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The authors have declared that no competing interests exist.

Conceived and designed the experiments: DC WTA EB. Performed the experiments: DC FW WTA. Analyzed the data: DC FW HX EB. Contributed reagents/materials/analysis tools: DC FW WTA. Wrote the paper: DC FW.

This work introduces a coordinate-independent method to analyse movement variability of tasks performed with hand-held tools, such as a pen or a surgical scalpel. We extend the classical uncontrolled manifold (UCM) approach by exploiting the geometry of rigid body motions, used to describe tool configurations. In particular, we analyse variability during a static pointing task with a hand-held tool, where subjects are asked to keep the tool tip in steady contact with another object. In this case the tool is redundant with respect to the task, as subjects control position/orientation of the tool, i.e. 6 degrees-of-freedom (dof), to maintain the tool tip position (3dof) steady. To test the new method, subjects performed a pointing task with and without arm support. The additional dof introduced in the unsupported condition, injecting more variability into the system, represented a resource to minimise variability in the task space via coordinated motion. The results show that all of the seven subjects channeled more variability along directions not directly affecting the task (UCM), consistent with previous literature but now shown in a coordinate-independent way. Variability in the unsupported condition was only slightly larger at the endpoint but much larger in the UCM.

Daily motor tasks typically involve more degrees-of-freedom than strictly required. For instance, pressing a button in the elevator only requires positioning the fingertip at a three-dimensional location in space. However, to move the arm we need to control many more degrees-of-freedom (at least seven, only considering the shoulder, elbow and wrist) than required by the task, each with its own variability due to physiological factors such as tremor. Variability at proximal joints (e.g. shoulder or elbow) is expected to be amplified and projected at the distal end (fingertip). Remarkably, inter-joint coordination reduces the final variability at the fingertip position. Recent theories, such as the uncontrolled manifold (UCM), distinguished between inter-joint variability that would not affect the finger position and variability that would affect the final task. A major issue is that traditional UCM methods rely on the coordinate system chosen to analyse the arm motion. Therefore, we introduce a coordinate independent UCM method for tasks performed with handheld tools, e.g. surgery. This paper describes a new method and demonstrates that it enables an accurate analysis of static pointing. The results clearly show that the subjects can channel variability in dimensions that do not affect the task outcome.

Although highly stereotyped, human movements performed with the same intention are never exactly the same, displaying large variability in consecutive trials. Rather than just ‘biological noise’, many studies have pointed out how variability may in fact provide important clues on the underlying neural strategies. Analysis of structure in variability, and its changes, has therefore become an important tool for researchers in neuromotor control and learning, especially in presence of redundancy

The problem of variability in redundant motor tasks was formulated by Bernstein

Redundancy and motion variability are important not only for blacksmiths but characterize virtually every daily activity, from grasping a cup to signing off a letter, where we typically have many more degrees-of-freedom (dof) than necessary to fulfil the task. We are particularly interested in tasks involving hand-held tools such as microsurgery, where noise induced by tremor, amplified by the visual magnification provided by the optical microscope, is a critical factor of performance

In this work, we consider static pointing tasks, such as the one in

The tool is grasped at a fixed position

A key aspect of our study is that

Scholz and Schoner

Despite its appeal, the computational procedures behind UCM (and principal component analysis in general) have been recently criticized for being coordinate-sensitive

More importantly, covariance-based analysis may reach different results if the coordinates are transformed. Even the linear transformation of joint angle coordinates from absolute to relative leads to different conclusions

More than a century ago, Physics undertook a geometrization process of its main theories in the effort to achieve descriptions of phenomena in a coordinate-independent way, using differential geometry. Computational modelling in motor control is still at an early stage although some attempts have been made in this direction, e.g.

Although internal representations are largely unknown, it is clear that the brain does take into account the geometry and physics of the external world. Early studies on reaching tasks in the horizontal plane showed how we consistently move along straight lines in the extrinsic, end-point space

Furthermore, a very recent study by Danziger and Mussa-Ivaldi

In this paper we study accuracy during pointing via a coordinate-independent analysis of variance based on a choice of metric structure suggested by the specific application, in our case manipulation via hand-held tools. The paper outline is as follows. Next section will present all the steps involved in the classical UCM approach, based on vector calculus. Then an overview of the Riemannian geometric framework required to extend such vector calculus steps to more general settings is presented, along with the detailed formulation required to compute intrinsic variance based on metric properties of rigid bodies motions. This approach is then applied to the analysis of data of healthy subjects performing static pointing tasks.

