**A Correlational Analysis of Shuttlecock Speed Kinematic Determinants in the Badminton Jump Smash**

#### **Mark King 1, Harley Towler 1,\*, Romanda Dillon <sup>1</sup> and Stuart McErlain-Naylor 1,2**


Received: 31 December 2019; Accepted: 10 February 2020; Published: 13 February 2020

**Featured Application: The findings suggest that players and**/**or coaches should focus on proximal segment movements: specifically, producing greater pelvis-thorax separation during the retraction phase and greater shoulder internal rotation at shuttle contact to increase shuttlecock speed.**

**Abstract:** The forehand jump smash is an essential attacking stroke within a badminton player's repertoire. A key determinate of the stroke's effectiveness is post-impact shuttlecock speed, and therefore awareness of critical technique factors that impact upon speed is important to players/coaches. Three-dimensional kinematic data of player, racket and shuttlecock were recorded for 18 experienced players performing maximal effort forehand jump smashes. Joint angles and X-factor (transverse plane pelvis-thorax separation) were calculated at key instants: preparation, end of retraction, racket lowest point, turning point and shuttlecock contact. Peak shoulder, elbow, and wrist joint centre linear velocities, phase durations and jump height were also calculated. Correlational analyses were performed with post-impact shuttlecock speed, revealing significant correlations to peak wrist joint centre linear velocity (r = 0.767), acceleration phase duration (r = −0.543), shoulder internal/external rotation angle at shuttlecock contact (r = 0.508) and X-factor at the end of retraction (r = −0.484). Multiple linear regression analysis revealed 43.7% of the variance in shuttlecock speed could be explained by acceleration phase duration and X-factor at the end of retraction, where shorter acceleration phase durations and more negative X-factor at end of retraction caused greater shuttlecock speeds. These results suggest that motions of the proximal segments (shoulder and pelvis–thorax separation) are critical to developing greater distal linear velocities, which subsequently lead to greater post-impact shuttlecock speed.

**Keywords:** velocity; technique; overhead; racket; swing; stroke

#### **1. Introduction**

The forehand smash is an effective attacking shot in badminton, accounting for 54% of "unconditional winner" and "forced failure" shots in international matches [1]. Success of the stroke is dependent upon two components: speed and direction, where speeds as high as 89.3 <sup>±</sup> 7.2 m·s−<sup>1</sup> have been reported in the literature for elite Malaysian players [2], whereas the competition world record is 118 m·s−<sup>1</sup> [3]. A shuttlecock with a greater post-impact speed will give an opponent less reaction time, while directing the shuttlecock away from the opponent requires them to make fast reactive movements in order to return the shuttlecock.

Several studies have determined that both linear and angular velocities of distal segments (hand and racket) are strong positive correlates with shuttlecock speed [2,4], which is unsurprising due to higher velocities causing greater transfer of linear momentum to the shuttlecock. Explaining what kinematic parameters determine this greater distal velocities may be more useful for players, coaches and practitioners, as the greater joint powers associated with distal segments are not fully generated by the muscles associated with these segments, but transferred between segments through reaction forces of a proximal-to-distal nature, e.g., shoulder–elbow–wrist [5]. Varied experimental set-ups and methodologies of quantifying joint contributions have generated conflicting results, where Rambely et al. [4] suggested that a distal-to-proximal order of wrist, followed by the elbow and shoulder are the major contributors to racket head speed (26.5%, 9.4% and 7.4%, respectively). In contrast, Liu et al. [6] reported that a proximal-to-distal order of shoulder internal rotation, forearm pronation and wrist palmar flexion contribute 66%, 17% and 11%, respectively, towards racket head speed. Rambely et al. [5] captured video data at 50 Hz and calculated the resultant linear velocities of the shoulder, elbow and wrist joint centres at impact, and expressed the segment contributions as these velocities represented as percentages of the racket head centre resultant velocity at impact. Conversely, Liu et al. [6] used a three-dimensional kinematic method developed by Sprigings et al. [7] and calculated the relative angular velocities of distal segments by removing contributions from the adjacent proximal segment. Data were captured using high-speed video (200 Hz) and processed using direct linear transformation [8]. Teu et al. [9] found no specific proximity order using a dual Euler angles method with high-speed video (200 Hz), and reported torso rotation, shoulder internal rotation, forearm pronation and wrist abduction contributed 57%, 3%, 27% and 10% to resultant racket head velocity, respectively.

In summary, previous research has reported that linear velocities of the distal segments best explain variation in shuttlecock speed/racket head speed, however it is unclear how the distal segment velocities and subsequent racket head and shuttlecock speeds are generated. Additionally, use of low frame rates and unclear methodology for defining both shuttlecock speed and racket head speed mean that it is difficult to compare results. The present study therefore aims to identify full-body kinematic parameters that best explain the generation of post-impact shuttlecock velocities in the badminton jump smash, such that coaches/practitioners can advise players how best to increase smash speeds through technique and/or strength training. It is hypothesised that positions (joint angles) of proximal segments and linear velocities of more distal segments will best explain variation in post-impact shuttlecock speed.

