# An Agent-Specific Stochastic Model of Generalized Reaching Task Difficulty
## 1 Introduction
Finally, the GIE is a deterministic measure, whereas possible trajectories can vary stochastically due to the limited repeatability of human actions. A different approach for evaluating the complexity of a motor task is the formulation of the Index of Difficulty (_ID_) originally proposed by [PERSON] [15], which quantifies the difficulty of a simple reaching motor task required to reach a target of width \(W\) placed at distance \(D\) from a starting point. The _ID_ has been derived by analogy with the Shannon theorem \(\mathrm{n}^{\circ}\) 17 [16], theorized within the information theory. Accordingly, a successive formulation of the _ID_ is proposed in [17] as:  
\[ID=\log_{2}\left(\frac{D}{W}+1\right) \tag{3}\]  
_ID_ is measured in [bit] and the above formulation has demonstrated a better fit with the [PERSON] empirical data [17]. The various formulations of the _ID_[15; 17; 18], and field studies [19; 20; 21; 22; 23; 24] refer to simple \"point-to-point\" reaching motor tasks without considering the travelled path. In such experiences, subjects can fail in meeting the final target and accuracy issues are addressed. The error rate of subjects in meeting the final target at different speed has been put into relation to the standard deviation of the position of the endpoint; to this purpose, the \"Speed-Accuracy\" model was discussed in [17]. The Index of Difficulty of a general trajectory (_ID_) spatially constrained along a path of any shape, defined in a two-dimensional plane, is proposed in [25]. Starting from the general _ID_ formulation, the \"Speed-ID-Accuracy\" model, which shows the effects of speed and difficulty of task on the accuracy, is proposed in [26]. In spite of a wide scientific literature on _ID_ formulations and experimental testing [19; 20; 21; 22; 23; 24], these experiences refer to motor tasks with negligible dynamics, i.e., they refer to reaching motor tasks at a slow speed or manipulating light objects (e.g., a computer device, stylus pen on a tablet, etc.). On the contrary, in many practical cases referable to reaching tasks in work environment or in biomechanics, the dynamics of the phenomenon cannot be neglected since mass and acceleration are not negligible. The ability of an agent to meet internal and external constraints is dependent on the dynamics that governs the motor task execution [27]. As often observed in work environments, an agent is required to execute spatially constrained complex trajectories. Perceived affordance in complex trajectories is affected by agent's features like geometry, mass distribution, force of the actuators (or physical strength), speed of execution. Modeling the dynamics of the phenomenon reveals a quite complex issue not yet addressed by research, also due to the stochastic variability of internal constraints of a human agent. In this study, a novel Index of Difficulty is proposed in order to overcome the above-mentioned limits. The model quantifies the Index of Difficulty as a probabilistic affordance-related measure of a generalized reaching task, based on the observed variability of the repeated trajectories executed by a given agent. The execution of a trajectory is considered as the reaching of a series of consecutive targets orthogonally distributed along a nominal path (i.e., the average trajectory). The current formulation is derived under the assumption of two-dimensional paths. Furthermore, the hits on targets are assumed as normally distributed. A major novelty of the proposed stochastic model relies in its intrinsic ability to take into account all the agent-specific features affecting the dynamics of the motor task under internal (agent) and external (task) constraints. Therefore, the model is effective in performing reliable affordance analysis of a specific individual or groups of individuals of given features accomplishing a reaching task for statistical analyses. The paper is organized as follows. In Section 2, the stochastic model to calculate the Index of Difficulty is proposed.