# An Agent-Specific Stochastic Model of Generalized Reaching Task Difficulty
## 3 Application of the Stochastic Model to an Example Motor Task
### Numerical Evaluation of the Index of Difficulty \(ID_{agent}\) for a Single Elderly Subject  
The proposed model of agent-specific Index of Difficulty is tested on one human subject (male, 61 years old) walking at fast speed level (average speed of 1.584 m/s). The spatial trajectories of the swing foot's ankle are recorded over 11 walking trials and projected onto the anatomical sagittal plane (\(x\), \(y\)), where the \(x\) coordinate indicates the direction of the forward gait progression and the \(y\) coordinate indicates the direction normal to the ground. The trajectory followed by the ankle during the second stride of each \(k\)-thwalking trial (\(k\) = 1-11) is sampled and averaged for the analysis of task difficulty (Figure 4). The stride motor task of a given leg is modelled as a generalized reaching task starting from the instant of \"flat foot\" (i.e., the foot is still in full contact with the ground, right before the swing phase) and terminating at the instant of \"helest strike\" (i.e., when the same foot makes the new contact with the ground, terminating the swing phase). Each trajectory is observed from a local reference frame located at the ankle of the foot at the initial time (i.e., at the flat foot condition). This allows for the quantification of the trajectory variability's effect on the net increase of affordance limits in the direction of forward progression, which leads to the variability in the reaching target x-position (i.e., stride length variability). The subject's Index of Difficulty at his swing foot during the walking motor task, \(ID_{agent}(x,\ 0.96)\), is calculated at every point of the stride, discretized over the \(x\) coordinate, at a probability level set at \(\varphi=0.96\) (Figure 5). The local affordance limits \(W(x,0.96)\) demonstrate that the small variability observed during the initial phase of the stride and right before the heel strike is associated with a reduced end-effector affordance and, therefore, a higher rate of increase of task difficulty. By analyzing the trend of \(ID_{agent}(x,\varphi)\), three phases of the stride are identified. In the first phase (\(0<x<400\) (mm)), the end-effector (swing foot) is lifted from the ground and follows a trajectory with high difficulty: it can be observed that 50% of the total task difficulty is allocated in the first quarter of the trajectory, from flat foot to the point of maximum swing foot height (Figure 4). Towards the end of this phase, the end-effector affordance of the subject improves, as shown by the decreasing derivative of \(ID_{agent}(x,\varphi)\). In the second phase (\(400<x<1200\) (mm)), 25% of the total task difficulty is realized during the central portion of the trajectory, in which the swing foot is moving from a point of maximum to minimum height, while traveling about 50% of the total stride length. In this phase, the rate of increase of \(ID_{agent}(x,\varphi)\) is approximately constant, with affordance limits \(W(x,\varphi)\) first increasing, reaching a maximum at about half of the stride, and then decreasing. In this phase, the subject's affordance in motor task execution is on average constant, demonstrating his potential to control his movements in a more flexible (redundant) way. The combination of the variability range of trajectories (\(W(x,\varphi)\)), that is constant on average, and the linear increase in the forward path (\(dx\)) results in an effectively constant task difficulty. Figure 4: Ankle spatial trajectories (\(n=11\)) in the sagittal plane and related average trajectory of a 61 years old healthy subject walking at an average speed of 1.584 m/s. The initial position of each sampled path represents the position of the ankle at the instant of flat foot, preceding each swing phase. Each sampled path terminates at a variable ankle final position, recorded at the instant of heel strike. Finally, in the last quarter of the stride (\(1200<x<1600\) (mm)), i.e., the last 25% of the travelled distance, the subject is employing 25% of the total task difficulty.