The starting point for classical uncontrolled manifold (UCM) analysis is the definition of a forward kinematic model

The UCM analysis is simplified by linearizing the nonlinear forward kinematics about the average posture, hereafter reference posture, computed across

Riemannian geometry allows generalizing to nonlinear spaces traditional concepts and tools from vector calculus, e.g all the steps behind the classical UCM approach. In this section, we shall try to build some intuition to help relating these new geometric tools with the classical ones. For a more comprehensive and detailed description, the reader is referred to

With reference to

The central element in Riemannian geometry is the introduction of a

A metric does not come with the manifold, it is extra structure that should be defined based on the application. In this lies the connection to Mechanics:

if our manipulator moves from an initial posture

among all the possible smooth trajectories between two points, the geodesics are the ones of minimum length in the sense of

There is an important map from the tangent space to the manifold itself which is known as

There is also its inverse map, known as

if reaching a new position for a starting point in a unit time requires a large initial velocity, despite the choice of the shortest path, then the two points are probably far apart. The initial velocity, for a geodesic, is therefore a good measure of

Given a set of

Once we have a mean value

As mentioned previously, a metric does not come with a configuration manifold, it is extra structure which is typically defined by the application. Therefore

The general Riemannian geometric approach starts from the definition of the configuration manifold. When it comes to rigid body motions, the configuration manifold is more structured than the general case. The space of rigid body configurations is in fact a

With reference to

The 3D orientation of a rigid body can be described by means of a

The rotation matrix

For a rigid body pivoting about a hinge,

The

If we were to describe a physical motion

The kinetic energy (

In dynamics, the inertia matrix can be determined once the shape and the material properties such as density of an object are known. Clearly, when kicking an object, it makes a lot difference for the subsequent motion whether it is a round soccer ball or an elliptical rugby ball.

However, for kinematic purposes, inertial properties may not be of interest. For example, in our application, if the hand-held tool is very lightweight and our motion is relatively slow, its shape and material properties might be neglected at a first level of analysis. In such cases, we can abstract the inertial properties and consider an isotropic inertia, rather than a general ellipsoid, by setting

The focus of our analysis is on pointing tasks performed via hand-held tools. The key aspect is that

As we seek a formulation which is spatial frame invariant, we will try to express everything in body coordinates. Body velocities

The nullspace

Similarly, it can be verified that

Given a set of

In the classical UCM approach

For nonlinear spaces as for rigid body motions, this is not possible and we will extend the classical UCM approach with the concept of geodesics as proposed by Fletcher et al.

Recalling (22), the ‘difference’ between a pose

This section first introduced the general steps involved in the classical UCM approach and then derived an intrinsic definition of each of these steps for the case of static pointing with hand-held tools. In particular, an intrinsic definition of deviations

To analyze variability during static pointing tasks with hand-held tools, experiments were conducted with 7 healthy subjects without any known history of neuromuscular impairment. All of them declared to be right-handed and gave their informed consent prior to the experiment. The study was approved by the institutional review board of Nanyang Technological University and was conducted according to the principles expressed in the Declaration of Helsinki. Each subject was asked to hold a sensorized stylus of a Polhemus Liberty system (

The experimental protocol consisted of 20 consecutive trials. In each trial, the subject was asked to make a 15 seconds, steady contact between stylus and target tips separated each time by a large movement of the stylus approximately 20 cm away from the body. Only the inner most 10 seconds between two large movements were analyzed (thick solid lines in

The ‘ * ’ marking the minima correspond to the positions furthest away from the target. The thick solid lines are the data of interest, within 5 seconds before and after the midpoint between two minima.

One minute rest was given every 5 trials. No visual magnification was provided to the subject. The protocol was performed in two different experimental conditions: in Exp I the elbow of the right arm was supported on the table, while in Exp II the arm was unsupported, resulting in different noise conditions

For each trial, only the inner most 10 seconds (2400 samples) of steady contact were analyzed.

To detect physiological tremor, the power spectral density (PSD) was estimated. To this end, the velocity components along each axis were estimated by numerically differentiating the tip position, component-wise. Then, for each (10 seconds) trial, the pwelch() function in the MATLAB environment was used to estimate the average PSD over ten non-overlapping time windows (1 second each). Finally, for each subject, the PSD estimates obtained for each trial were averaged.

