**Optimization Models**

A popular method for generating control signals in musculoskeletal models is to use some form of optimization. In some cases, when the optimization criterion is derived from motor control principles, this is tantamount to developing a theoretical control model. In other cases, the optimization criterion may result in a coordinated movement pattern yet have little physiological basis. In general, the optimization approach is used in an effort to determine which set of model control signals will produce a result that optimizes (minimizes or maximizes) a given criterion measure. The criterion measure is known as a cost function, objective function, or performance criterion. The cost function can be relatively simple (e.g., find the solution that yields minimal muscle force) or complex (e.g., determine a nonlinear combination of maximal muscle force at minimal metabolic cost). It can be directly related to muscle function (e.g., minimize muscular work) or to some aspect of the motion under study (e.g., maximize vertical jump height). Alternatively, the cost function may be formulated to minimize the differences between model outputs (e.g., joint angles, ground reaction forces) and corresponding experimental measures. This latter case is often referred to as a tracking problem in that the goal is to find a solution that causes the model to follow, or track, the experimental data.

In all cases, the cost function serves as a guiding constraint that determines the selection of one particular set of optimal muscular controls from among many different possible solutions. Two major optimization approaches that have been used to control musculoskeletal models are referred to in the literature as static optimization and dynamic optimization. The meaning of static in this context is that the cost function is evaluated at each time step during a movement, independent of any prior or subsequent time steps. In contrast, dynamic optimization is dynamic in the sense that the entire movement sequence must be simulated to determine the value of the cost function.

**Static Optimization**

Initial attempts to predict individual muscle forces used static optimization models in conjunction with inverse dynamics analysis of an actual motor performance. The inverse dynamics analysis permits the calculation of net joint moments at incremental times throughout the movement (see chapter 5). In most applications, numerical optimization is used to find the set of muscle forces that balances these joint moments while also satisfying a selected cost function (figure 11.9). Thus, with this approach the muscle forces themselves, rather than neural signals, are the controls. The time-independent nature of static optimization allows solutions to be obtained with relatively little computational cost, but there are some drawbacks. One issue in early applications was sudden, nonphysiological switching on and off of muscle forces caused by the independence of solutions for sequential time increments (i.e., the optimization model balanced the joint moments separately for each time interval using very different sets of muscles). This problem can be avoided by careful selection of the initial guess in the optimization problem, such as by using the solution from one time step as the initial guess for the next time step. A stronger approach to address this issue is to use muscle models that invoke physiological realism through force-length and force-velocity relations and time-dependent stimulation-activation dynamics (chapter 9). When more detailed muscle models are included in a static optimization model, muscle activations become the control variables, rather than muscle forces.

Early studies using static optimization were plagued also by two other kinds of nonphysiological results. The first was the prediction of model forces that were too high for actual muscles to produce. This difficulty was easily addressed by defining physiologically valid maximal force constraints for each muscle in the musculoskeletal model. The second problem was that solutions would often select only one muscle to balance the net joint moment rather than choose a more realistic muscular synergy. Depending on the exact formulation, the muscle with the largest moment arm (if minimizing muscle force) or a favorable combination of moment arm and muscle strength (if minimizing muscle stress) would be selected to fully balance the joint moment, with zero force predicted in other synergist muscles. Mathematically, synergism can be produced by using nonlinear cost functions (e.g., minimizing the sum of squared or cubed muscle forces), although the physiological rationale for specific nonlinear cost functions is not always clear. A widely used nonlinear cost function proposed by Crowninshield and Brand (1981a) involves minimizing the sum of cubed muscle stresses. It was originally argued that this particular cost function would lead to solutions that maximize muscle endurance, making it appropriate for predicting muscle forces in submaximal tasks such as walking. However, the Crowninshield and Brand criterion has since been used to solve for muscle forces in a range of activities, some of which are unlikely candidates for maximizing muscle endurance (e.g., jumping, landing).

An additional issue with static optimization models is that they are essentially a decomposition of experimental joint moments into individual muscle forces. Therefore, the predicted muscle forces will be subject to any errors in the experimental joint moments in addition to any shortcomings associated with the approximation made in creating the musculoskeletal model. To complete this section, we present a static optimization example motivated by a classic review article by Crowninshield and Brand (1981b), which demonstrates the process of distributing an empirically determined elbow joint moment across a set of muscles spanning the elbow joint.

The net joint moment to be balanced in example 11.1 was a flexor moment, and only muscles with flexor moment arms were included in the model (figure 11.10). If an elbow extensor muscle had been included, there is no cost function of the type presented here that would predict force in an antagonist muscle. Any force in an elbow extensor would itself contribute to a higher cost function value and would also require greater elbow flexor forces to balance the target 10 N·m joint moment, further increasing the cost function value. The total lack of coactivation in this example is contrary to the common observation that during heavy activation of the elbow flexors, there is some activity in the triceps muscle group. The extensor coactivation likely helps stabilize the joint during heavy exertion but will not be predicted using traditional static optimization techniques in simple one DOF models. Although the example presented here focuses on obtaining numerical results for muscle forces, the interested reader is referred to the review by Crowninshield and Brand (1981b), which presents an interesting graphical interpretation of the solution to this static optimization problem.