#### **2. Materials and Methods**

#### *2.1. Participants*

Eighteen male badminton players (mean ± SD: age 24.3 ± 7.1 years, height 1.84 ± 0.08 m, mass 79.6 ± 8.8 kg) of regional (n = 9), national (n = 4) and international (n = 5) standards participated in this study, each performing a series of twelve forehand jump smashes from a racket-fed lift via an international coach/player, representative of match conditions. A range of abilities were used to provide a variety of maximal smash speeds, facilitating an investigation into causal factors associated with this variation. Testing procedures were explained to each participant, and informed written consent was obtained in accordance with the guidelines of the Loughborough University Ethical Advisory Committee (SSEHS-1959).

#### *2.2. Data Collection*

An eighteen-camera Vicon Motion Analysis System (400 Hz; OMG Plc, Oxford, UK) was used to collect three-dimensional kinematic data of the participant, racket and shuttlecock on a mock badminton court within a hall of sufficient height. Forty-seven 14 mm retroreflective markers were attached to the participant (Figure 1), where joint centres were calculated from a pair of markers placed across the joint so that their midpoint coincided with the joint centre [10]. Hip, thorax, neck and head joint centres were calculated according to Worthington et al. [11]. A further marker was placed on the bottom of the racket handle, seven pieces of 3M Scotchlite reflective tape were attached to the racket frame and a single piece of reflective tape was attached around the base of the cork of the shuttlecock (Figure 2). Participants used their own racket and new Yonex AS40 shuttlecocks throughout, where misshapen or broken shuttlecocks were removed.

**Figure 1.** Participant marker locations.

**Figure 2.** Racket and shuttlecock marker locations.

#### *2.3. Data Reduction*

Position data were labelled within Vicon Nexus 1.7.1 where gaps within marker trajectories were filled using the "pattern-filled" function where possible, and "spline-filled" function thereafter. All position data were then imported into Matlab v.2018b (The MathWorks Inc., Natick, MA, USA) for all further processing. Position data of all body markers were filtered using a fourth-order, zero-phase, low-pass Butterworth filter with a cut-off frequency of 30 Hz, determined through residual analysis [12]. Racket and shuttlecock markers remained unfiltered to avoid double-filtering during subsequent curve-fitting methodologies.

Joint angles were calculated using three-dimensional rotation matrices, defining the rotation applied to the proximal segment coordinate system to bring it into coincidence with the coordinate system of the distal segment [12]. An XYZ rotation sequence was used, representing flexion–extension, adduction–abduction and longitudinal axis rotation, respectively. When describing humerothoracic motion, a YZY rotation sequence was used as recommended by ISB [13], using a different coordinate system for the humerus segment where xyz represented adduction–abduction, longitudinal axis rotation and flexion–extension, respectively. Wrist angles were normalised based on the player adopting their normal grip within a static trial, which was considered the neutral position. The mean offset was 28.8 ± 8.4◦ and 7.1 ± 7.4◦ of palmar extension and ulnar deviation, respectively. X-factor referred to the separation angle between vectors connecting the right and left shoulder joint centres and, the right and left hip joint centres, respectively, in the transverse plane [14]. Centre of mass was calculated using segment inertial values from de Leva [15], where the body was modelled as fourteen segments. Jump height was defined as the difference between the maximum centre of mass height and the height of that during a static standing trial. Table 1 details the joint angles calculated.


**Table 1.** Calculated joint angles, with their relative zero positions and positive directions.

† For further details on motion of the shoulder joint as a result of the ISB rotation sequence recommendations, see Wu et al. [13]. ‡ Positive direction relates to the direction of rotation during the forward swing of the movement, i.e., anticlockwise for a right-handed player. Note transverse plane is viewed from above.

Instantaneous post-impact shuttlecock speed and racket–shuttlecock contact timing were determined using a logarithmic curve-fitting methodology [16] with minor adjustments for the application to the badminton smash. The time of impact was derived from the intersection of preand post-impact shuttlecock displacement curves in the global anterior–posterior direction (dominant direction of the smash), and an intermediate 1 ms contact period [17] was added between the preand post-impact shuttlecock curves, where the racket face and shuttlecock velocity were assumed equal. Post-impact shuttlecock speed was determined via differentiation of the post-impact logarithmic shuttlecock displacement curve. Racket head speed was the component of the linear racket head centre linear velocity acting perpendicular to the racket stringbed. The pre-impact racket head speed data was then interpolated to the calculated time of initial shuttlecock contact.

