University of Groningen Synergies and end-effector kinematics in upper limb movements Tuitert, Inge

University of Groningen

Synergies and end-effector kinematics in upper limb movements

Tuitert, Inge

DOI:

10.33612/diss.98793947

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Tuitert, I. (2019). Synergies and end-effector kinematics in upper limb movements. University of Groningen.

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Synergies and end-effector

kinematics in upper limb movements

Colophon

This PhD-thesis was an international cooperation, a ‘co-tutelle’, between the University

of Groningen (the Netherlands) and Aix-Marseille University (France).

The experiments in chapters 2, 3, 4, and 5 were conducted at the Center for Human

Movement Sciences, University Medical Center Groningen, the Netherlands.

The project was financially supported by funding from the Excellence Initiative of

Aix-Marseille University - A*MIDEX, a French “Investissements d’Avenir” program.

PhD-training was facilitated by the research institute School of Health Research

(SHARE, Groningen) and l’Ecole Doctorale Sciences du Mouvement Humain (EDSMH,

Marseille).

The printing of this thesis was financially supported by:

•

University of Groningen

•

University Medical Center Groningen

Paranymphs:

Anniek Heerschop

Laura

Golenia

Cover and layout:

Studio Anne-Marijn (www.studioanne-marijn.com)

Printed by:

Netzodruk, Groningen

ISBN printed version:

978-94-034-1924-4

ISBN digital version:

978-94-034-1925-1

©Copyright 2019, Inge Tuitert

All rights reserved. No part of this publication may be reproduced or transmitted

in any form or by any means, electronic and mechanical, including photocopying,

recording, or any information storage retrieval system, without written permission

from the author.

Synergies and end-effector kinematics in upper limb movements

PhD thesis

to obtain the degree of PhD of the University of Groningen

on the authority of

Rector Magnificus Prof. C. Wijmenga and in accordance with the decision by the College of Deans

and

to obtain the degree of Doctor of Aix-Marseille University

on the authority of the President Prof. Y. Berland

and in accordance with

the decision by the Vice President for Research. Double PhD degree

This thesis will be defended in public on Wednesday 6 November 2019 at 14.30 hours

by

Inge Tuitert

born on 11 October 1990 in Deventer

Supervisors Dr. R.M. Bongers Prof. R.J. Bootsma Prof. E. Otten Assessment Committee Prof. A. Roby-Brami Prof. R.G.J. Meulenbroek Prof. J.J. Temprado Prof. T. Hortobágyi

TABLE OF CONTENTS

General

Introduction

6

Comparing different

methods to create a

linear model for the

uncontrolled manifold

analysis

14

Does practicing a

wide range of joint

angle configurations

lead to higher flexibility

in a manual obstacle-

avoidance target-

pointing task?

28

The effect of the

height to which the

hand is lifted on

hori-zontal curvature in

horizontal point-to-

point movements

44

Task constraints act at

the level of synergies

and at the level of

end-effector kinematics

in manual reaching

and manual lateral

interception

58

General

Discussion

76

References

86

Appendices

Summary 98

Samenvatting 102

Résumé étendu 105

About the author 108

Dankwoord 110

GENERAL

INTRODUCTION

9

The degrees of freedom problem

When humans perform movements repeatedly, they are never completely the same. This is possible because many degrees of freedom (DOF) of the human motor system are involved when performing a motor action. In most cases, the number of DOF involved exceeds the minimum necessary to complete the motor task at hand. This results in many possible solutions for a given task, which is the so-called redundancy problem [1–3]. The latter appears at different levels of the human motor system. For example, at muscle level [1,4,5], where more than one muscle is available for a specific joint rotation, or at end-effector level, where there are ample 3D position solutions to move the end-effector from one place to another [6,7]. An important level that also depicts redundancy is the joint level. The Russian physiologist Nicolai Bernstein described redundancy at the joint level in his famous example investigating professional blacksmiths [1]. Using a motion analysis system, Bernstein captured blacksmiths hitting a chisel with a hammer and analyzed both the joint level and the end-effector (i.e., the hammer) level during the execution of the task. He revealed that each hitting repetition was slightly different, that is, the motions of all joints were different, while the hammer hit the chisel at almost the exact same location with each repetition [8]. Bernstein called this the ‘repetition without repetition’ phenomenon [1]. He was intrigued by this finding and one of his main questions was how the redundant DOF are coordinated to accomplish a motor task with high precision.

Motor coordination

Coordination of redundant DOF has been a major topic of research during the past decades (e.g., [1,9–13]). The questions studied concerning the redundant DOF (also called DOF problem; [1]) range from how and at what levels the DOF should be selected (e.g. [10]), and whether the redundancy should in fact be called abundancy (e.g. [2,11]), to ideas on how the DOF are coordinated, such as, muscle synergies (e.g. [12,14]), optimal control theory (e.g. [15–17]), or dynamical systems theory [10,13,18–20]. In the discussion about how the DOF are coordinated, variability in DOF has received a large amount of attention and has been studied from different perspectives. For example, from an optimal control perspective one could argue that variability is equivalent to noise because there is only one optimal solution to the DOF problem for a given task and all deviations from that solution are the result of sensorimotor noise [15]. In contrast, from a dynamical systems perspective, variability in motor behavior is considered to characterize a motor system [3,13,18,20,21]. The latter perspective will be followed in the present thesis and will be further introduced below.

The dynamical systems perspective on motor coordination

Motor coordination from a dynamical systems perspective can be described within a system not merely restricted to the DOF that need to be coordinated, but it includes the full perception action cycle [22,23]. This cycle comprises the environment and the agent, where interactions amongst environment, organism, and task constraints regulate the motor behavior that emerges [10,24,25]. These constraints can be exemplified by returning to the

10

blacksmith example. Here, the ranges of motion in the joints of the arm are an example of organism constraints, the location of the chisel is one of the task constraints, and the gravity working upon the arm is one of many environmental constraints. Accordingly, the hitting behavior emerges from the interactions amongst the range of motion of the joints, the location of the chisel, the gravity working upon the arm, and many other constraints. Within the dynamical systems perspective, it is suggested that due to these interacting constraints, synergies emerge that temporarily link the DOF into task-specific units [3,13,18,20]. Note that, synergies are closely related to the concept of coordinative structures (e.g. [18,20]). In such a task-specific unit, potentially independent DOF are temporarily linked [10,13,26] into a unit with respect to a certain function or task. That is, DOF co-vary to stabilize the specific task performance, which implies that variations in (one) DOF are compensated for in other DOF in such a way that the performance remains constant, this is the so-called flexibility of a synergy [18,26–28].