For each trial, the relative UCM components

Similarly to the UCM components and their ratio, also the variances of

According to the UCM theory

To test whether the average UCM ratio

A series of analysis of variance (ANOVA) tests with repeated measures was conducted to test the effect of experimental conditions and of UCM component on the variance-per-dof. The dependent variables are variance-per-dof,

Analysis of the power spectral density of the variability of motions, revealed a frequency profile containing at least two major peaks in the frequency ranges 0–7 Hz and 7–15 Hz, consistent with the spectral density of physiological tremor

For both experimental conditions, there was much more variability in the UCM subspace than in the orthogonal subspace (i.e.

For both experimental conditions, each subject shows a statistically significant difference (

A three-way repeated measures ANOVA (experimental condition

Effects due to experimental condition (

Bottom figures show the

Similar analysis as for the UCM components and ratio, revealed, in general, more variability in

Vertical lines represent the standard errors. Statistically significant differences are highlighted for

As shown in

Bottom figures show the

Complexity of the human body typically leads to an excess of degrees of freedom for virtually every motor task we are routinely involved with. Redundancy is also adopted in the design of artificial systems, e.g. articulated robots, as extra dof can increase dexterity and robustness. However, redundancy also requires sophisticated control strategies, for example in devising control laws which guarantee repeatable postures

From an analytical perspective, repeatability and variability of movement have traditionally been distilled from experimental data via statistical approaches, by computing average and standard deviation estimates of movement properties derived from repeated trials. As pointed out by Newell and Slifkin

In the last decade, various researchers have started exploring the structure of variability rather than just its amount. Structure in variability has been so far explored along two major avenues: its temporal or its geometric features. These two aspects are by no means exclusive and, in general, a combined temporal and geometric analysis is likely to provide more insight into human motor control.

In this paper, we considered a pointing task and we focused on the geometric structure of variability, and also estimated the power spectral density to verify the frequency signature of physiological tremor. Consistent with previous literature on physiological tremor

The geometric structure of variability was underlined by the pioneering work of Scholz and Schoner

Similarly to Morrison and Newell

We performed a similar analysis comparing variability at the gripping point and at the tool-tip. Although leading to qualitatively similar results, UCM analysis leads to ‘crisper’ results in terms of statistical significance. This was expected since the goal of the task, clearly defined in pointing tasks, is fully captured by the UCM analysis.

One of the most appealing aspects of the UCM method is the possibility to distinguish ‘good’ variance (not affecting the task success) from ‘bad’ variance (affecting task performance) and, therefore, to identify skillful performance. Subjects who are able to channel physiological tremor into movements which do not affect the task, might be deemed more skillful. Therefore one might be tempted to relate skills to the UCM ratio. However, our results suggest another possible explanation.

From our experiments, there appear to be two groups of subjects: those who show a statistically significant improvement in terms of UCM ratio (

Despite its appeal, a weakness in the UCM analysis has been recently pointed out in relation to its coordinate dependence

As also mentioned by Sternad et al.

Differently from the UCM approach, Sternad et al.

From this perspective, a main contribution of this paper is the use of task-specific features to construct an appropriate metric, which leads to a frame-invariant and objective analysis in the sense of

In particular, manipulation via hand-held tools suggests the use of the scale-dependent left-invariant metric (18), a particular type of kinetic energy metric especially suitable for kinematic rather than dynamic analysis, initially proposed by Park and Brockett

From a mathematical perspective, it should be noted that one could have chosen a right-invariant metric to guarantee body-fixed frame indifference and forgo the left-invariance, i.e. bearing a dependence on the spatial frame. This would not be acceptable in our case, as the results would be dependent on the choice, for example, of the measuring system.

Previous mathematical arguments are very general and do not take into account the specific requirements of the task. To describe the pose of a tool, the experimentalist needs to choose two coordinate frames: a fixed frame

In this paper, we focused on static pointing tasks. However, starting from the original work by Scholz and Schoner

In general, this approach might raise concerns in dynamic tasks which are not appropriately timed. When a task is self-paced, there is no guarantee that events occurring in different trials at the same normalized time are necessarily related.

In any case, the method would still be applicable to

In a recent study, Danziger and Mussa-Ivaldi

As a final note, the type of statistical analysis needs not to be restricted to simple analysis of variance, as done in this paper along the line of classical UCM approach. Once a Riemannian framework is in place, as in our paper or in the work of Biess et al.

Our analysis does not depend on the orientation of the moving frame

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The authors are grateful to Dr Frank C. Park for very useful discussions as well as to the anonymous reviewers for their insightful feedback.