# An Agent-Specific Stochastic Model of Generalized Reaching Task Difficulty
## 1 Introduction  
Studies on human or robot motor performance pertain to an interdisciplinary field of research with several applications ranging from robotics to biomechanics to ergonomics. This performance is typically measured in human or robot agents by analyzing a trajectory executed during a motor task. Reaching motor tasks (i.e., a movement in which the end-effector aims at a reaching target starting from a given position) are widely studied when assessing human motor performance and control [1, 2]. These motor tasks are subject to external constraints imposed by the environment which include, for example, the spatial constraints as well as the geometrical features of the object to be moved by the agent. Factors, such as age, sex, and other physical features of a human agent affect the physical status and, in turn, the resulting trajectories that can be executed during a motor task. Reaching tasks typically consist of repetitive movements that the agent performs by executing a set of possible trajectories, from initial point to target, each of them being characterized by different features (e.g., path length and shape, energy consumption and speed of execution). The set of possible trajectories are limited by the agent's internal constraints, as well as environmental constraints. Internal constraints are of different nature and include the number of degrees of freedom available to control the musculoskeletal system of a human or bio-inspired robot agent, its available physical strength to move objects at a specific speed, and its experience and skill in executing the task. The higher theagent's motor abilities, the higher the possibilities for action that the agent perceives for a given movement and environment, also known as affordances [3]. The concept of \"affordance\" has been firstly theorized in [4]. Information-based affordance measures have been proposed for various tasks of both robot and human agents. The \"Labelling entropy\" measure or \"binary cross entropy\" loss function have been adopted to investigate the affordance of a robot in object recognition tasks [5; 6]. Information-based measures have been also proposed for motor tasks [7; 8; 9] to investigate the affordance of human climbing with assigned environment constraints. In climbing tasks, external constraints include the width of the path, the wall inclination, and the shape and orientation of the handles, which determine the climber's selection of the sequence of poses and resulting path. An expert climber tends to travel straighter paths, which are less complex and more affordable trajectories; on the contrary, when a novice climber faces with a challenging route that force him to reach the affordance boundaries of his abilities, he tends to perform more complex trajectories [10; 11]. A complexity measure of a general motion path is proposed in [12]; here, the entropy of a curve (i.e., a spatial trajectory) is defined as:  
\[H=\ln\left(\frac{2\cdot L}{c}\right) \tag{1}\]  
where \(L\) is the length of the curve, \(c\) is the convex hull (perimeter) of the curve, and \(\frac{2\cdot L}{c}\) is the average number of intersections between the curve and a straight line [13]. The \(H\) formulation, measured in bit, is also known as the Geometric Index of Entropy:  
\[GIE=\frac{H}{\ln(2)} \tag{2}\]  
It has been applied in human climbing [8; 9]. The GIE can measure the difficulty of a trajectory observed during general reaching tasks. Since the GIE is sensitive to external and internal constraints of an agent [14], it has been adopted as an affordance-related measure of the agent [8; 9]. However, the GIE shows limits in capturing the complexity of a trajectory as it refers to the overall trajectory neglecting local criticalities that a trajectory could show and that could be of interest to identify which internal or external constraints limit his action. Moreover, the GIE does not consider the accuracy the subject shows in executing a trajectory.