Kay [10] described the emergence of a synergy as the first step of a two-step constraining process (see also [26,29,30]). In the first step, the interactions amongst environment, organism, and task constraints temporarily link the independent DOF into a synergy (see Figure 1). In the second step, the constraints act on the synergy, resulting in the specific behavior (see Figure 1). This approach explicitly states that after the assembly of the synergy, a further constraining process must come into play, to produce one particular movement of the subset of solutions [10]. Kay [10] analyzed the outcome of both steps of the two-step constraining process at once using dimensionality analysis in a rhythmic task. In most other dynamical systems accounts on coordination by other authors, the differentiation into two steps has not often been described and examined as such. An example of an exception is a description of the two-step process in a perspective article on interpersonal coordination by Riley et al. [26], accompanied by an analysis of predominantly the first step of the process by Romero et al. [31]. However, to examine whether a two-step process occurs, I think that the interaction of the two steps of the process should be investigated. That is, to be able to grasp how the interactions of constraints lead to the emergence of behavior, both steps of the constraining process should be analyzed.

Therefore, I aimed to gather more understanding on how the redundant DOF are coordinated by focusing on synergies and their role in specific behavior. To do so, I focused on the influence of task constraints on the two steps of the process of emergent behavior and the interaction of these two steps. That is, I examined the influence of task constraints on synergies and specific behavior in discrete upper extremity movements. Investigating this in discrete upper extremity movements, such as goal-directed reaching and interception, is of major importance because these actions are involved in many activities in daily life. Because in previous research on discrete upper extremity movements the level of synergy and the level of specific behavior have not been analyzed as separate steps, a different methodology is needed in the present thesis, which can be outlined as follows. I assessed the synergies that are hypothesized to emerge in the first step of the two-step process in discrete upper extremity movements by examining structure in variability of DOF, using the uncontrolled manifold (UCM) analysis which will be explained below [4,32,33]. The specific behavior that is hypothesized to emerge from this synergy is quantified by means

11

of end-effector kinematics in the present thesis. The outlined innovative methodology is tested separately for each level, before looking at the interactions of the two levels. The present thesis assessed the influence of task constraints on the following aspects of goal-directed actions: 1) synergies, 2) end-effector kinematics, and 3) the interaction of synergies and end-effector kinematics. In the subsequent section, I will discuss the influence of constraints on the separate levels of synergies and end-effector kinematics, because, to my knowledge, the interactions of the two have not received much attention in the past.

The influence of constraints on emergent behavior

In the present thesis, the two-step constraining process of, first, the emergence of synergy and, second, the emergence of specific behavior will be assessed by looking at the influence of task constraints on synergy and end-effector kinematics. A selection of previous research on these topics will be outlined below.

Before addressing synergies, I will explain the method of analysis applied in the present thesis. The UCM analysis will be used to quantify the structure in variability of individual DOF across repetitions of trials [4,32,33]. To explain this analysis, I use manual pointing as an example. In pointing, the DOF selected at the joint level (i.e. elemental variables) are the shoulder, elbow, wrist, and finger joint angles, and the DOF at the end-effector level it is the 3D fingertip position. All different joint angle configurations which maintain the fingertip position compose the solution space for the task. Using this solution space, the variability observed in joint angles over repetitions can be parsed into two types of variability: V_ucm and V_ort. The former is the variability within the solution space that does not affect the position of the fingertip (see the green stick figures in Figure 2), and the latter is the variability outside the solution space, which does affect the position of the fingertip (see the red stick figures in Figure 2; [4,32,33]). These two types are used to examine the structure of variability of the

Figure 1. Two-step con-straining process where the DOF (squares, triangles, and circles at upper panels) are linked into a synergy at the upper arrows and the synergy is constrained to specific behavior at the lower arrows. The green arrows represent environ-ment, organism, and task constraints.

12

Figure 2. The UCM analysis partitions variability in V_ucm and V_ort. V_ucm is depicted by the stick figure at the left and V_ort at the right.

DOF. Variability within the solution space should be larger than outside the solution space such that the performance remains close to constant. In the present thesis, this structure in variability of DOF is interpreted as the consequence of a synergy and previous research will be presented as such. The UCM method also allows for the quantification of flexibility. This is quantified by the ratio of V_ucm and V_ort, where a larger V_ucm with respect to V_ort reflects a larger flexibility. Additionally, I also aim to take the UCM analysis to a higher level by making it more suitable for multi-joint tasks (chapter 2) and obtaining different measures (chapter 5) from this analysis that enables direct comparisons between synergies. These measures will also be applied in the present thesis.

In manual pointing, sit-to-stance, or finger-force production, it is revealed that V_ucm is larger than V_ort, indicating that there is structure in variability [31,32,34–43]. We interpreted this as the emergence of a synergy, which organizes the DOF to perform those tasks. Additionally, synergies’ hallmark flexibility has received much attention in previous research. Several studies suggest that if task constraints are more demanding, flexibility is exploited [39,40,44– 46]. For example, when participants perform a pointing task in the context of a potential change of target location, flexibility increases. Moreover, the flexibility of a synergy is also influenced by organism constraints [47,48]. For example, flexibility is reduced in particular groups, such as in patients with Parkinson’s disease [47,49], which probably makes it more difficult for those groups to maintain task performance in more demanding task conditions. Also, in a visuomotor adaptation paradigm, it has been shown that participants with high flexibility at baseline have higher learning rates [50].

Synergies are hypothesized to be constrained to specific behavior in the second step of the process, which is quantified by end-effector kinematics in the present thesis (cf. [10]). I outline previous research portraying the influence of task constraints on end-effector kinematics in the upper extremity tasks selected in the present thesis: manual reaching and manual lateral interception. Generally, the differences in task constraints between manual reaching and manual lateral interception seem to lead to different velocity patterns. That is, in manual reaching velocity patterns are bell-shaped [51,52], while in manual lateral interception the patterns are often skewed to the right [53] and expose an angle of approach effect [53–56]. This latter effect indicates that the angle of approach of the goal target influences the velocity profile of the end-effector during the interception movement

13

in a systematic way. Additionally, a generally known kinematic feature in manual reaching is the slightly curved trajectory of the end-effector in the horizontal plane (the so-called horizontal curvature; see [6,7,57–59]). This feature has also been shown to be affected by task constraints [6,7,57–59]. More precisely, horizontal curvature has been shown to be larger for unconstrained reaching movements, where the fingertip is lifted from the table top, compared to constrained reaching movements, where the fingertip is constrained to the table top [6,7,57–59].

