EXSC 550 Liberty University PCA Vertical Jumping Article Review assignments.pdf

INSTRUCTIONS

Length of assignment: Minimum length is 3 pages.

*Each of the 6 sections (explained below) must contain no fewer than 100

words.

*Format: The assignment must be double spaced using Times New

Roman font in size 12 and written in APA format.

*Citations: No additional citations other than the article chosen to review

is required.

*Acceptable sources: Each article must be a primary research study

published within the last 10 years in a peer-reviewed research journal.

*Important: Article Review assignment must contain the following:

I.The article title, author(s), journal title, volume, issue, year, and page

numbers, all written in current APA format.

II.Include the following 6 sections (include section headings, do not

include instructions).

a. Objectives: Include background information on the topic and the goal

or purpose of the article.

b. Methods:Include how many subjects participated (male/female, age,

medical history if applicable), what biomechanical tools were used in the

study, and what statistics were used by the author(s).

c. Results: Explain key observations and/or findings in the article.

d. Summary:Include a synopsis of the discussion and conclusions of the

article and was free of direct quotes.

e. Critique:Include an opinion, explanation, and biblical integration.

III. Question/Application: This section must include 1 question pertaining

to how the article relates to another research article performed on the

research topic. Secondly, provide two facts you have learned from your

article review that would be interesting to share with a coach or a

clinician who is an expert in their field.

ORIGINAL RESEARCH

published: 08 August 2019

doi: 10.3389/fbioe.2019.00193

Frontiers in Bioengineering and Biotechnology | www.frontiersin.org 1August 2019 | Volume 7 | Article 193

Edited by:

Veronica Cimolin,

Politecnico di Milano, Italy

Reviewed by:

Francesco Travascio,

University of Miami, United States

Andrew D. Nordin,

University of Florida, United States

*Correspondence:

Emily J. Cushion

emily[email protected]

Daniel J. Cleather

[email protected]

Specialty section:

This article was submitted to

Biomechanics,

a section of the journal

Frontiers in Bioengineering and

Biotechnology

Received: 20 December 2018

Accepted: 24 July 2019

Published: 08 August 2019

Citation:

Cushion EJ, Warmenhoven J,

North JS and Cleather DJ (2019)

Principal Component Analysis Reveals

the Proximal to Distal Pattern in

Vertical Jumping Is Governed by Two

Functional Degrees of Freedom.

Front. Bioeng. Biotechnol. 7:193.

doi: 10.3389/fbioe.2019.00193

Principal Component Analysis

Reveals the Proximal to Distal

Pattern in Vertical Jumping Is

Governed by Two Functional Degrees

of Freedom

Emily J. Cushion 1

*, John Warmenhoven 2, Jamie S. North 1and Daniel J. Cleather 1

1Faculty of Sport, Health and Applied Science, St Mary’s University, Twickenham, United Kingdom, 2Exercise and Sports

Science, Faculty of Health Sciences, The University of Sydney, Lidcombe, NSW, Australia

The successful completion of motor tasks requires effective control of multiple degrees

of freedom (DOF), with adaptations occurring as a function of varying performance

constraints. In this study we sought to compare the emergent coordination strategies

employed in vertical jumping under different task constraints [countermovement jump

(CMJ) with arm swing-CMJas and no arm swing-CMJnas]. In order to achieve this,

principal component analysis (PCA) was conducted on joint moment waveform data

from the hip, knee and ankle. This statistical approach has the advantage of analyzing

the whole movement within a time series and reduces multidimensional datasets to

lower dimensions for analysis. Both individual and group analyses were conducted.

For individual analysis, PCA was conducted on combined hip, knee, and ankle joint

moment data for each individual across both CMJnas (thirty-eight participants), and

CMJas (twenty-two participants) conditions. PCA was also performed comparing all

data from each individual across CMJnas and CMJas conditions. The results revealed a

maximum of three principal components (PC) explained over 90% of the variance in the

data sets for both conditions and within individual and group analyses. For individual

analysis, no more than 2PCs were required for both conditions. For group analysis,

CMJas required 3PCs to explain over 90% of the variance within the dataset and CMJnas

only required 2PCs. Reconstruction of the original NJM waveforms from the PCA output

demonstrates a greater loading of hip and knee joint moments to PC1, with PC2 showing

a greater loading to ankle joint moment. The reduction in dimensions of the original data

shows the proximal to distal extension pattern in the sagittal plane, typical of vertical

jumping tasks, is governed by only 2 functional DOF, at both a group, and individual

level, rather than the typically reported 3 mechanical DOF in some forms of jumping.

