Forces and Moments Generated by the Human Arm: Variability and Control

Home , Arm

The Pennsylvania State University

The Graduate School

College of Health and Human Development

FORCES AND MOMENTS GENERATED BY THE HUMAN ARM: VARIABILITY AND CONTROL

A Dissertation in

Kinesiology

Yang Xu

@2014 Yang Xu

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2014

The dissertation of Yang Xu was reviewed and approved* by the following:

Vladimir M. Zatsiorsky

Professor of Kinesiology

Dissertation Adviser

Chair of Committee

Mark L. Latash

Distinguished Professor of Kinesiology

Co-Chair of Committee

Andris Freivalds

Professor of Industrial and Manufacturing Engineering

Stephen J. Piazza

Professor of Kinesiology

Graduate Program Director

*Signatures are on file in the Graduate School.

ABSTRACT

To move and manipulate objects people exert forces and moments of force (further addressed as simply ―moments‖) on the environment. The five studies in this dissertation explored the accurate endpoint force vector production by the human arm in isometric conditions. In the first study, three common-sense hypotheses were proposed and falsified. The subjects exerted static forces on the handle in eight directions. The forces were of different magnitude levels. The torsion moment on the handle (grasp moment) was not specified in the instruction. The two force components and the grasp moment were recorded, and the shoulder, elbow, and wrist joint torques were computed. The following main facts were observed: (a) While the grasp moment was not prescribed by the instruction, it was always produced. The moment magnitude increased with force magnitude and moment direction changed with the instructed force direction. (b) The within-trial angular variability of the exerted force vector (angular precision) did not depend on the target force magnitude. (c) The time profiles of joint torques in the trials were always positively correlated, even for the force directions where flexion torque was produced at one joint and extension torque was produced at the other joint. (d) In contrast to the previously studied tasks, the analytical inverse optimization (ANIO) method failed to determine the optimization cost function. A hypothesis was formulated that this is a general property of the static tasks performed by the serial kinematic chains. In the second study, the preceding findings in the first study were confirmed in tasks where arm postures were systematically varied. Further, it was observed that the distribution pattern of endpoint force variability is dependent on the arm posture instead of the orientation of trunk. A following-up simulation was conducted to examine why the pattern of the joint torques distribution could not be explained by an optimization cost function additive with respect to the torques. The results suggested that the grasp capability might serve as the limit to a more optimal pattern of the joint torques distribution. In the third study, handle size was varied to test its effect on performance. Major results indicated: (a) There existed effects of handle size on the magnitude, but not on the distribution pattern of MVC end-point force. (b) Changing the

iii handle diameters in the range between 4.5 and 9.0 cm does not affect the maximal torque production. (c) Although systematic change in the coefficients of ANIO was observed, the method still failed to reconstruct an optimal cost function additive with respect to joint torques with large handle size. In the fourth study, two handles with different surface friction were used to characterize its effects on grasp moment and further on the endpoint force. It was found that: (a) No significant differences in the MVC forces existed between the low- and high- surface friction handles; (b) A higher surface friction led to a higher grasp moment magnitudes; (c)

Higher surface friction resulted in lower variability of force direction and higher enslaved moment at a specific direction. In the fifth study, grasp moments were prescribed or canceled out by using rotatable handle, and the results showed that: (a) Rotatable handle did not change the MVC forces significantly; (b) At the force level of 30% of MVC, using the rotatable handle didn‘t lead to different variability of force magnitude and directions at all prescribed directions; (c) Exerting of the prescribed grasp moment reduced the MVC force production; and (d) Exerting of the prescribed grasp moment led to higher variability of both force magnitude and direction. In conclusion, this dissertation demonstrated several attractive mechanisms and properties of arm control in isometric condition. Discovered unintentional grasp moment production is important for explaining mechanisms and ergonomics of arm control.

TABLE OF CONTENTS

LIST OF FIGURES ...... vii

LIST OF TABLES ...... x

ACKNOWLEDGEMENTS ...... xi

CHAPTER 1: Introduction ...... 1 1.1 Statement of the problem ...... 1 1.2 Statement of objectives ...... 4 1.3 Overview of the dissertation ...... 5

CHAPTER 2: Literature Review ...... 7 2.1 Anatomy and function control of upper extremity ...... 7 2.1.1 Anatomy of upper extremity ...... 7 2.1.2 Function control of upper extremity...... 10 2.2 Characteristics of isometric force production by upper extremity ...... 12 2.2.1 Finger interaction and ergonomics of grasping ...... 12 2.2.2 Tactile sensing and effect of friction on grasping ...... 16 2.2.3 Mechanics of multi-link bodies ...... 17 2.3 Motor redundancy and arm control synergy ...... 22 2.3.1 Motor redundancy ...... 22 2.3.2 Characteristics of arm control ...... 23 2.4 Variability ...... 25 2.5 Optimization ...... 28

CHAPTER 3: General Methods ...... 33 3.1 Equipment ...... 33 3.2 Subjects ...... 35 3.3 Data acquisition...... 35 3.4 Data processing and statistics...... 36

CHAPTER 4. Forces and moments generated by the human arm: variability and control39 4.1 Introduction ...... 39 4.2 Methods ...... 41 4.3 Results ...... 48 4.4 Discussion ...... 60 4.5 Conclusions ...... 75

CHAPTER 5. Effects of arm posture on the force and moment production ...... 76 5.1 Introduction ...... 76 5.2 Methods ...... 77 5.3 Results ...... 82 5.4 Discussion ...... 86 5.5 Conclusions ...... 93

CHAPTER 6. Joint torques prediction by optimization ...... 94 v

6.1 Optimization formulation ...... 95 6.2 Optimization results: Simulated joint torques vs experimental joint torques ...... 95 6.3 Conclusions ...... 100

CHAPTER 7. Effects of the handle size on force and moments generated by the human arm 101 7.1 Introduction ...... 101 7.2 Methods ...... 102 7.3 Results ...... 106 7.4 Discussion ...... 115 7.5 Conclusion ...... 119

CHAPTER 8 Effects of surface friction of handle on force and moments generated by the human arm ...... 121 8.1 Introduction ...... 121 8.2 Methods ...... 121 8.3 Results ...... 125 8.4 Discussion ...... 129 8.5 Conclusions ...... 133

CHAPTER 9. Effects of rotatable handles and prescribed grasp moment on force and moments generated by the human arm ...... 134 9.1 Introduction ...... 134 9.2 Methods ...... 134 9.3 Results ...... 138 9.4 Discussion ...... 144 9.5 Conclusions ...... 148

CHAPTER 10: Discussions and Conclusions ...... 149

10.1. General overview ...... 149

10.2 Limitations of this work ...... 152

10.3 Discussion of the main results and conclusions ...... 152

10.4 Future works ...... 157

BIBLIOGRAPHY ...... 159

APPENDIX A: Anatomy and Physiology ...... 166

APPENDIX B: Analytical Inverse Optimization...... 179

LIST OF FIGURES

Figure 2.1 Three-link chain with external force, F, and couple, C, applied to the endpoint. The

figure is representative of force production with the arm...... 18

Figure 3.1 Schematic drawing of the experimental setup ...... 34

Figure 4.1 Schematic diagram of the experimental setup ...... 42

Figure 4.2 Experimental setup ...... 44

Figure 4.3 MVC force vs. targeted force direction...... 49

Figure 4.4 The moments of force exerted on the handle (grasp moments), ...... 51

Figure 4.5 Dependence of the force magnitude variability (SDs) and force direction variability

on the target force level and direction (panels a and b, respectively)...... 54

Figure 4.6 Torque variability (SD) at the elbow and wrist joints in the trials at different target

force levels and directions, group average data...... 56

Figure 4.7 Intra-trial correlations between the joint torques at different force directions and

force levels in individual trials...... 60

Figure 4.8 Correlations between the grasp moment and the wrist torque. The figure is for a

representative subject...... 62

Figure 4.9 Upper panel. Joint torques and the grasp moment across the target force directions

(for 40% of MVC). Bottom panel. The joint torques multiplied by the coefficients from

equation 4.6b: –0.29T2 and 1.29T3, respectively...... 66

Figure 5.1 Schematic diagram of the experimental setup...... 79

Figure 5.2 Grasp moments for different arm postures and target force directions...... 83

Figure 5.3 Force magnitude (N, filled circle) and force direction (degrees, open circle)

variability across target directions...... 84

Figure 5.4 Coefficients of correlations between the time profiles of the shoulder and elbow

joint torques at different arm postures...... 85 vii

Figure 6.1 Simulated shoulder torques (T1) based on cost functions vs. observed shoulder

torques ...... 96

Figure 6.2 Simulated elbow torques (T2) based on cost functions vs. observed elbow torques

...... 97

Figure 6.3 Simulated wrist torques (T3) based on cost functions vs. observed wrist torques. 98

Figure 6.4 Simulated grasp moment (Mz) based on cost function vs. observed grasp moment

...... 99

Figure 7.1 Mean values of normalized MVC in 6 conditions ...... 106

Figure 7.2 Maximal grasp moments. Group average data and standard errors...... 108

Figure 7.3 ‗Enslaved‘ grasp moments at MVC ...... 109

Figure 7.4 Uninstructed (enslaved) grasp moment at 30% MVC across 6 conditions ...... 110

Figure 7.5 Forces exerted on the handles during the maximal grasp moment production

(positive, counter-clockwise)...... 111

Figure 7.6 Forces exerted on the handles during the maximal grasp moment production

(negative, clockwise)...... 111

Figure 7.7 Variability of force direction across 6 conditions ...... 112

Figure 8.1 MVC forces of low- and high-friction conditions across 8 directions of force

production (N). Means ± SEM are shown...... 125

Figure 8.2 Averaged grasp moment for two directions in two friction conditions (Nm). Means ±

SEM are shown...... 126

Figure 8.3 Variability(RMS) of force magnitude (N) and direction across 8 directions of force

production (degree) for the two handles...... 127

Figure 8.4 Enslaved grasp moment for the two friction conditions across 8 directions of force

production (Nm). Means ± SEM are shown...... 128

Figure 9.1 Normalized MVC forces across 8 directions ...... 138

viii

Figure 9.2 Average variability (RMS) of force magnitude (left) and force direction (right)

across 8 directions...... 139

Figure 9.3 Enslaved grasp moment at MVC force (blue) and 30% MVC trials (red) in

percentage of the maximal grasp moment; ...... 140

Figure 9.4 MVC forces at 7 conditions across 8 directions ...... 141

Figure 9.5 Variability of force magnitude across 8 directions (coefficient of variation)...... 142

Figure 9.6. Variability of force direction across 8 directions (degree) ...... 143

APPENDIX B Figure A.1 The predictions of the joint torques by an additive cost function.

...... 183

LIST OF TABLES

Table 4.1 The number of subjects (out of 10) who generated torque on the handle in the positive

(counterclockwise) direction ...... 52

Table 4.2 Intra-trials correlations between the force in the target direction and the joint torques

in the individual trials...... 58

Table 4.3 Intra-trials correlations between the grasp moment and the wrist torque...... 58

Table 5.1 Coefficients of cost function at six arm configurations across subjects ...... 86

Table 7.1 ANIO coefficients for the shoulder, elbow and wrist torques (k1,k2 and k3) across 8

subjects ...... 113

APPENDIX A Table 2.1.1 Muscles Anatomy and Innervation: upper arm 1 ...... 166

APPENDIX A Table 2.1.2 Muscles Anatomy and Innervation: upper arm 2 ...... 167

APPENDIX A Table 2.1.3 Muscles Anatomy and Innervation: upper arm 3 ...... 169

APPENDIX A Table 2.1.4 Muscles Anatomy and Innervation: upper arm 4 ...... 171

APPENDIX A Table 2.1.5 Muscles Anatomy and Innervation: forearm 1 ...... 172

APPENDIX A Table 2.1.6 Muscles Anatomy and Innervation: forearm 2 ...... 174

APPENDIX A Table 2.1.7 Muscles Anatomy and Innervation: hand 1 ...... 176

APPENDIX A Table 2.1.8 Muscles Anatomy and Innervation: hand 2 ...... 177

ACKNOWLEDGEMENTS

This dissertation would not have been possible without the support of my advisor,

Dr. Vladimir Zatsiorsky. He has been a mentor and role model for me both intellectually and personally and I‘ll never be able to adequately express my gratitude for all his guidance and encouragement throughout my doctoral process. It was the greatest time attending Dr. Zatsiorsky‘s courses in Strength Training and Advanced

Biomechanics.

I want to thank Dr. Mark Latash, my co-adviser. He introduced me to the wonderful world of motor control and especially helped me develop an understanding of motor control from the perspective of physics. The knowledge he taught me allowed finding meaningful results from my studies. Moreover, I would always be grateful to his immense helps during the past four years.

I was also very fortunate to have Dr. Steven Piazza and Dr. Andris Freivalds as my committee members. Dr. Piazza offered me his sincere guidance in many ways, especially with enriching my biomechanical knowledge, provide insightful input to my research, and also with improving my teaching skills. Dr. Freivalds expanded my understanding of human movement science and inspired my interest in ergonomics with his amazing wealth of knowledge in many areas of both kinesiology and engineering.

The pursuit of doctoral studies has been a very rewarding experience. The faculty and the staff in the department of Kinesiology have been encouraging and helpful at

xi every step of the way. Besides my committee members, I want to thank Dr. John

Challis and Dr. Jinger Gottschall in Biomechanics lab for their helps. My sincere thanks to Ms. Sharon Grassi for her time and helps throughout the process.

Finally, I also would like to thank my friends in Biomechanics and Motor Control lab for their time to help with my study and to share our knowledge. My special thanks go to Xun Niu, Joel Martin, Yen-Hsun Wu, Alexander Terekhov, Nick Phillips,

Jaebum Park, Satyaji Ambike, Stanislaw Solnik, Tao Zhou, Mu Qiao, Tarkeshway

Singh, Zheng Wang, Hang Jin Jo, Daniela Mattos, Sasha Reschechtko, Behnoosh

Parsa, Florent Paclet, Namanja Pazin, Herman Van Werkhoven, Huseyin Celik for all the wonderful time we spent together.

xii

CHAPTER 1: Introduction

The human arm has an excellent combination of two attractive features: a large movement range and a dexterous control. The large moment range is offered by the long segments and the multi-link chains structure; the dexterous control is the product of coordinated control by central nervous system (CNS). The arm is used in a vast range of motor tasks, such as pushing, grasping and reaching. One of the very fundamental functions is producing static force on objects or environment, which is also the result of coordinated torque production at each joint. Furthermore, the force production is affected by a variety of factors, such as the magnitude or direction of target force, the arm configuration, environment etc. A better understanding of isometric force production by arm will contribute to the knowledge of motor redundancy and optimization. Also, it will help improving the rehabilitation for patients with arm injury, as well as the performance in ergonomics and sports.

1.1 Statement of the problem

Static force and grasp moment produced by arm are crucial in human movement activity. Both force magnitude and its variability have been studied extensively

(Newell and Carlton 1988; Fujikawa 1997; Jones, Hamilton et al. 2002). In the research, endpoint force was mapped onto space of joint torques, shoulder, elbow and wrist, in order to investigate the coordination among joints. Friedman et al.(2011) studied the production of two-dimensional force vectors by a two-joint arm (the wrist

was braced). This design allowed mapping endpoint force vectors onto joint torques unambiguously. In natural three-joint tasks (the wrist is free) the arm is redundant with respect to the force vector production in a plane because it can also produce a moment of force on the grasped object, the grasp moment (Zatsiorsky 2002). While many studies explored redundant kinematic tasks performed by multi-joint serial chains (Domkin, Laczko et al. 2005; Yang, Scholz et al. 2007) and redundant kinetic tasks performed by several effectors acting in parallel (Li, Latash et al. 1998; Latash,

Scholz et al. 2002; Shim, Latash et al. 2004) performance in redundant static tasks performed by a serial chain has not been studied experimentally.

The presence of redundancy affords the system a possibility to use multiple solutions for any given task. In particular, when only the force magnitude and direction are specified, the performers can choose any moment magnitude and direction they prefer. Such preferences remain presently unknown. Common sense suggests that the performers — if not required—should exert no grasp moment at all to minimize effort.

In contrast to the moment production by a serial chain, the force production—especially the force magnitude variability — was an object of rigorous research. The dependence of the force magnitude variability on the target force level has been studied in detail (Carlton and Newell 1993). At low forces, standard deviation of force magnitude increases with the target force level (Schmidt, Zelaznik et al. 1979; Newell and Carlton 1988; Sherwood, Schmidt et al. 1988; Sherwood,

Schmidt et al. 1988; Slifkin and Newell 1999). At higher force magnitudes, the standard deviation peaks at about 65% of maximal force and then decreases at higher force levels (Sherwood and Schmidt 1980). Valero-Cuevas et al. (2009) examined muscle coordination using electromyograms during fingertip isometric force production and found that the variance was consistently lower for task-relevant variables than for muscle activation variables. On the other hand, there are no systematic data on the dependence of the force magnitude variability on the target force direction.

Studies on the variability of force vector direction are limited. These studies fall into one of the two groups dealing with the force production either in a single 2- or 3-D joint (Kutch, Kuo et al. 2008) or in planar kinematic chains. Studies on the effects of the force direction on the force variability in multilink tasks (which is a topic of the present study) have been mainly limited to fingertip force production in the flexion–extension plane. It was observed that during the fingertip force production the target direction significantly affected the variable error of the force direction, but not the systematic error (Gao, Latash et al. 2005). Fingertip force direction variability was shown to be larger for the force exerted downward and towards the body as compared to other directions (Kapur, Friedman et al. 2010).

In the mentioned study by Friedman et al. (2011) the hand force direction variability decreased with an increase in the target force magnitude. This finding was in sharp contrast to the well-established increase in the force magnitude variability

3 with the target force increase. In that study, the subjects grasped a handle and exerted forces of different magnitudes in various horizontal directions. The wrist joint was however braced, so the arm was mechanically reduced to the two-link kinematic chain.

It remained unknown whether the negative relation between the force magnitude and its angular variability is valid for natural tasks when the wrist joint is not braced.

The variability of force production is generally assumed to reflect noise at some level of the neuromotor hierarchy (Newell, Deutsch et al. 2006) . An increase in the amount of noise with the intensity of the neural signal is commonly referred to as the signal-dependent noise (Harris and Wolpert 1998). Peripheral sources may also contribute to the variability of the motor output, and attempts have been made to distinguish between these two sources (Wing and Kristofferson 1973).

Endpoint force variability is a function of the variability of the joint torques in the involved joints. There is a glaring gap in the literature on this topic.

1.2 Statement of objectives

The objectives of the current studies are :

(1) To investigate the grasping moment produced at isometric endpoint-force production tasks and the relations between the moments with other performance variables and parameters, such as MVC forces, arm postures, etc.

(2) To explore the variability of the end-point force magnitude and direction as a function of the target force magnitude and direction;

(3) To characterize the joint torques in isometric force production and the torque variability at the contributing joints, the shoulder, elbow and wrist.

(4) To evaluate the joint torques sharing pattern among the three joint of arms, the shoulder, elbow and wrist.

(5) To test the applicability of the recently developed analytical inverse optimization (ANIO) method in the arm force production tasks.

Isometric force production by arms will be tested in all experimental studies. The effects of a number of variables, relevant to force control performance and newly discovered in the studies, such as handle friction etc., will be also be examined.

1.3 Overview of the dissertation

This dissertation work is composed of ten chapters plus two appendices.

Chapter 1 is an introduction to the dissertation including statement of problem and objectives.

Chapter 2 is a literature review, which will cover anatomy, function control of the human arm; mechanics of multi-link chains; motor redundancy and optimization.

Chapter 3 is introduction to the general methods used in the experiments of this dissertation

Chapter 4 is a comprehensive study on the isometric force production and control by arms.

Chapter 5 is a study on the effect of systematic varied arm posture on the static force production and control by arm

Chapter 6 is a simulation based on the findings in Chapter 4 and Chapter 5 to investigate the cause of joint sharing pattern among the three joints of arm.

Chapter 7 is a study on the influence of handle size on the performance

Chapter 8 is a study on the effect of surface friction of handle on force and grasps moment control.

Chapter 9 is a study on the effect of rotatable handle and prescribed grasp moment on the endpoint force performance.

Chapter 10 includes a summary of the conclusions drawn from Chapter 4 to Chapter 9; a further discussion based on the summary, limitations of the current work and studies expected in the future.

APPENDICES include information on the muscles mentioned in the literature review

(Chapter 2) and detailed process of the Analytical Inverse Optimization (ANIO) analysis.

CHAPTER 2: Literature Review

Force production by arm is critical to human activity. Arm covers a wide range of space and can also be controlled dexterously. For this reason, the study of arm is a field of biomechanics and motor control that deserves a great deal of focus. The literature review will provide a basic understanding of what is already known about the biomechanics and control of isometric force production by arm. The review starts with basic anatomy and neurophysiology of the arm. Next, basic characteristics of arm force production and control will be discussed. The last portion of the review will focus on control theories as to variability and methods relating to optimization.

2.1 Anatomy and function control of upper extremity

2.1.1 Anatomy of upper extremity

The human upper extremity, excluding the hand, consists of five bones, which are clavicle, scapula, humerus, ulna and radius. A closed chain is formed by the thorax, clavicle and scapula though three joints: acromioclavicular joint, between the clavicle and scapula, sternoclavicular joint, between the thorax and clavicle, and the gliding of the medial border of the scapula on the thorax, in the scapulothoracic gliding plane

(Van der Helm 1994). The glenohumeral joint works as a ball-and-socket joint between the shallow glenoid cavity of the scapula and the head of the humerus. The elbow joint is the synovial hinge joint between the humerus in the upper arm and the radius and ulna in the forearm which allows the movement of flexion-extension. The radio-ulnar

7 joints are comprised of two separate pivot joint, the proximal and the distal, which allow the radius and ulna to rotate about each other.

The bones play important role in supporting upper limb and providing attachment points for the muscles. A larger number of muscles exist in the upper extremity, some of them crossing more than one joint. Furthermore, the cooperation of long and short muscles, as well as the structural feature of the shoulder and elbow joint, enables it to produce accurate force or delicate movement within wide amplitude.

Shoulder, one of the most complex joints, is of great flexibility and also reduced stability. Shoulder girdle, consisting of the clavicle and scapula, forms the attachment between the arm and the trunk. The clavicle allows the shoulder joint to move in circles by connecting the scapula to the sternum. The scapula and the humerus form the shoulder (glenohumeral) joint at the glenoid cavity on the lateral end of the scapula.

Many shoulder moving muscles attach to the scapula, such as the deltoid, trapezius and the rhomboids..

The humerus, the only bone of the upper arm, extends from the scapula of the shoulder to the ulna and radius of the forearm. The proximal head of the humerus is a round structure and work as the ball of the ball-and socket shoulder joint. The distal end of humerus, along with the ulna and radius, forms the inner hinge of the elbow.

Muscles such as the pectoralis, latissimus dorsi and rotator cuff muscles attach to the humerus contributing to the movement and torque production at the shoulder joint.

The insertions, origins and actions of these muscles are described in Appendix

A-Table 2.1.1-2.1.4

Forearm contains two long bones, the ulna and the radius. The ulna, residing on the medial side, is widest at the proximal end and narrow at the distal end. The proximal end of the ulna, known as the olecranon, forms the hinge of the elbow joint by extending past the humerus. The ulna also forms the wrist joint with radius and the carpal at its distal end. The radius, shorter and thinner than ulna, locates on the lateral side of the forearm. On contrary to the ulna, the radius is narrowest at the elbow and widest at the wrist. At elbow, the radius forms the pivoting part of the elbow joint which allows rotation of forearm. At the distal end, the radius is wider than the ulna and forms the wrist joint with the ulna and carpals. The distal end of the radius can also rotate around ulna when the forearm and hand rotate. The Muscles that contribute to the torque production of wrist are described in detail in Appendix A- Table

2.1.5-2.1.6

The hand contains twenty-seven bones and many flexible joint. The carpals are eight cube-shape bones in the proximal end of hand. They form the wrist joint with the ulna and radius and also form joints with the metacarpals. Many small gliding joints exist between the carpals giving extra flexibility to the wrist and hand. Five long metacarpals form the main area of the palm at the distal end. Each metacarpal also articulates with the proximal phalanx of a finger. Metatarsophalangeal joints permit the finger to perform the movement of flexion-extension and abduction-adduction. The

9 phalanges are a group of fourteen short cylindrical bones. Each digit contains three phalanges except for the thumb, which contains just two. The phalanges form hinge joints between and condyloid joints with the metacarpals. The hinge joints permit only the flexion and extension. Muscles located within hand, the so-called intrinsic muscles, are illustrated in Appendix A- Table 2.1.7-2.1.8

2.1.2 Function control of upper extremity

The primary motor cortex (M1) is the brain area that is responsible for producing central commands to activate muscles. The more control is needed for a given body part the greater the portion of M1 devoted to that part, i.e. the area representing upper extremity is larger than that representing the lower extremity in

M1. Pathways carrying control signals involve complex control loops. Commands are sent from neurons in the cortex down pathways to mononeurons in the spinal cord, which then send action potentials to activate the muscle fibers. A motoneuron and the fibers innervated by it are termed a motor unit. Feedback information, such as the length and contraction velocity of muscle fibers, are sensed by muscle spindles and the signals sent back ascending pathways to the brain which then regulates the descending commands with information of those feedbacks.

Two major descending pathways are the pyramidal and extrapyramidal tracts.

The pyramidal tracts are the major efferent pathways carrying information from the

CNS to the periphery. They are comprised of the corticobullar tract and the

10 corticospinal tract. The corticobullar tract fibers lead into motor nuclei of the cranial nerves and the corticospinal tract fibers go to the spinal cord. Neurons in the corticospinal tract produce excitatory or inhibitory effects on α-motoneurons of muscles on the contralateral side of the body. The excitatory effect can be direct on

α-motoneurons while the inhibitory effects are always mediated by interneuron.

(Latash 2008). The extrapyramidal tracts are chiefly found in the reticular formation of the medulla and pons, and lead to neurons in the spinal cord involved in locomotion, reflexes, and postural control. They also allow the afferent signals to travel back and provide feedback information to the central nervous system (CNS).

Regulatory components from various parts of the CNS, such as the cerebellum and basal ganglia, can as well be considered part of the extrapyramidal tracts.

Another important structure involved in voluntary movements is the cerebellum. The cerebellum has been speculated to contribute to a number of functions such as: multi-joint coordination, timing of muscle activation and serving as a comparator for identifying movement errors. The cerebellum receives information through the mossy fibers and the climbing fibers, from the vestibular, visual, somatosensory, auditory, the cerebral cortex, and proprioceptive system. The Purkinje cells are the only output of the cerebellum, which makes inhibitory synapses on neurons in the cerebellar and vestibular nuclei. It should be mentioned that no direct connections exists between the cerebellum and the spinal column.