Aim and outline of the thesis

In the present thesis, I aimed to gather more understanding on motor coordination by focusing on synergies and their role in specific behavior. To do so, I examined the influence of task constraints on synergies and on end-effector-kinematics. Finally, I studied the relation between these levels and the two-step process of emergent behavior.

Before doing so, I evaluated the UCM analysis in chapter 2. This chapter focused on how the linear model is created for UCM analysis and aimed to make the analysis more suitable for multi-joint tasks. Then I turned to the two-step process approach, where an innovative methodology is applied to analyze both steps of the two-step process approach of emergent behavior in discrete movements. In chapter 3, I assessed the influence of task constraints on synergies that are hypothesized to emerge in the first step of the two-step process by examining structure in variability of DOF. More specifically, it is examined whether changes in a task constraint during practice enhance the flexibility of a synergy in reaching. In chapter 4, I examined the influence of task constraints on end-effector kinematics that is hypothesized to emerge from the second step of the process. That is, I examined the relation between lifted height of the end-effector and horizontal curvature of the end-effector in both unconstrained and constrained reaching. If both the level of synergies and the levels of end-effector kinematics can separately be influenced by task constraints using the current methodology, I can test the influence of task constraints on both synergies and kinematic level concurrently. This is done in chapter 5, where I examined whether different constraints are involved in different steps of the process in manual reaching and manual lateral interception, by asking whether different synergies were used when task constraints are varied. When I find that some task constraints can be involved in the first step, while others can be involved in the second step, this would concur with the two-step process approach. Finally, chapter 6 summarizes and discusses the main findings of this thesis.

COMPARING DIFFERENT

METHODS TO CREATE A

LINEAR MODEL FOR THE

UNCONTROLLED MANIFOLD

ANALYSIS

Inge Tuitert, Tim A. Valk, Egbert Otten,

Laura Golenia, Raoul M. Bongers

Motor Control (2018), 23(2),189-204

Abstract

An essential step in the uncontrolled manifold analysis is creating a linear model that relates changes in elemental variables to changes in performance variables. Such linear models are usually created by means of an analytical method. However, a multiple regression analysis is also suggested. Whereas the analytical method includes only averages of joint angles, the regression method uses the distribution of all joint angles. We examined whether the latter model is more suitable to describe manual reaching movements. The relation between estimated and measured fingertip-position deviations from the mean of individual trials, the relation between fingertip variability and V_ort, V_ucm, and V_ort indicated that the linear model created with the regression method gives a more accurate description of the reaching data. We therefore suggest the usage of the regression method to create the linear model for the uncontrolled manifold analysis in tasks that require the approximation of the linear model.

17

Introduction

The uncontrolled manifold (UCM) method is a well-established approach to assess the coordination of multiple degrees of freedom (DOF) in synergies that stabilize performance in human actions. The method has been applied to a variety of actions, such as sit-to-stance, finger-force production, and goal-directed reaching [31,34–40,43,60]. The current paper focuses on computational aspects of the UCM method in goal-directed manual reaching movements to illustrate the argument. When performing a reaching movement, the DOF, i.e., the joint angles of the arm, have to be coordinated to stabilize the index-finger position. To assess how the joint angles are coordinated, the UCM method is applied to evaluate the variability in the joint angles across trials. Joint-angle variability is partitioned into variability that does not influence the index-finger position (V_ucm) and variability that does (V_ort). If there is more V_ucm than V_ort, it is assumed that the joint angles of the arm are coordinated into a synergy that stabilizes the index-finger position.

The computation of V_ucm and V_ort with the UCM method requires four steps [4]. The first two steps consist of selecting the elemental variables and the performance variable, respectively. In goal-directed manual reaching, elemental variables are usually the nine joint angles of the arm (shoulder, elbow, wrist, and finger-joint angles) while the 3D position of the tip of the index finger is the performance variable. Subsequently, small changes in the joint angles are related to small changes in the index-finger position by means of a linear model (third step). These relations have to be approximated in goal-directed manual reaching movements and are represented in a Jacobian matrix. Lastly, this matrix is used to partition the joint-angle variability across trials; variability within the null space of the Jacobian corresponds to V_ucm and variability orthogonal to the null space corresponds to V_ort. The current paper focuses on the creation of a linear model, which can either be done by means of an analytical method or by means of multiple regression (see below). Although the analytical method [32] is the most often used of the two [61], the regression method uses more information of the data to create the linear model, which can influence the accuracy with which the model describes the data. In this paper we compare the accuracy of the two methods in a manual reaching task.

In reaching movements, the analytical method to create the linear model [32,61] employs the computation of the fingertip position with respect to the trunk, using segment origins and rotation matrices of joint angles (i.e., the computation of forward kinematics). These calculations express the position of the end effector (e.g., the tip of the index finger) in the coordinate frame of the segment origin (e.g., the sternum). The resulting expression is a function of the joint angles and the segment lengths, i.e., the geometry of the kinematic chain. We refer to these calculations as geometric transformations, which are typically obtained from motion capture data. The rotation matrices used for geometrical transformations are computed from average joint-angle configurations across repeated trials. To generate the linear model using the analytical method, the model is composed of the partial derivatives of the geometric transformations; this approach is often used for UCM analysis in the literature [32,61].

18

When multiple regression analysis [61–63] is applied, the linear model is created by entering the joint angles as independent variables and the index-finger position as the dependent variable. Their relative relationships are described in the regression equations (see equation 1; a separate equation is used for each direction of the fingertip position). In equation 1, ŷ is the y value on the best-fit plane corresponding to x_k, where b_k are the coefficients, c is the constant, and k the number of joints [64,65]. To estimate the coefficients of the multiple regression equation, a least squares error solution is used (see equation 2). In equation 2, y is a vector with the y values of all trials (i.e., the position of the index finger), x is a vector with the x values of all trials (i.e., joint angles), resulting in a solvable equation with k unknowns (b_m is a vector including b₁-b_k) in j equations (maximum of j is k; independent counter; for a mathematical description of all steps to get from equation 1 to equation 2, see Zaiontz [64]). Note that to compute the coefficients, the covariance among all the joint angles and the covariance among all joint angles and the fingertip position are used. The coefficients (b₁-b_k) of the multiple regression analysis, representing partial derivatives, compose the linear model. The constant of the multiple regression equation (c) is not included in the Jacobian because this was the average of the end-effector position (c = y, see [64]). Until now, in UCM analysis the regression method to create the linear model has only been used when geometric transformations of the relations between elemental and performance variables were not available (e.g., when using EMG; [63]). However, we propose that the regression method to create the linear model should be considered, even when geometric transformations are available.

equation (1)

equation (2)

To understand why the accuracy of the two linear models described above might differ, the dissimilarities between the two need closer examination, especially since the two methods intuitively seem to be similar. The essence is that the regression method uses different information of the movement data than the analytical method does. The latter method only uses the averages of all joint angles and the averages of the 3D origins of the segments. The regression method, on the other hand, uses the (co)variance of the joint angles and the fingertip positions of all trials to estimate each of the coefficients (b₁-b_k), that make up the Jacobian. This implies that the regression method takes into account the distribution of the data, whereas the analytical method does not. To examine this we compared the two methods to describe goal-directed reaching movements, expecting that the linear model based on the regression method would be more accurate than that based on the analytical method.