Keywords: principal component analysis, vertical jumping, degrees of freedom, constraints, proximal to distal

pattern

Cushion et al. PCA of Vertical Jumping

INTRODUCTION

In both day-to-day and sporting contexts, human

beings consistently perform complex motor actions. Whether

this is walking up and down flights of stairs at home or

performing jumping actions in sport, people are able to perform

such skills with relative ease. Yet underpinning these observable

actions is a need to coordinate and control multiple degrees

of freedom (DOF), referred to as “the degrees of freedom

problem” by Bernstein (1967). With so many DOF within

the motor system, there are an almost limitless number of

coordination patterns available to the performer. However,

according to theoretical principles from ecological psychology,

and dynamical systems theory, it is proposed that the performer

seeks to satisfy the goal of the task whilst operating under

continually interacting task, environmental, and organismic

constraints, through which coordinated patterns of behavior

emerge (Newell, 1986; Hristovski et al., 2006; Orth et al., 2018).

Critical, therefore, in successfully completing such tasks is the

need to control the “redundant” DOF (Bernstein, 1967; Latash,

2000). These “redundant” DOF, however, provide important

functionality as they enable adaptation of emerging actions

to the varying performance constraints under which they are

operating. For example, Chow et al. (2008) demonstrated that

participants adapted their use of limb segments to continue

successfully performing a soccer chipping task whilst the task

constraints (which we define here as goals or rules which are

imposed that influence the movement) were manipulated.

Such a finding evidences how the redundant DOF allow for

functional variability and for performers to adapt to ever

changing performance constraints. Whilst there has been much

work within this area it is common for only kinematic changes

under varying constraints to be considered (e.g., Ketcham et al.,

2002; Seifert et al., 2007; Majed et al., 2017). However, analyzing

kinetic factors, such as forces and joint moments, would provide

a more complete understanding of how emerging behaviors are

shaped as a function of the task constraints under which the skill

is performed.

Explaining the control strategies adopted by performers has

been a topic within the motor control literature for many

years (see Dufek et al., 1995; Bates, 1996; James and Bates,

1997; James et al., 2003, 2014; Nordin and Dufek, 2019). It

has been suggested that certain movement features are more

important to satisfy the goal of the task, and therefore these

features are less likely to show variance between repetitions

or from person to person (Scholz and Schoner, 1999; Scholz

et al., 2000). This has been demonstrated using several tasks

(see Button et al., 2003; Seifert et al., 2011; Nordin et al.,

2017). For instance, elbow angle at ball release in a basketball

shooting task showed lesser variability between individuals of

varying skill levels, compared to other features such as wrist

angle at ball release (Button et al., 2003), which indicates that

elbow angle at ball release is an important movement feature

for successful performance in this task. Identification of such

movement features aids in developing our understanding of

motor control strategies employed to govern movements under

varying task constraints.

Traditional approaches to the study of human movement

have typically relied on selecting discrete data points for

analysis. However, these methods may not provide a complete

representation of the complexities of motor actions due to

data being discarded outside of those discrete points. Instead,

methods which analyse data across the full temporal cycle of the

movement, such as principal component analysis (PCA), may

prove more informative (Richter et al., 2014). The attraction

of PCA is that large amounts of data across movement cycles

can be analyzed whilst reducing the dimensionality of data to

features of importance (Daffertshofer et al., 2004). This statistical

approach has been used to determine key characteristics of

movement tasks (see for example Soechting and Flanders, 1997;

Kollias et al., 2001; Lamoth et al., 2006; Dona et al., 2009;

Witte et al., 2010; Federolf et al., 2013; Nordin and Dufek,

2016a) as well as determining similarities and differences between

groups through data filtering (see for example Deluzio et al.,

1997; Deluzio and Astephen, 2007; Laffaye et al., 2007; Lee

et al., 2009; Federolf et al., 2013; Verrel et al., 2013). Despite

the growth of PCA in sports sciences recently, its application

is still in its relative infancy. In particular, fewer researchers

have used this approach to determine coordination patterns

within movements or to compare movements under differing

task constraints (see Forner-Cordero et al., 2005; Lee et al.,

2009; Nordin and Dufek, 2016b; Majed et al., 2017 for some

exceptions). Therefore, applying a statistical approach such as

PCA can be useful in reducing large datasets to key movement

features and may be more effective in showing how movement

emerges under changing constraints.