Based upon previous literature within badminton, the movement was defined around two phases, defined as backswing and acceleration phases [18]. Five discrete instants were identified such that trials could be compared appropriately: preparation (P) was defined as the point at which centre of mass height was minimal [2]; end of retraction (ER) was defined as the point at which the racket was most medio-laterally positioned towards the non-dominant side of the participant within the global

coordinate system; racket lowest point (RLP) when the racket tip was at its lowest vertical point [19]; and turning point (TP) was defined as the point, after minimum (most negative), at which the racket head speed became positive [4] and shuttlecock contact (SC) was defined as the closest motion capture frame to the previously defined instant of racket-shuttlecock contact. The backswing phase (BSP) was defined as the time between PP and TP, whereas the acceleration phase (AP) was defined as the time between TP and SC.

Five joint angles (Table 1) were calculated for each trial, describing the elements of badminton smash technique which have previously been linked to shuttlecock velocity in literature or thought to be linked to shuttlecock velocity. Joint angles were defined at each key instant and their maximum range of motion through to SC calculated, e.g., maximum external rotation angle through to angle at contact. Furthermore, post-impact shuttlecock speed; racket head speed at impact; jump height; and peak shoulder, elbow and wrist joint centre linear velocities were calculated for each trial. Length of phases (BSP and AP), as well as total swing time, were also calculated for each trial. All kinematic variables for each player's trial with the greatest shuttlecock speed were entered into subsequent correlation analyses.

#### *2.4. Statistical Analysis*

All statistical correlational analyses were performed in Matlab v.2018b (The MathWorks Inc., Natick, MA, USA). Pearson product moment correlation analyses were performed between each kinematic (independent) variable and shuttlecock speed. Pearson product moment correlations (*r*) and their 95% confidence intervals (CI) were interpreted as negligible < 0.3; 0.3 ≤ low < 0.5; 0.5 ≤ moderate < 0.7; 0.7 ≤ high < 0.9; very high ≥ 0.9 [20]. An alpha value of 0.05 was used to determine significance. Correlates were then entered as "candidate" variables into a forwards stepwise multiple linear regression model to identify key kinematic parameters that best explain variation in shuttlecock speed. Entry requirements for inclusion of a parameter were *p* < 0.05, with a removal coefficient of *p* > 0.10. The regression model was rejected if the 95% CI coefficients included zero, the residuals of the predictor were heteroscedastic or if the bivariate correlations, tolerance statistics or variance inflation factors showed any evidence of multicollinearity [21–25]. The normality of the standardised residuals in the regression model was also confirmed using the Shapiro–Wilk test. The percentage of variance in the dependent variable explained by the independent variables (predictors) within the regression equation was determined by Wherry's adjusted R2 value [26]. Multiple linear regression analysis was performed in IBM SPSS Statistics 23 (IBM, Armonk, NY, USA).

#### **3. Results**

Maximal shuttlecock speeds for the cohort were 89.6 <sup>±</sup> 5.3 m·s−<sup>1</sup> (range: 80.1–99.8 m·s<sup>−</sup>1). Racket head speeds at SC achieved during these trials were 56.3 <sup>±</sup> 4.0 m·s−<sup>1</sup> (range: 46.7–64.6 m·s−1). Five kinematic variables were significantly correlated with shuttlecock speed, where a greater racket head speed, greater peak wrist joint centre linear velocity and shorter acceleration phase duration were associated with greater shuttlecock speeds. Likewise, greater shoulder internal rotation at SC and more negative X-factor at ER produced greater shuttlecock speeds. No other variables were significantly correlated to shuttlecock speed (Tables 2–4). Means and standard deviations, as well as a time-normalized comparison between the "fastest" and "slowest" participant for shoulder internal rotation angle, X-factor and racket head speed are shown in Figures 3–5, respectively.


**Table 2.** Pearson product moment correlation (r) between each of racket head speed, jump height, phase durations, joint centre linear velocities and ranges of motion, and shuttlecock speed.

Abbreviations: CI: confidence interval, BSP: backswing phase, AP: acceleration phase, TS: total swing, JC LV: joint centre linear velocity, IR: internal rotation, ROM: range of motion, PRO: pronation. \* Significant (*p* < 0.05) .

**Table 3.** Pearson product moment correlation (r) between joint angles of the shoulder and elbow at key instants and shuttlecock speed.


Abbreviations: KI: key instant, CI: confidence interval, INT: internal, EXT: external, P: preparation, ER: end of retraction, RLP: racket lowest point, TP: turning point, SC: shuttlecock contact. \* Significant (*p* < 0.05).


**Table 4.** Pearson product moment correlation (r) between joint angles of the wrist and X-factor at key instants and shuttlecock speed.

Abbreviations: KI: key instant, CI: confidence interval, P: preparation, ER: end of retraction, RLP: racket lowest point, TP: turning point, SC: shuttlecock contact. \* Significant (*p* < 0.05).

**Figure 3.** (**a**) Mean (solid line)±SD (shaded area) for the normalised racket arm shoulder internal/external rotation angle between preparation and shuttlecock contact key instants. (**b**) Normalised racket arm shoulder internal/external rotation angle between preparation and shuttlecock contact key instants, for participants with the fastest (solid line) and slowest (dashed line) participant's smash. Internal rotation (positive); external rotation (negative). A significant correlation between shuttlecock speed and shoulder internal rotation angle was present at shuttlecock contact.