Methods

Participants

The dataset used in the current paper is a subset of data presented in Valk et al. [66] and consisted of the data on the simple reaching condition obtained in 15 participants of whom seven were men (mean age: 21.3 years; standard deviation (SD): 1.4 years) and eight women

𝑦𝑦 = 𝑐𝑐 + 𝑏𝑏'𝑥𝑥'+ 𝑏𝑏)𝑥𝑥)+ ⋯ + 𝑏𝑏+𝑥𝑥+

𝑐𝑐𝑐𝑐𝑐𝑐 𝒚𝒚, 𝒙𝒙𝒋𝒋 = 𝒃𝒃𝒎𝒎∙ 𝑐𝑐𝑐𝑐𝑐𝑐(𝒙𝒙𝒎𝒎, 𝒙𝒙𝒋𝒋) /

19

(mean age: 20.5 years; SD 1.8 years). The study had ethical approval and all participants gave their informed consent.

Procedure

Participants were seated on a chair in front of a table. The backrest of the chair was extended with a plate to which the trunk of the participant was gently strapped to prevent movements of the origin (i.e., the sternum) while keeping the shoulder at approximately the same position in space without restricting shoulder motions. At the start of each trial, participants placed their index finger of the right dominant hand on the start location, a 1-cm diameter circle on the table, while resting the elbow on an arm rest to standardize the starting posture as much as possible across trials. Following a ‘go’ signal presented verbally by the experimenter, participants performed a forward movement in the sagittal plane to reach the 1-cm diameter target circle located 30 cm anterior of the start position. The experiment comprised a total of 50 trials. Participants were instructed to perform the movement as fast and as accurately as possible but were free to initiate the movement at their own convenience following the ‘go’ signal.

Materials and data collection

Movements were recorded using the Optotrak 3020 system (Northern Digital, Waterloo, Ontario, Canada). Using skin-friendly tape, six rigid PVC plates, each with three IREDs (infrared light-emitting diodes), were attached to the participant’s sternum, the acromion, on the left side of the right upper arm below the insertion of the deltoid, proximal to the ulnar and radial styloids, to the dorsal surface of the hand [67], and to the index finger [43]. Following the procedure described by Van Andel et al. [67], for each individual participant, the 19 anatomical positions were recorded together with the rigid bodies using a standard pointer device. A small aluminum plate was taped under the index finger to prevent flexion-extension in the interphalangeal joints while allowing for flexion-flexion-extension and adduction-abduction in the metacarpophalangeal joint [43].

Preprocessing

The position data of the rigid bodies and their relations to the 19 anatomical positions in the calibration trials were used to compute the positions of the 19 anatomical positions in the global reference frame in measurement trials. X-Y-Z velocities were derived using the three-point central difference method. Tangential velocity was calculated at each point in time as the square root of the sum of the three squared velocities. For each trial, movement termination was determined by searching forward from the moment at which peak tangential velocity was reached. The end of the movement was identified as the first data point where the tangential velocity fell below a speed of 2.5 cm/s and the position of the pointer tip fell within a radius of 1 cm around the target. The instant of movement termination was used in the analyses. Averages of variables were calculated across trials at movement termination. For more information about the data collection and analysis, see Valk and colleagues [66].

Uncontrolled Manifold Method

The UCM was calculated at movement termination, using the four steps introduced earlier. These four steps are described in more detail below. At step three, we explain both the analytical and regression method to create the linear model.

20

Selection of the elemental variables

The elemental variables selected were the nine joint angles of the arm (�_1―9): shoulder plane of elevation (�₁), shoulder angle elevation (�₂), shoulder endorotation-exorotation (�₃), elbow flexion-extension (�₄), forearm pronation-supination (�₅), wrist abduction-adduction (�₆), wrist flexion-extension (�₇), finger abduction-adduction (�₈), and finger flexion-extension (�₉). These joint angles were computed following International Society of Biomechanics (ISB) guidelines for the upper extremity [68].

Selection of the performance variable

The performance variable selected was the 3D fingertip position (r_X , r_Y , r_Z). According to the ISB guidelines, the coordinate system was defined as follows: positive X was the forward position, positive Y the upward position, and positive Z the rightward position.

Creating a linear model of the system

The deviation from the mean of the performance variable relates to the deviation from the mean of the joint configuration as:

equation (3)

where J is a Jacobian matrix (see equation 4) and j represents the trial. The Jacobian was computed as follows:

equation (4)

The elements of this matrix were the partial derivatives of the coordinates of the performance variable with respect to each joint angle.

Analytical method

The analytical partial derivative was calculated using geometric transformations of joint-angle means and segment-origin means. These transformations are shown in equation 5, where R_�1―�9 are the rotation matrices of each angle (number for each angle, see selection of elemental variables), D_1―5 the positions of the segments’ origins with respect to the sternum (i.e., the origin of the segment chain): D₁: glenohumeral, D₂: ulnar styloid, D₃: metacarpal 3, D₄: metacarpophalangeal 2, D₅: fingertip, and r the position of the fingertip in three directions. The Jacobian is obtained by differentiating equation 5 with respect to the independent variables (i.e., joint angles). The results of these computations are united the Jacobian matrix to create the linear model [32,61].

