The quantitative use of velocity information in fast interception

Abstract

We ask whether a target’s velocity is considered when planning a fast interceptive action. Human volunteers hit targets that could move at different velocities from across a tilted screen (the hand starting 40 cm away from the screen). We examined how the direction in which the hand initially moved depended on the target’s velocity, using various analyses. For slow targets, the initial movement direction was appropriate for the target’s velocity. This is evidence that velocity information was used quantitatively in directing the hand. A model analysis showed though that velocity information is probably not used to predict the future target position. For targets moving at a velocity above average, or above 12 cm/s, the initial movement direction did not depend on the target’s velocity. Similar behaviour is also known from pursuit eye movements.

Appendices

Appendix 1: the initial adjustment

Below we derive the initial adjustment. First, we describe how we estimated the slope of the relation between target position and initial movement direction for each subject (assuming that the subjects judge the difference in target position correctly). Subsequently we use this relation to calculate each subject’s initial adjustment. The relation between target position and movement direction can only be calculated for targets with the same velocity. Targets of each velocity appeared at several different positions in all experiments described in the present study. The initial movement direction was calculated from each subject’s average paths (see “General methods”). We used average paths instead of individual movements to obtain a better signal to noise ratio.

Let N be the number of pairs of targets (in a single experiment) with the same velocity but a different position of appearance. For example, in experiment 1 (Table 1, Fig. 2A) there were two conditions with 6 cm/s targets (conditions 1 and 2), three conditions with 12 cm/s targets (3, 4 and 5) and two with 18 cm/s targets (6 and 7). If we number these seven conditions, the N=5 combinations of conditions i, j with equal velocity would be i, j ∈ {1, 2; 3, 4; 3, 5; 4, 5; 6, 7}. Further, let θ _i be the initial direction of the rod’s average path in condition i, and let x _i be the average target position at that same moment (Fig. 2B). For each subject we derived a single constant S for the relation between x and θ (Fig. 2C):

$$ S = {{1}\over {N}}{\sum\limits_{i,j} {{{{x_{i} - x_{j} }} \over {{\theta _{i} - \theta _{j} }}} }} $$

(1)

There was a significant (t-test across the N combinations, α=.05) positive relationship between target position and initial movement direction in 75 out of a total of 78 experimental sessions (6 experiments, 12–14 subjects in each). In the three cases where this relation was not significant, the subject’s data had to be excluded from the quantitative analysis (though not from the qualitative analysis). The estimated slope S allows us to express differences in initial movement direction as differences in “aiming position” (x ^aim), even for targets of different velocities. Let k and l be conditions with targets of different velocity. Then

$$ x^{{aim}}_{k} - x^{{aim}}_{l} = S \cdot {\left( {\theta _{k} - \theta _{l} } \right)}. $$

(2)

The difference in aiming position is the combined effect of the differences in position at that moment (x _k −x _l ), and ones that are expected to arise because differences in velocity are taken into account ${\left( {x^{{adj}}_{k} - x^{{adj}}_{l} } \right)}$. Thus:

$$ x^{{aim}}_{k} - x^{{aim}}_{l} = x_{k} - x_{l} + {\left( {x^{{adj}}_{k} - x^{{adj}}_{l} } \right)}. $$

(3)

In order to express ${\left( {x^{{adj}}_{k} - x^{{adj}}_{l} } \right)}$ in terms of a difference in the velocity that is used to guide the hand (i.e. the initial adjustment, $v^{{adj}}_{{^{{k,l}} }}$) we have to divide it by the remaining movement time (rMT). Since the MT differed considerably between subjects, each subject’s average rMT was used for his or her data.

$$ v^{{adj}}_{{^{{k,l}} }} = {{{x^{{adj}}_{{^{k} }} - x^{{adj}}_{{^{l} }} }}\over {{rMT}}}. $$

(4)

Filling in (2) and (3) solves Eq. 4:

$$ v^{{adj}}_{{k,l}} = {{{S \cdot {\left( {\theta _{k} - \theta _{l} } \right)} - {\left( {x_{k} - x_{l} } \right)}}}\over {{rMT}}}. $$

(5)

We always present the initial adjustment with respect to the average of the three or four conditions with the reference velocity L (four in experiment 2, three in the other experiments). Thus, the initial adjustment for condition k becomes:

$$ v^{{adj}}_{{k,L}} = \frac{{{\left( {S \cdot \theta _{k} - x_{k} } \right)} - {\left( {\overline{{S \cdot \theta _{L} - x_{L} }} } \right)}}} {{rMT}}. $$

(6)

Targets of each velocity appeared at several (2–4) different positions. In order to obtain one value of the initial adjustment per target velocity per subject, these 2–4 values were averaged.