Vertical jumping is an interesting motor task to study as

it requires the effective coordination of the upper and lower

limbs to raise the center of mass vertically, with a clear goal

of jumping for height (Raffalt et al., 2016). Vertical jumping

also represents a task which can be easily manipulated to

provide insight into how emergent behaviors adapt to differing

conditions. Analysis of movements under constrained conditions

can lead to a better understanding of the key mechanisms for

efficient movement, such as the requirement to keep balance in

a sit to stand task (Scholz et al., 2001) or the accommodation

strategies with increased task demands (Nordin and Dufek,

2016b, 2017). With this in mind, this study used a vertical

jumping task and asked participants to complete this action

under two different task constraints (with arm swing and without

arm swing), with the aim of understanding how motor control

strategies, and emergent coordination patterns were affected

by manipulation of these constraints. To fully understand

these coordination strategies and overcome the limitations

associated with traditional analysis methods, we employed PCA

to determine the principal components (PCs) that characterize

vertical jumping movement patterns both between and within

individuals and across the two task constraints. Specifically, we

aimed to answer the question of the number of PCs expected

to describe either a single jump, repetitions of a jump from an

individual, and across the whole group. Typically, lower limb

coordination in jumping tasks is described as an extension of

the lower limbs in a proximal to distal fashion, with the peak

moment progressing temporally from hip, to knee, and then

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Cushion et al. PCA of Vertical Jumping

ankle (Bobbert and van Ingen Schenau, 1988). A moment curve

is typically bell-shaped and unimodal (examples can be seen in

Figures 6,7). If the timing of peak moments and the shape of

the moment curves are independent of one another it would be

expected that 3 PCs would be required to represent the 3 moment

curves of a particular jump (that is 3 functional DOF). If the peaks

of the 3 moment curves occur at different time points along the

curve, then there needs to be more than one PC score.

MATERIALS AND METHODS

Participants

Thirty-eight healthy individuals (males =24, females =14)

volunteered to take part in this study (mean ±SD; age =

27 ±5 years, height =174.4 ±9.0 cm, body mass 78.1 ±

14.1 kg). Participants were free from musculoskeletal injuries and

were provided with details of the study before written informed

consent was obtained. The experimental procedure was approved

by the ethics sub-committee at the institution where the research

took place.

Procedure

Participants were required to attend one data collection session.

This involved the collection of anthropometric measures (height

and weight), before each participant was provided with a

standardized shoe according to their shoe size. Eighteen

reflective markers were placed on the pelvis and on the right

lower limb. Data from the right limb was used for further

analysis in accordance with previous work from Cleather et al.

(2013). Markers were placed on the right and left anterior

superior iliac spine and posterior superior iliac spine, lateral

and medial femoral epicondyle, apex of lateral and medial

malleolus, posterior aspect of calcaneus, tuberosity of fifth

metatarsal, and head of second metatarsal. Three additional

markers placed on rigid plates were attached to the mid-

thigh and anterior tibial shaft, with an additional marker

attached to the top of the foot. Kinematic data was collected

using a Vicon motion capture system (Vicon MX System,

Nexus 2.2 software, Vicon Motion Systems Ltd, Oxford, UK)

with 14 LED cameras tracking the reflective markers at a

sampling frequency of 200 Hz. Kinetic data was collected via

two force plates positioned flush to the laboratory floor (Kistler

Type 9287BA, Bioware 3.24 software, Kistler Instruments Ltd,

Hampshire, UK), at a rate of 1,000 Hz and synchronized with the

Vicon system.

Participants completed a standardized warm-up (bodyweight

squats, lunges, inchworms, hip rotations, and vertical jumps)

prior to completing any vertical jumps. Thirty-eight participants

performed a vertical jump task (countermovement jump with no

arm swing—CMJnas). For this task, participants were instructed

to keep hands in contact with the hip throughout the jump,

and to land with one foot on each force plate. Participants were

required to complete five maximal effort trials with a self-selected

recovery period between each jump to reduce any effects of

fatigue. Twenty-two of the participants carried out an additional

jumping task. For this task, participants performed a vertical

jump with the use of an arm swing (CMJas). Participants were

instructed to use an arm swing to aid in jumping performance

and were required to land with one foot on each force plate. For

these participants the order of completion of the two tasks was

counterbalanced to prevent order effects. Participants were given

a 2 min recovery period between the two tasks.

