From Models of Cognition to Robot Control and Back Using SNNs
187
To compute the correlation between two trajectories, we use the method
described by Shadmehr et al. [22]. First, the hand position trajectories are transformed into hand velocity trajectories. The correlation between trajectory X and
trajectory Y is obtained through the following equations:
ρ(X, Y) =
Cov(X, Y)
σ(X)σ(Y)
(18)
Cov(X, Y) = E[XY] − E[X] − E[Y]
(19)
σ(X) =
E[(X i − E[X]) 2 ])
(20)
However, because these are not time series of scalars but rather 3D vectors,
the multiplications in the equations above should be seen as inner products,
resulting in scalar values.
We measured three runs in the experimental condition (i.e. with force field).
Each of these runs consisted of five consecutive trials, in which all nine targets
were presented for two seconds each, with the centre target presented between
two consecutive targets. For each of the three experiment runs we measured the
correlation between the trial trajectories and baseline trajectory.
4 Results
The results can be seen in Fig. 6. The maximum achieved correlation across all
the trials is 0.64. Although we see that none of the experiment runs achieve a
correlation of 0.9 such as seen in DeWolf et al. [10], we do see that the correlation
increases across the trials, indicating that trajectories get closer to the baseline.
We also see that the mean absolute error decreases across trial. To verify, we
compute a linear fit for each of the experimental runs and test for significant
effect of the resulting linear regression with trial number as the predictor (see
Table 1). We find statistically significant improvements in correlation with baseline in two of the three experimental runs. Similarly, we find that the hand space
error decreases significantly in two of three runs, as does the path length in one
run. These findings suggest that the adaptive CB component indeed improves
the reaching performance over multiple trials, showing that it is able to compensate the imperfect knowledge about arm parameters, and the inaccurate forward
dynamics. That said, given that the ranges in which these values increase or
decrease are relatively small (as is the number of total experimental runs), these
results should not be over-interpreted but be understood as a demonstration of
the concept.
187
To compute the correlation between two trajectories, we use the method
described by Shadmehr et al. [22]. First, the hand position trajectories are transformed into hand velocity trajectories. The correlation between trajectory X and
trajectory Y is obtained through the following equations:
ρ(X, Y) =
Cov(X, Y)
σ(X)σ(Y)
(18)
Cov(X, Y) = E[XY] − E[X] − E[Y]
(19)
σ(X) =
E[(X i − E[X]) 2 ])
(20)
However, because these are not time series of scalars but rather 3D vectors,
the multiplications in the equations above should be seen as inner products,
resulting in scalar values.
We measured three runs in the experimental condition (i.e. with force field).
Each of these runs consisted of five consecutive trials, in which all nine targets
were presented for two seconds each, with the centre target presented between
two consecutive targets. For each of the three experiment runs we measured the
correlation between the trial trajectories and baseline trajectory.
4 Results
The results can be seen in Fig. 6. The maximum achieved correlation across all
the trials is 0.64. Although we see that none of the experiment runs achieve a
correlation of 0.9 such as seen in DeWolf et al. [10], we do see that the correlation
increases across the trials, indicating that trajectories get closer to the baseline.
We also see that the mean absolute error decreases across trial. To verify, we
compute a linear fit for each of the experimental runs and test for significant
effect of the resulting linear regression with trial number as the predictor (see
Table 1). We find statistically significant improvements in correlation with baseline in two of the three experimental runs. Similarly, we find that the hand space
error decreases significantly in two of three runs, as does the path length in one
run. These findings suggest that the adaptive CB component indeed improves
the reaching performance over multiple trials, showing that it is able to compensate the imperfect knowledge about arm parameters, and the inaccurate forward
dynamics. That said, given that the ranges in which these values increase or
decrease are relatively small (as is the number of total experimental runs), these
results should not be over-interpreted but be understood as a demonstration of
the concept.