The basal ganglia participate in control of voluntary movement; however their exact function is not clear. The main input to the basal ganglia comes from the cerebral cortex, which is also the target of the main output from the basal ganglia.

People suffering with disorders of the basal ganglia tend to produce excessive involuntary movements and exhibit slower voluntary movements than the healthy. It appears that some of the functions of the basal ganglia are to prevent unwanted movement and disinhibit some areas of the movement system.

The pathways of the CNS can be divided into two categories: the efferent and the afferent pathways. The efferent pathways carry information from the CNS to the periphery while the afferent pathways carry information in the opposite direction.

2.2 Characteristics of isometric force production by upper extremity

2.2.1 Finger interaction and ergonomics of grasping

Finger interaction has been a lively area of research and demonstrated attractive features that help understanding the motor control. The features particularly include enslaving, force sharing and force deficit. The term enslaving describes involuntary force production by a finger when other finger(s) generate force.

(Zatsiorsky, Li et al. 1998; Slobounov, Johnston et al. 2002; Yu, van Duinen et al.

2010). Enslaving occurs in both isometric and non-isometric tasks(Kim, Shim et al.

2008). Several factors were suggested to result in enslaving: 1) passive mechanical connections among the digits(Kilbreath and Gandevia 1994); 2) motor units of the

12 extrinsic muscles that exert force on several digits(Kilbreath and Gandevia 1994); and

3) overlapping representation of the digits in the cortex (Schieber and Hibbard 1993).

Force sharing means that during multi-digit force production tasks, the forces are typically shared among the fingers (Kinoshita, Kawai et al. 1995). Finger sharing pattern are task-dependent and may reflect force production capabilities of fingers, task constrains or mechanical advantages of fingers. Force deficit refers to the fact that the maximum forces produced by individual fingers in multi-finger tasks are smaller than the maximum forces exerted in single-finger MVC tasks (Ohtsuki 1981).

This behavior is observed in both finger flexion and extension (Yu et al. 2010).

Finger forces in the flexion direction are produced isometrically in grasping.

The FDP has been shown to perform most of the flexion in low force tasks and the

FDS is recruited as force level increase. The complex anatomy of hand complicates the study the biomechanics of grasping. In particular, the flexor sheaths covering the finger flexor tendons work as the pulley systems to hold the tendons against the finger bones and prevent bowstringing. As a result, relatively constant movement arms are maintained. This pulley system is comprised of three cruciate pulleys and five annular pulleys. As a tradeoff, relatively large tendon excursions are required to rotate the finger joints. For a more comprehensive review of the biomechanics of the finger flexors see (Freivalds 2004).

Three tendon-pulley models were established to describe the relationship between joint and tendon displacement (Landsmeer 1963). The models show varied

13 complexity because of different assumptions they are based on. In the first model, the tendon is assumed to be held around the surface of the proximal bone of the joint. The equation is:

x = r (2.1) where x is the tendon displacement, r represents the distance from the tendon to the joint center, and θ is the joint angle.

In the second model, tendon is not assumed to be held securely and displacement from the joint can be allowed when finger is flexed. The relationship is described as:

x=2rsin ( ) (2.2)

In the third model, the tendon is assumed to go through a tendon sheath and be held securely against the bone, which allows the tendon to curve around the joint.

Tendon displacement is modeled by:

x = 2 * ( ) + (2.3)

where d represents the distance of tendon to the long axis of bone and y represents the distance from the point, where the tendon begins to bowstring, to joint axis measured along the long axis of bone.

Hand is the common interface between the body and the objects, thus numerous researches on grasp strength have been performed to characterize the force applied at the hand/handhold interface. These researches involved studies in the field of pull strength, push strength, torque strength, grip strength, and lifting capacity.

Results demonstrate that isometric grip strength vary with many factors, including posture of the arm and wrist (Hazelton, Smidt et al. 1975; Kuzala and Vargo 1992;

Kattel, Fredericks et al. 1996; McGorry and Lin 2007), diameter, or span of the object

(Amis 1987; Dvir 1997; Edgren, Radwin et al. 2004). Also, not only an active resistance from the fingers but also a complex interaction at the interface (i.e. friction, skin deformation, etc.) occurs when the object is pulled from the grasping hand.

Surface interactions on the interface between hand and the object have effect on other functional measures of grasp. An example of this is the ability of the person to create torque on a handle, which is indicated to vary with handle surface friction and area of contact (Pheasant and O'Neill 1975; Imrhan and Farahmand 1999; Yoxall and Janson

2008). Another finding is that the total manual torque output and comfort on a screwdriver also exhibited systematical changes with the cross-sectional size, orientation, as well as shape of its handle (Kong and Lowe 2005). In addition, greater torques output and normal force on a cylinder were observed in inward torque production on a cylinder (toward the finger tips) than in outward torque. As maximum grip torque was smaller than wrist torque, it was suggested that friction at the handle served as the strength limiting factor. However, these observations, that surface interaction are important when characterizing functional hand strength, are limited when applied to other situations because no directional external load was exerted on the object in these experiments.

Several studies attempted to examine the capability of subjects to pull or push a handle with various configurations of the arms, as well as upper body (Cochran and 15

Riley 1986; Kong and Freivalds 2003; Seo, Armstrong et al. 2008). It is still hard to achieve a direct measurement of functional hand strength as the hand/handle interaction would also be intervened by the strength of the elbow, shoulder, and torso.

2.2.2 Tactile sensing and effect of friction on grasping

The skins on the dorsal and palmar sides of the hand exhibit different physiological and mechanical properties. The dorsal skin is more pliable, with thinner epidermis, hair follicles and fewer sweat glands. In contrast, the skin on the palmar side has thicker epidermis and a denser concentration of sensory receptors. The cutaneous receptors detect various stimuli making the hand be able to gain tactile information about grasped objects or the surrounding environment. The three main types of receptors are thermoceptors, nociceptors, and mechanoreceptors.

Thermoceptors detect temperature, nociceptors detect pain or damaging stimuli, and mechanoreceptors detect mechanical stimuli. Mechanoreceptors are crucial to human movement – they sense tactile stimuli that may lead to responses. Four types of mechanoreceptors exist: Merkel disks, Meissner corpuscles, Pacinian corpuscles, and

Ruffini endings.

With the information provided by those tactile receptors, people make adjustment to the grip forces at digit-object interface in response to different weights and friction condition(Johansson and Westling 1984; Edin, Westling et al. 1992; Cole and Johansson 1993). The static grip force is reported to be proportional to the weight

16 of an object, and this proportional ratio is higher for the object with a more slippery surface (Johansson and Westling 1984; Aoki, Niu et al. 2006). The adaptive adjustment of modulating grip force (normal) and tangential force is essential to prevent slipping that only occurs when the grip is less than the minimal force required.

The Coulmb friction model is widely applied to define the relationship between normal force (nF) and tangential force (tF), with friction coefficient µ as a property of the materials of contact surface (eq. 2.4).

tF=µ×nF (2.4)

If tF is less than or equal to nFµ, no slipping occurs at the interface. It should be noted that the Coulomb model simplified the complexity of interaction between hand and handle. For instance, in human grasping the friction coefficient µ is dependent on the magnitude of the normal force and also the deformation of the palmar pad which is not taken into account.

2.2.3 Mechanics of multi-link bodies

The mechanics of force production with the arm can be modeled as a three-link chain (hand, forearm, and upper arm) with an endpoint force at the hand (Figure 2.1).

For a detailed review see Zatsiorsky 1998, 2002. The chain could be open or closed.

The open chain defined when the end is free to move. The closed chain is one where the end is constrained by a mechanical stop. An example of a closed chain would be isometric force production by arm against a fixed force sensor.

Further, in a multi-link chain, the total number of degrees of freedom (DOF) equals the mobility of the chain. The mobility of a chain can be calculated with

Gruebler‘s formula:

F =6(N-k) + ∑ (2.5) where F is the mobility of the chain, N represents the number of links, k is the number of joints and fi stands for the number of DOF of the ith joint. For the upper extremity the mobility, F, is 7 (F = 6 × (3-3) + (3 + 2 + 2= 7). This is assuming a chain of three joints (shoulder, elbow, wrist) with the shoulder, elbow, wrist joints having three, two, and two DOF, respectively.

Figure 2.1 Three-link chain with external force, F, and couple, C, applied to the endpoint. The figure is representative of force production with the arm.

The motion of the endpoint of a three-link chain relates to motion at all joints and this relation can be described with the following equation:

v=J ̇ (2.6)

where v stands for the velocity of the end-point ̇ ̇ , J represents the

Jacobian relating displacements of the endpoint to displacements of the joints, and ̇ is

a vector of joint angular velocities ̇ ̇ ̇ . Furthermore, the end-point acceleration is computed by differentiating equation 2.6:

a= ̇ ̇ + ̈ (2.7)

where a is the endpoint acceleration vector ̈ ̈ , and ̇ is found by differentiating J.

Generally, the motion of the joints and/or the end-point force is result of torques produced about the joints by muscles. In the framework of mechanical viewpoint, muscles are torque actuators that generate a moment about a joint with either passive or active force production. Existence of the two-joint muscles complicates the computation of the joint torques and their interpretation (see Zatsiorsky, 2002 for the discussion).

The general equation that relates the end-point force to joint torques is:

T= (2.8)

where T stands for the vector of joint torques, F represents a 6-component vector of forces at the end-point and is the transpose of the Jacobian that relates small joint displacements to end-point displacement.

For a planar three-link chain, as shown in Figure 2.2.3, the transpose of the

Jacobian is:

 l1S 1  l2S 12  l3S 123 l1C  l2C 12  l3C 123 1 J T  l S l S l C l C 1    2 12  3 123 2 12  3 123 

  l3S 123 l3C 123 1   (2.9)

where l1, l2, and l3 are the lengths of the links, C and S represent cosine and sine.

The subscripts 1, 12, and 123 following C and S stand for θ1, (θ1 + θ2), and (θ1 + θ2 + θ3) angles, respectively. In the planar case, the force vector becomes:

F = [ ] (2.10)

where Fx and Fy represent the x- and y-components of the external force. Ce stands for the external couple.

These equations indicate that the magnitude of the end-point force is determined by the magnitudes of the joint torques, whereas the direction of the end-point force is determined by the torque proportions of the three joint.

The forces and moments acting on the joints are of substantial implications and interest in studies of biomechanics. They can be computed using the inverse dynamics approach. While different methods of solving the inverse dynamic problem exist, e.g.

Lagrange equations, Kane equations etc., they all can be reduced to the basic mechanics

(Newton-Euler equations). In a nutshell, inverse dynamics is an iterative method that uses measured external forces and motion of the links to obtain information of the internal moments and forces at the joints using Newton‘s 2nd law equations of motion:

∑ 퐹=ma ∑ 푀=I ̈ (2.11)

For a static multi-link chain, the net joint torques is dependent on the external contact forces, as well as the weight of the links. However, if the link is moving (any of the joint) then additional forces get involved as a result of the angular velocities and accelerations of the joints. There is a normal (centripetal) force, Fn, directed along the link equaling to:

퐹푛 = 푚퐿 ̇ (2.12)

A tangential force due to the angular acceleration is computed by:

퐹푡 = 푚퐿 ̈ (2.13)

The normal, tangential forces and the weight are assumed to act at the center of mass (COM) of the link. In moving multi-link chains, the links- except for the most proximal link- experience a Coriolis force because of the rotating reference frames for the distal links with respect to proximal links. The Coriolis force is computed by:

퐹푐표푟 = m 푎푐표푟 (2.14)

where acor is given by:

푎푐표푟 = 2 퐿 ̇ ̇ (2.15)

where the subscripts 1 and 2 stand for the proximal and distal links, separately.

For a three- link planar chain there are two inertial forces acting on the first link, five on the second link, and eight on the third link. All of the forces also cause reaction forces, in the opposite direction, on the other links making the computation of joint reaction forces complex. The equations are normally simplified and written in the state-space form as the following:

̈ ̇ 푇 𝐼 𝐼 𝐼 푣 퐺 [푇 ] = [𝐼 𝐼 𝐼 ] [ ̈ ] [푣 ̇ ] [퐺 ] (2.16) 푇 𝐼 𝐼 𝐼 퐺 ̈ 푣 ̇

where [T] is the vector of joint torques, [v] is the vector of centrifugal and

Corliolis terms, [I] is the inertia matrix of the chain, and [G] is the vector of gravity.

For more detailed information of the terms see Zatsiorsky (2002).

2.3 Motor redundancy and arm control synergy

2.3.1 Motor redundancy

In human movement, the number of controlled variables typically exceeds the number of performance variables. For example, the number of degrees of freedom

(DoF) in the joints of a kinematic chain is greater than the DoF of endpoint. This discrepancy is commonly known as motor redundancy (Bernstein 1967) and reflects the complexity of the motor control system. Instead of being independently controlled by the CNS, the elements involved in the movement are controlled coordinately

(Latash, Danion et al. 2003). An example is the hitting movement of a blacksmith- a small variability in the hammer tip trajectory is present across trials compared to the 22 high variability in the individual limb joint. To guarantee the endpoint hit the target, the joints work cooperatively, instead of independently, to compensate for the endpoint error. This high variability in elemental space can be regarded as ―noise‖; however for researchers in motor control, motor variability is an attractive phenomenon informative to the understanding of motor control.

Bernstein (1967) suggested that the CNS uses synergies, which eliminate the redundant degree of freedom in motor control. Due to this elimination process, the central controller selects one solution from an infinite number of potential options to accomplish a motor task. Recently (Latash, Scholz et al. 2007), it has also been suggested that motor redundancy should be termed motor abundance; the suggestion is based on the fact that all elements take part in a motor task and no DoF is removed in movement. Such an adjustment is in line with real situation in motor control requiring flexibility and proper modulation of elements during the movement.

2.3.2 Characteristics of arm control

Both kinematic and kinetic variables in arm control have been studied extensively (Morasso 1981; Flash and Hogan 1985; Nakano, Imamizu et al. 1999). The joint moments seems to be fairly stereotyped in skilled arm reaching (Wada, Kaneko et al. 2001). And a co-variation of elbow and shoulder joint moments were also documented as a notable joint kinetic characteristic in arm control (Gottlieb, Song et al. 1996).

In postural tasks completed by arm, specific combination of apparent joint stiffness at a given arm configuration determine the endpoint resistance of the arm in response to perturbation, which can be defined in terms of shape, orientation and size of the stiffness ellipse (Mussa-Ivaldi, Hogan et al. 1985). The shape and orientation of the arm stiffness ellipse seem to be rather stereotypic when an arm posture is maintained, which also imply a kind of a kinetic invariant.

The study of arm control is complicated by the presence of bi-articular muscles that exert same level force with varied moment arm at each specific joint.

Numerous approaches have been attempted to investigate the mechanical functions of two-joint muscles at the joints. The approaches include analysis of length changes in the two-joint muscles (kinematic analysis) and analysis of muscle force contribution to the joint torques (kinetic analysis). A standard kinetics method requires: determination of the moment from a specific muscle and its component with regards to the joint axes at this given joint posture; determination of the resultant joint moment; and comparison of the direction of the joint moment from the muscle and the resultant joint moment. If the directions of the muscle and the resultant moment about an axis are the same, the muscle is termed a joint torque agonist. If the directions of a moment of force generated by a given muscle and the joint torque are opposite, the muscle is termed a joint torque antagonist. As a result, in certain situations, a bi-articular muscle could serve as a joint torque agonist at one joint and as a joint torque antagonist at the other joint. In researches on the control of arm (van Bolhuis et al. (1998) and leg force production (Jacobs and van Ingen Schenau (1992), it is 24 concluded that bi-articular muscle play a unique role in controlling the direction of the external force produced on the environment. The validity of these conclusions was questioned by other researches (Prilutsky 2000).

Another non-trivial phenomenon in multi-link chains control is that when large force production is expected, people show a marked tendency to exert forces in the direction where a larger force could be produced, rather than in the direction of expected target. For example, when pedaling a bicycle, even the most professional athletes do not exert force absolutely perpendicular to the crank throughout the complete circle. At certain leg positions, a large force component is produced along the crank, which tends to compress or extend it. This phenomenon, which is appropriate for pedaling, may be detrimental in tasks requiring accurately directed force production and not tolerant of deviation.

To assess the effectiveness of static force efforts, the concept of maximum advantage of using a component of exertion (MACE) was introduced (Grieve and Pheasant

1981). MACE is computed as the ratio of the maximum available force component in a given direction to the force intentionally exerted in this direction.

2.4 Variability

The variability of force production was introduced above in the section discussing a problem of motor redundancy. The variability of force production is suggested to reflect noise at several levels of the neuromotor hierarchy (Newell,

Deutsch et al. 2006). Researches, who were interested in the physiological mechanisms of variability, tried to find its origin in the central or peripheral systems

(Evarts, Wise et al. 1985; Brooks 1986). Also, researchers claimed that the standard deviation of force output is proportional to magnitude of force and called it impulse variability (Schmidt, Zelaznik et al. 1979; Newell, Carlton et al. 1980). These findings did not outline the physiological processes that drive variability but speculated that peripheral processes drive variability. The observations that the amount of noise increases with the intensity of the neural signal is commonly referred to as the signal-dependent noise (Harris and Wolpert 1998). The research suggested that noise is incorporated in the central command signal and vary with the level of force output.

It has been suggested that the attempts to minimize end-point variance under the influence of SDN could lead to the phenomena such as Fitts‘ speed-accuracy trade off, bell shaped velocity profile, and the two-thirds power law. In another study, significant differences in variability of force output between a cortical-electrical stimulation contraction and an intentional contraction were reported and interpreted as support for the idea that motor variability is due to both central and peripheral origins

(Jones, Hamilton et al. 2002). Additionally, attempts have been made to differentiate between these two sources (Wing and Kristofferson 1973). There are also a number of studies indicating that movement variability arises not at and after the onset of a movement but during motor preparation, before the movement begins (Churchland,

Afshar et al. 2006; Duarte and Latash 2007).

The relation between the force magnitude variability and the target force level has been examined in detail (Carlton and Newell 1993). At low force levels, standard deviation of force magnitude increases linearly with the target force level (Schmidt,

Zelaznik et al. 1979; Newell and Carlton 1988; Sherwood, Schmidt et al. 1988;

Sherwood, Schmidt et al. 1988; Slifkin and Newell 1999). At higher force magnitude, the standard deviation peaks at about 65% of maximal force and tends to decrease at higher force magnitude (Sherwood and Schmidt 1980). A study using electromyograms during fingertip force production estimated the variance of both force and the muscle activities; the variance was reported to be consistently lower for task-relevant variables than for the muscle activation variables (Valero-Cuevas,

Venkadesan et al. 2009).

In contrast to force magnitude variability, studies on the variability of force vector direction are limited. These studies consisted of two groups dealing with the force production either in planar kinematic chains or in a single 2- or 3- D joint

(Kutch, Kuo et al. 2008) . In the multilink chain studies, fingertip force production in the flexion-extension plane was used as the main task to investigate the effect of the force direction on the force variability. It was found that during the fingertip force production the variable error of the force direction, but not the constant error, varied with different target direction (Gao, Latash et al. 2005). Furthermore, fingertip force direction variability was demonstrated to be larger for the force exerted downward and towards the body as compared to other directions (Kapur, Friedman et al. 2010).

In a study based on arm force production (Friedman, Latash et al. 2011), the hand force direction variability decreased with an increase in the force magnitude.

This finding was in sharp contrast to the performance of force magnitude variability, which increased with the target force.

2.5 Optimization

As mentioned above, the problem of motor redundancy arises because for any given motor task the number of controlled variables (the elemental variables) is always greater than that of constrains (the performance variables). Although an infinite number of solutions exist to accomplish a task, the coordination of movement pattern is highly repeatable from trial to trial. Most people perform movements in relatively the same manner and thus variability across trials- or even across people- is also low. It is commonly hypothesized that people perform movement in an optimized way to minimize certain cost functions that might be applied by the CNS. The search for this possible cost function is one of the approaches for solving the problem of motor redundancy(Nelson 1983; Prilutsky and Zatsiorsky 2002). Such an optimization, or cost function, could be based on mechanical, physiological, or psychological variables(Seif-Naraghi and Winters 1990; Tsirakos, Baltzopoulos et al. 1997; Raikova and Prilutsky 2001). In order to test to what people might be optimizing, numerous motor tasks have been investigated, e.g. walking (Anderson and Pandy 2001; Pham,

Hicheur et al. 2007), jumping (Anderson and Pandy 1999), cycling (Prilutsky and

Gregor 2000), postural control (Kuo and Zajac 1993), sit- to-stand movement

(Kuželički, Žefran et al. 2005), force sharing among the muscles (Prilutsky and

Zatsiorsky 2002), reaching movements (Flash and Hogan 1985; Tsirakos, Baltzopoulos et al. 1997; Biess, Liebermann et al. 2007), and force sharing among fingers in grasping

(Pataky, Latash et al. 2004).

Typically, an optimization problem is defined as:

Let J: Rn → R1

: Minimize J(X) =f(x1, x2… xn) (2.17)

Subject to: g(X) = 0

H(X) ≤ 0

n where X=(x1, x2,…, xn) ∈R , n indicates the number of variables in the optimization procedures, f is the unknown cost function to be minimized, g and H is a group of functions defining the equality and inequality constraints that X are expected to satisfy.

The optimization problems can be divided into two categories: direct optimization and inverse optimization (Ahuja and Orlin 2001). The direct optimizations are applied extensively where the cost functions are known, or modeled.

Then optimal solutions are to be computed based on those cost functions, as well as known constrains. In contrast, the optimal solutions, instead of cost functions, are given in an inverse optimization problem in which the cost function is the unknown object to be deciphered.

A recently developed technique, called analytical inverse optimization (Park,

Zatsiorsky et al. 2010; Terekhov, Pesin et al. 2010; Niu, Terekhov et al. 2011; Park,

Zatsiorsky et al. 2011; Terekhov and Zatsiorsky 2011), allows for determination of the unknown cost function based on obtained experimental data. ANIO necessitates knowledge of the surface containing the experimental data. Also the objective cost function is assumed to be additive with linear constraints. To guarantee a unique solution to the inverse optimization problem, the constraints should also be non-splittable and have a dimension greater than or equal to two. A detailed description of the ANIO approach is available in Terekhov et al. (2010) and Terekhov,

Zatsiorsky (2011).

ANIO have been mostly applied in the finger studies where finger forces are assumed to be confined to a two-dimensional hyperplane, within the four dimensional space formed by the four finger forces (Park et al. 2010; Terekhov et al. 2010). To test the presence of this hyperplane, PCA (principle component analysis) has been used on the experimental data sets. The Kaiser Criterion was used to obtain the significant

PCs (Kaiser 1960). And the data were suggested to be confined to two-dimensional hyperplane as long as the amount of variance explained by the first two PCs was large in percentages, e.g. greater than 90% of the total variance was represented by the first two PCs. After the reconstructed cost functions were determined, several criteria are used to assess their performance. These criteria include the dihedral angle between the plane of predicted solutions from the reconstructed cost function and the plane of

30 experimental data, the root mean square difference between the predicted values and the actual measurements, etc.

Typically, the finger task tested has been to press with all four fingers to match a prescribed force and moment. As a result, there are four elemental variables (number of fingers) and two constraints (target total force and moment). The reconstructed function is assumed to be of the form shown in equation 2.18

Cost Function: J = ∑ 푘 퐹 +∑ 푤 퐹 (2.18) 푖 푖 푖 푖 푖 푖

The cost function contains finger force terms (Fi) multiplied by the first- (wi) and second-order (ki) coefficients.

ANIO have been successfully used for various subject and conditions, and significantly different results were found. For instance, a study (Park, Sun et al. 2011) applied ANIO to two groups of people, the elderly and the young. Subjects were asked to press to match nine levels of total force and nine levels of moment. Each of the eighty-one combination of force and moment was repeated three times and the average forces during a selected time interval of each trial were computed. The PCA of the data revealed that over 90% of the total variance was explained by the first two

PCs for young subjects while only 75% was explained by the first two PCs for the elderly subjects. The second-order finger force coefficients (ki) were all positive for both groups; however a significant difference were found between the groups. There existed negative coefficients in the case of first-order finger force while no significant difference was identified between the two age groups. As to the dihedral angle

(D-angle) between the optimal plane and experimental plane, significant smaller values were revealed for the young group compared to the elderly group. Likewise, in another study the dihedral angle increased when the index finger was fatigued before performing the pressing task. In addition, a significant increase in the coefficient at the quadratic term for the fatigued index finger was observed along with a decrease in the coefficients for the ring and middle fingers.

Note that ANIO was never applied to serial chains such as an arm where joint torques can be considered elemental variables.

CHAPTER 3: General Methods

This chapter presents the information on general methods used in this study. It includes experimental apparatus, procedures, processing of data, statistical analysis, and the parameters calculations.

3.1 Equipment

In the experimental studies of this dissertation, subjects were required to exert isometric force with the arm on the handle equipped with a 6-component force sensor.

In this way, two components of the force and the grasp moment were measured.

Subjects sat in a chair, grasped the handle and generated forces in different directions (Figure 3.1). The trunk was secured to the chair with seat belts. The arm was restricted to a horizontal plane at the shoulder height, with various arm configurations in different studies. This reduced the task to a system with three degrees of freedom that produced a two-dimensional force and a moment. Such an arrangement allowed reconstructing the joint torques from the recorded endpoint force and moment (explained below).

The handle was cylindrical and attached to a 6-DOF force/torque sensor

(Mini85, ATI, Apex, NC USA, resolution of Fx and Fy is 7/144 N and resolution of

Mz is 1/600 Nm). The arrangement allowed recording both the force vector and the moment (‗grasp moment‘) exerted on the handle. Surface of the handle was frosted, which provided greater static friction than a polished surface. The force sensor was mounted on an aluminium block, which could slide left and right along two poles.

These two poles were attached to blocks on the left and right side of the apparatus, which could slide forward and back on another two poles. Screws on the central block and on the two side blocks permitted fixing the handle at a desired location. The total workspace was 65 cm by 65 cm.

The subject sat on a large, heavy chair, and was strapped to the chair with two seatbelts to prevent movement of the trunk. The chair sat on a hydraulic lift, and its height was adjusted for each subject such that the bottom of the handle was at the shoulder height. The forearm was supported by a padded semicircular piece of plastic pipe, hanging down vertically from the ceiling. The location of the handle was adjusted for each subject such that the required arm configuration was met. The selected chair height ensured that the upper arm and forearm lay in a horizontal plane, at the height of the shoulder.

