of optimal control in developing therapies and assistive devices for clinical conditions that limit human mobility. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests The authors declare that they have no competing interests. Brian R. Data Deposition The following information was supplied regarding data availability:.
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National Center for Biotechnology Information , U. Journal List PeerJ v. Published online Jan Leng-Feng Lee and Brian R. Author information Article notes Copyright and License information Disclaimer. Corresponding author. Umberger: ude. Received Nov 3; Accepted Jan 7. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed.
This article has been cited by other articles in PMC.
Optimization Techniques in MATLAB
Abstract Computer modeling, simulation and optimization are powerful tools that have seen increased use in biomechanics research. Keywords: Predictive simulation, Dynamics, Musculoskeletal model, Optimization.
Introduction Dynamic models of the musculoskeletal system are powerful tools for studying the biomechanics of human movement. Materials and Methods We begin by outlining the general optimal control problem formulation and then describe the way in which the capabilities of OpenSim and MATLAB were combined to solve these types of problems using the DC approach. Open in a separate window. Figure 1. Figure 2. OpenSim models used in this project.
Lower limb model To evaluate the DC approach using OpenSim-MATLAB on a larger scale and more anatomically realistic model, we generated predictive simulations of lower limb movement using a sagittal plane, 3 DOF model of the human lower limb actuated by 9 muscles Fig. Results For the simple 1-DOF model, all node densities resulted in approximately sinusoidal motions Fig. Figure 3. Position of the block versus time for the simple model optimal control problem. Figure 4. Muscle activations versus time for the simple model optimal control problem.
Figure 5. Optimization algorithm performance for the simple model optimal control problem. Figure 6. Joint kinematics A—C and muscle activations D—L for the lower limb optimal control problem. Additional Information and Declarations Competing Interests The authors declare that they have no competing interests.
Optimality principles for model-based prediction of human gait. Journal of Biomechanics. Dynamic optimization of human walking. Journal of Biomechanical Engineering. Anderson Anderson E. Phildelphia: SIAM; Betts Betts JT. On the estimation of sparse Jacobian matrices.