Data Analysis

All data was filtered using a 5th order Woltring filter with a

cut off frequency of 10 Hz. The propulsive phase of the vertical

jump was used for analysis and was defined as being from

the point where the right anterior superior iliac spine marker

moved below stationary height until take-off (which was defined

as the point where the ground reaction force fell to zero).

Net joint moments (NJM) in the sagittal plane were calculated

for hip, knee, and ankle during this phase using a standard

inverse dynamics calculation (Winter, 2005) within the FreeBody

software (Cleather and Bull, 2015). Each jump trial therefore had

the 3 mechanical DOF described by the NJM.

As trial length varied across each participant, data was spline

interpolated and time normalized from 0 to 101 data points.

Maximal jump height was analyzed through calculation of the

change in displacement of the right anterior superior iliac spine

when standing, to the maximum height achieved during the jump

(Chiu and Salen, 2010).

PCA Calculation

PCA was used in this study to extract common patterns of

moment production during the vertical jump under two task

constraints. Using this approach has the advantage of retaining

the spatiotemporal pattern in the time series data whilst detecting

coordination patterns both within and between individuals. The

fundamental purpose of a PCA is to find a linear transformation

that maps the raw data described in its original coordinate

frame to a new coordinate frame with orthonormal bases. In

the context of data analysis, the coordinate frame for the raw

data is defined by the measured variables, but these variables

may have some degree of correlation with one another. The

new coordinate frame that is given by the PCA will be defined

by a set of new uncorrelated variables called the principle

components (PC). For instance, for a dataset consisting of p

variables observed at ndifferent time-points the raw data can

be described by the n×pmatrix Xwhere the columns of

Xare the individual variables, and the rows represent each

observation (time-point). The transformation Uthen maps the

raw data to the new coordinate frame defined by the PCs,

such that the raw data in the new coordinate frame Z, is given

by Z=XU.

In biomechanics, PCA has sometimes been used to compare

time-normalized waveforms. For instance, a waveform can be

described by the value of a variable sampled at 1% intervals over

the course of a movement. Typically, we have multiple waveforms

(cases) for one particular biomechanical variable which we want

to compare—for instance, if we want to compare the curves

of different individuals. In this instance, as outlined extensively

in Deluzio and Astephen (2007), the individual time-points

have often been considered as variables and cases (observations)

considered to be the individual curves being analyzed. Thus if

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Cushion et al. PCA of Vertical Jumping

we had pdifferent curves for a given variable, that each consists

of 101 time-points, we would perform the PCA on the p×101

raw data matrix XT. In the present study we have adopted a

more intuitive approach to study waveforms that is closer to

the traditional description of PCA. We have treated the time

normalized intervals as observations, and the individual curves

(cases) as variables. Thus, for our pdifferent curves for a given

biomechanical variable sampled at 101 time-points we perform

the PCA on the 101 ×praw data matrix X. This approach has

been used previously (see Borzelli et al., 1999).

It should be noted that in the method of Deluzio and Astephen

(2007), a separate PCA is performed for each of the measured

variables. In contrast, in this study we have included all of

the measured variables within the same PCA. For instance, in

this study, for an individual subject we have time-series for 3

biomechanical variables (hip, knee, and ankle NJM) measured in

5 different jumps. Using the method of Deluzio and Astephen

(2007) we would perform 3 PCAs on the 3 5 ×101 matrices

representing each joint. Instead, in this study we perform only

one PCA on the 101 ×15 matrix which includes all of the

individual’s data.

One common application of PCA is to reduce the

dimensionality of the data being analyzed—that is to reduce the

number of variables under consideration to a smaller set which

describe the majority of the variance in the data. The advantage

of the method we have adopted here is that the coordinate frame

in which the data is described is defined by all of the cases for all

of the measured joints (NJM). This means that the PCA produces

a reduced new coordinate frame that can be used to describe

all cases and joints. This therefore means that the reduced

coordinate system represents the number of functional degrees

of freedom that are present in the movement being analyzed.

Conventional approaches to using PCA would require each

variable (e.g., hip NJM, knee NJM, ankle NJM) to be analyzed

separately. This would limit the ability to effectively interpret the

functional DOF describing the movements. Thus, for the present

study PCA was used in this manner to extract common patterns

of moment production during the vertical jump under two task

constraints. Only those PCs that cumulatively explained over

90% of the variance in the data set were retained and used in

further analysis (Jolliffe, 2002).