Figure 3.1 Schematic drawing of the experimental setup

3.2 Subjects

Eight or ten subjects took part in each study. All subjects were healthy, with no known neurological or peripheral disorders. The subjects are all right-handed university students. Detailed information of subjects is provided in each specific study.

All of the subjects gave informed consent according to the policies of the Office for

Research Protections at the Pennsylvania State University.

3.3 Data acquisition

In general, the subjects were asked to produce maximal force (MVC) along each of the 8 directions or maximal moment at 2 directions for 5 s. The subjects were provided with feedback on the horizontal force vector or moment value exerted on the handle by the right arm using a monitor placed directly in front of the subject (blue arrow on the monitor of Figure 3.1). A blue arrow showed the amount and direction of force they were generating. Three concentric circles, marked in newtons, were subdivided by 8 directions.

A number of following trials were performed in each study to test the performance of isometric force production, covering various conditions. Typically, the force magnitude was set as certain percentage of MVC and the duration was from 15s to 25s. The target levels of force were presented in a random order. The force direction conditions were pseudo-randomized to prevent similar or same direction presenting in succession. Particularly, directions of the subsequent force exertion were

35 randomly selected at either 180 º, 225 or 135 º from the previous direction. Breaks were provided between trials to prevent fatigue.

3.4 Data processing and statistics

The force signals were conditioned (9105 IFPS-1, ATI, Apex, NC USA) and digitized using a 16-bit A/D converter (PCI-6225, National Instruments, Ausin, TX) at

1000 Hz. The data were collected using a custom program written in LabVIEW

(National Instruments). The data analysis was performed using a custom program written in Matlab (The MathWorks, Natick, MA). The first 3s of each trial were discarded.

Mechanical model and the joint torque computations

The joint torques was computed from the recorded values of the endpoint force and moment by employing the equation (see Zatsiorsky, 2002, Chapter 2):

 Fx  T  J T F  J T  F   y  M   z  (3.1)

where T is a 3×1 vector of joint torques, F is a 3×1 vector of the endpoint force

T and moment (grasp moment), and J is the transpose Jacobian of the kinematic chain

 l1S 1  l2S 12  l3S 123 l1C  l2C 12  l3C 123 1 J T  l S l S l C l C 1    2 12  3 123 2 12  3 123    l S l C 1  3 123 3 123  (3. 2)

36 where l1, l2, and l3 are the lengths of the upper arm, forearm, and the hand (from the wrist to the centre of the handle), respectively; S and C refer to the sine and cosine, respectively; and the subscripts refer to the angles: 1 – the shoulder angle 훼 , 12 –

훼 훼 , i.e. the upper arm angle with respect to axis X of the external system of coordinates, and 123 to the angle 훼 훼 훼 .

Data analysis

For all analyses the 3-s time periods at the beginning were discarded and the computations were performed for the remaining time periods.

Force variance was calculated over each trial. The variance of the force magnitude in the target direction was computed and then standard deviation (SD) and coefficients of variation (CV= SD/average) computed within a trial. The same was done for the computed joint torque time series. For easier comparison with the previously published research on force variability that was mainly performed with single-direction force sensors, we analyzed the force magnitude in the prescribed direction—essentially the projection of the actual force vector on this direction— rather than the force vector magnitude. Hence, the term ‗force magnitude‘ in the present study refers to its magnitude in the prescribed direction. The force orthogonal to the prescribed direction was also measured but it will not be discussed in detail here.

The instantaneous angle  ACT of force vector was calculated as

 Fy   = tan1  ACT  F   x  (3. 3)

where Fy and Fx are the force components along the coordinate axes, respectfully. The actual force direction was compared with the target force direction and the difference was treated as the angular deviation.

Calculation of the group mean values for the intra-trial standard deviations was performed by calculating the variances of the corresponding quantities for each subject, taking their group mean and then computing the square roots of the obtained values.

CHAPTER 4. Forces and moments generated by the human arm: variability and control

4.1 Introduction

To move and manipulate objects people exert forces and moments of force

(further addressed as simply ―moments‖) on the environment. Friedman et al.(2011) studied the production of two-dimensional force vectors by a two-joint arm (the wrist was braced). This design allowed mapping endpoint force vectors onto joint torques unambiguously. In the current study, more natural three-joint tasks was used (the wrist was free). Such a system is redundant with respect to the force vector production in a plane because it can also produce a moment of force by the hand on the handle, the grasp moment (Zatsiorsky 2002). The presence of redundancy affords the system a possibility to choose any moment magnitude and direction they prefer. Such preferences remain presently unknown.

In contrast to the moment production by a serial chain, the force production—especially the force magnitude variability — was an object of rigorous research (described in Chapter 2). Conversely, studies on the variability of force vector direction are limited. Studies on the effects of the force direction on the force variability in multilink tasks (which is a topic of the present study) have been mainly limited to fingertip force production in the flexion–extension plane. In the mentioned study by Friedman et al. (2011) the hand force direction variability decreased with an

39 increase in the target force magnitude. This finding was in sharp contrast to the well-established increase in the force magnitude variability with the target force increase. It remained unknown whether the negative relation between the force magnitude and its angular variability is valid for natural tasks when the wrist joint is not braced. In addition, Endpoint force variability is a function of the variability of the joint torques in the involved joints. There is a glaring gap in the literature on this topic.

In this work, we are specifically interested in the following:

(i) The moment produced by the hand (the grasp moment);

(ii) The variability of the end-point force magnitude and direction as a

function of the target force magnitude and direction; and

(iii) The torque variability at the contributing joints, the shoulder, elbow

and wrist.

The following main hypotheses were explored:

Hypothesis 1: When subjects do not receive an explicit instruction that requires moment production they exert a zero moment on the environment.

Hypothesis 3: The intra-trial correlations between the joint torques are positive in the tasks where the torques of similar sign are exerted (i.e. flexion-flexion or extension-extension) and the correlations are negative in the tasks that require the flexion (extension) torque at one joint and the extension (flexion) at another joint.

All these ‗commons sense‘ hypotheses were rejected in the experiments.

We were also interested in whether the central controller selected the joint torques in redundant static tasks by following an optimization cost function that is additive with respect to the joint torques. This issue is addressed in Appendix B.

Similar to the above three hypotheses, the answer to this question was also negative.

4.2 Methods

Overall description of the experiment

Subjects sat in a chair, grasped a handle and generated forces of different magnitudes in different directions (Figure 4.1). We restricted the right arm to a horizontal plane at the shoulder height, with the upper arm flexed at 45° from the frontal plane, and the elbow flexed at 65º (see Figure 4.1a). This configuration was selected to provide an insight into possible effects of combinations of the flexion or extension torques at the shoulder and elbow joints (the torque at the wrist joint always coincided in the direction with the elbow joint torque). The trunk was secured to the chair with seat belts. This reduced the task to a system with three degrees of freedom that produced a two-dimensional force and a moment. This allowed reconstructing the

41 joint torques from the recorded endpoint force and moment (explained below).

Figure 4.1 Schematic diagram of the experimental setup

Left panel (Fig 4.1a). The posture of the arm. The shoulder angle α1=45º, the elbow angle α2=65º, and the wrist angle α3=0º. l1, l2, and l3 are the lengths of the upper arm, forearm and hand (from the wrist joint to the handle center), respectively.

Right panel (Fig 4.1b). The directions in which forces were produced. The force production in the indicated directions requires the following combinations of the joint torques (under the assumption that the grasp moment is zero): (1) 45º - shoulder extension, elbow extension and wrist extension (EEE); (2) 90º- shoulder flexion, elbow extension, and wrist extension(FEE); (3) 135º - shoulder flexion, elbow flexion, and wrist flexion (FFF); (4) 180º - shoulder flexion, elbow flexion and wrist flexion (FFF); (5) 225º - shoulder flexion, elbow flexion, and wrist flexion (FFF); (6) 270º - shoulder extension, elbow flexion, and wrist flexion (EFF); (7) 315º - shoulder extension, elbow extension, and wrist extension (EEE). (8) 0º - shoulder extension, elbow extension, and wrist extension (EEE).

Subjects

Five male and five female right-hand dominant subjects took part in this study

(mean age 24.7 3.2 yr, mass 65.8  15.7 kg, height 1.70  0.09 m, shoulder to elbow length (upper arm) 27.4  2.3 cm, elbow to wrist length (forearm) 25.4  2.0 cm, wrist to the center of the handle distance 7.3  0.5 cm. All subjects were healthy, with no known neurological or peripheral disorders. All of the subjects gave informed consent according to the policies of the Office for Research Protections at the Pennsylvania

State University.

Apparatus

The apparatus is shown in Figure 4.2. An aluminium cylindrical handle (height 16 cm, diameter 5 cm) was attached to a 6-DOF force/torque sensor (Mini85, ATI, Apex,

NC USA, resolution of Fx and Fy is 7/144 N and resolution of Mz is 1/600 Nm). Such an arrangement allows recording both the force vector and the moment (‗grasp moment‘) exerted on the handle. Surface of the handle was frosted, which provided greater static friction than polished surface .The friction was not measured but the subjects never reported slippage during tests. The force sensor was mounted on an aluminium block, which could slide left and right along two poles. These two poles were attached to blocks on the left and right side of the apparatus, which could slide forward and back on another two poles. Screws on the central block and on the two side blocks permitted fixing the handle at a desired location. The total workspace was 65 cm

 65 cm.

Figure 4.2 Experimental setup

The subject sat on a large, heavy chair, and was strapped to the chair with two seatbelts to prevent movement of the trunk. The chair sat on a hydraulic lift, and its height was adjusted for each subject such that the bottom of the handle was at the shoulder height. The forearm was supported by a padded semicircular piece of plastic pipe, hanging down vertically from the ceiling. The location of the handle was adjusted for each subject such that the upper arm was at 45 degrees to the frontal plane and the elbow was flexed such that there was a 65° angle between the upper arm and forearm.

The selected chair height ensured that the upper arm and forearm lay in a horizontal plane, at the height of the shoulder.

Experimental Procedure 44

Initially, the subjects were asked to produce maximal force (MVC) for 5 s along each of the 8 directions, in a clock-wise order. The subjects were provided with feedback on the horizontal force vector exerted on the handle by the right arm using a monitor placed directly in front of the subject. A blue arrow showed the amount and direction of force they were generating. Three concentric circles, marked in newtons, were subdivided by 8 directions. No feedback on the moment exerted on the handle

(‗grasp moment‘) was provided.

A total of 32 trials were performed to cover the combinations of four force magnitude levels (10 %, 20%, 30% and 40% MVC) and eight force directions. No instruction on the moment (grasp moment) production was given. Each trial took 53 s.

The duration and composition of each trial was as follows: (1) Force production for 25 s → (2) Break 5s → (3) Force production without visual feedback for 25 s → (4) Rest

120 s. (The data on the force and moment production without visual feedback are not presented in this paper).

The visual feedback was adjusted for each trial such that an identical circle corresponded to varied force magnitude requirements. The target force level and direction were indicated by a red cross in visual feedback blocks. The subjects were instructed to keep the tip of the arrow as close as possible to the red cross for 25 s.

The target levels of force were presented in a random order. The force direction conditions were pseudo-randomized to prevent similar or same direction presenting in succession. Particularly, directions of the subsequent force exertion were randomly

45 selected at either 180 º, 225 or 135 º from the previous direction. In total, 320 trials were analysed (8 instructed directions × 4 target force magnitudes × 10 subjects).

Data Collection and Analysis

The data were collected using a custom program written in LabVIEW (National

Instruments). The data analysis was performed using a custom program written in

Matlab (The MathWorks, Natick, MA). The first and the last 3 s of each trial were discarded; hence only 19-s long segments were analysed.

Mechanical model and the joint torque computations

The joint torques were computed from the recorded values of the endpoint force and moment by employing the equation (see Zatsiorsky, 2002, Chapter 2):

 Fx  T  J T F  J T  F   y  (4.1) M z  where T is a 3×1 vector of joint torques, F is a 3×1 vector of the endpoint force and

T moment (grasp moment), and J is the transpose Jacobian of the kinematic chain

 l1S 1  l2S 12  l3S 123 l1C  l2C 12  l3C 123 1 J T  l S l S l C l C 1    2 12  3 123 2 12  3 123  (4. 2)  l S l C 1   3 123 3 123 

where l1, l2, and l3 are the lengths of the upper arm, forearm, and the hand (from the wrist to the centre of the handle), respectively; S and C refer to the sine and cosine,

respectively; and the subscripts refer to the angles: 1 – the shoulder angle 1  45 ,

12 – 1  2 , i.e. the upper arm angle with respect to axis X of the external system of coordinates, and 123 to the angle 1 2 3 .

Data analysis

The instantaneous angle  ACT of force vector was calculated as

 F  1 y   ACT = tan   (4.3)  Fx 

Calculation of the group mean values (n = 10) for the intra-trial standard deviations was performed by calculating the variances of the corresponding quantities for each subject, taking their group mean and then computing the square roots of the obtained values.

Linear regression was used to model the relations between the target force magnitude (FTASK), on the one hand, and the variability of the magnitude (FACT) and the direction (ACT) of the exerted forces, on the another hand.

To detect the effects of prescribed force direction (TASK ) on the variability of the actual force magnitude (FACT) and direction (ACT) the Levene‘s test for equality of variances was performed; p-value was set as 0.05.

Statistical analysis of the joint torques T1 , T2 and T3 included computation of the regression of the torque variability (the standard deviations, SD) on the average torque magnitude (i.e., on T1 , T2 , and T3 ).

Intra-trial correlation analysis of the time histories of the recorded signals was

47 performed at the zero-time lag between the two processes. Averaging of the correlation coefficients was done after Fisher‘s z-transform.

4.3 Results

The data are presented in the following sequence: (1) Maximal voluntary contractions. (2) Submaximal voluntary contractions: grasp moments. (3) Intra-trial performance variability: dependence on the prescribed force magnitude and direction:

3a. Force magnitude variability; 3b. Force direction variability; 3c. Grasp moment variability. (4) Joint torques: 4a. Correlation with the grasp moments; 4b. Joint torque variability; 4c. Intra-trial correlations.

Maximal voluntary contractions (MVC)

At the beginning of the test, we collected MVC data from all 10 subjects in all eight directions. MVCs of every subject were normalized by the maximum value across the directions. Then the mean values and standard deviation (SD) of the normalized forces across 10 subjects at each direction were computed. Results are shown in Figure 4.3.

Figure 4.3 MVC force vs. targeted force direction.

The numbers around the outer circle are the target force directions. The average force magnitudes (in percent of the maximal MVC across all eight directions for each subject) are printed in italics. Dashed lines are the group SDs.

The group average MVCs were largest at the direction of 90º (87.1011.05% of the maximal individual MVCs) and 270º (94.086.34%). In other words, the subjects tended to generate largest maximal forces in the ‗push‘ and ‗pull‘ directions. At these two directions, the group SDs of the normalized MVC values were the smallest (11.05% and 6.34%, respectively). These two directions are also the only two force directions which required torques in opposite directions at the shoulder and elbow (as well as the wrist), see Figure 4.1: At the direction of 90º, a joint torque combination of shoulder

49 flexion, elbow extension and wrist extension (FEE) is required, while at the direction of

270, a joint torque combination of shoulder extension, elbow flexion and wrist flexion

(EFF) is required. For other force directions, the joint torques were in the same direction, either EEE or FFF.

The MVC force distribution across the target directions (the force envelope) was approximately ellipse-like. This finding agrees well both with the previously published data (Fujikawa 1997) and with the mechanical analyses performed under the assumption that that the joint torque vector‘s (T) Euclidian norm stays constant at all targeted endpoint force directions (Valero Cuevas 1997; Zatsiorsky, Gregory et al.

2002).

Submaximal voluntary contractions: grasp moments

All subjects were able to complete the tasks successfully: they exerted force according to the prescribed values. The deviations of the trial-average data from the target values were relatively small, overall less than 4.3% of the target force magnitude.

While grasp moment production was not required by the instruction and no feedback on the grasp moment was provided, moments on the handle about its long axis (‗grasp moments‘) were regularly exerted (one-tail sign test p<0.01, Figure 4.4 and Table 4.1). Such a moment production represents a preference of the central controllers (the system is mechanically redundant and has certain freedom: three joint

50 torques should satisfy two constraints associated with two endpoint force components).

Figure 4.4 The moments of force exerted on the handle (grasp moments),

group averages. The arrows represent the moment magnitude and direction. Black circles – moments in counterclockwise direction; empty circles – clockwise moments. The locations of the circles correspond to the target force direction and magnitude. Counterclockwise grasp moments are in the direction of the flexion joint torques.

Table 4.1 The number of subjects (out of 10) who generated torque on the handle in the positive (counterclockwise) direction

Force level, % of MVC

Torques at the shoulder, elbow Target force direction and wrist joints (F- flexion, E – (angleº) extension) 10 20 30 40

0 EEE 8 9 10 9

45 EEE 9 10 10 10

90 FEE 9 10 10 10

135 FFF 3 1 3 1

180 FFF 1 1 2 2

225 FFF 0 0 0 0

270 EFF 0 0 0 0

315 EEE 5 4 4 5

When the forces were exerted in 0º, 45º and 90º directions, majority of subjects

generated counterclockwise torques. For force tasks in directions ranging from 135º to

270º clockwise torques were produced. At the angle of 315º, where the torque was close

to zero, different subjects generated torques in different directions (Table 4.1). For 10

subjects, the probability of observing 9 or more cases of torque production in one

direction by chance is 0.01 (p value for the one-tail sign test). Hence, for some force

directions the observed directional preferences of the grasp moment production were

statistically significant.

For all directions of force production, with the exception of the 315º (where grasp moments were around zero), the magnitude of grasp moments increased with the magnitude of the target force. Overall for the group r = 0.604 (p<0.01, n=32, correlation of group averages computed for each direction and force magnitude combination). At some force directions, e.g. 225º, the grasp moments at large target forces exceeded 1 Nm (approximately 20% of maximal grasp moments for this handle diameter and torque direction exerted by male subjects – Seo et al. 2007) .

As a rule, the direction of the grasp moments was opposite to that of the joint torques at the elbow and wrist (cf. the grasp moment directions in Figure 4.4 and the joint torque directions presented in Table 4.1). For instance, for force directions of 225º and 270º, where the flexion torques were required at the elbow and wrist joints, none of the subjects exerted the grasp moment in the counterclockwise direction.

Intra-trial performance variability: dependence on the prescribed force magnitude and direction

Dependence of FACT on FTASK. Force magnitude variability (SDs) increased monotonically with the target force level in all 8 directions (r = 0.91, p < 0.01, n = 32).

However, after normalization by the average force there was no dependence of CV on

FTASK (r = –0.29, p > 0.05). Paired t-test indicated that for some force ranges (from 10% to 20% MVC, and from 10% to 30%), CV decreased with FTASK (p value < 0.01) while there was no significant trend across other force levels (p value > 0.05).

Dependence FACT on TASK. Levene‘s test results showed that all subjects exhibited significantly different force magnitude variability (SD) across different directions

(Levene‘s statistic 6578, p<0.01), Figure 4.5a. The pattern of SD distribution across the force directions—especially at the levels 10 - 30%— was similar to the MVC pattern (see Figure 4.3), with maximal variance for the force directions of 90º and 270º.

There were no systematic changes in CV with the target force level and direction.

SD of force magnitude (N) 40% MVC

90 30% MVC 20% MVC 1.5 10% MVC

135 45

0.5

180 0

225 315

270

a b

Figure 4.5 Dependence of the force magnitude variability (SDs) and force direction variability on the target force level and direction (panels a and b, respectively).

In panel (a) note the systematic changes with the target force magnitude: outer contours always correspond to the larger target forces than the inner contours. This dependence

is not seen in (b)

Dependence of ACT on FTASK. Force direction variability (SD) did not show significant correlation with force level (r = –0.127, p > 0.05, n = 32). Figure 4.5b shows

54 no systematic changes in the SD contours with FTASK magnitude.

Dependence of ACT on TASK. Force direction variability depended on the targeted force direction. Levene‘s test yielded p < 0.01 across the four force levels. The variability was largest for the force vector directions of 180º (1.72±0.35º, average across the force levels) and 0º (1.73±0.38º). It was smallest for the directions of 90º

(0.66±0.09º) and 270º (0.53±0.14º) and relatively small for the directions of 45º

(1.31±0.13º), 135º (1.04 ±0.14º), 225º (1.30±0.54º), 315º (1.05±0.20º), Figure 4.5b.

Grasp moment variability. For all target force directions and levels combined, there was statistically significant correlation between the group averages of the grasp moment magnitude and its SDs (r = 0.80, p<0.01, n = 32). As the grasp moment averages depended on the target force magnitude and direction (see Figure 4.4), their variability (SDs) also depended on them. Systematic changes of the CVs with the force magnitude and direction were not observed.

Joint torques

The joint torques were computed using equation 4.1from the recorded endpoint forces and grasp moment and measured arm link dimensions. These results follow trivial mechanics and, as such, they will not be presented in detail here. The ‗joint torque-endpoint force direction‘ dependencies agreed well with the ones predicted theoretically from mechanical analysis (see Zatsiorsky 2002, Page 149, Figure 2.18).

Correlation with grasp moments

For the group averages, across force directions and levels (n=32), the grasp

55 moment values showed a strong negative correlation with the torques at the elbow (r =

–0.92, p<0.001) and wrist (r = –0.86, p< 0.001) joints and a weaker correlation with the shoulder torque (r = –0.55, p<0.001). This pattern was seen in the individual data of 9 out of 10 subjects.

Joint torque variability

The intra-trial torque SDs for the elbow and wrist joints are presented in Figure 4.6.

The SDs increase with the torque magnitude. The CVs do not change systematically.

The shoulder joint torque is not affecting the grasp moment (this fact will be explained below in the Discussion section) and is not presented here.

Figure 4.6 Torque variability (SD) at the elbow and wrist joints in the trials at different target force levels and directions, group average data. Note the different scales in the two panels. The ratios ‗SD (elbow joint)/SD (wrist joint)‘ for different target force levels varied between 3.1 and 3.4.

Intra-trial correlations

For each subject, the within-trials correlations of the force- and torque-time history processes —totally six: three endpoint force and moment components and three joint torques — at zero time lag were computed for all target force directions and levels. The obtained 4800 correlation coefficients (15 signal pairs × 32 tasks ×10 subjects) were then group averaged for individual force directions. The coefficients were computed for the signals that were digitized at 1000 Hz for 19 s. Hence, each coefficient was based on

19000 data pairs. Because consecutive observations in the continuous force recordings were not independent from each other and also due to the arbitrary selection of the digitization frequency, the classical methods of estimating statistical significance cannot be applied here.

The force component in the target direction showed close to zero correlation with both the force component in the orthogonal direction and grasp moment across all tasks.

The first finding agrees well with the results reported by Friedman et al (2011) who analyzed the forces along the prescribed directions and perpendicular to them. In the tasks that required torque production both at the shoulder and elbow joints (the wrist joint was braced) force variance was approximately equal in both directions as it must be expected for a close-to-zero correlation. The within-trial correlations involving joint torques are described below.

Intra-trials correlations with the endpoint force. Across force directions, the correlations differed in magnitude but their sign, positive or negative, always corresponded to the joint function in force production at a given direction (Table 4.2). In some directions the

correlations were very large, for instance for the zero degree direction the correlation

with the shoulder torque was 0.97.

Table 4.2 Intra-trials correlations between the force in the target direction and the joint torques in the individual trials. Group average data for different force directions (n=40; 4 force levels ×10 subjects).

Force 0.0 45.0 90.0 135.0 180.0 225.0 270.0 315.0 direction, º

Torque EEE EEE FEE FFF FFF FFF EFF EEE direction Shoulder -0.97 -0.46 0.41 0.92 0.96 0.51 -0.33 -0.91

Elbow -0.91 -0.88 -0.61 0.54 0.89 0.89 0.64 -0.51

Wrist -0.68 -0.47 -0.16 0.38 0.59 0.43 0.22 -0.38

Intra-trial correlations with the grasp moment. The correlations between the grasp

moment and the joint torques at the shoulder and elbow were low (around 0.1); hence,

they are not presented here. The correlations with the wrist torque were strong and

positive for all FTASK directions (Table 4.3). Note that in contrast to the positive

intra-trials correlations between the wrist torque and the grasp moment the correlation

between these two variables across the tasks was negative (see section ―correlation

with grasp moments‖ above).

Table 4.3 Intra-trials correlations between the grasp moment and the wrist torque. Group average data for different force directions (n=40; 4 force levels ×10 subjects).

Force 0.0 45.0 90.0 135.0 180.0 225.0 270.0 315.0 direction, º EEE EEE FEE FFF FFF FFF EFF EEE Joint torque

58 direction Correlation 0.65 0.65 0.76 0.70 0.68 0.71 0.80 0.77

Intra-trial correlations between the joint torques. The coefficients are presented in Figure

4.7. The coefficients were positive for all force levels and directions. However, for

TASK of 90º and 270º the correlations between the shoulder torque and the torques at

the elbow and wrist joints were smaller than for other force directions. For instance, for

TASK = 90º at 10%-force level; the shoulder-elbow correlation was only 0.43 while for

TASK = 270º it was 0.60 – smaller than for all other force directions. When comparing

group average data (1-way repeated measure ANOVA) the differences between the

z-transformed coefficients of correlation were found to be significant ( F7,24= 49.07;

p<0.01) and post hoc multiple comparisons also indicate significantly smaller

coefficients at the direction 90º and 270º than at other directions. Note that at the

directions of 90º and 270º the joint torques of the opposite signs are exerted: at 90º-

FEE; at 270º - EFF, while at other force directions the joint torques were all either in

flexion or in extension (see Figure 4.1).

r of SE 40% MVC

90 1 30% MVC 120 60 20% MVC 0.8 10% MVC

0.6 150 30 0.4

0.2

180 0

210 330

240 300 270

Figure 4.7 Intra-trial correlations between the joint torques at different force directions and force levels in individual trials. Group average data (n=10). Note that at the directions of 90º and 270º the intra-trial correlations are positive while the torques themselves are of opposite sense, FEE and EFF, respectively.

4.4 Discussion

All three main hypotheses formulated in the Introduction have been falsified.

Indeed:

Hypothesis 1: When subjects do not receive an explicit instruction that requires moment production they exert a zero moment on the environment. The data show that the subjects produced non-zero grasp moments of substantial magnitude.