**Figure 4.** (**a**) Mean (solid line) ± SD (shaded area) for normalised X-factor between preparation and shuttlecock contact key instants for all participants; (**b**) Normalised X-factor between preparation and shuttlecock contact key instants for participants with the fastest (solid line) and slowest (dashed line) smash. The end of retraction (ER) key instant for each participant is represented by a vertical line. X-factor at ER was significantly correlated with shuttlecock speed.

**Figure 5.** (**a**) Mean (solid line) ± SD (shaded area) for the normalised racket head speed between preparation and shuttlecock contact key instants for all participants; (**b**) Normalised racket head speed (normal to the stringbed plane) between preparation and shuttlecock contact key instants for participants with the fastest (solid line) and slowest (dashed line) smash. The acceleration phase begins at the point at which the racket head speed becomes positive, following the initial negative peak, and the duration between this turning point and shuttlecock contact was significantly negatively correlated with shuttlecock speed.

Multiple regression analysis revealed that the duration of AP alone explained 25.0% of the variation in shuttlecock speed, with a standard error of the estimate (SEE) of 4.4 m·s<sup>−</sup>1. The addition of X-factor at ER explained 43.7% of the variance in post-impact shuttlecock speed (SEE = 3.8 m·s<sup>−</sup>1), where a shorter acceleration phase and a more negative X-Factor at ER caused greater shuttlecock speeds (Table 5; Figure 6).


**Table 5.** Multiple regression equations explaining variance in shuttlecock speed.

Abbreviations: CI: confidence intervals, SEE: standard error of the estimate, AP: acceleration phase, ER: end of retraction.

**Figure 6.** Predicted shuttlecock speed against actual shuttlecock speed for the two-parameter stepwise regression equation (Table 5; Model b). With a higher percentage of the variation in shuttlecock speed explained the closer the data points lie to the line y = x (predicted shuttlecock speed = actual shuttlecock speed).

#### **4. Discussion**

Racket head speeds and shuttlecock speeds achieved by the participants showed good agreement with previously reported values by elite players [2,27,28]. Shuttlecock speed was greatest in the international players (94.4 <sup>±</sup> 3.2 m·s−1), followed by national players (91.2 <sup>±</sup> 2.6 m·s−1) and then regional players (86.4 <sup>±</sup> 4.6 m·s<sup>−</sup>1), suggesting that the ability to achieve greater post-impact shuttlecock speeds is a good indication of playing level [29]. A one-way ANOVA revealed a significant difference in shuttlecock speed between groups (*F*(2,15) = 7.29, *p* = 0.006). Bonferroni post-hoc tests revealed the international group smashed significantly faster than the regional group (mean difference <sup>=</sup> 8.06 m·s<sup>−</sup>1, CI: 2.22, 13.90; *p* = 0.006). The national group were not significantly different to either the international or regional groups. Five kinematic variables correlated significantly with shuttlecock speed: racket head speed at impact, peak wrist joint centre linear velocity, length of the acceleration phase, shoulder internal rotation angle at SC and X-factor (transverse) at ER. These results are indicative of proximal joint motions (shoulder and trunk) causing greater linear velocities in the distal segments (wrist and racket).

Racket head speed at impact, normal to the stringbed plane, correlated very highly (r = 0.903; CI: 0.753, 0.964; *p* < 0.001) with instantaneous post-impact shuttlecock speed. This is a larger effect size than those previously reported [2,27], which may be due to the more accurate calculation of post-impact shuttlecock speed i.e., at a precise time of impact, and an appropriate racket head speed (normal to the stringbed plane) interpolated to the time of initial contact with the shuttlecock in the present study. The non-perfect relationship between racket head and shuttlecock speed may be due to differences in racket specifications used by each player, where impact efficiency (location and coefficient of restitution), as well as mass properties of the racket, will affect the amount of linear momentum transferred to the shuttlecock, for a given impact velocity. Longitudinal impact locations, calculated using the previously described methodology of Peploe et al. [16], ranged between −29.4 and 24.9 mm from the racket head centre, whereas medio-lateral impact locations ranged between −33.2 and 17.1 mm from the racket head centre (medial–negative, lateral–positive; relative to the player). Modelling the racket as two segments (handle and frame) and assuming a frame transverse angular velocity of 80 rad·s−<sup>1</sup> at impact, consistent with experienced players [28], the difference in racket head speed at the most proximal and distal longitudinal impact locations would be 4.3 m·s−1, suggesting that impact location can have a large effect on subsequent shuttlecock speed, even given identical input in terms of racket angular velocities.