∆𝑟𝑟#= 𝐽𝐽 ∗ ∆𝜃𝜃# ∆𝑟𝑟#= 𝐽𝐽 ∗ ∆𝜃𝜃# 𝐽𝐽 = 𝛿𝛿𝛿𝛿& 𝛿𝛿𝛿𝛿& ⋯ 𝛿𝛿𝛿𝛿& 𝛿𝛿𝛿𝛿) ⋮ ⋱ ⋮ 𝛿𝛿𝛿𝛿, 𝛿𝛿𝛿𝛿& ⋯ 𝛿𝛿𝛿𝛿, 𝛿𝛿𝛿𝛿) 𝑟𝑟 = 𝐷𝐷$+ 𝑅𝑅($𝑅𝑅()𝑅𝑅(*𝐷𝐷)+ 𝑅𝑅($𝑅𝑅()𝑅𝑅(*𝑅𝑅(+𝑅𝑅(,𝐷𝐷*+ 𝑅𝑅($𝑅𝑅()𝑅𝑅(*𝑅𝑅(+𝑅𝑅(,𝑅𝑅(-𝑅𝑅(.𝐷𝐷++ 𝑅𝑅($𝑅𝑅()𝑅𝑅(*𝑅𝑅(+𝑅𝑅(,𝑅𝑅(-𝑅𝑅(.𝑅𝑅(/𝑅𝑅(0𝐷𝐷, equation (5)

21

Regression method

Contrary to the previous method where the mean across trials was used to create the linear model, in the multiple regression method [61–63] the (co)variance across trials was included. In the multiple regression analysis, the dependent variable was the fingertip position and the independent variables were the joint angles. The multiple regression equation and the least square error solution equation are shown in equations 1 and 2. Three separate multiple linear regression analyses were run for each dimension of the fingertip position. The constants of the regressions were excluded from the model because these were the averages of the end-effector positions (c = y, see [64]); note that this was the case because the regressions were not run ‘mean-free’ as done by de Freitas and colleagues [61,62]. The coefficients of the regression analysis composed the linear model, which were equal to , i.e., the partial derivative of the regression formula to a certain joint angle, which makes these coefficients suitable as a linearized model (where n is the dimension of the fingertip position and m is the number of the joint angle).

Partitioning of variance into V

and V

The variance per DOF was partitioned into two components: V_ucm and V_ort (see [32]). The nullspace of J represents those changes in the joint-angle configurations that do not cause any changes in the performance variable: ∆�Vucm_{. Variance that does not affect the} performance variable (V_ucm) and corresponds to the variance per DOF, which lies within the nullspace of J, was defined as:

equation (6)

Here, DF is the number of involved DOF; in our reaching example, DF was 9 and DV, the dimension of the performance variable, was 3. The variance affecting the performance variable (∆�Vucm _{; V}

ort) and corresponding to the variance per DOF of the orthogonal component was defined as:

equation (7)

Testing the Linearized Models

To test these linearized models we used three measures: 1) the estimated fingertip-position deviations from the mean of individual trials, 2) the relation between the fingertip variability and V_ort, and 3) V_ucm and V_ort.

Estimated Fingertip-Position Deviations from the Mean of Individual Trials

We computed the difference between the estimated fingertip positions and the mean fingertip positions for the two methods, after which we compared the differences to the measured fingertip-position deviations from the mean for all individual trials. We computed the relations between these two dependent variables for the two methods separately (i.e., based on the analytical and the regression method, respectively). The method for which this relation was strongest describes the data better.

The estimated fingertip-position deviations from the mean of the linearized models were calculated using equation 3 (following [32]). For each of the two linearized models, the

joint-𝛿𝛿𝛿𝛿# 𝛿𝛿𝜃𝜃% 𝑉𝑉"#$= ∆𝜃𝜃 ()*+, 𝐷𝐷𝐷𝐷 − 𝐷𝐷𝑉𝑉 𝑉𝑉"#$=∆𝜃𝜃 ()*+, 𝐷𝐷𝑉𝑉

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angle deviations from the mean were computed for each trial and were subse-quently multiplied with the Jacobian matrix. Two sets (each based on one of the linear models) of three vectors (one vector for each dimension of the index finger) were obtained, representing the estimated deviation of individual trials from the mean fingertip position . .

To compare these estimated fingertip-position deviations from the mean o the measured position deviations from the mean , we calculated the Pearson correlation coefficient (PCC) between and for each individual participant and dimension. A correlation was valued as high if PCC was greater than 0.6 and as medium if PCC was greater than 0.4 and less than 0.6 [69]. A MANOVA on PCC with the three directions as dependent variables and Method (Analytical Method and Regression Method) as within-subject variable was conducted to compare the PCCs of the linear model created with the analytical method and the linear model created with the regression method for all directions. Furthermore, we fitted a regression line through the data of two participants, one participant with a low end-effector variability and one with a high end-effector variability, to visualize the relation between and for each method in different situations.

Relation between Fingertip Variability and V

We examined the relation between the SD of the measured fingertip position at movement termination (we refer to this as the fingertip variability) and V_ort for the two methods. If the data is described appropriately by the linear model, then there should be a relation between the fingertip variability and V_ort.

Fingertip variability was computed as the SD of the tangential fingertip positions at movement termination. The tangential position was calculated as the square root of the sum of the 3D position. To examine the relation between fingertip variability and V_ort, we calculated the PCC between these two variables for each method. A regression line was fitted through the data to illustrate this relation for each method.

V

and V

We compared the V_ucm and V_ort of the linear models created using the two methods. The manual reaching task is a simple task of which it has been repeatedly shown that the position of the index finger is stabilized, showing that V_ucm is larger than V_ort [35,43,70]. Moreover, the data used in the current study showed a low variability of the index finger at the end of the movement [66]. This underscores the notion that, if the linear model is a good description of the data, then the stabilization of the index finger, as reflected by a high V_ucm and a low V_ort, is stronger.

To compare V_ucm and V_ort of the two linear models, we conducted a MANOVA with V_ucm and V_ort as dependent variables and Method (Analytical Method and Regression Method) as within-subject variable. To correct for non-normal data distributions, V_ucm and V_ort were log-transformed prior to statistical analysis [71], as indicated by the subscript log.

Additionally, we quantified the difference between the two Jacobians by comparing the nullspace and the orthogonal space of the Jacobians of the two methods through the cosine of principal angles [72]. This analysis reveals the shared dimensions by the subspaces. The threshold for similarity was set at 0.9 [73].

𝜃𝜃"− 𝜃𝜃 (∆𝑟𝑟$). (∆𝑟𝑟$). (∆𝑟𝑟$). (∆𝑟𝑟$). (∆𝑟𝑟$). (∆𝑟𝑟$). (∆𝑟𝑟$).

23

For all statistical analyses the level of significance was set at α = 0.05. All variables that were subjected to statistical analyses were normally distributed according to the Kolmogorov-Smirnov test (p’s < 0.05).