Hypothesis 2: The variability indices of force magnitude and direction show similar dependencies on the target force magnitude and direction, e.g., in tasks where the force magnitude variability is maximal (minimal) the force direction variability is also maximal (minimal). The data show that the dependences of force magnitude and direction variability indices on task variables were opposite: In the directions where

the variability of FACT was maximal directional of ACT was minimal and vice versa.

Hypothesis 3: The intra-trial correlations between the joint torques are positive in 60 the tasks where the torques of similar sign are exerted (i.e. flexion-flexion or extension-extension) and the correlations are negative in the tasks that require the flexion (extension) torque at one joint and the extension (flexion) at another joint. The data show that in the tasks requiring the joint torque production in opposite directions the within-trial correlations between the joint torques were positive.

The following discussion covers the following topics: (1) grasp moments, (2) the end-effector force variability, and (3) joint torque variability and intra-trial correlations.

Grasp moments

The subjects generated the grasp moment in a very systematic manner (Figure 4.4 and Table 4.1): the moment direction (clockwise or counterclockwise, as seen from the top) depended on the end-effector force direction and the moment magnitude depended on the force magnitude.

Across the tasks, i.e. across the directions and force levels, the grasp moments correlated negatively with the joint torques (see subsection ―correlation with grasp moments‖ in the Results section). With increasing magnitude of the grasp moments the joint torques magnitudes decreased. It could be thought that the grasp moment production assisted in decreasing the joint torques. At first glance, the observed negative correlations can be explained by tendency to minimize the muscle efforts.

However the actual situation is more complex. As well known, ‗correlation does not imply causation‘ and at least some of the mentioned correlations can be false (spurious).

It seems that they are.

Consider the correlation between the grasp moment and the wrist torque. As mentioned above, the negative correlation between these two variables across the tasks was observed (subsection ―correlation with grasp moments‖ in the Results section) while positive correlations within the trials were found (Figure 4.8, see also Table 4.3).

Figure 4.8 Correlations between the grasp moment and the wrist torque. The figure is for a representative subject. Correlation across the target force levels and directions is negative (the large ellipse, r = –0.86, n= 32, p< 0.001) while the intra-trial correlations (small clouds of points) are all positive. For instance, for 315º direction the coefficients are: at 10% force r = 0.94, at 20% r = 0.92, at 30% r = 0.90 and at 40% r = 0.83.

We consider first biomechanical aspects of the grasp moment production and then the motor control issues.

Biomechanical aspects. Due to the peculiarities of multi-link mechanics the effects 62 of the external moments (in the present setup it is the grasp moment) on the joint torques and the opposite effects of the joint torques on the grasp moment are quite different. As follows from analysis of the transpose Jacobian of the chain (equation 4.2), the grasp moment is simply added to the joint torques (all elements of the last column of the transpose Jacobian equal 1) while the individual joint torques affect the grasp moment in a more complex way. For a three-link arm chain the relation is:

 C 12  l2 C 12  l1C 1 C 1    l1 S 2 l1l2 S 2 l2 S 2 FX    T1  T 1    S 12  l2 S 12  l1 S 1 S 1    (4.4) F = J  T = FY =  T2   l S l l S l S     M   1 2 1 2 2 2 2  T     3  l3 S 3  l2 l3 S 3  l1l3 S 23 l3 S 23+l2 S 2    l1 S 2 l1l2 S 2 l2 S 2 

1 where F is a 3×1 endpoint force vector, J T  is the inverse of the transpose Jacobian,

T is a 3×1 joint torque vector, T1, T2 and T3 are the torques at the shoulder, elbow and wrist joints, respectively, and other symbols have been defined previously. The grasp

(endpoint) moment M equals the dot product of the third row of the matrix and the joint torque vector. It is

l3S3  l2l3S3  l1l3S23  l3S23  l2S2 M  T1   T2  T3 (4.5) l1S2  l1l2S2  l2S2 i.e. the grasp moment is an additive function of all three joint torques with the coefficients depending in a complex way on the joint angles and link lengths.

In the chain under consideration the wrist angle is 0º and hence S3=0 and S23=S2.

Therefore equation 4.5can be simplified:

l3 l3 M  (1 )T3  T2 (4.6a) l2 l2

which for the average link length values in this research (l2= 25.4 cm, l3= 7.3 cm) yields

M=1.29T3–0.29T2 (4.6b)

Hence joint torque changes at the elbow and at the wrist of the same signs, i.e. both positive or both negative, induce the grasp moment changes in the opposite directions.

Because equation 4.6b predicts opposite effects of the wrist torque T3 and elbow torque T2 on the grasp moment M (the effects are positive for the wrist torque and negative for the elbow torque), it does not agree easily with the data presented in Table

4.1. For instance for the force vector directions of 225º and 270º, where the flexion—i.e. positive— torques were required at the elbow and wrist joints all subjects exerted grasp moments in the clockwise, i.e. negative, direction. Hence the expected positive effects of the wrist torque T3 on the grasp moment M were not seen. Similar facts were observed for the 45º and 90º force directions where the negative (in extension) torques were produced at both joints while at least 9 subjects generated positive

(counterclockwise) grasp moments. At the same time, there was excellent correspondence between the grasp moments predicted from equation 4.6b and the actual moments recorded in the experiment (r >0.99).

The disagreement between the negative ‗wrist torque-grasp moment‘ correlation across the tasks and positive correlations within the trials is explained by the different magnitudes of the variations of T2 and T3, such that in different conditions effects of either T2 (negative) or T3 (positive) dominate. This is illustrated in Figure 4.9 64 that shows how the joint torque and the grasp moment change across the directions

(upper panel) and the same changes multiplied by the coefficients from equation 4.6b

(bottom panel).

20 elbow component wrist component 15 grasp moment

-5 Joint Torque (Nm) Joint Torque

-10

-15

-20 0 EEE 45 EEE 90 FEE 135 FFF 180 FFF 225 FFF 270 EFF 315 EEE Direction (degrees)

6 elbow component wrist component 4 grasp moment

Joint Torque (Nm) Joint Torque -2

-4

-6 0 EEE 45 EEE 90 FEE 135 FFF 180 FFF 225 FFF 270 EFF 315 EEE Direction (degrees)

Figure 4.9 Upper panel. Joint torques and the grasp moment across the target force directions (for 40% of MVC). The data are for a representative subject. Note that the grasp moment changes in the opposite directions to both joint torques (negative correlation). Bottom panel. The joint torques multiplied by the coefficients from

equation 4.6b: –0.29T2 and 1.29T3, respectively. The grasp moment changes in the same direction as the elbow joint component (–0.29T2) and opposite to the wrist joint component (1.29T3). This should result in the negative across tasks correlation between

the T3 and M despite their positive relation seen in equation 4.6.

From Figure 4.9 and the analysis presented in the figure caption it follows that the negative across trials correlations between T3 and M are spurious. They are due to two factors: the inter-trial correlations between the joint torques and the much larger range of the elbow torque changes than the wrist torque changes (the upper panel in

Figure 4.9). The same factors explain the negative correlation between the shoulder

66 torque and grasp moment (r = –0.55) mentioned above in section ―correlation with grasp moment‖ although, according to equation 4.6a, 4.6b, shoulder torque does not immediately affect the grasp moment at all. This is an exemplary case of a spurious correlation.

The positive within-trials correlations between the grasp moment and wrist torque can be explained by two mechanisms. First, the positive relation is due to straightforward mechanics (see equation 4.6a,4,6b). Second, the negative (spurious) effect of the intra-trial T2 variations is smaller in this case: the T3 coefficient (1.29) is

4.45 times larger than the T2 coefficient (1.29/0.29=4.45) while the variations of T2 are larger than those of T3 only 3.1-3.4-fold (see the caption to Figure 4.6). Hence the positive effect of T3 dominates over the negative effect of T2.

Motor control aspects. Generating the non-required grasp moment is in accord with the reported in the literature tendency of the subjects to produce the static endpoint forces in the direction different from the instructed direction — when the visual feedback is not provided and the external object is mechanically constrained. For instance, during pedaling the athletes exert forces not only in a tangential direction but also in the normal directions along the crank thus either compressing or extending it

(Cavanagh PR 1986). Such a pattern of pedaling is evidently suboptimal —the athletes spend efforts and energy on generating forces that are not necessary for the task.

However, even the best athletes do this. Similar patterns of force production are reported for the manual wheelchair propulsion (van der Woude, Veeger et al. 2000).

Examples of producing force in a ‗wrong‘ direction are abundant in the literature.

Grieve and Pheasant (1981) called these directions ‗naturally preferred‘ and introduced a measure for estimating the effectiveness of static force efforts – the maximum advantage of static force efforts (MACE). Pan et al. (2005) in an elegant study have shown that these results can be interpreted in terms of an unknown optimization used by the central controller. While the authors were not able to reconstruct the cost functions used by the central controller they derived from the experimental data the so-called "isocost" contours of objective functions. On the whole, these results suggest that the observed force patterns are due to some kind of optimization used by the central controller.

None of the above studies analyzed the grasp moment or any torque exerted on the environment. Application of the optimization methods in this case is however unusual: as follows from equation 4.6a,4.6b, the grasp moment is a function of the opposite

—positive and negative—influences from the involved joint torques. Consider a task where both torques are of the same sign, for instance force should be exerted at 135º where both the elbow and wrist joint act in flexion. Suppose that the subject is also exerting a non-specified grasp moment while minimizing an unknown cost function of the joint torques. For simplicity assume that the shoulder joint is immobilized and its torque does not affect the grasp moment. If the performer decreases the wrist torque, the grasp moment will decrease but if the elbow torque is decreased the grasp moment will increase. The effect of the simultaneous increase or decrease of both joint torques on the grasp moment is unclear in this case. 68

In motor control studies addressing the issue of optimization additive cost functions have been commonly used (Nubar and Contini 1961; Yeo 1976;

Crowninshield 1981; An, Kwak et al. 1984; Herzog and Leonard 1991; Pandy and

Zajac 1991; Tsirakos, Baltzopoulos et al. 1997; Anderson and Pandy 2001; Raikova and Prilutsky 2001; Prilutsky and Zatsiorsky 2002; Zatsiorsky, Gregory et al. 2002;

Ackermann and van den Bogert 2010; Park, Zatsiorsky et al. 2010). A newly developed analytic inverse optimization (ANIO) method that allows reconstructing the unknown cost function from experimental data—rather than assuming the function a priori—is also based on the presumption that the sought cost function is additive with respect to some ‗elemental‘ variables, for instance individual finger forces in multi-finger tasks

(Terekhov, Pesin et al. 2010; Terekhov and Zatsiorsky 2011). The ANIO method was successfully applied to many multi-finger prehension and pressing tasks (Park,

Zatsiorsky et al. 2010; Niu, Terekhov et al. 2011; Park, Sun et al. 2011; Park, Zatsiorsky et al. 2011; Niu, Latash et al. 2012; Park, Singh et al. 2012).

In search of the optimization cost function that could explain the findings of the present study we applied the ANIO method to the obtained experimental data (see

Appendix). The outcome was unambiguous: the central controller does not use for the control of the endpoint force and moment a cost function additive with respect to the joint torques. This finding is in stark contrast with the arm movement control where such an additive cost function as the minimum torque change (the sum of the squared values of the time derivatives of the joint torques) has been suggested (Nakano,

Imamizu et al. 1999) and validated (Wada, Yamanaka et al. 2006). Developing optimization methods for static serial chains will be a challenge for researchers.

Future research. We are going to further test the mechanisms of the grasp moment generation in three sets of experiments: (a) the handle will be placed in a freely rotating housing (such that the grasp moment cannot be exerted), (b) the handle diameter will differ, and (c) the subjects will have to produce the grasp moments of different magnitude in clockwise and counterclockwise directions while simultaneously exerting the end-effector force of various magnitude in different directions such that the task becomes non-redundant.

The end-effector force variability

The increase of the force magnitude variability (SD) with the force level is a well known phenomenon discussed in detail in many publications (for a recent review see e.g. Friedman et al. 2011). This discussion will not be repeated here.

In contrast, the data on the force direction variability are relatively new. So far the

dependence of SD(ACT) on FTASK was addressed only in a recent study by Friedman et al. (2011) who found a counter-intuitive relation: SD(ACT) decreased with an increase in FTASK. We were not able to confirm this finding quantitatively (in the present study the coefficient of correlation was only –0.127, p > 0.05), but taken together the results from the two studies provide sufficient evidence to conclude that force direction variability does not increase with the force level (see Figure 4.5b). The differences

70 between the results of the two studies may be due to experimental details: In the

Friedman et al. experiments the wrist was braced and hence the arm acted as a two-link system while in the present study more natural conditions of the arm functioning were preserved.

We would like to bring up one observation that can lead to a hypothesis that may

explain the opposite trends of SD(FACT) and SD(ACT) as functions of FTASK. Equation

4.3 is written in the external, X and Y, coordinates (explained in Figure 4.1) but it can be also written in the local coordinates, along the target force direction and normal to it.

The force in the direction perpendicular to the target one is the main contributor to the angular deviation of the actual force from the target direction. For instance, if the target force is at 90º (in the ‗vertical‘ direction in Figure 4.5), the angular deviation is a function of the force component in the ‗horizontal‘ directions, i.e. in 0º and 180º directions. For 0º target force direction, the angular deviations of the force vector are determined by the ‗vertically‘ oriented force components, i.e. the force components at

90º and 270º.

Comparing the data presented in Figures 4.5a and b suggests that across the

targeted directions the variability of FACT and ACT exhibit opposite trends. In the directions where the variability of FACT is maximal (90 º and 270 º) directional of ACT is minimal and in the directions where ACT variability is maximal (0 º and 180 º) the

FACT variability is minimal. Across the targeted force directions and levels, the coefficient of correlation between the FACT and ACT variability was r = –0.67 (n = 32; α

<0.001). This observation, if confirmed, leads to a hypothesis that the force direction control is determined (or at least affected) by neural processes functioning in the body centered system of coordinates, i.e. in the system of coordinates with

‗forward-backward and left-right‘ axes (and not, for instance, in the task relevant systems of coordinates with the axes along the target force direction and normal to it).

This hypothesis agrees with the earlier postulate that the control is organized within the shoulder-centered referent frame (Soechting 1992). The accuracy of the control along the two shoulder-centered axes of coordinates is naturally different and this difference is manifested in the results shown in Figure 4.5. If this hypothesis is accepted as a starting point, a large body of experimental evidence would be necessary to either validate or compromise it.

Future research. We expect that systematic variations in the arm position can provide additional information on the mechanisms behind the data presented in Figure 4.5a and b.

Joint torque variability and intra--trial correlations

Intra-trial correlations were computed for the zero time lag between the two processes and hence these correlations characterize the level of synchronization between the processes.

We are specifically interested in the two tasks—with the force directions of 90º and

270º—in which the joint torques of the opposite signs were exerted: at 90º- FEE; and at

270º - EFF. As expected, for these force directions the within-trial correlations of the

72 joint torques with the end-point force of the opposite signs were found (Table 4.2).

Hence when the end-effector force increased or decreased, the torques at the shoulder and other two joints changed in opposite directions, and the negative torque-torque correlations could be expected (Hypothesis 3).

In contrast, the torque-torque within-trial correlations were positive for the 90º and

270º force vector directions, similarly to all other directions (Figure 4.7). [Positive torque-torque correlations were also reported by Friedman et al. (2011) however in their study there were no tasks requiring the joint torque production in opposite directions.] The correlation coefficients at the 90º and 270º force directions were, however, smaller than at other directions. The discussion below attempts to address this controversy.

Let us assume that the endpoint force magnitude variations are due to the variability of a central, neural drive. If the drive ‗intensity‘ (whatever it physiologically is) increases, the endpoint force also increases accompanied by an increase of the joint torque magnitudes. Because the torques are of opposite signs, a negative correlation between them should be expected following changes in the neural drive. Instead a positive correlation was observed..

It seems that more than one mechanism (the ‗neural drive‘ variability) affect the result. Decreased intra-trial correlations for the 90º and 270º force directions, as compared with other directions, where either flexion-flexion or extension-extension torque combinations were required, speak in favor of this possibility. Such a second

73 mechanism could be the activation of two-joint muscles—the long heads of the biceps brachii and triceps brachii—that serve both the shoulder and elbow joints. Activation of either one of these muscles results in producing flexion-flexion or extension-extension torque sets. In the tasks requiring the joint torques of the same sign the effects of the two abovementioned central drives are summed up (and very large intra-trial correlations are observed), while for the force directions requiring joint torques of opposite signs, the two drive effects are subtracted (and we observe smaller correlation).

Study limitations. The main limitation of this study was that only one arm position was used, and it is not known to what extent the observations depended on the arm posture.

Future research. We are planning to perform similar experiments with arm configurations that allow using a larger number of force directions for which the joint torques of opposite signs are required. We plan to concentrate on coordination of only two joints (the wrist will be braced). For the two-link arm, the end-point force can be represented by two force components (explained in Zatsiorsky 2002, Chapter 2, see

Figure 2.19) that are due to: (a) shoulder torque – this force component is along the pointing axis – the axis along the forearm, and (b) elbow torque—this force component is along the radial axis – the axis along the line from the shoulder center to the endpoints. The forces that require the torques of opposite signs are located in the sectors limited by the two above axes. Due to the selected arm configuration (see Figure

4.1) these sectors were narrow in the present research. By changing the arm posture we will able to offer many force directions requiring the torques of opposite sign. We expect that this will allow shedding more light on the mechanism of the end-point force control.

4.5 Conclusions

The following main facts were observed: (a) While the grasp moment was not prescribed by the instruction, it was always produced. The moment magnitude and direction depended on the instructed force magnitude and direction. (b) The within-trial angular variability of the exerted force vector (angular precision) did not depend on the target force magnitude (a small negative correlation was observed). (c) Across the target force directions, the variability of the exerted force magnitude and directional variability exhibited opposite trends: In the directions where the variability of force magnitude was maximal, the directional variability was minimal and vice versa. (d) The time profiles of joint torques in the trials were always positively correlated, even for the force directions where flexion torque was produced at one joint and extension torque was produced at the other joint. (e) The correlations between the grasp moment and the wrist torque were negative across the tasks and positive within the individual trials. (f) In static serial kinematic chains, the pattern of the joint torques distribution could not be explained by an optimization cost function additive with respect to the torques. Plans for several future experiments have been suggested.

CHAPTER 5. Effects of arm posture on the force and moment production

5.1 Introduction

During static accurate force production tasks the forces fluctuate both in magnitude

(Newell and Carlton 1988) and direction (Friedman, Latash et al. 2011; Xu, Terekhov et al.

2012). Such force fluctuations require concomitant variations in the joint torques. In our previous study (Xu et al. 2012, also Chapter 4 of the dissertation) we analyzed static force production on a hand-grasped cylindrical handle. The subjects exerted forces on the handle in eight directions at one fixed arm posture (the angles at the shoulder, elbow and wrist were

45-65-0 degrees, correspondingly). The following four main facts have been reported:

(a) While the grasp moment on the handle was not prescribed by the instruction, it was always produced. The moment magnitude and direction depended on the instructed force magnitude and direction.

(b) Across the target force directions, the variability of the exerted force magnitude and directional variability exhibited opposite trends: In the directions where the variability of force magnitude was maximal, the directional variability was minimal, and vice versa.

(c) The time profiles of joint torques in the trials were always correlated positively, even for the force directions where the joint torques had different signs: e.g. flexion torque was produced at one joint and extension torque was produced at another joint.

At such postures, positive correlations between the torques signify negative correlations between the torque magnitudes.

(d) For the human arm, the patterns of the joint torque distribution could not be explained by optimization of a cost function additive with respect to the joint torques (this analysis is presented in Appendix B).

The goal of the present study has been to test whether the above findings are valid for a range of arm postures. We varied the arm posture in different trials in a systematic way.

5.2 Methods

Subjects: Eight male right-hand dominant subjects took part in the study (mean age

26.1 4.0 years, mass 70.9  8.3 kg, height 1.74  0.08 m, shoulder to elbow length (upper arm) 28.0  1.4 cm, elbow to wrist length (forearm) 25.4  1.0 cm, wrist to the centre of the handle distance 7.5  0.5 cm. All subjects were healthy, with no known history of neurological or peripheral disorders and injuries. All the subjects gave informed consent according to the policies of the Office for Research Protections at the Pennsylvania State

University.

Apparatus: The apparatus used in the experiment included an aluminium cylindrical handle (height 16 cm, diameter 5 cm) attached to a 6-DOF force/torque sensor (Mini85, ATI,

Apex, NC USA). Such an arrangement allows recording both the 2-D force vector (resolution about 0.048 N) and the moment exerted on the handle, the grasp moment (resolution 1/600

Nm). The handle was secured at different positions corresponding to the arm postures shown in Figure 5.1, left panel. A detailed description of the apparatus can be found in Chapter 3

77 and Xu et al. (2012).

Subjects sat in a chair, grasped the handle and generated forces of prescribed magnitudes in eight directions (Figure 5.1, Right panel). The trunk was secured to the chair with seat belts. The right arm was in a horizontal plane at the shoulder height, with the upper arm in various postures flexed at 10°, 20°, 30°, 40°, 50° and 60°, and the elbow flexed at 120º

(Figure 5.1, Left panel). For these arm postures four or six (depends on the arm configuration) out of the eight directions required shoulder and elbow torques to have the same sign

(flexion-flexion or extension-extension), while other directions required them to be of opposite signs (flexion-extension or extension-flexion). The signs of the joint torques were estimated under the assumption that the grasp moment was always small as compared to the joint torques, like it was done in the previous study (Xu et al 2012). A posture 60-120-0° possesses a specific feature: Force directions 0° and 180° are along the arm pointing axis, i.e. along the forearm-hand line. In an ideal world, in such a task, elbow torque should be zero

(in reality, the elbow torques were very close to zero).

Figure 5.1 Schematic diagram of the experimental setup.

Left panel. The arm postures. The shoulder angle α1 varied in different test sessions from 10° to 60°, the elbow angle α2=120°, and the wrist angle α3=0°. l1, l2, and l3 are the lengths of the upper arm, forearm and hand segments (from the wrist joint to the handle centre), respectively. Right panel. The directions of force production.

Experimental procedure. In the preliminary trials, at each arm posture, the maximal voluntary contraction (MVC) forces were recorded under the instruction ‗push the handle as strongly as you can‘. The duration of each trial was 5 s. In a separate test, the subjects were asked to produce the maximal grasp moment on the handle, ‗try to turn the handle as strongly as you can‘. The MVC data were used for the normalization of the prescribed forces in the main experiment and will not be presented here.

79 feedback on the horizontal force vector exerted on the handle. No instruction or feedback on the grasp moment was provided. A total of 108 trials were performed to cover the combinations of six arm configurations with 18 trials in each configuration; 30-s breaks were provided between the trials and a 2-min break between blocks of trials with different arm configurations.

Data analysis. The force signals were conditioned (9105 IFPS-1, ATI, Apex, NC USA) and digitized using a 16-bit A/D converter (PCI-6225, National Instruments, Ausin, TX) at

1000 Hz. The data were collected using a custom program written in LabVIEW (National

Instruments). The data analysis was performed using Matlab (The MathWorks, Natick, MA).

The joint torques were computed from the recorded values of the endpoint force and moment by employing the equations of statics (described in Chapter 3) .

The 4-s time periods at the beginning and 1-s at the end of each accurate force production trial were discarded to avoid edge effects and the computations were performed using the central 10-s time periods. The preferred direction of the grip moment production in the group, clockwise or counter-clockwise, was estimated with the sign test. For 8 subjects the significance level p<0.5 was attained when at least 7 subjects exerted a moment in the same direction (p=0.035).

Intra-trial correlation analysis of the time series of the recorded signals was performed at zero-time lag. For further analyses, the correlation coefficients were subjected to the Fisher‘s z-transform.

Inverse optimization (ANIO) analysis.

We employed ANIO analysis (Terekhov, Pesin et al. 2010; Terekhov and Zatsiorsky 2011) to assess the possibility that the subjects selected the joint torques in order to minimize a total cost G, which we assumed to be an additive function of the joint torques, i.e.

G (T1, T2, T3) = g1(T1) + g2(T2) + g3(T3) (5.1)

where g1, g2, g3 are unknown functions of the shoulder, elbow and wrist torques, respectively.

The ANIO method was used for each configuration of the arm. The analysis was performed according to the procedures described earlier (Park, Zatsiorsky et al. 2010; Terekhov, Pesin et al. 2010; Niu, Terekhov et al. 2011; Park, Sun et al. 2011; Terekhov and Zatsiorsky 2011; Park,

Singh et al. 2012).

First, we tested whether the values of the joint torques exerted by each subject were distributed over a two-dimensional surface. This was done by performing the principal component analysis (PCA) both on raw torque data and on the torques, normalized by their maximal magnitudes computed among all directions. The data were considered planar if the two first PCs accounted for more than 90% of the total variance. Indeed, the joint torque data formed two-dimensional planes for each subject (see Results). In this case, gi must be second-order polynomials of their arguments (Terekhov et al. 2010, Xu et al. 2012):

2 gi(Ti) = ki (Ti) + wiTi, i = 1,2,3. (5.2)

For G to be a valid cost function, all ki must be positive. Otherwise, the polynomial does not have a global minimum. It was shown in (Xu et al 2012, Appendix) that, if at least one of the ki coefficients is negative, then no cost function of the form (1) can explain the choice of the joint torques.

5.3 Results

Grasp moments.

The subjects exerted a moment on the handle in all arm postures, Figure 5.2. The moment magnitudes ranged from –1.40 Nm to 1.05 Nm, in relative units they were up to

46.69% of the maximal value of the grasp moment (measured separately).

The moment direction, clockwise or counterclockwise, depended on the target force direction. At 90° and 135° force directions, the subjects exerted a counterclockwise (positive) grasp moment in all studied arm postures, with the exception of one subject in the 10-120-0° posture and one subject in the 20-120-0° posture. When the target force direction was 225°,

270° or 315°, the subjects generated grasp moments in the clockwise direction. It happens in all arm postures with the exception of the 10-120-0° posture. Overall, there was a tendency to exert counterclockwise grasp moments during pushing efforts and clockwise moments during the pulling arm efforts.

Repeated-measures ANOVA indicated significant differences in grasp torque magnitudes across different configurations at target directions of 180°, 315° and 360° only (p<0.05).

Figure 5.2 Grasp moments for different arm postures and target force directions.

The arrows represent the moment magnitude (group average) and direction. Black and open circles show moments in counter-clockwise and clockwise direction, respectively. Only the cases when at least 7 subjects out of 8 exerted moments in the given direction are marked in this way. The grey circles correspond to the cases when less than 7 subjects exerted moments in one direction.