As well as racket head speed, peak linear velocity of the wrist joint centre, was found to be a strong predictor of shuttlecock speed (r = 0.767; CI: 0.467, 0.908; *p* < 0.001), indicating that players should ultimately aim to achieve high linear velocities within distal components of the kinetic chain. This correlation was stronger than previous reported by Rambely et al. [4] in elite international players, who reported a low positive relationship (r = 0.454) despite mean values in the present study showing good agreement (14.2 vs. 11.7 m·s−1) with differences attributable to the variety in player ability in the present study and differences in capture frequency (400 vs. 50 Hz), where 50 Hz may be inadequate for the badminton smash [28]. Greater shuttlecock speeds were also produced when AP was shorter in duration (r = -0.543; CI: -0.805, −0.101; *p* = 0.020), demonstrated in Figure 5b, where the racket head speed becomes positive much closer to impact, when comparing the "fastest" to the "slowest" participant.

Furthermore, two kinematic technique factors significantly correlated with shuttlecock speed. First, players who were in a position of greater shoulder internal rotation at impact produced greater shuttlecock speeds (r = 0.508, moderate; CI: 0.054, 0.788; *p* = 0.031), where a clear difference in position between the "fastest" and "slowest" player is shown in Figure 3b, despite very similar minimum positions. This suggests that greater shoulder internal rotation allows more work done by this joint rotation, which Liu et al. [6] found contributed 66% towards racket head velocity. However, correlational analysis revealed a non-significant relationship between shoulder internal rotation range of motion and shuttlecock speed (r = 0.403, low; CI: −0.079, 0.732; *p* = 0.097). When accounting for the time in which this range was completed, i.e., average angular velocity, the correlation came closer to significance (r = 0.444, low; CI: −0.029, 0.755; *p* = 0.065). Second, a low negative correlation was found between X-factor at ER and shuttlecock speed (r = 0.484; CI: −0.755, −0.022; *p* = 0.042), where players who produced a greater pelvis–thorax separation angle in the transverse plane at ER produced greater shuttlecock speeds. Again, a clear difference is seen between the "fastest" and "slowest" participants in Figure 4b. X-factor (maximum pelvis-thorax separation) has previously been found to be a strong correlate (r = 0.60, *p* < 0.01) with shuttlecock speed [12]. A greater rotational countermovement of the torso presumably allows greater trunk rotation contribution to racket head speed, which Teu et al. [9] found to contribute 57% to racket head speed.

It has been previously reported that both peak vertical ground reaction force and jump height significantly correlate with shuttlecock speed (r = 0.548 and 0.508, respectively) [2]. The present study found a non-significant correlation, with CI marginally crossing zero, between jump height and shuttlecock speed (r = 0.454; CI: −0.017, 0.760; p = 0.059); however, a similar effect size. Perhaps greater jump heights are a characteristic of more able players, who typically produce greater shuttlecock speeds [12,29], and serves as a tactical factor allowing players to produce steeper smash strokes as opposed to producing greater shuttlecock speeds. Note that jump height was significantly correlated with racket head speed in the present study (r = 0.494; CI: 0.035, 0.781; *p* = 0.037).

The fact that distal linear velocities (wrist and racket) best explain variation in shuttlecock speed, yet proximal angles of the trunk and shoulder best explain variation in shuttlecock speed, is suggestive of the kinetic link principle whereby movements of a proximal-to-distal nature generate and conserve angular momentum to produce high distal end-point velocities [5,30–32]. Important longitudinal axis rotations, typically difficult to measure and observe, may not always follow this strict sequence with regards to timing, however the proximal-to-distal nature of overhead strokes provides a good general understanding of how high distal end-point velocities can be generated [33,34]. A more negative X-factor at ER causing greater shuttlecock speed endorses the idea of the stretch–shortening cycle, whereby more elastic energy is stored and recovered to enhance the concentric phase when X-factor is more negative at ER. The stretch–shortening cycle has been linked to greater velocities in throwing actions due to enhancement of the concentric phase [35,36]. Finally, no ranges of motion were found to significantly correlate with shuttlecock speed. Lees et al. [32] previously suggested that increasing the range of motion can improve performance (racket head speed) by increasing the acceleration path of the racket, allowing more muscular force to be generated and applied to accelerate the racket.

A further relevant biomechanical principle, not explored in this study, is the velocity lever principle [32]. The angle between the forearm and racket longitudinal axes is of importance for generating racket head speed. For example, if the racket is held at 90◦ to the forearm, then the racket head will move through the greatest distance when the forearm pronates, increasing the racket head linear speed for any given angular velocity of pronation [31]. Likewise, if the elbow is flexed at 90◦, the contribution from shoulder internal rotation is maximised [30]. This principle is difficult to analyse within a complex motion such as the badminton smash, as multiple segmental rotations are responsible for producing the racket motion, as well as ensuring racket head orientation is optimal at impact, which may make certain joint angles, such as a racket–forearm angle of 90◦, undesirable. Tang et al. [37] reported that within their cohort of four elite players, the average racket–forearm angle was 147◦, which may represent a suitable compromise between the height and the speed at contact in practical play.