Prior to running the PCA, all data was normalized to the

peak hip joint moment of each trial (Jolliffe and Cadima, 2016).

Analysis of the data was performed on both an individual (termed

analysis 1), and group (termed analysis 2) basis (as has previously

been conducted within the literature, e.g., Nordin and Dufek,

2016a,b, 2017) and the PCs from the individual (PCi), and group

(PCg) analyses are indicated by subscripts as shown. For the

individual analysis a PCA was carried out for each individual

separately, which contained hip, knee and ankle NJM data

combined (101 ×15 matrix for each individual containing 15

columns representing 5 hip, 5 knee, and 5 ankle joint moment

time series with 101 data points). This was performed for both

CMJnas and CMJas conditions. For the group analysis, all NJM

data from each participant were combined and subjected to a

PCA (CMJnas: 101 data points ×567 joint moment time series,

CMJas: 101 data points ×330 joint moment trials). A further

FIGURE 1 | Variance explained by (A) PC1i, (B) PC2i, and (C) the sum of

PC1i&2i for CMJnas, and CMJas. Horizontal line represents group mean.

i=individual PCA analysis. *Indicates significant difference from CMJnas.

PCA was performed to compare CMJnas and CMJas data with

only the participants who had performed both jumps (101 data

points ×672 joint moment time-series). This is referred to as

analysis 3. Finally, an additional group level PCA was conducted

using only the first 2 PCs obtained from the individual PCA

(CMJnas: 101 data points ×76 PCs, CMJas: 101 data points ×

44 PCs; hereafter called analysis 4).

The output of each PCA was a matrix (henceforth named

the coefficients matrix) where each column gives the coefficients

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Cushion et al. PCA of Vertical Jumping

FIGURE 2 | PC1i scores for (A) CMJnas and (B) CMJas from 22 participants completing both conditions. Data is normalized to peak value. i =individual PCA analysis.

FIGURE 3 | PC waveforms along with percentage of explained variance for CMJnas (A,B) and CMJas (C,D) conditions. Data in (C) has been flipped to a negative

peak to more clearly show comparison between the shape of waveforms in (A,C). PC1, 2, and 3a represent data obtained from PCA using only the first two PCs from

each individual (analysis 4). PC1, 2, and 3b represent data obtained from group PCA for CMJnas, and CMJas (analysis 2).

of a PC, and the PCs are ordered in terms of the amount of

variance they explain. The coefficients matrix thus gives the

transformation of the raw data into the variable space defined

by the PCs. Thus, the matrix obtained by multiplying the raw

data matrix by the coefficients matrix is a matrix describing

the time-series of the values for each PC. These time series are

reported as the PC scores in this study. Similarly, the raw data

can be reconstructed based upon a limited number of PCs by

transforming the selected number of PCs back to the variable

space defined by the raw data. This was performed for CMJnas

data as an example. All PCA calculations were performed in

Matlab (The MathWorks, Inc., MA, version 2017a).

Statistical Analysis

Further statistical analysis was performed to compare the

variance explained by each PC attained from the individual

analysis of both CMJnas and CMJas. Group differences between

PC1, PC2, and the sum of PC1 and PC2 were assessed using an

independent sample t-test. The alpha level was set at p<0.05.

RESULTS

When considered individually, only one or two PCs were

required to describe at least 90% of the variance in the data.

Data from each individual PCA showed PC1i and the sum of

PC1i, and 2i explained a significantly greater amount of the

variance in CMJnas than CMJas (p=0.05, p=0.00, respectively;

Figure 1). The greater variation in PC1i scores for CMJas can be

seen in Figure 2.

When all joint moment data was combined in the same

PCA, three PCs were retained for the CMJas condition, and

two were retained for the CMJnas condition (Figure 3). Figure 3

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Cushion et al. PCA of Vertical Jumping

also illustrates waveforms from the group analysis PCA that just

retained the first two PCs for each individual (analysis 4).

To make a robust comparison between jumps at the group

level, data was reanalyzed with only those participants who

completed both CMJnas and CMJas conditions (Figure 4,

analysis 3). This highlighted the same trend with CMJas requiring

three PCs and CMJnas requiring only two PCs to describe over

90% of the variance. Waveforms for each jump condition were

similar to those observed in Figure 3.