Variability of the force magnitude and direction across the targeted force directions and arm postures.

In the study by Xu et al.(2012) it was found that at the directions where the force magnitude variability was maximal the force direction variability was minimal and vice versa. In the present study, these facts were not observed for all arm postures.

They were valid only for some of them, for instance for the 20-120-0 arm configuration.

The minimal force direction variability was observed at two target force directions. These force directions were (a) opposite to each other (180ºdifference) and

(b) close to the pointing line, i.e. along the longitudinal line of the forearm-hand segment, Figure 5.3.

Figure 5.3 Force magnitude (N, filled circle) and force direction (degrees, open circle) variability across target directions.

Vertical broken lines indicate the pointing axis (the forearm orientation). Group average±SE data. Note that the minimal force direction variability was observed at the tasks where the target force direction was close to the pointing axis.

Torque-torque correlations in the individual trials at different arm postures

The time profiles of joint torques in the trials were always correlated positively, even for the force directions where opposite torques were produced at the shoulder and elbow joints, flexion-extension or extension-flexion. This finding was confirmed for all arm postures tested in the present study (Figure 5.3). The correlation coefficients for these directions were however smaller than for the directions where the joint torques were in the same direction, i.e.

84 flexion-flexion or extension-extension. Repeated-measures ANOVA supported this claim (F

≥4.91,p<0.01 of all 6 configurations). Force directions 0° and 180° at the configuration

60-120-0° represent, however, a special case: at these conditions the forces should be exerted along the pointing axis, i.e. along the forearm-hand line and hence elbow torques should ideally equal zero. For these tasks, the smallest coefficients of correlations were observed

(see Figure 5.4).

Figure 5.4 Coefficients of correlations between the time profiles of the shoulder and elbow joint torques at different arm postures.

The thick diameters indicate the force directions, at which the shoulder and elbow exhibit the torques of opposite sign, flexion-extension or extension-flexion. The internal circle indicates the coefficient of correlation value of 0.50. In the configuration 60-120-0° the forearm was along the 0°-180° line, the elbow joint torques were close to zero and minimal coefficients of correlation were observed.

The torques data planarity and cost function analysis (ANIO)

In the 3D joint-torque space, the first two principal components (PCs) accounted for over

99% of the joint torque variance for both normalized and non-normalized joint torques. The

second PC accounted for about 10-25% of variance, thus confirming that the data tended to

form a plane and not a line.

The planarity of the data justifies the use of second-order polynomials as cost function

candidates (for details see Terekhov et al 2010 and Xu et al 2012 Appendix). The averages of

the second-order coefficients are provided in Table 5.1. Note coefficients of different sign for

all arm postures. This finding proves that no torque-additive cost function can explain the

distribution of the joint torques adopted by the subjects (see the formal proof in Xu et al 2012

and Appendix B).

Table 5.1 Coefficients of cost function at six arm configurations across subjects

Configuration 10-60-0° 20-60-0° 30-60-0° 40-60-0° 50-60-0° 60-60-0°

k1 -0.17±0.01 -0.13±0.01 -0.02±0.01 -0.02±0.01 -0.02±0.01 -0.02±0.01

k2 -0.20±0.02 -0.20±0.02 -0.19±0.02 -0.18±0.02 -0.18±0.02 -0.18±0.02

k3 0.98±0.00 0.98±0.00 0.98±0.00 0.98±0.00 0.98±0.00 0.98±0.00

The coefficients for the wrist torque (k1), elbow torque (k2) and shoulder torque (k3) are presented as across-subjects means ± standard deviations.

5.4 Discussion

Results of the current study have confirmed the main findings of the previous study (Xu et

al. 2012) and generalized them to various arm postures. In particular, non-instructed non-zero

grasp moments were observed for most of the combinations of arm posture and force direction.

The joint torque profiles correlated positively across all conditions including those that required shoulder and elbow torques in opposite directions. In spite of the planarity of the joint torque distribution, no additive cost function of joint torques was able to account for the observed data. A novel fact is that the minimal force direction variability (‗the highest precision of force exertion‘) was observed for the target force directions that were along or close to the arm pointing axis.

Grasp moments, causes for non-instructed actions

When an action is performed by a redundant system, its performance can be associated with producing mechanical variables not explicitly related to the task. The arm, viewed as a serial chain with three joints, by itself is not mechanically redundant: three joint torques generate three components of the endpoint force vector (two force components and grasp moment). However since the grasp moment was not prescribed, the task was formulated as a redundant one.

If one assumes that specific patterns of performance are selected based on some economy principle, such non-instructed forces and moments of force in static tasks are expected to be close to zero (cf. minimization of secondary moments, Li et al. 1998). In our experiment, however, non-zero grasp moment was produced in a majority of the force direction conditions (see Figure 5.2), similarly to Xu et al. (2012). The pattern of grasp moment production across the eight directions did not change systematically with the arm posture. Most subjects (at least 7 out of 8) produced positive (counter-clockwise) moments on the handle at force directions 90º and 135º independently of the arm posture and they

87 exerted clockwise moments when force directions were 225°, 270° and 315° (with the exception of the 10-120-0° posture).

One possible cause of the relative independence of grasp torques of the arm posture was that the elbow and wrist angles did not change across the studied arm positions. As follows from the analysis presented in Xu et al. (2012): when the wrist angle is zero (the hand and forearm are along the same line), the grasp moment is a function of only the elbow and wrist torques; the shoulder torque at such postures does not contribute to the grasp moment. The absence of variations of the elbow and wrist angles should be considered a delimitation of this study. We are going to explore the role of variations at the elbow and wrist angles on the grasp moments in our future studies.

Variability of the exerted force across the targeted force directions and arm postures.

An interesting novel fact is that the minimal force direction variability—it can be interpreted as the highest precision of force exertion—was observed when the target force directions were along or close to the pointing axis. At such directions, the force vector passes through the elbow joint center and does not generate a moment of force at the elbow (for a more detailed biomechanical analysis see Zatsiorsky, 2002,

Chapter 2). Therefore, at these force directions, the endpoint force and its variability is solely due to the joint torque at the shoulder. This is valid both for the pushing and pulling efforts along the pointing axis: at these directions two minimal values were observed.

It sounds reasonable that if only one joint torque is involved in the endpoint point 88 force production the force variability should be smaller than when the force is a function of two or three joint torques. The minimal variability was observed however for the force direction only. This was not true for force magnitude suggesting that involvement of more than one joint torque could be accompanied by error compensation among joint torques with respect to the endpoint force magnitude

(Latash 2008).

Torque-torque correlations and implications for the notion of signal-dependent noise

Positive within-trial correlations between the joint torques were observed for all endpoint force directions and arm postures (Figure 5.3). This observation was also valid for the directions at which the joint torques of the opposite sign—e.g. flexion at the shoulder joint and extension at the elbow joint—were required. For such force directions, positive correlations between the joint torques correspond to negative correlations between the joint torque magnitudes, i.e. when the torque at one joint increased the torque at the other joint decreased, and vice versa.

The torque-torque correlations potentially open a window for pinpointing the source of the ‘signal depending noise‘ that has attracted large interest in the motor control literature

(Harris and Wolpert 1998).

Consider a hypothetical hierarchical structure of neural elements involved in the control of the endpoint force and torque production. If the noise were generated independently at the lower levels of such a hierarchy (e.g. at the level of groups of muscles serving individual joints), there should be no correlation between the individual joint torques during steady-state 89 force production. In contrast, if the noise were generated at the highest level—and only at that level—then the efferent signals below the highest level should be highly correlated. We observed such correlations and, hence, can claim that the noise origin is not limited to the lowest levels of the control hierarchy but includes also its higher levels.

Changes in the magnitude of the hypothetical higher-level signals can induce larger/smaller variations in (a) endpoint force and/or (b) joint torques. When the joint torques are of the same sign and correlate positively, the torques and the endpoint force increase and decrease in-phase. When they are of opposite signs, a positive correlation between the torques corresponds to a negative correlation between their magnitudes. In such a case, opposite changes in the joint torques magnitudes may be expected, e.g., an increase in one joint and a decrease in the other joint. In contrast to these predictions, the torque-torque correlations for these arm positions were positive (though the coefficients of correlations were smaller than at other arm postures). A mechanism for such unexpected changes remains unknown.

Relation to optimal control of force production

Another unexpected observation was the surprisingly high planarity of the torque distributions, with more than 99% of the total torque variance explained by the first two PCs.

To the best of our knowledge, the only example of such a high planarity in motor control literature is the so-called ‗planar covariation law‘, according to which the elevation angles of the foot, shank and thigh segments are confined to a plane in the space of these angles

(Borghese Na Fau - Bianchi, Bianchi L Fau - Lacquaniti et al. 1996). It has been suggested

90 that the planar covariation law reflects the fact that the CNS uses two independent variables in the control of three elevation angles of the limb (reviewed by Ivanenko et al 2007, though see the criticism by Hicheur et al 2006 ). The planar covariation law is also claimed to be tightly related to optimization of the energy expenses in locomotion (Bianchi, Angelini et al.

1998).

If an analogy with the planar covariation law is allowed, the joint torque covariation

(planarity) could also signify the use of two control variables instead of three. This is in line with the commonly accepted hypothesis that the control of the arm action is organized at the level of the endpoint—its position on a plane is characterized by two coordinates—and not at the level of individual joints (Li, Latash et al. 1998).

Another possible explanation of the joint torque planar distribution is that it reflects a certain optimization process, responsible for the distribution of the efforts among the joint torques. Indeed, as reported in (Terekhov et al. 2010) the planarity can arise as a result of optimization of an additive quadratic cost function subject to two linear constraints, which in this case are the X and Y components of the target force.

Additive cost functions have been broadly used in motor control studies (Nubar and

Contini 1961; Anderson and Pandy 2001). Usually the candidate cost functions were suggested (guessed) by the researchers, and the results of their application were compared to the experimental data. A new analytic inverse optimization (ANIO) method that allows reconstructing an unknown cost function from experimental observations —rather than assuming the function a priori—was developed based on an assumption that the sought cost

91 function is additive with respect to some ‗elemental‘ variables. The ANIO was successful in reconstructing cost functions from experimental data in several studies (Park, Zatsiorsky et al.

2011; Niu, Latash et al. 2012)

In contrast to those studies—and in agreement with Xu et al. (2012)—the reconstructed second-order coefficients in the present study show both positive and negative values. As it has been proven in Xu et al (2012), this result does not only exclude the possibility of the reconstructed function to explain the data, but also proves that no additive cost function at all can explain the data if the joint torques are chosen as elemental variables. In other words, the experimentally observed torque distribution pattern among three joints could not be reproduced by any cost function of the form (5.1).

Our explanation of the observed discrepancy between successful application of the

ANIO method in some research (see above) and unsuccessful results of the present study as well as in study by Xu et al. (2012) lies in the different motor tasks explored.

From the biomechanical standpoint, the motor tasks previously studied with the ANIO corresponded to parallel kinematic chains (e.g. several fingers acting in parallel) while in the present study—as well as in the study by Xu et al. 2012—the human arm was modeled as a three-link serial kinematic chain. The parallel and serial chains exhibit different properties

(see Zatsiorsky 2002, Chapter 2 for detailed discussion). When the number (n) of the elemental variables, e.g. finger forces or joint torques, exceeds the number of the task-related performance variables, N (in our example this is endpoint wrench, i.e. total force and torque exerted by the human hand), n>N, the static parallel chains are redundant while the static

92 serial chains are over-determined.

From the results of all studies published so far with the ANIO as a cost function reconstruction tool it follows that in statics such a reconstruction is successful for the parallel kinematic chains and unsuccessful for the serial chains. It seems that the optimization of a cost functions additive with respect to the joint torques is not an appropriate model for the serial kinematic chains in static conditions. One possible approach worth exploring is constructing a muscle-based model that uses muscle forces as elementary variables. Because muscles work essentially in parallel, it is possible to expect that such a model in combination with the ANIO will be successful.

5.5 Conclusions

The following main facts were observed: (a) Non-instructed non-zero grasp moments were observed for most of the combinations of arm posture and force direction. (b) The minimal intra-trial variability of force direction was observed for the target force directions along or close to the arm pointing axis, i.e. the forearm-hand longitudinal axis. (c) The time profiles of joint torques within the trials were always positively correlated, even in the tasks where flexion torque was produced at one joint and extension torque was produced at the other joint. (d) After application of the analytical inverse optimization (ANIO) method it has been concluded that experimentally observed torque distribution patterns could not be explained by any optimization cost function additive with respect to the joint torques.

CHAPTER 6. Joint torques prediction by optimization

In the first two studies, we failed to find an additive cost function with respect to the joint torques using the analytical inverse optimization (ANIO). However; the reasons were not clear since all the mathematical premises were met in the studies. These premises were explored in (Terekhov and Zatsiorsky 2011) who performed computer simulations to test some of the assumptions underlying the ANIO method. It was concluded that the cost function approximation becomes significant worse when the data lie on a curve as opposed to a surface while the experimental observations were still distributed within a hyperplane. Also, it was shown that if the number of constraints equals one then there exists in infinite number of cost functions that explain the data (there are two constrains in the studies of arm control).

Lastly, if the constraints are splittable then not all of the terms of the cost function can be identified (in the first of the above studies involving the ANIO the constraints were proven to be not splittable). Hence, the reasons for the unsuccessful performance of ANIO in the studied tasks were still not clear. In contrast as it was already mentioned the ANIO effectively reconstructed the optimal cost functions for all the studies of fingers (Terekhov et al. 2010; Terekhov, Zatsiorsky, 2011; Park et al. 2010, Niu et al. 2011, Park et al. 2011). The purpose of this chapter is to attempt identify the reason that lead to the failure in searching for cost function with respect to joint torques in the study of arm control. With this aim we performed a computer simulation of the optimization task for the endpoint force production by a human arm. The arm model was planar, it included three joints and was similar to the model used in the above experimental studies.

6.1 Optimization formulation

Analytical inverse optimization failed to find the additive quadratic cost function. The optimization is based on the assumption that an additive cost function with respect to joint torque could be found. As a result, the function should be of the form given in equation 6.1:

Minimize

Cost Function: J = ∑ k T (6.1)

Subject to constraint function:

h (T ) = CT = F (6.2) where k is the coefficients, i is equal to 3 representing the three joints. T is the 3×1 vector of the joint torques, C is the 3×2 constrain matrix, and F stands for the 2×1 target force vector.

In order to cover different combinations of joints in terms of coefficients, 4 levels of coefficients were used for each joint, giving 64 simulated cost functions. These 4 levels are

0,0,3,0.6, and 0.9. The value of 0 is available to take into account the options of cost functions in which only one or two of the three joints are optimized.

6.2 Optimization results: Simulated joint torques vs experimental joint torques

The simulated joints were identified by optimizing the virtual cost functions and then the grasp moment was computed with the information on the external force and joint torques constrains. Simulated joint torques and grasp moment computed based on those 64 cost functions, as well as the experimental joint torques, are presented in Figures 6.1, 6.2, 6.3, and

6.4.

configuration:10-120-0 configuration:20-120-0 configuration:30-120-0 sim shoulder 20 20 20 observed shoulder 15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

shoulder moment (Nm) shoulder moment (Nm) shoulder moment (Nm) shoulder -10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

configuration:40-120-0 configuration:50-120-0 configuration:60-120-0 15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

shoulder moment (Nm) shoulder moment (Nm) shoulder moment (Nm) shoulder -10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

Figure 6.1 Simulated shoulder torques (T1) based on cost functions vs. observed shoulder torques

Green curve: T1 when k2 and k3 (coefficients of the cost functions for the elbow and wrist joints) equal 0; k1 > 0; Blue curve: T1 when k1 and k3 (coefficients for the shoulder and wrist) equal 0; k2 > 0; Cyan curve: T1 when k1 and k2 (coefficients for the elbow and wrist) equal 0; k3 > 0; Red curve: T1 when k1 =0;k2=k3>0; line with open circle---○---: observed T1; line with cross---+---: simulated T1 based on the cost functions with other combinations of k1,k2 and k3.

configuration:10-120-0 configuration:20-120-0 configuration:30-120-0 sim elbow15 15 15 observed elbow 10 10 10

5 5 5

0 0 0

-5 -5 -5

elbow moment (Nm) elbow moment (Nm) elbow moment (Nm) elbow

-10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

configuration:40-120-0 configuration:50-120-0 configuration:60-120-0 15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

elbow moment (Nm) elbow moment (Nm) elbow moment (Nm) elbow

-10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

Figure 6.2 Simulated elbow torques (T2) based on cost functions vs. observed elbow torques

Green curve: T2 when k1 and k3 (coefficients of elbow and wrist) equal 0; k2 > 0;

Blue curve: T2 when k2 and k3 (coefficients of shoulder and wrist) equal 0; k1 > 0;

Cyan curve: T2 when k1 and k2 (coefficients of elbow and wrist) equal 0; k3 > 0; Red curve: T2 when k2 =0 ; k1=k3>0; line with open circle---○---: observed T1; line with cross---+---: simulated T2 based on the cost functions with other combinations of k1,k2 and k3.

configuration:10-120-0 configuration:20-120-0 configuration:30-120-0 15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

wrist moment (Nm) wrist moment (Nm) wrist moment (Nm)

-10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

configuration:40-120-0 configuration:50-120-0 configuration:60-120-0 15 15 15 sim wrist 10 10 10 observed wrist

5 5 5

0 0 0

-5 -5 -5

wrist moment (Nm) wrist moment (Nm) wrist moment (Nm)

-10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

Figure 6.3 Simulated wrist torques (T3) based on cost functions vs. observed wrist torques.

Green curve: T3 when k1 and k2 (coefficients of elbow and wrist) equal 0; k3 > 0;

Blue curve: T3 when k2 and k3 (coefficients of shoulder and wrist) equal 0; k1 > 0; Cyan curve: T3 when k1 and k3 (coefficients of elbow and wrist) equal 0; k2 > 0; Red curve: T3 when k3 =0 ; k1=k2>0; line with open circle---○---: observed T1; line with cross---+---: simulated T3 based on the cost functions with other combinations of k1,k2 and k3.

configuration:10-120-0 configuration:20-120-0 configuration:30-120-0 15 15 15

10 10 10

5 5 5

0 0 0

-5 -5 -5

Mz (Nm) Mz (Nm) Mz (Nm)

-10 -10 -10

-15 -15 -15

-20 -20 -20 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

configuration:40-120-0 configuration:50-120-0 configuration:60-120-0 15 15 15 sim Mz 10 10 10 observed Mz

5 5 5

0 0 0

Mz (Nm) Mz (Nm) Mz (Nm) -5 -5 -5

-10 -10 -10

-15 -15 -15 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 direction (degrees) direction (degrees) direction (degrees)

Figure 6.4 Simulated grasp moment (Mz) based on cost function vs. observed grasp moment

Green curve: Mz when k1 and k2 (coefficient of shoulder and elbow) equal 0;k3 is positive Blue curve: Mz when k1 and k3 (coefficients of shoulder and wrist) equal 0;k2 is positive; Cyan curve: Mz when k2 and k3 (coefficients of elbow and wrist) equal 0;k1 is positive; Red curve: Mz when k1=k2=k3 > 0; line with open circle---○---: observed

T1; line with cross---+---: simulated Mz based on the cost functions with other combinations of k1,k2 and k3.

In Figures 6.1-6.4, areas surrounded by green, blue and cyan curves were all the possible solutions if an available cost function based on joint torques were followed. The results indicated that all three joint torques could be more or less reduced to certain extent by following some of these cost functions. However, as seen in figure 6.4, the observed grasp (a line with open circles) is much smaller than any of the simulated grasp moments. Therefore none of the studied cost functions would lead to decrease in grasp moment. Actually, much

99 greater grasp moment would be expected (more than 500%) if any of the studied optimal cost functions is applied. As a result, grasp moment capability is suspected to be the key factor preventing an optimal joint torque distribution pattern, which ultimately prevented the application of ANIO in arm control tasks. Thus, the future studies should address the effects of grasp moment on the performance of isometric force production by arms.

6.3 Conclusions

Observed values of the grasp moments were much smaller than the values predicted by the computer simulation based on the assumption that the force and grasp moment production is a consequence of using an additive cost function of the joint torques. It seems that the CNS is not trying to minimize a weighted sum of the joint torques whatever the weighting coefficients might be.

100

CHAPTER 7. Effects of the handle size on force and moments generated by the human arm

7.1 Introduction

The hand-handle interaction may serve as an extra factor influencing the production of endpoint force. For this reason, for current study, magnitude of the moment arm of the tangential force exerted at the handle-palm interface (the moment arm of the grasp moment) as well as the finger configurations was varied. This was achieved by using cylindrical handles of different size (diameter).

Size of cylindrical handle was reported to influence the MVC grip force, MVC grasp moment, contact area between hand and handle and EMG of wrist muscles (Ayoub and Presti

1971; Blackwell, Kornatz et al. 1999; Seo, Armstrong et al. 2007). It has been reported that the MVC grasp moment increases with the handle size significantly at the handle diameters smaller than 5 cm, while no significant differences are found when the size is larger than about 5.0cm. Also, when the handle size is larger than 5.0 cm in diameter, contact area between the hand and handle does not change significantly (Pheasant and O'Neill 1975).

We hypothesize that an increase in the diameter of a circular handle will lead to: (a) an increase in both the MVC moment and uninstructed grasp moment because of the increased moment arm, (b) change of force MVC and variability; the way of change will be explored, (c) ANIO will start to work on handles of larger size, because of the improved capability of grasp moment. Also, ANIO will still fail to work on smaller handles (3.0 cm and

4.5 cm in diameter).

101

7.2 Methods

Subjects

Eight male right-hand dominant subjects took part in the study (mean age 26.1 4.0 years, mass 70.9  8.3 kg, height 1.74  0.08 m, shoulder to elbow length (upper arm) 28.0 

1.4 cm, elbow to wrist length (forearm) 25.4  1.0 cm, wrist to the centre of the handle distance 7.5  0.5 cm. All subjects were healthy, with no known history of neurological or peripheral disorders and injuries. All the subjects gave informed consent according to the policies of the Office for Research Protections at the Pennsylvania State University.

Equipment

The apparatus used in the experiment included five aluminium cylindrical handles

(diameter 3.0, 4.5, 6.0, 7.5 and 9.0 cm, surface friction coefficient 1.08±0.12 , height 16 cm,) that could be attached to a 6-DOF force/torque sensor (Mini85, ATI, Apex, NC USA). Such an arrangement allows recording both the 2-D force vector (resolution about 0.048 N) and the moment exerted on the handle, the grasp moment (resolution 1/600 Nm). A detailed description of the apparatus can be found in Chapter 3 and Xu et al. (2012).

Subjects sat in a chair, grasped the handle and generated forces in eight directions and

MVC grasp moment in two directions. The trunk was secured to the chair with seat belts. The right arm was in a horizontal plane at the shoulder height, with the upper arm flexed at 20°,

102 the elbow flexed at 120º and the wrist at 0º. The handle was secured at a position corresponding to the arm posture. Hence, the aperture, estimated as the line between the fingers and thumb tips, and defined here as grasp aperture, with this arm configuration was at the angle of about 180º from the frontal plane of the body.

To measure the static friction coefficient between the skin and handle surface, we asked participants to push the handle to the left direction, with only palm, at a certain magnitude of force and then slip the palm upward as slowly as possible. We detected slips as a sudden decrease in the vertical force. We determined the ratio between the force in the left direction

(normal force) and vertical force at slip (slip ratio; Johansson & Westling, 1984b). We used the inverse of the slip ratio as an estimate of the static friction coefficient (μ).

Experimental procedure

In the preliminary trials, at each condition, the maximal voluntary contraction (MVC) forces were recorded under the instruction ‗push the handle as strongly as you can‘. The duration of each trial was 5 s. In a separate test, the subjects were asked to produce the maximal grasp moment on the handle, ‗try to turn the handle as strongly as you can‘. The

MVC data were used for the normalization of the prescribed forces in the main experiment.

103 feedback on the horizontal force vector exerted on the handle. No instruction or feedback on the grasp moment was provided. A total of 108 trials were performed to cover the combinations of six conditions with 18 trials per condition; 30-s breaks were provided between the trials and a 2-min break between blocks of trials with different conditions.

The six conditions differed by handle diameter: 1) 3.0 cm, 2) 4.5 cm, 3) 4.5 in diameter with the hand bandaged onto the handle, 4) 6.0 cm, 5) 7.5 cm and 6) 9.0 cm. In condition 3, a bandage was used to bind the hand onto the handle to provide more pulling force in the direction opposite to the grasp aperture and more normal force against the handle surface.

Data analysis

The force signals were conditioned (9105 IFPS-1, ATI, Apex, NC USA) and digitized using a 16-bit A/D converter (PCI-6225, National Instruments, Ausin, TX) at 1000 Hz. The data were collected using a custom program written in LabVIEW (National Instruments). The data analysis was performed using Matlab (The MathWorks, Natick, MA). The joint torques were computed from the recorded values of the endpoint force and moment by employing the equations of statics (described in Chapter 3) .

MVC grasp moment or force exertion. The preferred direction of the force or grip moment production in the group was estimated with the sign test. For 8 subjects the significance level

104 p<0.5 was attained when at least 7 subjects exerted a force directed in a quadrant of space or moment in the same direction (p=0.035).

Two-way repeated measures ANOVAs were performed to test the effect of handle size and force direction independently on the generated force and moment. Significance level was also set at p≤0.5.

Inverse optimization (ANIO) analysis.

We employed ANIO analysis to assess the possibility that the subjects selected the joint torques in order to minimize a total cost G, which we assumed to be an additive function of the joint torques, i.e.

G (T1, T2, T3) = g1(T1) + g2(T2) + g3(T3) (7.1)

where g1, g2, g3 are unknown functions of the shoulder, elbow and wrist torques, respectively. The ANIO method was used for each condition. The analysis was performed according to the procedures described earlier.

gi(Ti) = ki (Ti)2 + wiTi, i = 1,2,3. (7.2) 105

For G to be a valid cost function, all ki must be positive. Otherwise, the polynomial does not have a global minimum. It was shown in (Xu et al 2012, Appendix) that, if at least one of the ki coefficients is negative, then no cost function of form (7.1) can explain the choice of the joint torques.

