# An Agent-Specific Stochastic Model of Generalized Reaching Task Difficulty
## 5 Conclusions and Further Research  
This study deals with affordance measures of observed reaching motor tasks executed by subjects. The issue is of great scientific and industrial interest as it pertains the ability of a human agent with given dynamic features to accomplish different trajectories constrained by the environment (external constraints) and by internal constraints (features of the agent). To this purpose, the authors propose an analytical stochastic model to calculate a new Index of Difficulty \(ID_{agent}\) extending the deterministic formulation of the Index of Difficulty, \(ID_{t}\), proposed in the literature [25]; \(ID_{t}\) was originally defined as being dependent on only the geometry of a spatially constrained trajectory. On the contrary, stochasticity of the phenomenon has to be considered for taking into account of the natural variability of subjects in following different trajectories for an assigned reaching motor task in a given environment. The model proposed has been applied to an available dataset of differently aged subjects, each of them having known features such as mass and leg length, executing a trajectory in a given environment. We proved the proposed Index of Difficulty, \(ID_{agent}\), being an effective measure to capture the stochastic complexity of a general spatially constrained trajectory or parts of it. The model revealed a strong statistical dependency of \(ID_{agent}\) on age and average execution speed of the subjects. A statistical analysis (mixed design ANOVA and post hoc tests) provided significant differences in the average \(ID_{agent}\) for \"Young\" vs. \"Old\" subjects, and, for each age class, for \"Slow\" vs. \"Fast\" average speed. A regression model confirms the significant dependency of \(ID_{agent}\) of a subject on age and speed execution. The results obtained are strictly referable to the dataset considered. However, the approach and model proposed are of general applicability. A current limitation of the proposed model (Equations (7) and (8)) is the assumption of two-dimensional trajectories. This assumption is valid in many practical cases, such as walking, where the foot end-effector movement is mostly contained in the sagittal plane [34], or in object manipulation for industrial settings where the hands of the operator move in the horizontal plane (like the positioning of a component or a fixture on a horizontal work surface area). Future research will focus on extending the model to a three-dimensional trajectory. The extension will further enable the applicability of the model to real cases. In biomechanics, the model proposed could be adopted to appreciate differences in motion affordance of observed trajectories for different parts of the human body. Clinically, the model can provide quantitative estimations in affordance improvements of a subject participating in a rehabilitation program by executing the same reaching motor task, allowing for the analysis of the subject's improvement after the program. During a general task, humans are able to demonstrate variability while maintaining robust control over their motion due to redundancies that exist within these tasks [35]. In this regard, the Index of Difficulty suggests a novel additional perspective to understand the nature and effects of trajectory variability, which cannot only predict motor impairment as it is usually considered, but it can also be a measure of redundancy exploitation and be an indicator of optimal motor control. Within the working environment, the model could be usefully applied to investigate affordance of subjects in executing reaching motor tasks by upper limbs. The model proposed can provide quantitative affordance evaluations on capability of workers in carrying out working tasks and related improvements due to workers' training. **Author Contributions:** All authors contributed equally to the manuscript, in particular as regards conceptualization, methodology, software, investigation, writing--original draft, and writing--review, and editing.