The multiple linear regression analysis found that two predictor variables were able to explain 43.7% of the variance in shuttlecock speed (Table 5; Figure 6). Participants with the fastest smashes were found to have a shorter acceleration phase duration and more negative (greater separation) X-factor at ER. From a practical standpoint, this would suggest that players attempt to delay the onset of their forward swing (when the racket head velocity normal to the stringbed becomes positive) such that the forward swing can be completed in the shortest possible time. This would ultimately achieve a greater velocity of the racket, i.e., the same acceleration path of the racket for a given player but completed in a shorter time period [31]. Additionally, players should seek to utilise as much counter-rotation of the trunk as possible before reversing this rotation within the acceleration phase.

A potential limitation to the present study was use of each participant's own racket causing a lack of experimental control over the impact mechanics between racket and shuttlecock, including effective mass of the "racket particle" within the collision between racket and shuttlecock, and thus the momentum transferred [27]. The effect of racket properties on player kinematics is also a potential source of limitation, where Whiteside et al. [38] reported that increasing the "swingweight" of the racket caused a reduction in peak angular velocities of shoulder internal rotation and wrist flexion during the tennis serve, where both of these joint rotations have been reported to be the two greatest contributors to racket head speed at impact [39]. Conversely, players are accustomed to their own racket, and introducing a "control" racket may require a great amount of familiarization or cause suboptimal performances due to unfamiliarity with the racket. It must also be acknowledged that the present study relies on investigation of joint kinematics at discrete time points (i.e., key instants). Future studies may therefore extend the current work using methodologies to investigate the continuous time series of kinematic data, such as statistical parametric mapping [40], vector coding [41] or principal component analysis [42], as well as considering other biomechanical principles not explored within this study. Additionally, differences in anthropometric data were not accounted for, where the same angular velocities and joint angles may lead to different linear velocities. Retrospective power analysis revealed that for the lowest significant correlation coefficient (*r* = 0.484), with 80% power and a significance threshold (*p* = 0.05), a sample size of 31 would be required. The study was therefore underpowered; however, recruiting more participants meeting the minimum standard criteria was not achievable.

#### **5. Conclusions**

In conclusion, proximal kinematics explained the greatest proportion of variation in shuttlecock speed during the forehand jump smash stroke in a cohort of experienced male badminton players. From a practical standpoint, it is suggested that players and/or coaches attempt to produce high shuttlecock speeds by increasing racket and distal joint centre linear velocities. This should be achieved primarily by having a greater internal rotation angle at shuttlecock contact. Furthermore, greater pelvis–thorax separation during the backswing phase is likely to aid the concentric phase of the swing via the stretch–shortening cycle.

**Author Contributions:** Conceptualisation, methodology and data collection: all authors; data processing: H.T. and R.D.; formal analysis and writing—original draft preparation: H.T. and S.M.-N.; writing—review and editing: all authors; supervision, M.K. and S.M.-N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Whole-Body Dynamic Analysis of Challenging Slackline Jumping**

#### **Kevin Stein \* and Katja Mombaur**

Optimization, Robotics and Biomechanics (ORB), Institute of Computer Engineering (ZITI), Heidelberg University, 69117 Heidelberg, Germany; katja.mombaur@ziti.uni-heidelberg.de

**\*** Correspondence: Kevin.Stein@ziti.uni-heidelberg.de; Tel.: +49-6221-54-14869

Received: 31 December 2019; Accepted: 1 February 2020; Published: 6 February 2020

**Abstract:** Maintaining balance on a slackline is a challenging task in itself. Walking on a high line, jumping and performing twists or somersaults seems nearly impossible. Contact forces are essential to understanding how humans maintain balance in such challenging situations, but they cannot always be measured directly. Therefore, we propose a contact model for slackline balancing that includes the interaction forces and torques as well as the position of the Center of Pressure. We apply this model within an optimization framework to perform a fully dynamic motion reconstruction of a jump with a rotation of approximately 180◦. Newton's equations of motions are implemented as constraints to the optimization, hence the optimized motion is physically feasible. We show that a conventional kinematic analysis results in dynamic inconsistencies. The advantage of our method becomes apparent during the flight phase of the motion and when comparing the center of mass and angular momentum dynamics. With our motion reconstruction method all momentum is conserved, whereas the conventional analysis shows momentum changes of up to 30%. Furthermore, we get additional and reliable information on the interaction forces and the joint torque that allow us to further analyze slackline balancing strategies.