Results

Estimated Fingertip-Position Deviations from the Mean of Individual Trials

The MANOVA of the correlation between the estimated and measured fingertip-position deviations from the mean revealed a significant effect of Method (F(3,12) = 53.93, p < 0.001). The separate univariate ANOVAs on the X (F(1,14) = 53.34, p < 0.001), Y (F(1,14) = 112.57, p < 0.001), and Z (F(1,14) = 47.18, p < 0.001) directions indicated that PCCs were higher in the regression method compared with the analytical method in all directions (see Figure 1). Figure 2, which depicts the relations between and for a participant with low and a participant with high fingertip variability, elucidates that the linearized model created with the regression method revealed a higher correlation between and in all movement directions (PCCs > 0.42; see Figure 2, lower panels) than that of the analytical method (PCCs < 0.17; see Figure 2, top panels) for participants with low and high fingertip variability. To check whether the regression method had higher correlations than the analytical method for the complete movement, we also calculated the correlations between and at each instant of the (time-normalized) movement trajectory. Visual inspection of these correlations also revealed higher correlations for the linearized model created with the regression method compared to the analytical method in 100% of the instances in all movement directions. Taken together, these results support our expectation that the estimated position deviations from the mean of the index finger computed using the regression-based linear model describe the data better than those computed using the analytical linear model.

Figure 1. Means and standard errors of the correlations between (∆𝑟𝑟$). and (∆𝑟𝑟$). for the analytical and the

regression method to compute the linear model and each dimension of the position of the index finger (X, Y, and Z).

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Relation Between Fingertip Variability and V

The PCC of the relation between the fingertip variability and V_ort (see Figure 3) was medium to high in the linear model created with the regression method (r = 0.59), whereas it was very low in the model created using the analytical method (r = -0.01). This result suggested that in the linear regression model joint-angle variability was partitioned into V_ort when appropriate whereas this was not the case in the analytical model.

V

and V

The MANOVA of V_ucmlog _{and V}

ortlog comparing the models generated by the two methods revealed a significant Method effect (F(2,13) = 7.94, p = 0.006). Separate univariate ANOVAs on V_ucmlog _{and V}

ortlog indicated that in the regression-based model more joint-angle variability was partitioned into V_ucmlog _{(F(1,14) = 14.02, p = 0.002) and less into V}

ortlog (F(1,14) = 16.67, p =

Figure 2. Relations between of the linearized model and for each method in a separate plot and each direction in a different color (top row: analytical method; bottom row: regression method). The dashed grey line represents the hypothesis where and show a PCC of 1. The left panels show the least variable participant and the right panels the most variable participant (end-effector variability). The shades of grey for the different directions are as follows: X direction blue, Y direction orange, and Z direction yellow. Note that the axes of the upper panels are different from the axes of the lower panels.

(∆𝑟𝑟$).

(∆𝑟𝑟$).

(∆𝑟𝑟$).

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0.001) than in the analytical-based model (see Figure 4). Given the small variability of the position of the index finger, these results suggested that the linear model created through regression described the data best.

To examine the orientations of the Jacobians for the two methods, we compared the nullspace and orthogonal space of both Jacobians separately using the cosine of principle angles and found the averages across participants to be higher than the similarity criterion of 0.9 for the first four dimensions of the nullspace and for all three dimensions of the orthogonal space. This indicated that the differences between the Jacobians of the regression and analytical methods to create the linear model were in the fifth and the sixth dimension of the nullspace, implying that the subspaces of the two Jacobians are different, and, therefore, that the regression-based and analytical Jacobians were indeed different, showing that the differences between the two methods on the other measures were valid.

Figure 3. The relation between fingertip variability and V_ort for each method, where each triangle represents one participant. Note that the y axis of the upper panel is different from the y axis of the lower panel.

Figure 4. Means and standard errors of the means of Vucm and Vort for the analytical and the

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Discussion

When multi-joint coordination is studied with the UCM method, the linear model is usually created using the analytical method rather than a regression method. One major difference between the two approaches is that in the analytical method only the averages of joint angles are used, whereas in the regression method the distribution of angular values of the joints and positional values of the end-effector across repetitions are used. Comparing the linear models the two methods computed for the data obtained in a manual reaching task, we first found higher correlations between the estimated and the measured fingertip-position deviations from the mean in the regression-based linear model. Second, the relationship between fingertip variability and V_ort indicated that if the linear model is created with the regression method, an appropriate amount of joint-angle variability is partitioned into V_ort whereas this is not the case if the linear model is created analytically. Moreover, we showed that with the regression method more joint-angle variability was partitioned into V_ucm and less into V_ort. Taken together, these results demonstrate that the linear model created with the regression method provided a more accurate description of the data in our goal-directed reaching task, which is why we propose using the regression method to create the linear model for the UCM analysis when analyzing reaching data.

Although we only examined goal-directed reaching, we argue that this recommendation could be extended to other actions, such as in sit-to-stance or walking. We hypothesized that the regression method described the data better than the analytical method because it incorporated the distribution of the joint angles and fingertip positions across repetitions into the linear model, whereas the analytical method considered only the averages of the joint angles and origins of the segments. Confirming our assumption, we have shown the added value of including these changes across repetitions into the analysis. Given that the distribution of the data across repetitions of the performance variable and the elemental variables also plays a role in other motor tasks, including this distribution in the creation of the model may also improve the linear model for these tasks. Note that this recommendation only applies to tasks in which the linear model has to be approximated. In tasks that are by definition linear, the regression method should not be considered. This is, for instance, the case in a finger force-production task where a certain amount of force needs to be exerted with four fingers (DOF) at a certain point in time [74,75]. Here, the production of the total force is a linear combination of the four DOF and hence the exact linear model, which makes the use of the regression method superfluous. It is for tasks that require a linear approximation (i.e., tasks that are not exactly linear) that we suggest using the regression method to describe the data more accurately.