**Dynamic Optimization**

Ongoing work with static optimization models has led to the development of dynamic optimization or optimal control models, which are applied in conjunction with forward dynamics models of human motion. In contrast to the inverse dynamics approach, which uses experimental data from an actual performance to calculate net joint moments, forward dynamics analyses simulate the motion of the body from a given set of joint moments or muscle forces (see chapter 10). Dynamic optimization models are therefore often designed to find the muscle stimulation patterns that result in an optimal motion (figure 11.9). As mentioned earlier, the optimal motion may be one that maximizes (or minimizes) a performance criterion, such as maximizing jump height; or, the optimal motion may be defined as one that best reproduces a set of experimental data. Regardless, the variables that are optimized are usually the muscle stimulation patterns that control the motion of the musculoskeletal model. These stimulation patterns are used as the inputs to muscle models that predict individual muscle forces, which are then multiplied by the appropriate moment arms to compute the active muscle moments. The active muscle moments are combined with passive moments to compute the net joint moments, which actuate the skeletal model and produce movement. Thus, dynamic optimization leads to the synthesis of body segment kinematics associated with optimal performance, which is fundamentally different from the static optimization approach.

The optimal kinematics predicted from a dynamic optimization can be compared with experimental motion data; however, the experimental data are not required to obtain the solution, as they are with static optimization. This feature of dynamic optimization permits one to address questions for which no experimental data exist, such as testing the feasibility of possible forms of locomotion in extinct species (e.g., Nagano et al. 2005). Moreover, because dynamic optimization uses forward dynamics models that simulate the whole movement performance and provide complete muscle force time histories (rather than solutions at independent time increments), many of the problems associated with static optimization models are overcome. However, these advantages come at a computational cost, because dynamic optimization often requires a more detailed musculoskeletal model and the entire movement must be simulated for each possible solution. Thus, solving a dynamic optimization problem can easily require an order of magnitude more time than is required to solve a comparable static optimization problem. Although computational costs are difficult to compare in an objective manner, it is not uncommon for static optimization solutions to take seconds or minutes to obtain, whereas dynamic optimization solutions can take hours, days, or even weeks, when run on standard computer workstations.

In many cases, users of dynamic optimization must define the goal of the movement mathematically. This definition is easiest for movements in which the performance criterion to be optimized is clear in a mechanical sense. In vertical jumping, for example, the cost function can be stated as maximization of the vertical displacement of the body center of mass during the flight phase. If the model is constructed with appropriate constraints (realistic muscle properties and joint range of motion), jump heights and body segment motions approximating those of human jumpers can be attained (Pandy and Zajac 1991; van Soest et al. 1993). However, for many human movements the performance criterion is not as clear. In walking, for instance, the goal may seem to be getting from point A to point B in a certain amount of time. Unfortunately, this does not provide enough information to solve for a set of muscle stimulation patterns that will result in realistic walking. Successful simulations of walking have been generated using a cost function that minimizes the metabolic cost of transport (Anderson and Pandy 2001; Umberger 2010); however, the cost function formulation is considerably more complicated than for vertical jumping. Minimum-energy solutions also require the inclusion of an additional model for predicting the metabolic cost of muscle actions (e.g., Umberger et al. 2003). In cases where the performance criterion is not clear, dynamic optimization can serve as an effective approach for testing various theoretical criteria to see how well each can produce the desired movement patterns.

Movements for which the underlying criterion is difficult to define, such as walking or pedaling, have often been simulated by formulating and solving a tracking problem. This approach has a similar appeal to the EMG-based techniques described earlier, in that the resulting motion should closely approximate the way in which humans actually move. However, it is unclear which of several possible experimental measures (kinematics, kinetics, EMG) are the most important for the model to track. Also, because of errors in the experimental data and differences between the musculoskeletal model and the experimental subjects, it may be impossible for the model to track the data perfectly. This approach also limits much of the predictive ability of the dynamic optimization approach, as it is only possible to consider conditions for which experimental data are available.

The tracking approach has seen frequent use in hybrid solution algorithms, which seek to retain the advantages of forward dynamics and dynamic optimization while achieving the computational efficiency of static optimization. Two examples are computed muscle control (Thelen et al. 2003) and direct collocation (Kaplan and Heegaard 2001). Computed muscle control works by solving a static optimization problem at each time step within a single forward dynamics movement simulation. By using feedback on the kinematics and muscle activations at each time step of the numerical integration, computed muscle control can generate a forward dynamics simulation that optimally tracks a set of experimental data without solving a dynamic optimization problem that would require thousands of forward dynamics simulations. In contrast, direct collocation works by converting the differential equations of motion into a set of algebraic constraint equations and treating both the control variables (muscle stimulations) and state variables (positions and velocities) as unknowns in the optimization problem. Although they differ considerably in implementation, both techniques appear to be able to produce results that are similar to dynamic optimization tracking solutions, with a computational cost closer to that of static optimization.

Although static and dynamic optimization have both become popular means for controlling musculoskeletal models, several issues concerning their use must be addressed. Do humans actually produce movements based on one given performance criterion, or does the performance objective change during the movement? If the optimized model results differ from those of a human performer, is it because the model is too simple or not constrained properly, or is the human not performing optimally? Finally, EMG data have illustrated various degrees of simultaneous antagonistic cocontraction during some movement sequences. Optimization models tend not to yield solutions predicting muscle antagonism around a joint, although some antagonism is predicted when models contain muscles that contribute to more than one joint moment. However, optimization models that seek to minimize or maximize specific cost functions will not predict antagonistic cocontraction associated with joint stability or stiffness. In the case of walking, for example, there is no single cost function that could simultaneously predict the relatively minimal muscle coactivation in healthy subjects and the substantial antagonism exhibited by patients with cerebral palsy. Despite these drawbacks, optimization models have increased the understanding of human biomechanics and will continue to do so in the future.