An analysis was also conducted with only those participants

who completed both CMJnas and CMJas conditions with all joint

moment data combined (Figure 5). Only two PCs were required

to explain over 90% of the variance.

Figure 6 presents an illustrative example of the reconstruction

of a typical jump from PC1g and PC2g—and shows how the

sum of the time series for PC1g and PC2g describe such a high

proportion of the variance in the raw data.

Figure 7 presents an illustrative example of the reconstruction

of a series of five jumps from a representative participant

performing CMJnas. The figure shows the sum of the time series

for PC1i and PC2i describe a high proportion of the variance in

the raw data.

DISCUSSION

The purpose of this study was to use PCA to characterize

movement strategies, through reducing a large data set of

hip, knee, and ankle NJMs measured during vertical jumping.

The main results of this study were that for all comparisons

considered here, 90% of the variance in the raw data could be

described by at most 3 PCs, and in many cases by 2. Whilst

there are multiple sources of variation within the dataset (e.g.,

joint, trial, individual), we suggest that the PCs here largely

describe the variance in joint moments during jumping, and

suggest that the joint moment production both within, and

between individuals and within each task, share similar structural

patterns. Further to this, the dimensionality of sagittal plane

lower limb extension in some forms of vertical jumping can be

reduced to only 2 functional DOF. This is an important result—

it indicates that simply providing the participants with the task

constraint “jump as high as you can” produced a markedly

similar pattern, and that between and within participant variation

amounted to <10%. This is a highly non-trivial result and

provides insight to the nature of motor control within this task—

although it has previously been suggested that during jumping,

people exhibit a characteristic proximal to distal pattern of joint

moment production (Bobbert and van Ingen Schenau, 1988), the

degree of similarity has not been emphasized.

When considering the group and individual analysis for

CMJnas only 2 PCs are required to explain over 90% of

the variance (Figure 3). This would suggest the relationships

between curves are not just unique to specific individuals but are

demonstrated by the whole group. If this was not the case further

PCs would be required to describe the movement. For CMJas an

additional PC was required at the group level in order to capture

90% of the variance in the data (see Figure 3), although it was

FIGURE 4 | Comparison of PC waveforms for CMJnas and CMJas for the 22

participants who completed both jumps: (A) PC1, (B) PC2, and (C) PC3.

CMJas waveforms have been flipped and peaks matched to show differences

in temporal pattern.

not required at the individual level. These results are consistent

with previous research findings which have also looked to

determine how the high dimensionality of human movement can

be reduced to low dimensional states (see Soechting and Flanders,

1997; Borzelli et al., 1999; Balasubramaniam and Turvey, 2004;

Witte et al., 2009; Nordin and Dufek, 2016a; Majed et al., 2017),

For example, (Santello et al., 1998) have shown that in a hand

grasping task only two PCs were required to describe over 80% of

the variance in the data, demonstrating few synergies govern the

production of hand motion in the specific grasping task. Equally,

in a sit to stand task only two PCs were required to describe

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Cushion et al. PCA of Vertical Jumping

FIGURE 5 | Comparison of PC waveforms for CMJnas and CMJas for the 22

participants who completed both jumps along with PC waveform of combined

CMJnas and CMJas analysis (analysis 3): (A) PC1, (B) PC2. CMJas

waveforms have been flipped and data peaks matched to show differences in

temporal pattern.

the task from six components of ground reaction force data (see

Borzelli et al., 1999).

It is interesting to note that at only the group level an

additional PC was required to describe over 90% of the variance

within the dataset for CMJas. The need for the third PC is

indicative of the greater variability in the CMJas data. This

greater variability is seen both within individuals (see Figure 1)

and between individuals (see Figure 2), where a significant

difference in explained variance for PC1 was observed between

CMJnas and CMJas. In contrast, in analysis 4 (which essentially

eliminates the effect of within individual variation on the group

scores) only 2 PCs are required to explain 90% of the variance.

This improvement in predictive value is gained in the variance

explained by PC1g. It is interesting to consider the shape of

the PC3g score and think about the linear sum of the PC1g

score with PC3g score. In particular, the effect of adding PC3g

to PC1g is to make the curve less flat and to make it more

peaked toward the end of the jump. This type of variation is also

seen between individuals (Figure 2). It is also worth noting that

the requirement for a third PC for the group analysis provides

further support for the fact that the reduction to 2 PCs seen

for CMJnas is not trivial. The increase in variability within

the CMJas data could be related to unfamiliarity with the task

or complexity of the task for the participants involved. It has

previously been demonstrated an increase in complexity of a

task results in an increase in dimensionality and a reduction in

explained variance within the first principal component (Zago

et al., 2017). Whilst participants within this study were familiar

with jumping, the requirement to use an arm swing may have

added complexity to the task such that they were required

to explore movement options to a greater extent than in the

constrained CMJnas condition.