Note that for S, θ, x and rMT we used the measured values for each individual subject in the experiment that is analysed.

Appendix 2: models with and without use of target velocity

The modelling approach that we followed is described in de Lussanet et al. (2002b), where we predicted the hand’s path with use of target velocity (velocity model) and without the use of target velocity (position model). Here we will first briefly describe the two models and then how we implement them to predict the initial adjustments for the data of experiment 3 in Smeets and Brenner (1995) and for the data of the six current experiments. In all cases, we applied the model to movement paths averaged over those of all subjects.

The hand’s path was modelled as continuously accelerating towards the screen on which the target was presented and as a damped linear oscillator in the perpendicular component, the direction in which the target moved. The acceleration towards the screen was chosen such that for each condition the modelled movement time was the same as the real average movement time. In the position model, the oscillator describes the differences between the paths towards targets at different positions and of different velocities. With x being the hand’s position and q being the equilibrium point (in which the hand’s acceleration is 0), the general differential equation for such an oscillator is:

$$ v^{{adj}}_{{k,L}} = \ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + b \cdot \ifmmode\expandafter\dot\else\expandafter\.\fi{x} + k \cdot {\left( {x - q} \right)} = 0 $$

(7)

(each dot over a variable represents the derivative with respect to time). Stiffness, k, and damping, b, were fitted for movements towards targets that only differed in the position on the screen where they appeared, with respect to the hand’s starting position. We used the same values of b and k for the position model and the velocity model. For the data from experiment 3 in Smeets and Brenner (1995) we used paths towards stationary targets, averaged over all subjects (as in de Lussanet et al. 2002b). The fitted stiffness b=7.96 s⁻¹ and the damping k=61.0 s⁻². In the present study, the movement times were much shorter than in Smeets and Brenner (1995) so we had to fit a new b and k. For this we used the movements towards the 12 cm/s targets in experiment 1, again averaged over all subjects. The fitted stiffness b=22.7 s^–1 and the damping k=146.0 s^–2. These values were used to predict the initial adjustments of all six experiments.

The equilibrium point is not necessarily stationary. We assumed that it is the continuously updated position where the subject expects to hit the target. The rate at which the equilibrium point changes its position depends on the subject’s expectation of the target’s velocity, v. We used the average velocity (de Lussanet et al. 2001). With t being the time from which the hand moves (the reaction time, RT), s _RT the target’s position at the average RT and $\ifmmode\expandafter\dot\else\expandafter\.\fi{s}$the target’s velocity, q can be calculated as:

$$ q{\left( t \right)} = s_{{RT}} + \ifmmode\expandafter\dot\else\expandafter\.\fi{s} \cdot t + v \cdot {\left( {MT - t} \right)} $$

(8)

(where t <MT, the movement time).

In the position model, target velocity thus only influences the hand through the changes in position. In the velocity model, the hand is damped relative to the equilibrium point, rather than to space. The damping coefficient thus acts on the difference between the target’s velocity and the average velocity: $\dot{q} = \dot{s} - v$. The differential equation for the velocity model is thus:

$$ \ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + b \cdot {\left( {\ifmmode\expandafter\dot\else\expandafter\.\fi{x} - \ifmmode\expandafter\dot\else\expandafter\.\fi{q}} \right)} + k \cdot {\left( {x - q} \right)} = 0 $$

(9)

For the (imaginary) case that the MT would be independent of target velocity, the velocity model would predict that the initial adjustment scales proportionally with target velocity. As the MT really decreases with target velocity, we expect that the relation flattens off with increasing velocity. The solutions for differential Eqs. 7 and 9 are given in de Lussanet et al. (2002b) as well as further details on the modelling results.

To predict the initial adjustment we first predicted the hand’s path, using the velocity model with the fitted values for b and k given above, the target’s velocity, the target’s average position at the RT (Table 2), and the average MT (Table 2). In some simulations, we incorporated a delay of 62 ms before velocity information was used. In these cases, the first 62 ms of the path was predicted using zero velocity (which is in effect the same as the position model), whereas the rest of the path was predicted using the actual target velocity. The initial adjustment was calculated from the predicted paths in the same way as it was calculated from the measured paths (see “Appendix 1”).