# From Knowledge to Leverage: How to Use Musculoskeletal Simulation to Design Exoskeleton Concepts
## 2 Methods  
In the inverse dynamics paradigm, the dynamic equilibrium equations take the form of a mathematical program [22]:  
Minimize  
\[\sum_{i=1}^{n_{m}}\left(\frac{f_{i}^{(M)}}{N_{i}}\right)^{p}, \tag{1}\]  
subject to  
\[\mathbf{C}\left\{ \begin{matrix}\mathbf{f}^{(M)}\\ \mathbf{f}^{(R)}\end{matrix} \right\}=\mathbf{r} \tag{2}\]  
\[f_{i}^{(M)}\geq 0,\text{ for }i=1...n_{m} \tag{3}\]  
where vector \(\mathbf{f}=\{\mathbf{f}^{(M)},\mathbf{f}^{(R)}\}^{\mathrm{T}}\) contains the unknown forces in the system, separated into \(n_{m}\) muscle forces (\(M\)) and joint reactions (\(R\)). The right-hand side, \(\mathbf{r}\), contains externally applied forces and velocity-dependent forces that are known when the kinematic analysis is completed; \(p\) is the power of the convex combination that forms the objective function, and \(\mathbf{C}\) is the coefficient matrix of the equilibrium equations. Normalization factors, i.e., \(N_{i}\), express the strength of the \(i\)th muscle. Depending on the value of \(p\), the mathematical programming problem can be solved efficiently by a variety of convex programming algorithms. We notice that forces of the mathematical program (1)-(3) separate into three categories:  
1. The right-hand side forces, \(\mathbf{r}\): these forces can be specified as any time-varying function, including exoskeleton actuator forces, and they can be applied anywhere in the system without further complicating the problem;
2. The reaction forces (and moments), \(\mathbf{f}^{(R)}\): these are dependent variables determined by equilibrium, and they can take on any positive or negative value necessitated by the equilibrium;
3. The muscle forces, \(\mathbf{f}^{(M)}\): these are affiliated with perceived effort and restricted in sign as shown in (3) because muscles can only pull. Forces from categories 1 and 2 do not appear in the objective function, so they are costless. However, they do influence the objective function indirectly because they influence the equilibrium for better or worse from the point-of-view of the muscles. Muscle forces (category 3), on the other hand, directly influence the cost of the objective function. In order of priority, the algorithm (i) honors the specified values of right-hand side forces in category 1 while (ii) minimizing muscle forces in category 3 at the expense of (iii) costless reaction forces from category 2. In other words, the algorithm returns the values of reaction forces that minimize the muscle effort. The optimization problem (1)-(3) can therefore be perceived as a minimum effort or laziness criterion [22; 23]. Although the optimization problem is formulated in terms of muscles, the circumstances of most ergonomic situations are such that the moment arm of the external loads, for instance, the weight of the box in Figure 1, exceeds the moment arms of the muscles around the joints. This relationship attributes the majority of the joint reaction forces to muscle actions rather than direct load application from the external forces, so reduction in muscle effort will also in most cases reduce the joint reaction forces. In the formulation of the biomechanical system, the user can place forces in the different categories, and this opportunity can be exploited to gain knowledge about favorable exoskeleton design decisions. Specifically, if exoskeleton actuator forces are placed in category 2, they will automatically attain the most favorable values in terms of reduction in muscle effort. These values can subsequently be studied and used to select an actual actuator and its control strategy. Once selected, its characteristics can be placed in category 1 for verification of the effect. This technique is illustrated in the subsequent sections by means of an example. We herein consider a typical warehouse task of lifting a box, as illustrated in Figure 1. The box dimensions are 0.53 m wide by 0.40 m high by 0.36 m deep, and its mass is 10 kg. The box is held by hand slots cut out in its sides.