To do a UCM analysis in a reaching task, a linear approximation of the relations between changes in joint angles and changes in fingertip position is used, although these relations are actually non-linear [4,76]. This is exemplified by a 3D plot with a curved (non-linear) solution manifold where three joint angles (three axes) keep the fingertip at one specific location at an instant in time. If the joint-angle ranges across repetitions are small, only a small part of this curved manifold is exploited, allowing this part to be approximated by a linear model because in a small part of the manifold the deviations from linearity are small. Given that in our simple reaching task the ranges of the joint-angle rotations

27

across repetitions are small and the estimated deviations from the mean of the fingertip do not differ much from the measured deviations from the mean, the non-linear behavior is suitable for approximation using a linear model [77–79], facilitating the UCM analysis. While in the current task linearization is appropriate, in other tasks where the ranges of joint angles (or other elemental variables) across repetitions is larger and thus also the scattering of trials on the curved solution manifold, a linearized manifold to approximate the solution manifold is less suitable. In such cases, one might consider using a non-linear method to assess variability across trials. Müller and Sternad [80], for example, proposed a surrogate non-linear data analysis that was adapted and applied by Ambike and collegues ([78]; see also [81]) for an inverse piano finger-force task. Having created a surrogate data set by randomizing the original data set, they found the surrogate data set to show a much larger variance than the original one, implying that there was less co-variance in the surrogate data set and indicating that in the original data the variance was mainly co-variance along the non-linear solution manifold. In short, if the joint-angle ranges in a multi-joint task are small, the usage of a linearized model is appropriate, whereas if the joint-angle ranges are larger, a non-linear method would be the better option. Note that there are also other discussions regarding the analysis of variability in redundant tasks [5,82], but these are beyond the scope of this paper.

In conclusion, our results show that in goal-directed reaching the regression method to create the linear model is preferred to the analytical method. We argue that if the UCM analysis is applied to tasks that require the approximation of the linear model, the use of the regression method to create the linear model should be considered.

DOES PRACTICING A WIDE

RANGE OF JOINT ANGLE

CONFIGURATIONS LEAD TO

HIGHER FLEXIBILITY IN A

MANUAL OBSTACLE-AVOIDANCE

TARGET-POINTING TASK?

Inge Tuitert, Reinoud J. Bootsma,

Marina M. Schoemaker, Egbert Otten,

Leonora J. Mouton, Raoul M. Bongers

Abstract

Flexibility in motor actions can be defined as variability in the use of degrees of freedom (e.g., joint angles in the arm) over repetitions while keeping performance (e.g., fingertip position) stabilized. We examined whether flexibility can be increased through enlarging the joint angle range during practice in a manual obstacle-avoidance target-pointing task. To establish differences in flexibility we partitioned the variability in joint angles over repetitions in variability within (V_ucm) and variability outside the solution space (V_ort). More V_ucm than V_ort reflects flexibility; when the ratio of the V_ucm and V_ort is higher, flexibility is higher. The pretest and posttest consisted of 30 repetitions of manual pointing to a target while moving over a 10 cm high obstacle. To enlarge the joint angle range during practice participants performed 600 target-pointing movements while moving over obstacles of different heights (5-9 cm, 11-15 cm). The results indicated that practicing movements over obstacles of different heights led participants to use enlarged range of joint angles compared to the range of joint angles used in movements over the 10 cm obstacle in the pretest. However, for each individual obstacle neither joint angle variance nor flexibility were higher during practice. We also did not find more flexibility after practice. In the posttest, joint angle variance was in fact smaller than before practice, primarily in V_ucm. The potential influences of learning effects and the task used that could underlie the results obtained are discussed. We conclude that with this specific type of practice in this specific task, enlarging the range of joint angles does not lead to more flexibility.

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Introduction

Skilled behavior is characterized by flexibility [1]. When attempting to avoid spilling coffee from a hand-held cup, such flexibility is for instance seen in the adaptations of the joint angles of the arm following a slight perturbation. Conceptually, flexibility may be defined as deploying a range of different solutions to solve a given motor problem. This can be operationalized to the observation of variability in the use of elemental degrees of freedom (such as joint angles) over repetitions of trials while task performance (such as holding the cup without spilling coffee) is maintained [4,32].

Because the availability of multiple solutions to a given motor problem is thus a prerequisite for flexibility, the latter capitalizes on the redundancy of elemental degrees of freedom observed at many levels of the movement-production system [1,83]. While from a computational point of view motor redundancy is generally considered a problem, requiring additional constraints to harness the system [16,84], the principle of motor abundance suggests that it is in fact a bliss [83,85,86]. Indeed, the organization of elemental variables (i.e., individual degrees of freedom) into functional units (i.e., synergies) allows variations in solutions, not only when a task is performed repeatedly within the same context [32], but also when the context changes. An increase in variability in the use of degrees of freedom not affecting performance (i.e., flexibility) has for instance been observed when participants perform a target-pointing task in the context of potential changes in target location, or in the presence of a secondary task, or new constraints, or when stabilizing movements to multiple task goals simultaneously [39,40,44–46,77,87,88].

Practice under specific conditions has also been reported to lead to the deployment of a larger range of solutions [34,89–91]. Particularly interesting for the present purposes was the finding that practicing a target-pointing task in the presence of an unfamiliar force-field resulted in more flexible target-pointing behavior in a test condition without such an additional force-field [35]. If, as suggested by these authors, the observed increase in flexibility in the test condition resulted from practice under a condition requiring participants to explore a wider range of joint configurations, we hypothesized that flexibility could be increased in a manual obstacle-avoidance target-pointing task in which the hand had to move over an obstacle to reach its target location. Practicing the task with obstacles of varying height was expected to lead to variable movement trajectories of the fingertip during the reach [77,92,93]. This was hypothesized to stimulate participants to use an enlarged range of joint angles (i.e. degrees of freedom) during pointing to the (unchanged) target. The idea was thus that practicing such an enlarged range of joint angles would give rise to an increase in flexibility for pointing movements over an obstacle of middle-range height which was not included in the practice phase.

Flexibility was quantified by the uncontrolled manifold (UCM) method, a method often used in studies that start from the principle of abundance [83], which parses the total variability observed in individual degrees of freedom over repetitive trials [32,33]. When this method is applied to manual pointing movements the degrees of freedom are the joint angles of the upper limb [37,43,77,94]. All possible arm postures reached by moving the shoulder, elbow, wrist and finger can be captured in a multi-dimensional space where each

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axis corresponds to a different joint angle; this space is the joint space. Within this joint space lies a solution space for the pointing task; this solution space consists of all different joint angle configurations where the fingertip reaches the target location. Goal equivalent variability (V_ucm) in joint angles is defined by variability of joint angle configurations within the solution space whereas non-goal equivalent variability (V_ort) is defined by variability outside the solution space. The former type of variability does not affect the position of the fingertip, while the latter does [4,32,33]. When the variability within the solution space is larger than the variability outside the solution space, task performance over repetitions reflects flexibility. A larger V_ucm with respect to V_ort reflects a larger flexibility.

Thus, the rationale of the present study was the following. If practicing with obstacles of different height gives rise to the use of an enlarged range of joint angle configurations during practice, this may be expected to result in the continued deployment of multiple solutions after practice. Because the to-be-reached target position is kept similar, flexibility would thus be increased. Operationally, this corresponds to a larger increase in V_ucm than in V_ort after practice.

