#### 2.2. Task and Measurements

All archers completed 100 shots to an 18-m indoor target with their own bows. The target was a standard FITA (World Archery federation) circle of 40-cm diameter, located 130 cm above the ground. The archers were instructed to try to hit the center of the target on each shot. Accuracy of the shot was marked according to the standard rules as 10, 9, 8, 7, 6, or 0, starting from the center (10) to the outer ring (6) or miss (0). The first 10 shots were used to accommodate the archers to the room and to the measurement equipment, and the final 90 shots were used for data analysis. Archers shot 10 rounds of 10 arrows with a 3-min rest between rounds to prevent potential effects of fatigue. They were equipped with eight retroreflective markers (14 mm) on the (bilateral) acromia, epicondylus lateralis humeri, processus styloidueus radii, and caput metacarpale II, which were captured at 50 Hz with a 5-camera VICON MX F-20 system. Marker trajectories were labeled and filtered (low-pass Butterworth filter, fourth-order zero-lag, 6-Hz cut-off frequency) in the VICON Nexus 1.8.2 software. The global reference frame had its origin at the line of shooting and was oriented with the y-axis toward the target, the x-axis to the right, and the z-axis upward.

#### 2.4. Uncontrolled Manifold (UCM) Analysis

The UCM concept [

12] assumes that the variability in a system with abundant degrees of freedom is structured in a specific way to stabilize a particular performance variable. In the present study of archery shooting, the accuracy of the shot depends critically on the relative orientation of both hands which will, therefore, be used as hypothetical performance variable. This variable was previously successful in explaining motor control in a pistol shooting task [

18]. The geometric model of

Figure 1 shows the dependency of this performance variable on several elemental variables, namely the orientation of the trunk (described by the position of the acromion markers) and the shoulder, elbow, and wrist angles. For the UCM analysis in this study, we hypothesized that the performance variable was the relative orientation between both end-effectors (metacarpal markers), as a proxy for the true orientation of the arrow. The end-effector positions of the left (

LH) and right (

RH) hand were written as geometric models in the horizontal plane (Equations (1) and (2)) and in the vertical plane (Equations (3) and (4)) using the joint angles (

θ_{i}) and the acromion positions (

RACR,

LACR) as the elemental variables (

l_{i} represents the segment lengths of the upper arm (1), lower arm (2), and hand (3).

The relative orientation (

PV =

LH −

RH) was then used as the hypothesized performance variable (

PV) in the UCM analysis in both planes. This is clearly a redundant system because the number of degrees of freedom of the

PV (

m = 2) is much smaller than that of the elemental variable (

n = 10). For the horizontal plane, the Jacobian matrix (

J) which related differential changes in the elemental variables (

EV),

$dE{V}^{T}=(\begin{array}{cccccccccc}dLAC{R}_{x}& dLAC{R}_{y}& dRAC{R}_{x}& dRAC{R}_{y}& d{\theta}_{1}& d{\theta}_{2}& d{\theta}_{3}& d{\theta}_{4}& d{\theta}_{5}& d{\theta}_{6}\end{array})$, to differential changes in the performance variable, (

$dPV=J\cdot dEV$), was as follows:

Notation used to save space: $cos({\theta}_{1:2})=cos({\theta}_{1}+{\theta}_{2})$, $cos({\theta}_{1:3})=cos({\theta}_{1}+{\theta}_{2}+{\theta}_{3})$, etc.

Similarly, for the geometric model in the vertical plane,

$dE{V}^{T}=(\begin{array}{cccccccccc}dLAC{R}_{y}& dLAC{R}_{z}& dRAC{R}_{y}& dRAC{R}_{z}& d{\theta}_{7}& d{\theta}_{8}& d{\theta}_{9}& d{\theta}_{10}& d{\theta}_{11}& d{\theta}_{12}\end{array}),$ and

To analyze the structure of the movement variability, we decomposed the total variability into two components, one component in the UCM (

V_{UCM}, variability that results in a net zero change of the performance variable, i.e., “goal-equivalent variability”) and the second component orthogonal to the UCM (

V_{ORT}, variability which results in a changed orientation of the arrow, i.e., “non-goal equivalent variability”) [

13]. The equations below provide a dense notation in linear matrix algebra to compute the total variability and the components along the UCM and orthogonal to it [

20] (

**C** is the covariance matrix of the elemental variables, and

null(

J) is an orthonormal basis of the Jacobian which can be computed with singular-value decomposition).

These equations were solved at every time point in the dataset (the Jacobian and covariance matrix are time-dependent according to the changing posture), resulting in a time series of V_{UCM} and V_{ORT}. This calculation was performed separately for the trials with different accuracy scores. The normalizations in the denominator by the dimensions of the space (n = 10: dimension of the elemental variable, and m = 2: dimension of the performance variable) and the number of trials in a particular condition were to be able to compare the values across participants and conditions.

For each plane of motion and accuracy score, we formulated two hypotheses about the control of the performance variable. The first hypothesis took the time dependence of the UCM into account, i.e., the covariance matrix was calculated around the time-dependent mean of the elemental variables, i.e., ${C}_{ii}\left(t\right)=1/N\cdot {{\displaystyle \sum}}_{Ntrials}{\left(EV{\left(t\right)}_{i}-{\overline{EV\left(t\right)}}_{i}\right)}^{2}$. This hypothesis assumes that the motor control system tries to attain different positions at different percentages of the aiming phase, for instance, a more closed position toward the point of release. The second hypothesis assumed a simplified version, namely a time-independent control, and we used the overall mean around which the covariance matrix was calculated, i.e., instead of ${\overline{EV\left(t\right)}}_{i}$, we used ${\overline{EV}}_{i}$ (mean across trials and throughout time). This hypothesis was motivated by visual inspection of the time series of the elemental variables in the aiming phase revealing no clear trends in the data.

For both hypotheses, planes of motion and accuracy conditions, we calculated

V_{TOT},

V_{UCM}, and

V_{ORT}, and used these to define the index of motor abundance (IMA) [

21]:

This process up to the calculation of the index was performed for every subject individually and per accuracy score. A positive IMA (

V_{UCM} >

V_{ORT}) signifies that the elemental variables are controlled together in a specific way so as to stabilize the performance variable (kinematic synergy), while a negative IMA (

V_{ORT} >

V_{UCM}) indicates the reverse, i.e., destabilization of the performance variable. An IMA of zero would mean that there is no structure in the covariance of the elemental variables with respect to the chosen performance variable [

21]. Similar to Reference [

18], we took a criterion of a minimum of 10 trials per accuracy score to calculate a stable mean and covariance matrix around which the Jacobean could be linearized. However, the analysis of the accuracy (see

Table 1) showed that the distribution was too skewed and unequal across subjects and did not meet this criterion of a minimum of 10 trials per score. We, therefore, concatenated all trials with scores of 6, 7, or 8 into one category (“low accuracy”) and all trials with scores of 9 or 10 into another category (“high accuracy”).