## 1 Introduction
The 'Speed-Accuracy' tradeoff model was proposed to capture the variability of the effective endpoint position expressed in terms of standard deviation, SD(x) that is put into linear relation with speed [18]:  
\[\mathrm{SD(x)}\ =\ \mathrm{a+b*}\left(\frac{\mathrm{A}}{\mathrm{MT}}\right) \tag{4}\]  
Despite the similarity with the [PERSON] law, Equation (4) is related to the stochasticity of the phenomenon: the higher the velocity of covering a trajectory is, the higher is the uncertainty of meeting the target. Equation (4) refers to the [PERSON] experience of a reaching task between an initial position and a final target of width W at distance A. We propose to extend the 'Speed-Accuracy' tradeoff model (4) by considering the standard deviation of the endpoint position depending on both speed and ID as:  
\[\mathrm{SD(x)}\ =\ \mathrm{a+b*v^{\beta}+c*}\int_{\mathrm{t}}\frac{\mathrm{ds }}{\mathrm{W(s)}} \tag{5}\]  
where v is the average speed (v = A/MT) of a subject in following a nominal trajectory 't' while continuously meeting targets with variable width W(s) along the trajectory, and a, b, c, and \(\beta\) are the model parameters. In [10], a discussion of [PERSON]'s law experiments pointed out as the variability of the endpoint position follows a normal distribution meeting the nominal target width with an average error rate of 4%: in other words, individuals actually meet the target within an effective width We = W in 96% of hits observed. Consistently, the range of the standardized variable, z, of the unit-normal distribution was identified as: \(-2.066<\mathrm{z}<+2.066\). In case of an error rate observed other than 4%, We is obtained as:  
\[\mathrm{W_{e}}\ =\ \mathrm{W*}\ \frac{2.066}{\mathrm{Z_{e}}} \tag{6}\]being \(\rm z_{e}\) the standardized variable of the unit-normal distribution corresponding to the observed percentage error rate, \(\rm err_{e}\), other than 4%. Since the endpoint position follows a normal distribution, its standard deviation can be obtained as:  
\[\rm SD(x)\ =\ \ \frac{W_{e}}{2*z_{e}} \tag{7}\]  
In the next sections, the model (5) along with relations (6) and (7) are applied to two different reaching tasks: the 'Tunnel Traveling Task' and the 'Circular Traveling Task' whose experimental data are available in [19]. In case of the Tunnel Traveling Task (TTT), a rectilinear trajectory of length A and an infinite sequence of target of width W has to be met (see Figure 1). The corresponding ID formulation was obtained in [13] starting from Equation (3) as:  
\[\rm ID_{TTT}\ =\ \frac{A}{W*ln(2)} \tag{8}\]  
Following the same approach, in case of a semicircular trajectory reaching task the ID formulation has been obtained as follows. The task is represented in Figure 2 where a system of polar coordinates is considered. It is assumed a target width decreasing with the curvilinear abscissa, s, according to the following equation:  
\[\rm W(s)\ =\ W(0)-k*s\ =\ W(0)-k*r*\theta \tag{9}\]  
where:  
* W(0) is the initial target width (\(\theta\ =\ 0\))
* k is a constant tuning the reduction of the target width with the polar angle, \(\theta\)
* r is the radius of the nominal trajectory constrained by variable target width, W(s), along the trajectory  
Figure 1: Representation of a Tunnel Traveling Task. \[\text{C}\ =\ \text{B}*\log_{2}\left(\frac{\text{S}+\text{N}}{\text{N}}\right) \tag{2}\]0 Semicircular trajectory with a decreasing width (W(s)). Being:  
\[\text{C}\ =\ \text{B}*\log_{2}\left(\frac{\text{S}+\text{N}}{\text{N}}\right) \tag{2}\]1  
by substituting Equations (9) and (10) in Equation (3), the index of difficulty of a circular trajectory of target width W(s) can be obtained as:  
\[\text{C}\ =\ \text{B}*\log_{2}\left(\frac{\text{S}+\text{N}}{\text{N}}\right) \tag{2}\]2  
where \(\theta_{\text{f}}\) is the final polar angle of the trajectory.