**Keywords:** slackline balancing; dynamics reconstruction; contact force modeling; optimal control; subject-specific modeling

#### **1. Introduction**

Slackline balancing is a recreational sport where the athlete tries to stand, walk or jump on a spring-like elastic ribbon band that is mounted between two anchor points as shown in Figure 1. Unlike balancing on a stiff beam or when performing a tandem walk on a regular surface, the slackline can swing both sideways and vertically, which increases the difficulty of maintaining an upright position [1]. In ongoing work, we investigate how humans maintain balance on a slackline and, to date, motion captured over 20 subjects of different skill levels. This led to a big data set containing standing, walking and jumping motions from beginners, intermediate and expert subjects. We previously analyzed different performance indicators for standing and walking based on kinematic measurements and subject-specific rigid body modeling [2] and found that experts show reduced angular momentum around the vertical axis and reduced Center of Mass (CoM) acceleration due to adjusted stance leg compliance. The analysis, however, lacks information on the interaction forces with the slackline. They are fundamental for understanding human locomotion [3] and would allow us to further investigate the data. The measurement of Ground Reaction Forces (GRF) enables us to compute joint torques, assess stability parameters in balancing tasks and is therefore part of many gait analysis protocols [4,5]. Pressure sensors and insoles are undergoing rapid development and are applied in various setups to measure interaction forces [6], however, it is not always possible to measure all forces directly for every task. Slackline balancing is usually done

barefoot as experts report higher contact friction with the slackline which is crucial when landing from jumping. Beginners also prefer barefoot balancing over wearing shoes because they get a better feedback and control of the contact force. In the literature there are no reports on instrumented slacklines that can measure all interaction forces between the slackline and the subject. Previous work by Karatsidis et al. [7] estimated GRF for walking based only on inertial measurement data, a dynamic subject model and a heuristic on how the GRF are distributed during the double support phase. In previous work, we applied optimization methods and rigid body modeling to reconstruct GRF from kinematic data for sprinting motions with prostheses [8] and for a gymnastic cartwheel motion [9]. A dynamic model of the subject and a specific model of the interaction was crucial to obtain meaningful results. We now propose a new contact model specifically for balancing on a slackline and demonstrate its application by formulating and solving an optimal control problem (OCP) that reconstructs joint torques and forces of the slackline from marker-based motion capture and purely kinematic data. Newton's equations of motions are implemented as constraints to the optimization and therefore the optimized motion is physically feasible. When analyzing jumping motions with the conventional kinematic method we found inconsistencies in the CoM and angular momentum dynamics during the flight phase. In this work, we show the advantage of the proposed analysis for a jump motion with rotation as an example.

**Figure 1.** Left: A subject balancing on a slackline. An elastic ribbon band is mounted between two anchor points. The foot contact is able to move sideways and in up and down direction. Right: Kinematic motion capture data of slackline balancing. The coordinate system definition is used throughout this work.

#### **2. Materials and Methods**

#### *2.1. Experimental Approach to the Problem*

We present an overview of the slackline measurements and the data acquisition in Section 2.2. The conventional way of analyzing the data is described in Section 2.3. In a pre-study we investigated the differences between a regular planar contact and the contact with the slackline using sensor insoles (Section 2.4). The new analysis method is described in Section 2.5. It is based on the new contact model (Section 2.5.1) that takes the contact force, the contact torque and the position of the Center of Pressure (CoP) position into account. The subject-specific modeling and the formulation of the rigid body system dynamics with external contact forces are described in Section 2.5.2. In Section 2.5.3 we formulate and solve an OCP that tracks the recorded kinematic data by employing the proposed model. As a result we obtain a fully dynamic description of the motion, including joint torques and contact forces. We apply our approach to reconstruct the dynamics of one example motion where the subject jumps on the slackline and performs a rotation of approximately 180◦. A visualization of the resulting motion and the contact forces and torques is shown in Section 3. We validate the OCP

result by computing the CoM and angular momentum dynamics to show that the motion is physically feasible and follows Newton's mechanics (Section 2.5.4). We then compare the proposed method to the results of a conventional analysis that is often used in the literature.

#### *2.2. Slackline Study and Data Acquisition*

The slackline was installed using the Gibbon Slackrack 300 (ID Sports GmbH, Gibbon Slacklines, Stuttgart, Germany) as shown in Figure 1. The slackline was 3 m in length, 5 cm in width and mounted 31 cm above the ground. The motions were recorded using the marker-based motion capture system Qualisys (Qualisys, Goeteborg, Sweden) consisting of 8 Oqus 500 cameras at a frame rate of 150 Hz. The subject was prepared with 49 spherical infrared-reflective markers of 14 mm diameter, following the bone landmarks of the Gait-IOR marker set [10]. This marker set has originally been designed for gait analysis and was extended by two additional markers on the Medial Epicondyle of the Humerus to allow for better upper arm tracking and shoulder angle reconstruction. A static pose was recorded to create the subject-specific rigid body model. Six static markers were removed after this pose following the Gait-IOR guidelines, since the static markers were located at the medial side of the leg segments and are often occluded, thus tending to collide with other body parts and eventually hinder free movement. All dynamic motions were recorded using the remaining 45 markers. The slackline motion capture experiments were approved by the ethics committee of the Faculty of Behavioral and Cultural Studies of Heidelberg University according to the Helsinki Declaration (AZ Mom 2016 1/2-A1, 2016 with amendment 2019). Written informed consent was obtained from the subject. before the measurements. The subject was 1.9 m tall, weight 86 kg and was 27 years old and had considerable experience in slackline balancing.