7.3 Results

MVC Force and Grasp Moment

Figure 7.1 Mean values of normalized MVC in 6 conditions

To determine the effects of Conditions and Direction on MVC, the repeated measures

ANOVAs were performed with Bonferroni pairwise comparisions. MVCs were affected by

106 the factor of Direction (p<0.01) and Conditions (P<0.05). Across the force directions the

MVC force distribution followed approximately ellipse-like pattern (Figure 7.1) with the maximal values at 135 and 315 degrees and minimal values at 45 and 225 degrees. The distribution did not depend on the handle size. Further, condition 1 exhibited significant higher value of MVC than Condition 6. No significant effect of interaction was found

(p=0.129). A follow-up paired t test found difference of MVCs across conditions at directions of 0º, 45º and 315º(p<0.01), which are at about 180º from the orientation of grasp aperture (at about 180º). Smaller MVC forces were produced on the larger handles

(condition 5 and condition 6) than on the smaller handles (condition 1 and condition 2, p<0.05). Significantly greater MVC force was produced in condition 3 than in condition 1 at direction of 45º. No differences were found in the rest of paired comparisons.

The data on the maximal grasp moments (MVC conditions) are presented in Figure 7.2.

107

mean normalized max grasp moment 1.5

0.5

-0.5

-1

-1.5 - - negative max normalized mean moment grasp positive + condition 1 2 3 4 5 6

Figure 7.2 Maximal grasp moments. Group average data and standard errors.

A repeated measures ANOVA was performed separately across conditions with post hoc analysis. Results indicated that MVC grasp moment in condition 1 was significantly smaller than in other 5 conditions (p<0.01). No significant differences were found among those 5 conditions. Hence we may conclude that changing the handle diameters in the range between

4.5 and 9.0 cm does not affect the maximal torque production.

Enslaved grasp moment production during the arm force production

Maximal forces. When the subjects exerted forces on the handle they also generated

108 grasp moments, i.e. the turning efforts on the handle. We will call these involuntarily produced moments the ‗enslaved moments‘. Both the enslaved moment direction and magnitude depended on the arm force direction (Figure 7.3). These findings agree well with those previously reported (Xu, Terekhov et al. 2012), see also Chapter 4.

condition :1 condition :2 condition :3

1 Nm 1 Nm 1 Nm

condition :4 condition :5 condition :6

1 Nm 1 Nm 1 Nm

Figure 7.3 ‗Enslaved‘ grasp moments at MVC

During the maximal arm force production, the enslaved grasp moment in condition 1 was smaller than in other conditions (p<0.01). From condition 2 to condition 6 magnitudes of the grasp moment at MVC stayed unchanged or increased with increasing handle size.

Magnitude of grasp moment in condition 6 was larger, in the directions of 45º,135º,275º and 315º, than in condition 2. Directions of the enslaved grasp moment at MVC did not change with the handle size. When comparing condition 2 with condition 3, the only

109 difference was found in the direction of 275º, where condition 3 demonstrated a greater magnitude of the grasp moment (p<0.05). Effects of handle size on the grasp moment production on the whole were similar for the maximal grasp moments and the enslaved moments during the maximal arm force production: the moments were smaller for the handle diameter of 3.0 cm and showed only a small tendency to increase or stayed for the diameters from 4.5 cm to 9.0 cm.

We conclude that —in the contrast to moment magnitude— the pattern of the enslaved grasp moment production during the maximal force production does not depend on the handle diameter.

Forces of 30% of the MVC. Magnitude of the grasp moment at MVC increased with handle size at 30% MVC (Figure 7.4). Directions of grasp moment were more consistent across 8 subjects for larger handles than for smaller ones. In addition, the bandage increases the magnitude of grasp moment for all force directions except 0º and 180º (p<0.05).

1.5 condition 1 condition 2 condition 3 1 condition 4 condition 5 condition 6 0.5

Average grasp moment grasp Average (Nm) -0.5

-1 0 45 90 135 180 225 270 315 Direction (degree)

Figure 7.4 Uninstructed (enslaved) grasp moment at 30% MVC across 6 conditions

110

Uninstructed (enslaved) force during maximal grasp moment production

When subjects exerted maximal turning efforts on the handles they also generated uninstructed (enslaved) force on them. The force direction depended on the grasp moment direction, clockwise or counterclockwise (Figures 7.5 and 7.6).

condition :1 condition :2 condition :3 20 40 60 sub 1 sub 2 15 sub 3 30 40 sub 4 10 sub 5 20 sub 6 5 20 sub 7 0 sub 8 10 -5 0 0 -10

-15 -10 -20 -50 -40 -30 -20 -10 0 10 -150 -100 -50 0 50 -150 -100 -50 0 50

condition :4 condition :5 condition :6 70 60 100

60 50 80 50 40 60 40 30 30 40 20 20 20 10 10

0 0 0 -150 -100 -50 0 50 -150 -100 -50 0 50 -150 -100 -50 0 50

Figure 7.5 Forces exerted on the handles during the maximal grasp moment production (positive, counter-clockwise).

condition :1 condition :2 condition :3 sub 1 20 20 20 sub 2 15 sub 3 0 0 sub 4 10 sub 5 -20 -20 sub 6 5 sub 7 0 sub 8 -40 -40 -5 -60 -60 -10

-15 -80 -80 -60 -50 -40 -30 -20 -10 0 -100 -80 -60 -40 -20 0 20 -150 -100 -50 0 50

condition :4 condition :5 condition :6 0 0 0

-20 -20 -20

-40 -40 -40

-60 -60 -60

-80 -80 -80

-100 -100 -100 -100 -80 -60 -40 -20 0 -80 -60 -40 -20 0 -80 -60 -40 -20 0

Figure 7.6 Forces exerted on the handles during the maximal grasp moment production (negative, clockwise).

111

7 out of 8 subjects were producing force at a quadrant of space – within an angular range of 90º– in all conditions except condition 3. Subjects tend to produce –Fx and +Fy force when exerting maximal positive grasp moment and –Fx and –Fy forces when exerting negative moments in conditions 2, 4, 5 and 6.

Also, resultant force directions at maximal grasp moment could be consistent across handle sizes for a specific subject. For example, subject 5 was always producing a small magnitude of +Fx & +Fy forces while others tended to produce –Fx&+Fy forces.

Variability of force magnitude and direction

condition90 2:1 condition90 2:2 condition90 2:3 120 60 120 60 120 60

150 1 30 150 1 30 150 1 30

180 0 180 0 180 0

210 330 210 330 210 330

240 300 240 300 240 300 270 270 270

condition90 2:4 condition90 2:5 condition90 2:6 120 60 120 60 120 60

150 1 30 150 1 30 150 1 30

180 0 180 0 180 0

210 330 210 330 210 330

240 300 240 300 240 300 270 270 270

Figure 7.7 Variability of force direction across 6 conditions

Across the 8 directions, variability of force direction changed in an ellipse-like way

112

(Figure 7.7). The ellipse orientation was approximately at 90 degrees to the orientation of the

MVC force magnitude ellipse (cf. Figure 7.1).

Across the handle sizes, force magnitude variability, calculated as the coefficients of variations of force magnitude, did not show significant differences across conditions. The variability of force direction in condition 6 was significant larger than in conditions 2 ,3, 4, and 5 (p<0.05). Similarly, condition 5 also showed a greater variability of force direction than condition 2 (p<0.05). No differences were found between other conditions and at other directions. Hence, there exists a slight tendency for increasing the force direction variability for large (>7.5 cm) handles.

The torques data planarity and cost function analysis (ANIO)

In the 3D joint-torque space, the first two principal components (PCs) accounted for over

99% of the joint torque variance for both normalized and non-normalized joint torques. The planarity of the data justifies the use of second-order polynomials as cost function candidates

(for details see Terekhov et al 2010 and Xu et al 2012 Appendix). The second-order coefficients are provided in Table 7.1. Coefficient for k1 in condition 6 is always smaller than that in condition 1 and 2 for each subject. Coefficients of different sign for all conditions were found. We have to conclude that the estimated functions do not have a single minimum and hence cannot serve as optimization cost functions.

Table 7.1 ANIO coefficients for the shoulder, elbow and wrist torques (k1,k2 and

113 k3) across 8 subjects

cond. 1 cond. 2 cond. 3 cond. 4 cond. 5 cond. 6

subject1 k1 -0.0291 -0.0290 -0.0290 -0.0314 -0.0214 -0.0009

k2 -0.1820 -0.1901 -0.1720 -0.1791 -0.1727 -0.1809

k3 0.9829 0.9813 0.9847 0.9833 0.9847 0.9835

subject2 k1 -0.0165 -0.0283 -0.0041 -0.0055 0.0068 0.0202

k2 -0.1668 -0.1663 -0.1705 -0.1838 -0.1865 -0.1966

k3 0.9859 0.9857 0.9853 0.9829 0.9824 0.9803

subject3 k1 -0.0170 -0.0281 -0.0278 -0.0185 0.0064 0.0377

k2 -0.1763 -0.1755 -0.1367 -0.1796 -0.1963 -0.2056

k3 0.9842 0.9841 0.9902 0.9836 0.9805 0.9779

subject4 k1 -0.0035 -0.0200 0.0021 -0.0170 0.0048 0.0390

k2 -0.1857 -0.1837 -0.1350 -0.1654 -0.1650 -0.1844

k3 0.9826 0.9828 0.9908 0.9861 0.9863 0.9821

subject5 k1 -0.0077 0.0050 -0.0165 -0.0022 -0.0195 0.0104

k2 -0.2051 -0.2087 -0.1998 -0.2333 -0.2280 -0.2067

k3 0.9787 0.9780 0.9797 0.9724 0.9735 0.9783

subject6 k1 -0.0174 -0.0248 -0.0406 -0.0115 -0.0250 -0.0085

k2 -0.2263 -0.2247 -0.1966 -0.2273 -0.2157 -0.2303

k3 0.9739 0.9741 0.9797 0.9738 0.9761 0.9731

subject7 k1 -0.0066 -0.0120 0.0036 -0.0170 -0.0093 0.0171

k2 -0.1750 -0.1819 -0.1716 -0.1758 -0.1684 -0.2161

k3 0.9845 0.9832 0.9852 0.9843 0.9857 0.9762

subject8 k1 0.0202 -0.0078 0.0328 0.0032 0.0337 0.0416

k2 -0.1785 -0.1820 -0.1978 -0.1965 -0.2005 -0.2287

k3 0.9837 0.9833 0.9797 0.9805 0.9791 0.9726

average k k1 -0.0097 -0.0181 -0.0099 -0.0125 -0.0029 0.0196

k2 -0.1870 -0.1891 -0.1725 -0.1926 -0.1916 -0.2062

k3 0.9820 0.9816 0.9844 0.9809 0.9810 0.9780

std of k k1 0.0145 0.0122 0.0235 0.0109 0.0197 0.0189

k2 0.0194 0.0189 0.0258 0.0249 0.0228 0.0185

k3 0.0039 0.0038 0.0045 0.0051 0.0046 0.0039

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7.4 Discussion

The present study aimed at answering the question of whether the handle diameter, and hence lever length of grasp moment, would affect the arm force and grasp moment production. We formulated three hypotheses: increasing handle size will lead to (1) an increase in both MVC and non-instructed grasp moment because of the increasing moment arm, (2) change of force variability in certain directions due to varying configurations of fingers, (3) change of coefficients in the inverse optimization analysis (ANIO). In the experiments, the first of the three hypotheses could not be confirmed, while the other two received a support.

MVC force and grasp moment

The significantly smaller magnitudes of MVC force in directions of 0º, 45º and 315º reflect that handle size does impact the production of end-point force, particularly in the direction opposite to the hand aperture, i.e. to the gap between the thumb and fingers tips. At the same time, MVC force was less influenced in the direction opposite to the palm, where fingers were less involved in the end-point force production.

Both the MVC grasp moment and enslaved grasp moments exhibited a significant increase from the smallest handle, 3.0 cm in diameter, to larger handles. However no systematic regularities across handles larger than 4.5 cm in diameter were observed. This result agrees with those reported previously (Pheasant and O'Neill 1975; Seo, Armstrong et al.

2007). Preceding studies have suggested that the MVC grasp moment on a handle is related to the handle surface friction, cross-section area of the handle and area of contact (Pheasant

115 and O'Neill 1975; Imrhan and Farahmand 1999; Seo, Armstrong et al. 2007; Yoxall and

Janson 2008). Cross-section area, in this study, increases with handle sizes; area of contact remain almost the same from 4.5 cm to 9.0 cm since palm and fingers are in full contact with handle when grasp aperture, i.e. the gap between the thumb and fingers tips, emerged.

It is possible that the friction between the hand and handle surface limited production of grasp moment. A relatively low friction coefficient makes the MVC tangential forces depend on the MVC grasp forces, since MVC tangential forces equal the product of friction coefficient and grasp force when it was reaching the slipping threshold. Hence, the MVC of grasp force will determine the performance of MVC grasp moment. According to studies of grasp force, handle size has an optimum diameter between 50 and 60 mm (Oh and Radwin

1993; Blackwell, Kornatz et al. 1999). The decrease in grasp force at larger handle sizes might contribute to the non-significant changes in MVC grasp moment from handle size 4.5 cm to 9.0 cm. A study on grip force distributions (Seo, Armstrong et al. 2007) has suggested that a small handle allows both fingertip and thumb forces to act against the palm, resulting in a high reaction force on the palm, and, therefore, a high grip force. On a larger handle, fingertip force and thumb force act against each other, resulting in lower reaction force against the palm and, thus, a lower grip force.

Variability of force magnitude and direction

Variability of force magnitude and direction were analysed and compared across direction and conditions. The results agree well with previous findings (described in Chapter

5), that minimal force direction variability (‗the highest precision of force exertion‘) was

116 observed for the target force directions that were along or close to the forearm-hand axis.

For the force direction of 45º, MVC force decreased for larger handles, while the variability of force direction increased. In contrast, for the force directions of 0º and 315º, a decrease in the MVC force magnitude comes with no significant change in variability of force direction. On the whole, these data agree well with the previously reported findings

(Friedman, Latash et al. 2011; Xu, Terekhov et al. 2012) that the variability of force magnitude (which is a well-studied phenomenon) and variability of force direction (where studies are limited) are manifested in different ways. This finding suggests existence of different mechanisms controlling the force magnitude and force direction variability.

ANIO and causes for non-instructed grasp moment

Grasp moment was the only non-instructed variable in the task formulation. However, in spite of the absence of an instruction regarding the grasp moment, the subjects showed a regular pattern of grasp moments across the eight directions. To investigate whether this regular pattern could be explained by employing an optimal cost function—an unknown additive function of the joint torques—inverse optimization analysis (ANIO) was applied. As the distribution of the three joint torques was planar it was concluded that the function, if exists, should be a quadratic polynomial of joint torques. Such a function can get a global minimum only if all its second order coefficients are of the same sign, positive or negative.

Hence, the focal point in the analysis was to check whether all coefficients of the second-order terms were of the same sign.

117

In agreement with the data from our previous experiments (Xu et al. 2012, 2013), the reconstructed functions had second-order coefficients with both positive and negative values, which proves that no additive cost function can explain the data if the joint torques are chosen as elemental variables.

An additional result is that coefficient, k1, is consistently larger in condition 6 than in conditions 1 and 2. Condition 6 used a larger handle size than conditions 1 and 2, as well as significantly larger magnitudes of grasp moment. In previous simulation results, it was indicated that grasp moment served as the factor limiting the application of optimization in arm force production (Chapter 6). Hence, the improvement of the grasp moment capability, due to the increased handle size, might lead to a changed orientation of the plane formed by data observations in joint-torque space. Consequently, the coefficients changed in a regular way. The predicted grasp moments, although much larger in magnitude, were always in the same direction as the experimentally recorded grasp moments. This might suggest that the grasp moments— being limited in magnitude—were produced actively in order to approach the optimum.

Another interpretation of non-instructed grasp moments comes from the observations presented in Figure 7.4: The magnitudes of the grasp moment increased with the handle size while the direction of grasp moment kept unchanged. Significant differences in the magnitude of grasp moment were found between small (3.0 cm and 4.5 cm) and large size

(6.0cm,7.5cm and 9.0cm) in all directions except 0ºand 180º(p<0.05), where magnitudes of grasp moment were small across all conditions . Nevertheless, after being normalized by the

118 handle diameter, non-instructed grasp moments differ only in the direction of 225º(p<0.05).

Hence, the same level of tangential forces was produced on the handle regardless of its size.

It would be hard to maintain the same level of submaximal tangential forces across handle conditions if the force is produced actively, or if the force level is not limited by some constraints. Such a constraint can arise from the tendency to minimize the level of muscle activity. A previous study indicated that EMG amplitude of wrist muscles is lower if grasp moment production is allowed when producing a prescribed level of grip force (Seo,

Armstrong et al. 2007). This suggests that non-instructed grasp moments were produced to minimize the level of muscle activity.

A comparable phenomenon, windlass mechanism in lower extremities, works in a similar way. The plantar fascia simulates a cable attached to the calcaneus and the metatarsophalangeal joints. Dorsiflexion during the propulsive phase of gait winds the plantar fascia around the head of the metatarsal. This winding of the plantar fascia shortens plantar fascia and enhances its tension (Kwong, Kay et al. 1988; Fuller 2000; Bolgla and

Malone 2004; Erdemir and Piazza 2004). We suggest an analogous interpretation of producing a non-instructed grasp moment: performers tend to decrease their muscle activity, utilizing more of the passive components within muscles, by winding hands around the wrist joint. This tendency is restrained by the friction between hand and handle, which results in the grasp moment. In other words, such a strategy, to reduce muscle activity, should lead to a regular pattern of grasp moment production.

7.5 Conclusion

119

The following main facts were observed: (a) There existed effects of handle size on the magnitude, but not on the distribution pattern of MVC end-point force. (b) Changing the handle diameters in the range between 4.5 and 9.0 cm does not affect the maximal torque production. (c) The enslaved moments were smaller for the handle diameter of 3.0 cm and showed only a small tendency to increase or stayed put for the diameters from 4.5 cm to 9.0 cm. (d) There was a slight tendency for increasing the force direction variability for large

(>7.5 cm) handles while force magnitude variability, calculated as the coefficients of variations of force magnitude, did not show significant differences across conditions. (e)

Although systematic change in the coefficients of results of ANIO was observed, it still failed to reconstruct an optimal cost function additive with respect to joint torques with large handle size.

120

CHAPTER 8 Effects of surface friction of handle on force and moments generated by the human arm

8.1 Introduction

Friction at the hand-object interface was reported to affect the tangential component of grip force; the ability for workers to create torque on a handle is limited by handle surface friction (Pheasant and O'Neill 1975; Imrhan and Farahmand 1999; Yoxall and Janson 2008).

The grip force normal to surface also increases in lower-friction conditions (Johansson and

Westling 1984). Also, the maximum force that can be exerted on an object before it is pulled or slips from the grasp of the hand (―breakaway strength‖) was found to be higher when the surface of handhold object was less slippery (Young, Woolley et al. 2009).

The aim of this study is to obtain insights into the effect of surface friction between the hand and the handle on the endpoint force and grasp moment. We hypothesize that exerting isometric force on a higher friction handle will result in: (a) an increase in both MVC force and maximal grasp torque; (b) lower variability in force direction and magnitude; and (c) The change in enslaved grasp moment. At higher friction conditions, the same level of grip force is expected to lead to higher enslaved grasp moment while a decrease in grip force may result in no change in enslaved grasp moment.

8.2 Methods

Subjects:

Eight male right-hand dominant subjects were recruited in the study (mean age 25.3

2.6 years, mass 67.5  4.6kg, height 1.73  0.06 m, shoulder to elbow length (upper arm)

121

27.9  1.5 cm, elbow to wrist length (forearm) 25.5  1.0 cm, wrist to the centre of the handle distance 7.4  0.5 cm. All subjects were healthy, with no known history of neurological or peripheral disorders and injuries. All the subjects gave informed consent according to the policies of the Office for Research Protections at the Pennsylvania State

University.

Apparatus:

The apparatus used in the experiment included two cylindrical handles which are of same size but different surface friction (diameter 6.0 cm, height 16 cm, surface friction coefficient 0.97±0.11 for the low friction handle and 1.45±0.09 for the high friction one) that could be attached to a 6-DOF force/torque sensor (Mini85, ATI, Apex, NC USA). Such an arrangement allows recording both the 2-D force vector (resolution about 0.048 N) and the moment exerted on the handle, the grasp moment (resolution 1/600 Nm). A detailed description of the apparatus can be found in Xu et al. (2012).

Subjects sat in a chair, grasped the handle and generated forces in eight directions and

MVC grasp moment in two directions. The trunk was secured to the chair with seat belts. The right arm was in a horizontal plane at the shoulder height, with the upper arm flexed at 45°, the elbow flexed at 80º and the wrist at 0º. The handle was secured at a position corresponding to the arm posture. Hence, the aperture, estimated as the line between the fingers and thumb tips, and defined here as grasp aperture, with this arm configuration was at the angle of about 180º from the frontal plane of the body.

122

To measure the static friction coefficient between the skin and handle surfaces, we asked participants to push the handle to the left direction, with only the palm, at a certain magnitude of force and then slip the palm upward as slowly as possible. We detected slips as a sudden decrease in the vertical force. We determined the ratio between the force in the left direction

(normal force) and vertical force at slip (slip ratio; Johansson & Westling, 1984b). We used the inverse of the slip ratio as an estimate of the static friction coefficient (μ).

Experimental procedure.

In the preliminary trials, the maximal voluntary contraction (MVC) forces were recorded on both low- and high-friction handles under the instruction ‗push the handle as strong as you can‘. The duration of each trial was 5 s. The subjects were also asked to produce the maximal grasp moment on the handle under the instruction ‗try to turn the handle as strong as you can‘.

The MVC data were then used for the normalization of the prescribed forces in the main experiment.

After recording the MVC trials on both handles, in the main experiment, the subjects were asked to produce 30% of the mean value of MVC forces on both low- and high-friction handles in the eight directions, keeping the force magnitude and direction as stable as they could for 15 s. The mean value of MVC forces on both handles, rather than corresponding

MVC forces on each condition, was applied in order to exclude the influence of different absolute force levels, and thus to better identify the differences between conditions. The force directions were pseudo-randomized to prevent similar or same directions presented in succession. The monitor, placed approximately 80 cm in front of the subject, provided visual

123 feedback on the horizontal force vector exerted on the handle. No instruction or feedback on the grasp moment was provided. A total of 36 trials were performed to cover the combinations of two conditions with 18 trials per condition; 30-s breaks were provided between the trials and a 2-min break between blocks of trials with different conditions.

Data analysis.

MVC grasp moment or force exertion. All descriptive statistics are displayed in the text and figures as mean and standard errors unless stated otherwise. Two-way repeated-measures

ANOVA were used to test effect of Direction and Surface friction on performances of MVC forces, variability of endpoint force, and the enslaved grasp moments. Paired t-test was performed to detect the differences of MVC grasp between conditions. The significance was set at p=0.05 for are statistical tests, which were performed with SPSS 20.0 (IBM

Corporation) and Matlab (Mathworks Inc, MA,USA).

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8.3 Results

MVC forces and grasp moment

The MVC forces exhibited systematic change across 8 directions (Fig. 8.1). No significant differences were discovered between low- and high-friction handles in repeated measures ANOVA (p>0.05). Also, comparisons between the two handles by paired-t test at each specific direction also led to no significant differences. Additionally, the MVC forces at directions of 135º(131±25.0 N), 270º(138.2±33.7 N) and 315º(118.7±32.8) were higher than those at the directions of 45º(74.0±25.1 N), 225º and 360º (81.6±21.8, p<0.05).

180 low friction 160 high friction

140

120

100

80 MVC forces 60

0 45 90 135 180 225 270 315 360

Figure 8.1 MVC forces of low- and high-friction conditions across 8 directions of force production (N). Means ± SEM are shown.

125

In the second session of MVC collection trials, subjects were asked to produce maximal grasp moment on both handles. The average values as well as the standard deviation across subjects are displaced in Fig. 8.2. According to the repeated measures ANOVAs, MVC grasp moments on high-friction handle (9.85±2.37 counter-clockwise and 9.86±1.12 clockwise) were consistently greater compared to those on low-friction (9.29±2.52 counter-clockwise and 8.76±2.12 clockwise) handle. No significant difference between counter clockwise and clockwise directions was identified with both repeated measures ANOVAs.

14 low friction high friction 12

4 magnitude of magnitude MVC torque grasp

0 counter-clockwise clockwise

Figure 8.2 Averaged grasp moment for two directions in two friction conditions (Nm). Means ± SEM are shown.

Variability of force magnitude and direction

126

low friction high friction

variability of force magnitude variability of force direction 90 1 90 2 120 60 120 60

150 0.5 30 150 1 30

180 0 180 0

210 330 210 330

240 300 240 300 270 270

Figure 8.3 Variability(RMS) of force magnitude (N) and direction across 8 directions of force production (degree) for the two handles.

For this specific arm configuration (45º-80º-0º), significant differences across directions were found in both force magnitude (p<0.01) and direction variability (RMS, p<0.01). Across the 8 target force directions, opposite trends were observed between the variability of the exerted force magnitude and force direction: In the directions where the variability of force magnitude was maximal (90°, 120°, 270°, and 315°), the directional variability was minimal, and vice versa (Fig. 8.3).

Repeated measures ANOVAs revealed significant differences between low- and high-friction conditions in force direction variability (p<0.01). Further, paired t-tests found greater force direction variability occurred at direction 225° for the low-friction handle (0.71

±0.13 vs 0.52±0.12, p<0.01). For force magnitude variability, no significant differences 127 were discovered between friction conditions.

Enslaved grasp moment

2 low friction high friction 1.5

0.5

0 enslaved grasp enslaved moment grasp

-0.5

-1 45 90 135 180 225 270 315 360

Figure 8.4 Enslaved grasp moment for the two friction conditions across 8 directions of force production (Nm). Means ± SEM are shown.

The distribution of the enslaved grasp moment across directions showed a regular pattern

(Fig 9.4). On average, subjects produce positive (counter clock-wise) grasp moment at directions about 90° and negative (clock-wise) grasp moment at the directions about 270° despite the relatively high inter-individual differences (standard deviation) across subjects.

No significant difference in enslaved grasp moment between the low- and high- friction

128 condition (p=0.41) was found by the two-way repeated measures ANOVA, however a significant interaction between friction and direction was indicated (p<0.05). To further detect the difference between friction conditions, the paired-t test was performed in each direction. The results showed that the magnitude of enslaved grasp moments were greater on high-friction handle at the direction of 225° (p<0.05). For the other directions, no significant differences were found (all p values > 0.23)

8.4 Discussion

This study was trying to characterize the performance of endpoint force and moment production by arm under different handle friction conditions. Three hypotheses were proposed. Under high friction conditions we expected to find: (a) an increase in both MVC force and maximal grasp torque (b) lower variability in force direction and magnitude; and (c) the change (increase) in the enslaved grasp moment. All three hypotheses were partially confirmed, and the details will be discussed.

