# Aging Affects Multi-Objective Optimal Control Strategies during Obstacle Crossing
## 3 Results  
There were no statistical interactions between group and obstacle height factors for any tested variables, so only the main effects are reported here. The older group was found to have significantly greater leading toe-clearance, energy expenditure, and longer trailing toe-obstacle distances, but shorter leading heel-obstacle distances (Table 2 and Figure 2). With increasing obstacle height, the energy expenditure was significantly increased linearly (Table 2).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Obstacle Height (\% Leg Length)**} & \multicolumn{2}{c}{**Group Effect**} & \multicolumn{2}{c}{**Height Effect**} \\ \cline{2-5}  & **10** & **20** & **30** & \multicolumn{2}{c}{**\(p\)-Value**} \\ \hline Leading heel-obstacle distance (\% Leg length) & & & & \\ \hline Older & 17.8 (4.9) & 17.3 (3.5) & 16.9 (3.9) & & \\ Young & 20.3 (2.6) & 20.2 (2.4) & 19.5 (2.2) & & \\ \hline Trailing toe-obstacle distance (\% Leg length) & & & & \\ \hline Older & 24.9 (4.4) & 24.6 (4.2) & 24.2 (4.8) & & \\ Young & 21.0 (2.7) & 20.4 (3.1) & 20.6 (2.9) & & \\ \hline Leading-toe clearance (mm) & & & & \\ \hline Older & 156 (34) & 169 (33) & 176 (40) & & \\ Young & 149 (25) & 144 (26) & 128 (20) & & \\ \hline Leading-heel clearance (mm) & & & & \\ \hline Older & 130 (31) & 137 (34) & 142 (39) & & \\ Young & 135 (27) & 137 (35) & 135 (33) & & \\ \hline Energy expenditure (\% body weight\(\times\)leg length\(\times\)total time) & & & & \\ \hline Older & 174.4 (41.7) & 172.8 (30.1) & 183.2 (35.9) & & \\ Young & 149.3 (22.6) & 145.4 (20.8) & 156.3 (19.3) & & \\ \hline \hline \end{tabular}  
*: significant difference between groups. \(\uparrow\): significant linearly increasing trend.  
\end{table}
Table 2: Means (standard deviations) of the end-point variables and energy expenditure for the older and young control groups when crossing obstacles of three different heights.  
The older group showed significantly increased weightings to the heel- and toe-clearances, but decreased energy expenditure weightings (W1: 0.533; W2: 0.233; W3: 0.233) when compared with the young group (W1: 0.68; W2: 0.16; W3: 0.16) (\(p<0.01\), Figure 3). There were no significant height effects for the best-compromise weightings (\(p>0.05\), Figure 3).  
With the best-compromise weighting sets, the RMSEs of the leading ankle trajectories over the swing phase between MOOC model predictions and experimental measurements  
Figure 3: Means (standard deviations) of the best-compromise weighting sets (W1, W2, W3) for the older and young groups when crossing obstacles of three different heights: 10% (white), 20% (grey), and 30% (black) leg length (LL). \(p\)-values for the main effects are also given (Pg: group effects; Ph: height effects). Asterisks indicate statistically significant differences (\(p<0.05\)).  
Figure 2: Sagittal trajectories of the ankle, toe, and heel of the leading swing limb of typical subjects of the older (**red curves**) and young (**black curves**) groups when crossing obstacles (**solid blue lines**) of 10%, 20%, and 30% of leg length (LL).  
were less than 8 mm for all test conditions, without significant group or height effects (\(p>0.05\), Table 3). Similarly, the RMSEs for the swing hip and knee angles were less than \(2^{\circ}\) for all test conditions, without significant group or height effects (Table 3). The RMSEs for the joint moments of the trailing stance limb were less than about 5 Nm for both groups and all obstacle conditions, without significant group or height effects (Table 3).