# Aging Affects Multi-Objective Optimal Control Strategies during Obstacle Crossing
## 2 Materials and Methods
### Multi-Objective Optimal Control (MOOC) of Obstacle Crossing  
For the current MOOC of obstacle crossing, some of the model variables were assigned with experimentally measured data, namely, the trailing foot (\(\theta_{TF}\)), ankle (\(\theta_{TA}\)), knee (\(\theta_{TK}\)), and hip (\(\theta_{TH}\)), and the leading ankle (\(\theta_{LA}\)) (Figure 1). The measured swing ankle position (\(y_{1}\)) at the toe-off immediately before crossing (\(t_{1}\)), as well as the swing ankle position (\(y_{n}\)) at the initial contact immediately after crossing (\(t_{n}\)), were also taken as model input. Given these measured input data, the swing ankle trajectories would be determined by the MOOC laws. Considering the linkage of the leading swing limb, the angles of the leading hip (\(\theta_{LH}\)) and knee (\(\theta_{LK}\)) were determined once the positions of the swing hip and ankle were known (Figure 1). According to [PERSON] et al., the MOOC of the lower extremities during obstacle crossing involves trade-offs between the conflicting objectives of minimizing mechanical energy and maximizing the swing heel and toe clearances [12]. Therefore, the MOOC problem of obstacle crossing in the current study was to search for the swing ankle trajectory, and thus the angular trajectories of the joints of the swing limb, and the torques of all the lower limb joints, that minimized the energy expenditure (\(f_{1}\)) and maximized the heel-obstacle (\(f_{2}\)) and toe-obstacle clearances (\(f_{3}\)) above the obstacle. This MOOC problem was converted to a non-linear programming problem using a parameterization approach. Figure 1: The sagittal seven-link model of the body during obstacle crossing with seven degrees of freedom, corresponding to the angular displacements: hip (\(\theta_{LH}\)), knee (\(\theta_{LK}\)), and ankle (\(\theta_{LA}\)) of the leading swing limb, and foot (\(\theta_{TF}\)), ankle (\(\theta_{TA}\)), knee (\(\theta_{TK}\)), and hip (\(\theta_{TH}\)) of the trailing stance limb. Toe-obstacle clearance is also indicated, defined as the vertical distance of the toe above the obstacle. The X-axis indicates the direction of progression. The spatial trajectory of the swing ankle was discretized to \(n\) positions over the total time of the swing phase, i.e., from toe-off (\(t_{1}\)) to initial contact (\(t_{n}\)) of the swing limb, at equal-timed intervals, \(\Delta t=\frac{t_{n}-t_{1}}{n}\). The discretized swing ankle positions, i.e., \(\overline{y}=(y_{1},y_{2},\ldots,y_{n})\), were the design variables of the non-linear programming problem. The resulting parameterized MOOC problem can be described as follows:  
Problem MOOC: Find a set of design variables, \(\overline{y}=(y_{1},y_{2},\ldots,y_{n})\) to minimize the following objective functions:  
\[f_{1}\left(\overline{y}\right)=\sum_{j=1}^{7}\sum_{i=1}^{n}T_{i}^{i}\omega_{j }^{i}\Delta t \tag{2}\]  
\[f_{2}\left(\overline{y}\right)=-d_{h} \tag{3}\]  
\[f_{3}\left(\overline{y}\right)=-d_{t} \tag{4}\]  
subject to the equations of dynamics of the seven-link model defined in Equation (1), where \(f_{1}\) is the mechanical energy from the muscular works at the lower limb joints calculated as the sum of the products of the joint torques (\(T_{j}^{i}\)) and joint angular velocities (\(\omega_{j}^{i}\)) over all time intervals (\(\Delta t\)) for all joints (\(j\)), \(f_{2}\) is the negative of the toe-obstacle clearance (\(d_{t}\)), and \(f_{3}\) is the negative of the heel-obstacle clearance (\(d_{h}\)) (Figure 1). Basically, the larger the value of \(n\), the better the representation of the ankle trajectory. However, considering the dimensionality, and thus the computational effort, for solving the non-linear programming problem, a value of 20 was selected for \(n\) in the current study [12].