MVC forces and grasp moment

Overall, the MVC force pattern was similar to that in earlier studies. However, the finding that no significant difference exists between frictions conditions does not agree with the previous findings (Young et al. 2009). Three major differences in the measurement might lead to this discrepancy in the results. 1) The strength measured in the study of 2009 was breakaway strength, which is defined as maximal force that can be exerted on object before it

129 is pulled or slips from the grasp of the hand. Body weight of the subjects was used to produce the breakaway strength. In contrast, in the current study, the MVC forces was self-produced actively and did not reach the threshold of breakaway; 2) The study of 2009 used a configuration where the shoulder and elbow joints were placed in full overhead extension.

The ligaments and stabilizing tissues, other than hands, all can bear the traction strength. The hand was the only link that contributed to the breakaway strength; 3) The handle size was much smaller and the friction coefficient was not specified in the earlier study. In the current study, handle size was as large as 6.0 cm in diameter and even low-friction handle used a frosted surface, which could provide greater friction than polished surface.

The first hypothesis has been partially confirmed that increasing the surface friction significantly improves subjects‘ ability to produce grasp moment. Friction could limit the grasp moment production in study of Chapter 7 and results in relatively equal maximal grasp moment when handle size was larger than 4.5 cm.

The ratio of high- to low- friction coefficient was about 1.5 while the ratio of maximal grasp moment on high- to that on low-friction handle was only about 1.1, which implies that the friction might no longer be the bottleneck factor for generating greater grasp moment on high-friction handle. Other links in the serial chains, such as fingers or wrist, and their flexion capability might began to prevent a higher grasp moment production. Thus, our hypothesis from the perspective of ergonomics is that for a cylindrical handle of any size, a surface friction threshold affects the grasp moment production. For a handle with a friction coefficient above this threshold, the maximal grasp moment will be subject to other factors.

130

Based on the results of the previous and this study, the threshold friction coefficient is expected to be within the range from 1.1 to 1.4 if the handle size is within the range from 4.5 to 9.0 in diameter.

Measuring the maximal grip force exerted on a circular handle is not standardized and can be performed in different ways. The problem is that the local forces exerted on force sensors, e.g. on the elements of a pressure mat, are directed in variuos directions and can cancel each other. Computing the radial force distribution can be a cumbersome procedure

(Pataky, Slota et al. 2012). A new more simple method to measure maximal grip force could be proposed as follows: 1) Maximal power grip force is applied on a cylindrical low-friction

(e.g. with the friction coefficient around 0.2) handle, which is rotated by a motor; 2) Maximal moment is measured and the scalar sum of tangential forces is calculated by dividing the maximal moment by the radius of handle; 3) the ratio of the scalar sum of tangential forces to the friction coefficient is the power grip force.

The maximal grasp moments showed no differences between the counter-clockwise and clockwise directions in this study. This result may be configuration-dependent. For other arm configurations, this equal capability of grasp moment production may not be observed. A previous study indicated that grasp moment in the counter-clockwise directions was slightly larger than that in the clockwise direction when the arm configuration was not constrained

(Shih and Wang 1997).

Variability of force magnitude and direction

In line with previous findings, the variability of force magnitude was maximal in the

131 direction approximately perpendicular to the forearm long axis and minimal in the direction along the long axis of forearm. The distribution of force direction variability showed an opposite trend.

Compared to the low-friction condition, directional variability decreased for the direction of 225ºand no changes in the variability of force magnitude were found. In the present research the surface of the handle at the low-friction condition was frosted, friction coefficient was about 0.97. In future studies the friction could be further lowered to investigate its effect on variability of force.

Enslaved grasp moment

Enslaved grasp moment on low- and high-friction handles only differed for the direction of 225º, which is also the only direction where variability of force direction decreased for the high friction handle. This may have an implication on the relation between grasp moment and endpoint force variability, especially the directional variability. It should be noted that, in the configuration of 45-80-0, a relatively larger contact area of hand-handhold interface arise when producing force at direction of 225ºas compared to other directions where only fingers or part of palm act firmly against the handle. Coincidently, this direction of 225° also corresponded to a high force direction variability. Hence, for this specific force direction, it may be speculated that the higher friction surface provided a more significant influence on the endpoint force production by sufficiently interacting with fingers and palm, which ultimately contributes to the decrease in the variability of force direction. At other directions

132 where the hand was barely in full contact with handle, the advantage of high friction could not be seen. A future study should investigate the effect of handle with much lower friction on the variability of endpoint force.

8.5 Conclusions

The following main facts were observed: (a) No significant differences in the MVC forces was found between the low- and high- surface friction handles; (b) A higher surface friction led to a higher grasp moment magnitudes; (c) Higher surface friction resulted in lower variability of force direction and higher enslaved moment at a specific direction.

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CHAPTER 9. Effects of rotatable handles and prescribed grasp moment on force and moments generated by the human arm

9.1 Introduction

In order to obtain better understanding of the effect of grasp moment on force production and control, the current study was conducted by changing the grasp moment with the following two approaches: 1) The friction about rotating axis of the handle was eliminated by using a handle that could rotate freely about the vertical central axis; and 2) Level of grasp moment was prescribed.

We hypothesized that: 1) The MVC forces would decrease when acting on rotatable handles. This hypothesis was based on the preceding findings as mentioned above (Seo et al.,

2008); 2) The variability of isometric force would increase when acting on rotatable handle because of the less stable handle; 3) MVC forces would decrease when grasp moment is prescribed/controlled, since an extra task is introduced; and 4) Change in variability of isometric force will be explored. On one hand, the extra task to control grasp moment might interfere with the performance of force production; on the other hand, grasp moment is a part of the isometric performance of the serial kinematic chain; a controlled grasp moment might lead to less flexible performance of other variables, such as the endpoint force.

9.2 Methods

Subjects:

Eight male right-hand dominant subjects took part in the study (mean age 27.5 4.4

134 years, mass 70.0  8.4 kg, height 1.75  0.08 m, shoulder to elbow length (upper arm) 28.1 

1.0 cm, elbow to wrist length (forearm) 25.8  1.0 cm, wrist to the centre of the handle distance 7.3  0.15 cm. All subjects were healthy, with no known history of neurological or peripheral disorders and injuries. All the subjects gave informed consent according to the policies of the Office for Research Protections at the Pennsylvania State University.

Apparatus:

The apparatus used in the experiment included two aluminium cylindrical handles, one normal fixed and the other one rotatable about the centre axis (both diameter 4.5 cm, surface friction coefficient 0.98 ± 0.11, height 16 cm,). They could be attached to a 6-DOF force/torque sensor (Mini85, ATI, Apex, NC USA). Such an arrangement allows recording both the 2-D force vector (resolution about 0.048 N) and the moment exerted on the handle, the grasp moment (resolution 1/600 Nm). A detailed description of the apparatus can be found in Xu et al. (2012).

Subjects sat in a chair, grasped the handle and generated forces in eight directions and

MVC grasp moment in two directions. The trunk was secured to the chair with seat belts. The right arm was in a horizontal plane at the shoulder height, with the upper arm flexed at 20°, the elbow flexed at 120º and the wrist at 0º. The handle was secured at a position corresponding to the arm posture. Hence, the aperture, estimated as the line between the fingers and thumb tips, and defined here as grasp aperture, with this arm configuration was at the angle of about 180º from the frontal plane of the body. 135

(normal force) and vertical force at slip (slip ratio; Johansson & Westling, 1984). We used the inverse of the slip ratio as an estimate of the static friction coefficient (μ).

Experimental procedure.

In the preliminary trials, at each condition, the maximal voluntary contraction (MVC) forces were recorded under the instruction ‗push the handle as strongly as you can‘. The duration of each trial was 5 s. In a separate test, the subjects were asked to produce the maximal grasp moment on the fixed handle, ‗try to turn the handle as strongly as you can‘.

The specific MVC data for each condition and the general maximal grasp moment for the first condition (fixed handle, uninstructed grasp moment) were then used for the normalization of the prescribed forces in the main experiment.

In the main experiment, the subjects were asked to produce 30% MVC force in the eight directions, keeping the force magnitude and direction as stable as they could for 15 s. The force directions were pseudo-randomized to prevent similar or same directions presented in succession. The monitor, placed approximately 80 cm in front of the subject, provided visual feedback on the horizontal force vector exerted on the handle. A gauge displaying the instant grasp moment was displayed at the side of target on the monitor. Such an arrangement allows subjects to keep track of their performance of force and grasp moment simultaneously.

136

For the rotatable handle, no instruction on the grasp moment was required; for the fixed handle, there were six conditions: no instruction on grasp moment, -20% maximal negative grasp moment, -10% maximal negative grasp moment, 0 moment, 10% of maximal positive grasp moment, and 20% of maximal positive grasp moment. A total of 114 trials were performed to cover the combinations of seven conditions with 16 or 18 trials per condition;

30-s breaks were provided between the trials and a 2-min break between blocks of trials with different conditions.

Data analysis.

MVC grasp moment or force exertion. All descriptive statistics are displayed in the text and figures as mean and standard errors unless stated otherwise. Two-way repeated-measures

ANOVA were used to test effect of Direction, and Friction or Controlled grasp moment on performances of MVC forces, as well as the variability of endpoint force. Paired t-test was performed to detect the differences of MVC grasp between conditions. The significance was set at p=0.05 for are statistical tests, which were performed with SPSS 20.0 (IBM 137

Corporation) and Matlab (Mathworks Inc, MA,USA).

9.3 Results

Effect of rotatable handle on MVC forces

fixed handle: no instruction on grasp moment rotatable handle 90 90

180 0 180 0

270 270

Figure 9.1 Normalized MVC forces across 8 directions

The numbers around the outer circle are the target force directions. The average force magnitudes normalized by the maximal MVC across all eight directions for each subject. Dashed lines are the group SDs.

At the beginning of the test, MVC data were collected from all 8 subjects in all eight directions on the fixed and rotatable handles. MVCs of every subject were normalized by the maximum value across the directions. Then the mean values and standard deviation (SD) of the normalized forces across 8 subjects at each direction were computed. Results are shown in Figure 9.1. No significant difference between fixed handle and rotatable handle was revealed using 2-way repeated measures ANOVAs.

138

Effect of rotatable handles on variability of force

variability of force magnitude (N) variability of force direction (degree) 90 1 90 2 120 60 120 60

150 0.5 30 150 1 30

180 0 180 0

210 330 210 330

240 300 240 300 270 270

fixed handle rotatable handle

Figure 9.2 Average variability (RMS) of force magnitude (left) and force direction (right) across 8 directions.

No significant differences were found in the variability of force magnitude between the fixed and rotatable handle groups (p=0.837). Further, paired t-test did not showed significant differences between conditions at each of the eight directions neither. Similarly, for directional variability, no significant differences between conditions were discovered by the two-way repeated measures ANOVAs (p=0.17) and the paired t-test (p values>0.14).

Effect of controlled grasp moment on MVC

139

The text below does not correspond to this title: The title is about the effect of the controlled moment on the MVC and the text is about the effect of the generated force on the enslaved moment.

-20

-40

-60

enslaved moment in percentage enslaved ofmoment in MVC percentage moent grasp (%) -80 45 90 135 180 225 270 315 360

Figure 9.3 Enslaved grasp moment at MVC force (blue) and 30% MVC trials (red) in percentage of the maximal grasp moment;

Enslaved grasp moment at MVC forces and 30% isometric MVC forces trials shared a similar distribution pattern and the mean values are linearly correlated (Fig 9.3, r=0.94, p<0.01). Also, the distribution agreed with previous findings. In 30% MVC trials condition, the maximal enslaved grasp moment reached 8% at direction of 135. Moreover, the enslaved grasp moment could be as large as more than 42% maximal grasp moment at MVC forces

140 trials.

300 no instruction -20% grasp moment -10% grasp moment 250 0% grasp moment 10% grasp moment 20% grasp moment 200

150 MVC forces

100

0 45 90 135 180 225 270 315 360

Figure 9.4 MVC forces at 7 conditions across 8 directions

MVC forces of in the grasp moment controlled conditions were significantly smaller than those in the moment-uncontrolled conditions at each direction; however, no significant differences were found within the grasp moment controlled conditions by ANOVAs. Paired t-test between each pair out of 5 conditions revealed some significant differences as following: the MVC force at the direction of 90º was greater in +10% grasp moment condition than those in -20% and -10% conditions; The +20% and +10% conditions also had a higher MVC forces than -10% condition at the direction of 270º;+20% condition showed a higher value than +10% at direction of 315 º; and -20% condition exhibited less MVC force than -10%, 0%, and 10% at the direction of 360 º.

141

Effect of controlled grasp moment on variability of forces

0.08 no instruction -20% grasp moment 0.07 -10% grasp moment 0% grasp moment 0.06 10% grasp moment 20% grasp moment 0.05

0.04

0.03

0.02 variability of force (norm) magnitude

0.01

0 45 90 135 180 225 270 315 360

Figure 9.5 Variability of force magnitude across 8 directions (coefficient of variation).

Across 8 directions, significant differences in variability of force magnitude were revealed by ANOVAs(p<0.01). In terms of the tests across conditions, equal or higher variability were observed in uncontrolled grasp moment condition if the variability of magnitude is computed as root mean square of magnitude (RMS). However, if the variability is computed as the coefficient of variation, cancelling out the influence of the large average magnitude, the variability of force magnitude in grasp moment uncontrolled groups is lower than those in uncontrolled ones (p values<0.05).

Within the five conditions where grasp moment was controlled, several significant differences were found: 1) the variability of force magnitude in -20% grasp moment 142 condition is higher than those in -10%,0,10%, and 20% conditions at direction of 45º; 2) the variability of force magnitude in -20% grasp moment condition is higher than those in -10%,

0, and 10%conditions at direction of 90º. The 10% condition showed a lower value than that in 20% condition. 3) Variability of force magnitude is higher at zero grasp moment condition than that at 10% grasp moment at direction of 180º; 4) Lower variability of force magnitude was observed in -20% condition than that in 20% condition at direction of 225º; and 5) variability of force magnitude is higher in condition of -20% than that in condition of 10% grasp moment at direction of 270º.

4.5 no instruction -20% grasp moment 4 -10% grasp moment 0% grasp moment 3.5 10% grasp moment 20% grasp moment 3

2.5

1.5

1 variability of force direction (degrees)

0.5

0 45 90 135 180 225 270 315 360

Figure 9.6. Variability of force direction across 8 directions (degree)

Significant differences in directional variability were found both across directions (p<0.01) and conditions (p<0.05) by two-way repeated measures ANOVAs. Directional variability in

143 uncontrolled moment condition was significantly lower than in other conditions. Furthermore, the paired comparison indicated that: 1) Directional variability in -20% condition is higher than those in condition -10%, 0, and 10% grasp moment at directions of 45º,90º, and 135 º.

Meanwhile, the variability at -10% conditions also had higher values than those in 10% grasp moment condition; 2) 0 and 10% grasp moment conditions at direction of 270 º had significant lower directional variability than those in condition of -20% and 20%; 3)

Directional variability was significantly higher in condition of -20% than in conditions of 10% and a zero grasp moment at the direction of 360º.

9.4 Discussion

This study was to examine the effect of rotatable handle and controlled grasp moment on performances of endpoint force production by arm. Three hypotheses included: 1) The MVC forces will decrease when applied on rotatable handles. 2) The variability of isometric force will increase on rotatable handle because of the presence of one more degree of freedom, possible rotation about the long axis of the handle; 3) MVC forces will decrease when grasp moment is prescribed/controlled; and 4) Variability of isometric force will be explored in grasp moment controlled conditions. In the experiments, the first two predictions and last one did not receive support, while the third one was confirmed.

Effect of rotatable handle on MVC forces

No differences between fixed and rotatable handle conditions were found in MVC forces

144 production. Still, it did not demonstrate the function of hand-handle interface interaction, such as frictions, in improving MVC endpoint force as the study of 2009 (Young et al., 2009).

The differences in the results can be attributed to the discrepancies between studies, such as force level, arm configurations and size of handle etc. It should be noted that one feature of the rotatable handle makes it not a perfect representation of a friction-free handle. Despite of the fact that both handles allow rotation of the hand about the central axis without resistances, the real low-friction handle also allows for slides between hand and handle on the interface and, therefore, a much higher chance of slipping the handle out of the hand. As a result, on the current handle, subjects were still able to produce their maximal force without detachment of the hand from the handle. Nevertheless, same MVC values for the rotating handle (zero moment) showed that enslaved moments are not necessary to reach high forces.

So, these are true choices by the CNS unrelated to the explicit task (producing MVC). For instance, it might provide some kind of sensory signals or make the force production more efficient.

Effect of rotatable handle on force variability

Variability of force magnitude and directions on rotatable handle demonstrated no significant differences from that on fixed handle. The intrinsic differences between a rotatable handle and a handle with low surface-friction as discussed above may contribute to these non-significant results. A firm attachment of the hand onto rotatable handle is still possible because of the relatively high surface friction. Subjects could adjust the force by operating the arm as a whole without manipulating the tangential force to achieve the same performance as long as the stability of the handle on the rotation axis was ensured. At current 145 force level of 30% MVC, the accurate control of endpoint force could still be matched by operating the arm as a whole; however, a higher activity level of arm muscles were suspected to arise, and this hypothesis should be tested in the future. Moreover, if the force level is increased to a higher level, such as 40% or 50% MVC, significant difference between the fixed and rotatable handle may emerge since a small percentage of deviation in force from the target direction will have to be compensated for by a greater amount of arm action. In other words, controlling the force without utilizing the tangential forces could not be as effective and efficient.

According to the idea of control with referent configurations (Feldman,AG 2010), the action has two major referent values: the subjects had to specify, one for force

(two-dimensional, direction and magnitude or X and Y coordinates) and the other for rotational action (referent angle). At a relatively low force level (30% MVC), the referent arm position (force) was not significantly influenced by the rotational action and these two tasks, force and grasp moment production, are independent of each other.

The observed unchanged force variability supported the idea of independent control of the force and moment production despite the apparent mechanical coupling of the hand force and moment (they both depend on the three joint torques).

Effect of controlled grasp moment on MVC

The MVC forces decreased when grasp moments were prescribed, even in the condition where an approximately the same grasp moment was to be generated when grasp moment was not prescribed. This observation is parallel to the well-known phenomena in the studies

146 of fingers or bilateral limbs control, the force deficit (Ohtsuki 1981). One of the hypotheses offered to explain the observed force deficit was based on the idea of double-tasking, in which a second task introduced during a motor performance leads to a decrease in either task.

An alternative explanation is that the central neural drive gets distributed among all the controlled elements in a given task. There is a limit to how much central neural drive can be delivered and thus the more elements involved in a task the more the neural drive has to be divided among the performance variables, resulting in a force deficit compared to tasks involving fewer controlled elements. In the case of current task, the maximal endpoint force was attenuated by the extra control of grasp moment while the force drop did not increase with higher instructed moment (increased drive in other elements), and thus the double-tasking was suggested to lead to this results.

Effect of controlled grasp moment on force variability

Higher variability of both force magnitude and direction was demonstrated in the conditions where grasp moment got controlled. Intuitionally, a complete control of all the three performance variables, two force components in the horizontal plane and grasp moment along vertical axis, was supposed to minimize the intra-trial variability of performance.

However, the results displayed an opposite trend. This unexpected performance might be interpreted as following: For daily movement activities, force productions or grasp productions are commonly practised separately while, in contrast, the synchronous control of both force and moment is seldom used. As a result, people tend to set one target variable as the highest priority and neglect the other one as long as it does not interfere with the target. In the currently studied task that is unfamiliar to subject, the highest priority of controlling force 147 individually was disturbed by the usually ignored variables, grasp moment, and as a result the variability of force increased. By the way, this observation provides an additional evidence for the role of central command signals in generating peripheral motor variability.

The task in current study, requiring control of all three performance variables within a plane, was rarely studied; hence not much data are available to be compared with the results.

In general, the variability of both magnitude and variability of force direction had a tendency to decrease when the prescribed grasp moment was about the same level as the enslaved moment if the instruction on the moment production was cancelled out. Generally, the larger the discrepancy between the prescribed grasps moment and the naturally enslaved grasp moment, the higher the variability of force, especially variability of force direction (Fig

9.5,9.6).

9.5 Conclusions

The following main facts were observed: (a) Rotatable handle did not change the MVC forces significantly; (b) At the force level of 30% of MVC, using the rotatable handle didn‘t lead to different variability of force magnitude and directions at all prescribed directions; (c)

Exerting of the prescribed grasp moment reduced the MVC force production; and (d)

Exerting of the prescribed grasp moment also led to higher variability of both force magnitude and direction.

148

CHAPTER 10: Discussions and Conclusions

10.1. General overview

The dissertation includes six studies devoted to the arm force production and its variability at different conditions. The individual studies focus on the effect of force levels and arm positions on general force production (Chapter 4-5), modelling sharing pattern of joint torques (Chapter 6) and effect of hand/handhold interface on isometric force and grasp moment production by arm (Chapter 7). The following summarizes the individual studies.

Study #1(Chapter 4) is a comprehensive study on the isometric force production and control in different directions with different magnitude. The following main facts were observed: (a) While the grasp moment was not prescribed by the instruction, it was always produced. The moment magnitude and direction depended on the instructed force magnitude and direction. (b) The within-trial angular variability of the exerted force vector (angular precision) did not depend on the target force magnitude (a small negative correlation was observed). (c) Across the target force directions, the variability of the exerted force magnitude and directional variability exhibited opposite trends: In the directions where the variability of force magnitude was maximal, the directional variability was minimal and vice versa. (d) The time profiles of individual joint torques in the trials were always positively correlated, even for the force directions where flexion torque was produced at one joint and extension torque was produced at the other joint. (e) The correlations between the grasp moment and the wrist torque were negative across the tasks and positive within the individual trials. (f) In static serial kinematic chains, the pattern of the joint torques distribution could

149 not be explained by an optimization cost function additive with respect to the torques.

Study #2 (Chapter 5) focused on the effect of systematically varied arm posture on the static force production and control by the human arm. The following main facts were observed: (a) Non-instructed non-zero grasp moments were observed for most of the combinations of arm posture and force direction. (b) The minimal intra-trial variability of force direction was observed for the target force directions along or close to the arm pointing axis, i.e. the forearm-hand longitudinal axis. (c) The time profiles of joint torques within the trials were always positively correlated, even in the tasks where flexion torque was produced at one joint and extension torque was produced at the other joint. (d) After application of the analytical inverse optimization (ANIO) method it has been concluded that experimentally observed torque distribution patterns could not be explained by any optimization cost function additive with respect to the joint torques. On the whole, the results of this study performed at different arm posture confirmed the results reported in Chapter 4 for one arm posture.

Study #3 (Chapter 6) is a computer simulation based on the findings reported in Chapter

4 and Chapter 5 to investigate the cause of joint sharing pattern among the three joints of the arm. The results suggested that grasp moment capability is suspected to be the key factor preventing application of ANIO in arm control tasks. Thus, effects of grasp moment on the performance of isometric force production by arms should be addressed.

Study #4 (Chapter 7) investigated the influence of handle size on the force performance by arm. The following main facts were observed: (a) There existed effects of handle size on

150 the magnitude, but not on the distribution pattern, of MVC end-point force. (b) Changing the handle diameters in the range between 4.5 and 9.0 cm does not affect the maximal torque production. (c) The enslaved moments were smaller for the handle diameter of 3.0 cm and showed only a small tendency to increase or stayed put for the diameters from 4.5 cm to 9.0 cm. (d) There was a slight tendency for increasing the force direction variability for large

(>7.5 cm) handles while force magnitude variability, calculated as the coefficients of variations of force magnitude, did not show significant differences across conditions. (e)

Although systematic change in the coefficients of results of ANIO was observed, it still failed to reconstruct an optimal cost function additive with respect to joint torques with large handle size.

Study #5 (Chapter 8) examined the effect of surface friction of at the handle-hand interface on the force and grasp moment control. The following main facts were observed: (a)

No significant differences in the MVC forces were found between the low- and high- surface friction handles; (b) higher surface friction led to higher magnitudes of grasp moment production; (c) higher surface friction resulted in lower variability of force direction and a higher enslaved moment at a specific direction of the force production .

Study #6 (Chapter 9) explored the effects of rotatable handle and prescribed grasp moment on the endpoint force performance. The following main facts were observed: (a)

Rotatable handle did not change the MVC forces significantly; (b) At the force level of 30%

MVC, rotatable handle didn‘t lead to different variability of force magnitude and directions at all directions; (c) Exerting of the prescribed grasp moment reduced the MVC force

151 production; and (d) Exerting of the rescribed grasp moment also led to higher variability of both force magnitude and direction to various extent.

10.2 Limitations of this work

(1) Only young, healthy male subjects were recruited in the experiments. The effects of age, injury, and disease on the results were not examined.

(2) Only the right hands of right arm dominant subjects were tested. Testing the non-dominant arm may have different results in force and its variability distribution in spaces.

(3) The tested population was limited to the surrounding community, which may not be an accurate representation of larger populations.

(4) Subjects were only tested a limited number of trials for each condition. Collecting more trials should be necessary to provide more accurate depictions necessary for ergonomics and other application fields.

(5) Only the additive quadratic function was used in direct and inverse optimization.

10.3 Discussion of the main results and conclusions

A number of findings was discovered in the dissertation and can mainly be categorised into three groups, which provide three different perspectives on the isometric force control by the human arm.

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Variability of force and coordination among joints

Counter-intuitive observations of variability of force direction were reported in the first study (Chapter 4): 1) The within-trial angular variability of the exerted force vector (angular precision) did not depend on the target force magnitude (a small negative correlation was observed); while the increase of the force magnitude variability with the increasing force level is a well-known phenomenon; and 2) Across the target force directions, the variability of the exerted force magnitude and directional variability exhibited opposite trends: In the directions where the variability of force magnitude was maximal, the directional variability was minimal, and vice versa. Other studies also demonstrated several discrepancies between variability of force magnitude and directional variability. For example, in Chapter 7 where handle size was varied and Chapter 8 where surface friction of handle was varied, the variability of force direction was consistently more sensitive to the change in the handle characteristics. Moreover, in the study where grasp moment was prescribed (Chapter 9), the variability of force direction was more likely to be lower when the value of prescribed grasp moment was close to the value of preferred enslaved grasp moment, as compared to the variability of force magnitude.