Results from the present study suggest the first PC represents

the hip and knee joint moment whereas the second PC represents

the ankle. Reconstruction of the raw joint moments from the

PCA (Wootten et al., 1990; Troje, 2002) can aid in visualizing this.

The reconstruction of the moment curves for a representative

individual (see Figure 6), shows the score for PC2i featuring

heavily in the linear sum of the reconstructed curve for the

ankle moments. Similarly, the first PC curve shares a similar

pattern to either the hip or knee joint moment curve—this varied

depending on the individual (for the individual in Figure 6 it

was hip-like). The relationship between the two PCs can also

be seen in Figure 7. Consider, for instance, the ankle moments

and compare jump1 to jump4. Here jump4 shows a flatter peak

compared to jump1 and a much wider curve. Consideration of

the 2 PC scores shows how these 2 quite different ankle moment

curves can still be constructed. In particular, the scores for PC1i,

and PC2i are out of phase, however the difference in timing

between them is such that they can still be summed to produce

all the different ankle curve combinations. To put this another

way, given the differences in the shape of the ankle curves we

know that at least 2 PCs would be needed to reconstruct them.

However, the 2 PCs that we obtain not only allow all of the

ankle moment curves to be created, but also all the hip and knee

moment curves. This synergy implies a relationship between the

different joint moment curves that persists across jumps.

This concept of synergy is not unique to this study and

demonstrates an approach of the system to optimize control of

motor tasks (see Santello et al., 1998; Jaric and Latash, 1999;

St-Onge et al., 2004; Todorov and Ghahramani, 2004; Latash

et al., 2007; Latash, 2010; Kipp et al., 2012; Nordin and Dufek,

2017). This organization of movement to synergies may occur

due to mechanical constraints (defined as limits in the options of

motion that can be achieved due to the mechanical configuration

of the human body) on the system that determine the movement

pattern—so called mechanical intelligence (Blickhan et al., 2006).

Within this study it is suggested that the proximal to distal

pattern of moment production (in the sagittal plane) during

vertical jumping might be governed by only 2 functional

DOF and this may be related to mechanical constraints. For

instance, Cleather (2018) has recently shown how the geometry

of the patellofemoral joint alone can explain over 90% of the

variance in the relative timing of femoral and tibial segment

moments (Cleather, 2018). To put that observation in terms

of the findings reported here, the patella could provide a

mechanical constraint that reduces 2 functional DOF to 1—

the hip and knee moments are largely dependent on one

another. This would then represent a potential explanation for

the remarkably consistent timing of moment production here. It

seems more credible that such consistency would be produced

by a mechanical feature of human anatomy that is shared by

all (a shared organismic mechanical constraint), than it being

Frontiers in Bioengineering and Biotechnology | www.frontiersin.org 7August 2019 | Volume 7 | Article 193

Cushion et al. PCA of Vertical Jumping

FIGURE 6 | Original NJMs (raw data) for a typical CMJnas compared with reconstructed data using only the first two principal components. The reconstructed curve

in (C) is the linear sum of the curves in (A,B).

the result of everyone learning a pattern of motor coordination

during their development that is so markedly consistent. A

similar conclusion of mechanical constraints reducing the

dimensionality of movement patterns has been proposed in a

similar, albeit much slower task of sit to stand (Borzelli et al.,

1999) as well as in drop landings (Nordin and Dufek, 2017).

An additional comparison between CMJnas and CMJas was

made between only those individuals who had completed both

forms of the jump (Figures 4,5, analysis 3). This comparison

demonstrates that the PC2g scores are effectively the same. This

can be interpreted as suggesting that the ankle is being used in

the same way, and that the potential effect of the ankle on the

relative timing of hip and knee NJM is also the same. However,

there is a notable difference in the PC1g scores. Specifically,

Figure 4A shows PC1g curves between each jump. This shows

a PC1g curve for CMJas to exhibit a flatter peak along with a

greater area under the curve compared to the CMJnas condition.

This is likely to be reflective of the use of the arm swing to slow

joint extension and thus provide greater time for the musculature

to develop force (Feltner et al., 2004; Domire and Challis, 2010).