Methods

Participants

Thirteen right-handed participants (6 males and 7 females, mean age 23.6 years, SD 1.4 years) participated in the experimental group and nine right-handed participants (3 males and 6 females, mean age 22.8 years, SD 2.3 years) participated in the control group. All participants had normal or corrected to-normal vision.

Ethics statement

The study was approved by the local Ethical Board of the Center for Human Movements Sciences (University Medical Center Groningen). Participants gave written informed consent before the start of the experiment.

Materials and Procedure

Participants sat on a chair placed at a table. The backrest of the chair was extended with a plate to which the trunk of the participant was gently strapped to prevent movements of the trunk and kept the shoulder at approximately the same position in space. At the beginning of each trial, participants placed the fingertip of the right hand on the start location (i.e., a 1 cm diameter dot on the table) and held the elbow against a stand to standardize starting posture as much as possible across trials (see Figure 1). Following a ‘go’ signal delivered by the experimenter, participants performed a forward movement in the sagittal plane to reach the target (i.e., a second 1 cm diameter dot on the table) located at a 30 cm distance in front of the start position. During the movement they had to lift the index finger over an obstacle, placed at 10 cm in front of the starting position (see Figure 1). Participants were instructed to perform the movement as fast and accurate as possible but could initiate the movement at their convenience after the ‘go’ signal.

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The obstacles were 11 cuboid wooden stands (3 x 1 cm ground surface dimensions) varying in height from 5 to 15 cm in steps of 1 cm. Kinematic data were collected at a sampling frequency of 100 Hz with two Optotrak 3020 system sensors (Northern Digital, Waterloo, Canada). Six rigid plates, made of PVC, with each three markers were attached to the sternum, to the acromion, at the left side of the right upper arm below the insertion of the deltoid, proximal to the ulnar and radial styloids, to the dorsal surface of the hand [67], and to the index finger [43] with skin friendly tape. Following the procedure described by Van Andel et al. [67], for each participant the positions of the six rigid bodies were linked to the 19 local anatomical positions using a standard pointer device. A small aluminum plate was taped under the index finger to prevent flexion-extension in the interphalangeal joints while allowing for flexion-extension and adduction-abduction in the metacarpal phalangeal joint [43,66].

Design

The experiment took place over 3 consecutive days, with participants performing a pretest followed by 200 practice trials on day 1, 200 practice trials on day 2, and 200 practice trials followed by a posttest on day 3. The pre- and posttests consisted of 30 movements over a 10 cm high obstacle. Practice consisted of 600 trials over obstacles of various other heights (5, 6, 7, 8, 9, 11, 12, 13, 14, 15 cm), presented in blocks of 30 trials. With the order of presentation of the different obstacles during practice being quasi-random, participants performed 60 trials with each of the 10 practice obstacles. There was one minute of rest between the tests and practice blocks. A no-practice control group only performed the pretest on day 1 and the posttest on day 3.

Data analysis

The data were analyzed using customized programs written in Matlab (Mathworks, Natick, MA). Converting the collected kinematic data for the 6 rigid bodies (marker clusters) to the positions of 19 anatomical landmarks [67] allowed extraction of the time series of the fingertip position and 9 relevant joint angles (see below).

End-effector

X-Y-Z velocities were derived using the three-point central difference method. Tangential velocity was calculated at each point in time as the square root of the sum of the three squared velocities. For each trial, the start and the end of the movement were determined by searching backward and forward, respectively, from the moment at which peak tangential

Figure 1. Bird’s-eye view of the set-up of the experiment where the participant is in starting position. The elbow of the participant is on the grey stand and the fingertip of the participant is on the start target. The second blue dot is the end target and the green rectangle is the obstacle. The participant is strapped to the brown chair with the yellow bandage.

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velocity was reached. The start of the movement was identified as the first data point (working backwards from the moment of peak velocity) where the tangential velocity fell below a threshold of 5 cm/s, and the position in the forward direction and the position in the sideward direction was within the boundary of 1.5 cm from the center of the start point. The end of the movement was identified as the first data point where the vertical position of the fingertip was at table level. Movement time was determined by the time period between the start and the end of the movement. Peak velocity was defined as the maximum value in the tangential velocity profile. The symmetry index of velocity was defined as the time from movement onset to moment of peak velocity divided by total movement time where 0.5 denotes a symmetric velocity profile. The moment the fingertip crossed the obstacle was identified as the first data point where the position of the fingertip in the forward direction passed the obstacle location at 10 cm in front of the start position. End-point precision, was calculated as the within-participant SD per condition (pretest, practice, posttest) of the position of the fingertip in the forward and sideward directions at the end of the movement.

Joint Angles

Joint rotations were calculated following the orientations as proposed in the ISB standardization proposal for the upper extremity by Wu et al. [68]: shoulder plane of elevation (SPE), shoulder elevation (SE), shoulder inward–outward rotation (SIO), elbow flexion–extension (EFE), forearm pronation–supination (FPS), wrist flexion–extension (WFE), wrist abduction–adduction (WAA), index finger flexion–extension (FFE), and index finger abduction–adduction (FAA). We determined the arm’s postural configuration on each individual trial (i) at the moment the fingertip passed over the obstacle and (ii) at the end of the reaching movement. We specifically chose to analyze the postural configurations at these instants because moving over the obstacle and reaching the target were the two prominent constraints of the task. From the variations in postural configurations observed at these instants over repeated trials during the pretest, practice, and posttest we derived the within-participant ranges and variances for each of the nine joints. The measure of joint angle range was operationally defined as the within-participant mean over the observed ranges of the nine joints (collapsed over obstacles in the practice phase). The measures of joint angle variance were calculated per obstacle, and averaged over obstacles and joints when analyzing the practice phase (not collapsed over obstacles).

Uncontrolled Manifold Analysis

The Uncontrolled Manifold (UCM) method based on the covariance matrix (C) [71] was used to determine flexibility when the fingertip was at the obstacle and at the end of the movement. The elemental variables selected in this task were the joint angles of the shoulder, elbow, wrist, and finger (9-DOF). The performance variable selected was the fingertip position of the index finger (3-DOF). The relations between changes in joint angles (i.e., postural configurations) and fingertip positions were computed and united in a Jacobian matrix (J) [37,71]. The null-space of the J was used as a linear approximation of the UCM. The variance components V_ucm and V_ort were computed by projecting the total variance in joint space onto the null-space of J and the orthogonal complement, respectively. Each UCM (V_ucm and V_ort) component was normalized by its number of DOFs.