All these findings imply some inherent differences between variability of force magnitude and direction. One of the interpretations attributed it to the mechanical structure of the arm. An interesting novel fact is that the minimal force direction variability—it can be interpreted as the highest precision of force exertion—was observed when the target force directions were along or close to the pointing axis (hand-forearm axis). At such directions, the force vector passes through the elbow joint center and does not generate a moment of 153 force at the elbow (for a more detailed biomechanical analysis see Zatsiorsky, 2002, Chapter

2). Therefore, at these force directions, the endpoint force and its variability is solely due to the joint torque at the shoulder. This interpretation also explains the systematic shift of distribution pattern of directional variability when arm posture was varied.

Moreover, a perspective from motor control was also introduced. Intuitionally, it sounds reasonable that if only one joint torque is involved in the endpoint point force production the force variability should be smaller than when the force is a function of two or three joint torques (this ‗common sense‘ conclusion does not agree well with the known in motor control facts that due to inter-compensation between the elements of a complex system variability of the performance variables can be smaller than the variability of the contributing elements, e.g. variability of the total force produced by several fingers is smaller than the variability of the single finger forces; discussed above in Chapter 1) . The minimal variability was observed however for the force direction only. This was not true for force magnitude suggesting that involvement of more than one joint torque could be accompanied by error compensation among joint torques with respect to the endpoint force magnitude (Latash et al. 1998).

To further the investigation in the coordination among three arm joints, analysis in the space of joint torques was performed. In order to minimize the force variability, the joint torques should be exerted in a coordinated way. Further, to exert force at certain directions, the shoulder and elbow torques in opposite directions (flexion and extension) have to be produced. Hence when the end-effector force increased or decreased, the torques at the

154 shoulder and elbow changed in opposite directions, and the negative torque-torque correlations could be expected. In contrast, the torque-torque within-trial correlations at these directions were always positive, though smaller than at other directions. Activation of bi-articular muscles was suggested to be responsible for the positive (but smaller) correlations between the shoulder and elbow torques. In the future a biomechanical model taking into account the bi-aritcular muscles might be developed to simulate the endpoint-force performance..

Optimization of force production and grasp moment

The regular distribution of grasp moment paved the way for attempts to reconstruct an optimal additive cost function with respect to joint torques using analytical inverse optimization (ANIO). However, all the attempts failed (Chapter 4 and Chapter 5). A simulation was performed to address these failed attempts (Chapter 6). Simulation results indicated that the capability of grasp moment production prevented the optimal distribution of joint torque and thus lead to the failed application of ANIO. One possible approach worth exploring is constructing a muscle-based model that uses muscle forces as elementary variables. Because muscles work essentially in parallel, it is possible to expect that such a model in combination with the ANIO will be successful.

As to the enslaved grasp moment, studies showed that it could result from the passive components in force production. A previous study indicated that EMG amplitude of wrist muscles is lower if grasp moment production is allowed when producing a prescribed level of

155 grip force (Seo, Armstrong et al. 2007). Also, generating the non-required grasp moment is in accord with the reported in the literature tendency of the subjects to produce the static endpoint forces in the direction different from the instructed direction — when the visual feedback is not provided and the external object is mechanically constrained. Grieve and

Pheasant (1981) called these directions ‗naturally preferred‘ and introduced a measure for estimating the effectiveness of static force efforts – the maximum advantage of static force efforts (MACE). Pan et al. (2005) in an elegant study have shown that these results can be interpreted in terms of an unknown optimization used by the central controller. While the authors were not able to reconstruct the cost functions used by the central controller they derived from the experimental data the so-called "isocost" contours of objective functions.

On the whole, these results suggest that the observed force patterns are due to some kind of optimization used by the central controller. An interference with the non-instructed grasp moment, for example prescribing grasp moment, as in Chapter 9, would lead to a decrease in performance variables, such as MVC and variability of force.

Hand-handle interaction:

Preceding studies have suggested that the MVC grasp moment on a handle is related to the handle surface friction, cross-section area of the handle and area of contact (Pheasant and

O'Neill 1975; Imrhan and Farahmand 1999; Seo, Armstrong et al. 2007; Yoxall and Janson

2008). As the simulations in Chapter 6 illustrated the importance of grasp moment, the hand-handle interface was investigated in the following studies. The effect of handle size was demonstrated to be subject to the surface friction (Chapter 7) while the surface friction affects the endpoint –force performance. To further examine the effect of friction, a rotatable 156 handle was introduced which allow for rotating about the central axis (Chapter 9). The findings for the rotatable handle were different from the findings for the low-friction handles.

Despite the fact that both handles allowed free rotation about the central axis freely without resistances, the real low-friction handle also allows for slides between hand and handle on the interface and, therefore, a much higher chance of slip out of hand. As a result, on the current handle, subjects were still able to try producing their maximal force without real fear for detachment of hand from the handle. The correlation between the friction of different types and levels, and arm muscles EMG could be estimated to offer more implication on the problem of optimization and ergonomics.

Overall, two merits, a large movement range and a dexterous control, make the arm a tool to be used hundreds of times a day. Such a huge amount of practices make people to be able to operate arm in an accurate, efficient and effective way. Muscle coordination within arm was applied to ensure an accurate control. The enslaved grasp moment was incorporated to optimally and efficiently achieve an activity target. The properties of environment or object, such as the surface frictions, would also be utilized effectively to better accomplish the desired task. This dissertation demonstrated various features of arm in force production and control. Hopefully the future work with advanced technology could lead to better understanding of this delicate and complex movement system.

10.4 Future works

157

(1) The 3D pressure mat could be introduced to help characterizing the grasp moment and finding its origin. The hypothesis is that when grasping the handle and generating isometric force against the handle, either the thumb side or the fingers side serves as the main producer of the grasp moment. The other side of hand facilitates attaching hand on the handle, which might generate little, or even conflicting, grasp moment.

(2) The proposed experiment in Discussion of Chapter 8 could be performed to measure the maximal isometric grip force in the following conditions: 1) Maximal power grip force is applied on a cylindrical low-friction (i.e., friction coefficient of 0.2) handles which is rotated by a motor; 2) Maximal moment is measured and the scalar sum of tangential forces is calculated by dividing the maximal moment by the radius of handle; 3) The ratios of the scalar sum of tangential forces to the coefficient is the power grip force.

(3) The analysis of force magnitude and variability could be performed in the space of about 30 arm muscles. Such a biomechanical model may provide insight into the characteristics of endpoint force and its variability. It should be noted that the results would be quite sensitive to the accuracy of approximation during each step of the model building.

158

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APPENDIX A: Anatomy and Physiology

APPENDIX A Table 2.1.1 Muscles Anatomy and Innervation: upper arm 1

Muscle Origin Insertion Action Inner vation

Those inserted on The Humerus

(Dorsal Group)

Supraspinous Greater tubercle Abduction of the arm; Suprascap fossa of scapula & of humerus & stretching of the ular nerve Supraspinatus supraspinous capsule of articular capsule fascia shoulder joint

Infraspinous fossa Greater Lateral rotation of the Suprascap of scapula & tubercle of arm ular nerve Infraspinatus infraspinous humerus fascia

Lateral margin of Greater tubercle Lateral rotation of the Axillary scapula & of humerus arm nerve Teres minor infraspinous fascia

Acromial end of Deltoid Whole muscle abducts Axillary clavicle, acromion tuberosity of the arm to horizontal nerve & scapular spine humerus level: the clavicular Deltoid part flexes the arm; spinous part extends the arm

Subscapular fossa Lesser tubercle Pronates the arm & Subscapul Subscapularis & lateral margin of the humerus adducts it to the body ar nerve

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of scapula & its crest

Inferior angle of Crest of the Medial rotation & Subscapul scapula and lesser tubercle of adduction of the arm ar nerve Teres major infraspinous humerus fascia

Spinous process Cresit of the Adduction of arm, Thoracodo of lower T6 and lesser tubercle of lowers a raised arm as rsal nerve all lumbar humerus well as pronation & Latissimus vertebra, iliac extension in the dorsi crest, medial shoulder. sacral crest & 3-4 lower ribs

APPENDIX A Table 2.2.2 Muscles Anatomy and Innervation: upper arm 2

Muscle Origin Insertion Action Inner vation

(Ventral Group)

Coracoid process Humerus, below Flexion & adduction Musculoc Coracobrachial of scapula the crest of the in the shoulder joint uteneous is lesser tubercle nerve

Outer surface of Medial border & Draws shoulder Medial ribs 3,4 & 5 near upper surface of forward & pectoral Pectoralis their anterior the coracoid downwards. It nerve (C8 minor ends. process of the depresses the & T1) by (exception) scapula shoulder. piercing the muscle

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Latera l pectoral nerve (C5,6,7) that communic ates with pectoral nerve.

Clavicular head: Lateral lip of the A powerful adductor Lateral from front of the intertubercular and medial rotator of pectoral medial half or 2/3 sulcus of the the humerus nerve of the anterior humerus by a (C5,6,7) surface of the bilaminar clavicle tendon.

Sternocostal head: Fibres of Main climbing Medial infront of clavicular head muscles and pulls the pectoral Pectoralis manubrium sterni are attached body up to the flexed nerve major & body of infront & distal arm. (C8,Th1) sternum close to to the the midline. sternocostal part.

Abdominal head: Acts as an from aponeurosis accessory respiratory of the external muscle oblique muscle of the abdomen.

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Head & Neck Muscles Inserted on The Shoulder Girdle

External occipital Lateral third of By simultaneous Accessory protuberance, the clavicle, the contraction, it moves nerve & mediall part of the acromion and the scapula to the branches upper nuchal line, the scapular vertebral column. of cervical Trapezius he nuchal lig. and spine plexus the spinous processes of the C7 & all thoracis vertebrae.

Anterior surface Mastoid process During bilateral Accessory of manubrium of the temporal contraction it bends nerve sternocleidoma sterni & medial bone the head back. stoid end of clavicle. Accessory muscle in respiration

APPENDIX A Table 2.1.3 Muscles Anatomy and Innervation: upper arm 3

Muscle Origin Insertion Action Innerv ation

Those inserted on The Shoulder Girdle

(Dorsal Group)

Spinous process of Medial margin of the Move the Dorsal Th2-Th5 vertebrae scapula below the scapula scapular Rhomboideus scapular spine towards the nerve major vertebral column

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Spinous process of Medial margin Move the Dorsal C7 & Th1 vertebrae of scapula above the scapula scapular Rhomboideus scapular line. towards the nerve minor vertebral column

Tendon fascicles Upper part of the Lifts the Dorsal from the transverse medial margin of the scapula scapular processes of 3-4 scapula. Its upper 1/3 bringing it nerve Levator upper cervical is covered by closer to the scapulae vertebrae sternocleidomastoid spine. (SCM) & lower 1/3 by trapezius

Spinous processes Back surfaces of ribs Lifts the ribs Intercostal Serratus of C6, C7, Th1 & 1-5 lateral to their nerves anterior Th2 vertebrae angles

Ventral Group

Cartilage of rib 1 Acromial end of Pulls the Subclavian clavicle clavicle nerve Subclavius medially & downwards

Superior margin of Hyoid bone Pulls hyoid Cervical scapula. bone ansa Omohyoid downwards & stretches the cervical fascia.

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APPENDIX A Table 2.1.4 Muscles Anatomy and Innervation: upper arm 4

Muscle Origin Insertion Action Innerv ation

MUSCLES OF THE ARM

Anterior Compartment (Flexors)

Coracoid process of Humerus, below the Flexion and Musculocut Coracobrachial scapula crest of the lesser adduction eneous is tubercle in the shoulder nerve joint

Long head: Radial tuberosity & Flexion and Musculocut superglenoid bicipital aponeurosis supination of eneous tubercle of scapula forearm in the nerve Biceps brachii elbow joint; Short head: flexion in the coracoid process shoulder joint

Humerus, distal of Tuberosity of ulna Flexion of Musculocut Brachialis the deltoid forearm eneous tuberosity nerve

Posterior Compartment (Extensors)

Long head: Olecranon process of Extension of Radial infraglenoid ulna forearm in nerve tubercle elbow joint; long head Triceps brachii Medial & lateral extends and heads: posterior adducts the arm surface of the body of humerus

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Lateral epicondyle Olecranon & Extension of Radial ne Anconeus of humerus posterior surface of forearm in the ulna elbow

APPENDIX A Table 2.1.5 Muscles Anatomy and Innervation: forearm 1

Muscle Origin Insertion Action Innerv ation

FOREARM

(Anterior Compartment)

Lateral Radius, above the Flexion of arm Radial supracondylar crest styloid process and setting it in nerve of humerus and a middle lateral position intermuscular between

` septum of arm supination & pronation

Media epicondyle Lateral surface of Pronation & Median of humerus and radius flexion of the nerve coronoid process of forearm ulna

Medial epicondyle Palmar surface of Flexion of the Median of humerus, medial bases of the II-III wrist, nerve Flexor carpi intermuscular metatarsal bones abduction of radialis septum of arm and the hand, and antebrachial fascia flexion of the forearm

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Medial epicondyle Palmar aponeurosis Stretches the Median of humerus, medial palmar nerve Palmaris intermuscular aponeurosis longus septum of arm and flexes the forearm and hand

Medial epicondyle Pisiform and hamate Flexion of the Ulnar nerve of humerus, medial bones and base of wrist, Flexor carpi intermuscular fifth metacarpal abduction of ulnaris septum of arm, the hand and olecranon and flexion of the antebrachial fascia forearm

Medial epicondyle Four tendons insert Flexion of the Median of humerus, on palmar surface of middle nerve coronoid process of middle phalanges of phalanges of ulna and II-V fingers. Near the II-V fingers, Flexor antebrachial fascia shafts of proximal flexion of the digitorum phalanges each hand and superficialis tendon splits into two forearm parts, beneath which pass the tendons of the deep flexor

Anterior surface of Four tendons insert Flexion of Median & ulan, interosseous on the distal distal ulnar nerves Flexor membrane of the phalanges of fingers phalanges of digitorum forearm II-V fingers II-V profundus and flexion of the hand

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Flexor pollicis Anterior surface of Palmar surface of Flexion of the Median longus radius intersseous distal phalanx of the thumb and nerve membrane of the first finger hand forearm

Anterior margin Anterior surface of Pronation of Median Pronator and medial anterior radius (its lower the forearm nerve quadratus surface of ulna quarter) and hand

APPENDIX A Table 2.1.6 Muscles Anatomy and Innervation: forearm 2

Muscle Origin Insertion Action Innerv ation

POSTERIOR COMPARTMENT

Lateral epicondyle Dorsal surface of Extension and Radial of humerus, lateral base of the 2nd abduction of nerve Extensor carpi intermuscular metacarpal bone the hand; radialis longus septum of arm flexion of the forearm

Lateral epicondyle Dorsal surface of Extension and Radial Extensor carpi of humerus and base of the third abduction of nerve radialis brevis antebrachial fascia metacarpal bone the hand

Lateral epicondyle Four tendons insert Extension of Radial of humerus and on dorsal surfaces of fingers II-V nerve Extensor antebrachial fascia middle and distal and extension digitorum phalanges (dorsal of the hand aponeurosis of fingers)

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Lateral epicondyle Dorsal surface of Extension of Radial of humerus and middle and distal the little finger nerve Extensor digiti antebrachial fascia phalanges of the little minimi finger (into its dorsal aponeurosis)

Lateral epicondyle Dorsal surface of the Extension and Radial Extensor carpi of humerus and base of the 5th abduction of nerve ulnaris antebrachial fascia metacarpal the hand

Lateral epicondyle Proximal third of the Supination of Radial Supinator of humerus, ulna lateral surface of the forearm nerve radius

Dorsal surface of Dorsal surface of the Abduction of Radial radius and ulna, base of the first the thumb and nerve Abductor and interosseous metacarpal hand pollicis longus membrane of forearm

Dorsal surface of Dorsal surface of the Extension of Radial radius, base of the proximal the proximal nerve Extensor interosseous phalanx of thumb phalanx of the pollicis brevis membrane of thumb forearm

Dorsal surface of Dorsal surface of the Extends the Radial Extensor ulna, interosseous base of the distal thumb nerve pollicis longus membrane of phalanx of the thumb forearm

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Dorsal surface of Dorsal surface Extension of Radial ulna, interosseous (aponeurosis) of the the index nerve Extensor indicis membrane of proximal phalanx of finger forearm index finger

APPENDIX A Table 2.1.7 Muscles Anatomy and Innervation: hand 1

Muscle Origin Insertion Action Innerva tion

MUSCLES OF THE HAND

Muscles of The Thenar

Scaphoid and Lateral border of the Abduction of Median Abductor trapezium bones, base of the proximal the thumb nerve pollicis brevis flexor retinaculum phalanx of the thumb of the hand

Trapezium and Anterior surface of Flexion of the Median and trapezoid bones, the base of the thumb ulnar nerves Flexor pollicis flexor retinaculum proximal phalanx of brevis of the hand, second the thumb metacarpal bone

Trapezium bone, Lateral border and Opposition of Median Opponens flexor retinaculum anterior surface of the thumb to nerve pollicis of the hand the first metacarpal the little finger

Capitate bone, Base of the proximal Adduction of Ulnar nerve Adductor bases and anterior phalanx of the thumb the thumb pollicis surfaces of II and III metacarpals

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Muscle of The Hypothenar

Flexor retinaculum Skin of the medial Wrinkles skin Ulnar nerve of the hand border o the hand of the Palmaris brevis hypothenar region

Flexor retinaculum Medial border of the Abducts the Ulnar nerve Abductor digiti of the hand and base of the proximal little finger minimi pisiform bone phalanx of the little finger

Hook of the hamate Palmar surface of the Flexion of the Ulnar nerve Flexor digiti bone and flexor proximal phalanx of little finger minimi brevis retinaculum of the the little finger hand

Hook of the hamate Medial border and Opposition of Hook of the bone and flexor anterior surface of little finger to hamate bone Opponens retinaculum of the the fifth metacarpal thumb and flexor digiti mnini hand bone retinaculum of the hand

APPENDIX A Table 2.1.8 Muscles Anatomy and Innervation: hand 2

Muscle Origin Insertion Action Innerv ation

Middle group of muscles

Tendons of the Dorsal surface Flexion of the First and Lumbricals deep flexors of (aponeurosis) of the proximal second fingers proximal phalanges phalanges; lumbracals

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of II-V fingers extension of – median the middle and nerve; third distal and fourth – phalanges ulnar nerve

Medial border of II Dorsal surface Adduction of Ulnar nerve Palmar and lateral border (aponeurosis) of fingers II, IV interossei of IV and V proximal phalanges and V to the metacarpal bones of fingers II, IV, V third finger

Adjacent surfaces Dorsal surfaces Abduction of Ulnar nerve Dorsal of I-V metacarpals (aponeurosis) of fingers II, IV interossei proximal phalanges and V from the of II-IV fingers third finger.

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APPENDIX B: Analytical Inverse Optimization

In this Appendix we demonstrate that the sharing of the joint torques cannot be explained by any additive cost function with respect to joint torques. An exact definition of such function will be given in the next paragraph. The general idea of the proof is as follows. We first show that if the experimental data are explained by an additive cost function then this cost function must be quadratic. Then we show that the data cannot be explained by any additive quadratic cost function.

From these statements it follows that the data cannot be accounted by any additive cost function.

Below we present elements used in the proof, which follows after.

Optimization problem

As emphasized in the main text, the problem of joint torques distribution is redundant if the instruction is given only with respect to the endpoint force. It is reasonable to assume that the distribution of the joint torques adopted by the subjects minimized a certain cost function.

Mathematically this can be formalized as:

G (T1, T2, T3) → min subject to the constraints on the end point force

[C]T = f, where the matrix [C] comprises the first two rows of the inverse transpose Jacobian matrix [JT]-1

T defined in equation (4) and f = [FX, FY] is the endpoint force vector.

Uniqueness Theorem

Recently we formulated and proved the Uniqueness Theorem for inverse optimization problems (Terekhov et al 2010). The theorem suggests the conditions under which the cost function can be identified uniquely from experimental data. The main assumption of the Uniqueness

Theorem is that the cost function is additive with respect to a known set of variables. If joint torques are chosen as such variables, this assumption means that the cost function is

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G (T1, T2, T3) = g1(T1) + g2(T2) + g3(T3),

where g1, g2, g3 are unknown functions of individual torques.

For the current problem, the Uniqueness Theorem requires that, for different task values FX and

FY, the experimental data be distributed over a 2-d surface S. In addition it requires that the matrix

[Č] = [I] – [C]T([C][C]T)-1[C] cannot be made block-diagonal by simultaneous swapping of the columns and rows with the same indices (i.e. the problem is not-splittable). The reader can verify that the latter requirement is satisfied. Then, according to the Uniqueness Theorem, for every two ~ functions and G , such that

[Č] grad = 0 and [Č] grad = 0 (A1) on the experimental surface S, follows

T (T1, T2, T3) = r (T1, T2, T3) + [q] [T], (A2) where grad stands for gradient, r is a non-vanishing scalar value, [q] is a 3-d vector, for which [Č][q]

= 0.

Planarity of the data

A particular implication of this theorem is that if the experimental data are distributed along a plane, then the cost function — if it exists — must be a quadratic polynomial (for details see

Terekhov et al. 2010; Terekhov, Zatsiorsky 2011). In line with this fact we begin with checking whether the data have planar distribution. We performed the principal component analysis (PCA) and found that the first two PCs accounted for 99.98±0.01% of the total variance. Since the wrist torque was significantly smaller in magnitude than the shoulder and elbow torques, such a high degree of planarity could result from the uneven spread of the magnitudes. To verify that it was not the case we normalized the torques by its standard deviation over all trials (SD) and repeated the

PCA. For normalized data the planarity was still very high: the two first PCs accounted for

99.49±0.45% of the total variance. This finding suggests that the experimental data have planar distribution, i.e. they can be fitted by equation 180

[A]T([T] – [ T ]) = 0, (A3) where vector [A] is the normal to the plane formed by the data. The vector [A] coincides with the last PC (the one accounting for the smallest percentage of total variance); [ ] is the average of the joint torques across the trials.

The planarity of the data suggests that the cost function — if it exists —must be searched on the class of quadratic polynomials.

Cost function coefficients

According to the Uniqueness theorem and the previous paragraph, if the central controller uses a cost function G, additive with respect to the joint torques, then this cost function must be quadratic:

0 0 0 G  k1(T1 T1 )  k2 (T2  T2 )  k3 (T3 T3 ) (A4)

According to the Lagrange principle for the inverse optimization problems proved in

(Terekhov et al. 2010), if the experimental data minimize G , then satisfies (A1) on the experimental plane (A3).

Now let us define a function

~ ~ ~0 ~ ~0 ~ ~0 G  k1(T1  T1 )  k2 (T2  T2 )  k3 (T3 T3 ) (A5)

~ ~ ~ such that the coefficients k1 , k2 , k3 satisfy

~ [Č][ K ] = [A], (A6)

~ ~0 ~0 ~0 T [ ] is a diagonal matrix with , , on the diagonal and T[T1 ,T2 ,T3 ]  T.

~ One can verify that such a function G also satisfies (A1) on the experimental plane (A3).

Hence, according to Uniqueness Theorem, (A2) holds, and in particular the second-order coefficients of (A4) and (A5) coincide up to normalization:

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~ ~ ~ k1  rk1,k2  rk3,k3  rk3 . (A7)

We determined the second order coefficients k1, k2, k3 of the presumed ―true‖ cost function G using (A6) and (A7). Since a cost function can be determined only up to multiplication by a scalar

2 2 2 value, we arbitrary choose r such that k1 + k2 + k3 = 1. The computed coefficients were k1: –

0.02±0.01, k2: –0.14±0.02, k3: 0.99±0.01 (the units of the second-order coefficients are arbitrary).

The signs of the coefficients were the same in all subjects. Clearly, the function with negative second-order coefficients cannot be a cost function, because it does not have any local minima for the available constraints.

Proof

Assume that the experimental data can be explained by a function additive with respect to the torques. Then, it must satisfy equation (A1) on the experimental data (A3). At the same time, ~ there exists an additive quadratic function G which satisfies (A1) on (A3) and its second-order coefficients have different signs. According to Uniqueness theorem, for the functions and holds (A2) and hence is also quadratic and its second-order coefficients also have different signs.

But then cannot be the cost function explaining the experimental data. Hence, is not additive with respect to the joint torques. It proves that no function additive with respect to the torques can explain the joint torque sharing observed in the experiments.

Quality of estimation of the experimental plane

One can suggest that since the experimental plane is not ideal (the first 2 PCs account for less than 100%), its orientation is known imprecisely and hence even if the plane (A3) cannot be explained by an additive cost function, it may happen that some other plane, very close to (A3), can.

This suggestion would be valid if the percentage of variance explained by the first 2 PCs was not so high. To illustrate this, we determined a plane, which could be explained by an additive cost function and which was the closest to the experimental plane in terms of the dihedral angle between

182 the two planes (Niu, Terekhov et al. 2011; Niu, Latash et al. 2012). For such a plane we determined the cost function and then for each combination of FX and FY we computed the torques this function predicted. The results for a representative subject are shown in the Figure A1.

APPENDIX B Figure A.1 The predictions of the joint torques by an additive cost function.

The predictions were good for the shoulder and elbow torques and they fell short

for the wrist torque.

One can see that, though such a cost function could rather well predict the shoulder and elbow torques, it failed completely for the wrist torque. Hence, it is not just a question of prediction quality: any cost function additive with respect to the joint torques will fail to reproduce the general tendency of the torque sharing. This, however, does not exclude that the data could be explained by another cost function, which would be, for example, additive with respect to the muscle forces, but not to the joint torques.

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Curriculum Vitae

Yang Xu

Email: [email protected]

29 Recreation Building,

Pennsylvania State University

University Park, PA-16802

EDUCATION:

The Pennsylvania State University - University Park, PA Ph.D. Biomechanics (ABD; expected graduation: December.2014)

Beijing Sport University M.S Sport Biomechanics, June 2010 B.S Kinesiology, June 2007

PEER-REVIEWED PUBLICATIONS

1. Xu Y, Terekhov AV, Latash ML, Zatsiorsky VM (2012) Forces and moments generated by the human arm: variability and control. Exp Brain Res. 223:159-175.

2. Xu Y, Terekhov AV, Latash ML, Zatsiorsky VM (2014) Effects of arm posture on the force and moment production. Journal of Motor Behavior (submitted)

3. Xu Y, Latash ML, Zatsiorsky VM (2014) Effects of size of handle on force and moments generated by the human arm. Currently being drafted.

4. Xu Y, Latash ML, Zatsiorsky VM (2014) Effects of surface friction of handle on force and moment generated by human arm. Currently being drafted.

5. Xu Y, Latash ML, Zatsiorsky VM (2014) Effects of prescribed grasp moments on force and moment generated by human arm. Currently being drafted.