Previous literature (e.g., Lees et al., 2004; Hara et al., 2006)

has suggested that the use of the lower limb in CMJnas and

CMJas is distinct. The results reported here suggest that although

this is true, the motor control strategy only differs in terms of

the first PC, and that there is still a striking similarity between

the movements.

Whilst interesting and novel findings have been identified

within this study, limitations of this work should be highlighted.

Although statistically rigorous procedures have been employed,

the analysis of the results is in some parts interpretative. This is a

limitation of trying to describe the mechanics of movement from

statistical techniques—although it should be noted that we are

not alone in using this technique to tease out different sources

of variation (e.g., Daffertshofer et al., 2004). In particular, there

are multiple sources of variation—joints (hip, knee or ankle),

participant, trial, and type of jump (with or without arms). Our

contention in this paper is that the PCs primarily represent the

variation in moment production between different joints. This

can be inferred based upon the following reasoning. Firstly, the

PCA analysis was first performed considering each individual

separately—thus eliminating inter-individual variability—this

analysis produced similar PC1i and PC2i for each participant

suggesting that the PC1g and PC2g are not produced by inter-

individual variability. Secondly, the group PCA analysis was

performed on both the whole data set, and on a reduced data

set comprising the resulting PCs from the individual analysis.

Inter-trial variation for each individual would be to a large part

mitigated in the latter data set, yet there was little difference in

the results of these two analyses (Figure 3). This suggests that

inter-trial variability is not a major factor in the group PCs.

Thirdly, the variation based upon type of jump was quantified

by the comparison of CMJas and CMJnas in Figure 4, and the

Frontiers in Bioengineering and Biotechnology | www.frontiersin.org 8August 2019 | Volume 7 | Article 193

Cushion et al. PCA of Vertical Jumping

FIGURE 7 | Original NJMs (raw data) for a typical CMJnas compared with reconstructed data using only the first two principal components, for a representative

participant across five jump trials.

same PCs were found for both jump types. Having eliminated 3

of the 4 sources of variation, we suggest that the PCs from the

group analysis describe variation in joint moment production.

Of course, this contention has not been “proven” statistically,

but, we argue, is the only reasonable explanation. Furthermore,

Figures 6,7demonstrate the reconstruction of joint moments

from the PCA. This shows the similarity of the reconstructed

curves to the original raw joint moment curves. If variance was

predominantly from individual differences in jumping style, the

ability to so closely reconstruct the moment curves would likely

not occur. In addition, whilst the focus of this research was

on the analysis of the lower limb during jumping, based on

their large contribution to the movement, future research would

benefit from conducting a whole-body analysis to determine the

dimensionality of the system.

In conclusion, the current study has shown the applicability of

using a PCA to analyse complex multi-joint tasks and successfully

compare between similar movements. As discussed, there are

several key outcomes from the use of a PCA within this study.

This analysis method was effective in reducing the statistical

and mechanical dimensionality of the data in jumping tasks and

showed a maximum of three PCs were required to describe 90%

of the variance in the original data set for a group analysis.

Specifically, 3 PCs were required to explain the variance within

CMJas group data and only 2 PCs were required for CMJnas

group data. The first PC has a greater loading for hip and knee

Frontiers in Bioengineering and Biotechnology | www.frontiersin.org 9August 2019 | Volume 7 | Article 193

Cushion et al. PCA of Vertical Jumping

joint moments, whereas the second PC has a greater loading

for the ankle joint moment. The results suggest the proximal to

distal extension pattern in the sagittal plane typical of vertical

jumping tasks is governed by only 2 functional DOF rather than

the previously reported 3 mechanical DOF. This study adds to the

motor control literature suggesting that despite the redundancy

within the system coordinated movements are produced based

on task and mechanical constraints. Importantly, the results

suggest vertical jumping is controlled by 2 functional DOF at

both an individual and group level. We suggest the reduction

in dimensionality of the movement may be mechanically driven

by human anatomy.

ETHICS STATEMENT

All subjects gave written informed consent in accordance with the

Declaration of Helsinki. The protocol was approved by the ethics

sub-committee of St Mary’s University.

AUTHOR CONTRIBUTIONS

EC, JN, and DC were involved in the design of the

study. EC carried out all data collection. All authors

were involved in the interpretation and final write

up of the study.

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Conflict of Interest Statement: The authors declare that the research was

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