#### *2.3. Conventional Analysis*

In conventional motion analysis the joint angles of the rigid body model are computed using whole-body inverse kinematics for every frame individually. We used the open source sofware tool Puppeteer by Martin Felis [11] for this task. It employs a damped Levenberg-Marquardt algorithm, described by Sugihara [12], to solve for the whole-body posture on a frame-by-frame basis. Afterwards, a low-pass filter was applied to generate smooth and differentiable joint trajectories. Joint velocities and joint accelerations are computed using finite central differences. The resulting trajectories represent the measured kinematics and we can compute different mechanical properties such as CoM acceleration or angular momentum when employing a dynamic subject model. The model is usually based on anthropomorphic measurements, e.g., by de Leva [13] or Jensen [14]. This kind of analysis is widely spread in the sports and biomechanics community and is also part of commercial motion analysis software like Visual3d (C-Motion, Germantown, MD). However, a frame-by-frame analysis in combination with an approximated model does not necessarily follow the laws of physics. Noise in the measurement, filtering and numerical derivation of joint velocities and accelerations, can result in violation of Newton's Equations of Motion (EoM). Additionally, it does not provide information on the acting joint torques or any interaction forces. We used a conventional analysis of the example motion as a comparison to show the advantages of the proposed method.

#### *2.4. Experimental Foot Contact Analysis and Modeling—A Pre-Study*

The Center of Pressure (CoP) is often used to analyze balance capabilities [5]. In this pre-study we performed CoP measurements using Moticon Pressure Insoles (Moticon GmbH, Munich, Germany) as we expected additional insight into the interaction from these measurements. We recorded standing on one leg in three different situations and for each foot individually. The following results are visualized in Figure 2:


• **Right:** Balancing on the slackline with turned stance foot

We see a clear difference between the cases: On flat ground the CoP moves in medial-lateral and in anterior-posterior direction, as is established in the literature. It is different on the slackline: we observe that the CoP does not move within the whole contact plane, but only on a single line. From the aligned and turned foot positioning we can conclude that the direction of this line is determined by the direction of the slackline. Even though the slackline measures 5 cm in width, we see from the measurement data that the CoP does not deviate from the very center of the slackline. We explain this with the fact that the ribbon band of the slackline is able to freely rotate around the center axis between the anchor points. This rotation is barely damped and therefore highly sensitive to shifting the CoP away from the slackline center. Rotating a contact surface, however, greatly reduces the normal force and therefore reduces the non-slipping threshold. We conclude that a small shift of the CoP away from the center of the slackline can already lead to the contact foot slipping off. The very narrow constraints on the CoP together with the continuous vertical and sideways movement of the stance foot explains the difficulty of slackline balancing especially in contrast to walking on a beam.

**Figure 2.** Center of Pressure recording of different single leg balancing tasks. We recorded both legs individually using Moticon Pressure Insoles. Left: Single leg balancing on regular surface. The CoP is moving in the anterior-posterior and in medial-lateral direction. Middle: Balancing on the slackline with aligned stance foot. Right: Balancing on the slackline with turned stance foot. The CoP is constrained by the direction of the slackline and to a single line.

#### *2.5. Proposed Analysis Method*

#### 2.5.1. Slackline Contact Modeling

A foot contact model is required to perform a dynamic motion analysis. A regular planar, non-slipping contact can be described using a total of 8 variables [15]: The contact force *F* acting in three directions, two variables for the CoP position *p* in the contact plane and three contact torques *M*. The forces are constrained by the friction cone to prevent slipping. Taking the measurements presented in Section 2.4 into account, we propose a new contact model for slackline balancing that includes 6 variables. Based on the slackline coordinate system shown in Figure 1 they are:

$$F = [F\_{\mathbf{x}\_{\ell}} F\_{\mathbf{y}\_{\ell}} F\_{\mathbf{z}}] \text{: The contact forces} \tag{1}$$

$$p = [p\_x, 0] \text{: The CoP position in along the slackine} \tag{2}$$

$$\mathcal{M} = [0, M\_{\mathcal{Y}}, M\_{\mathbb{Z}}] \text{: The contact torques in the slackine coordinate system} \tag{3}$$

Zeros are placed where the regular contact model would have had an additional variable. The contact force can still act in all three directions and is still subject to the friction cone. As a consequence of the measurements presented in Figure 2, the CoP position is described by only one free variable instead of two. The Y-coordinate is always zero. The position can no longer be defined in the local coordinate system of the foot, but has to be transformed in the coordinate system of the slackline. Furthermore, we allow only two instead of three contact torques. Again, these are defined by the global position of the slackline and cannot be modeled locally. They act around the vertical axis and are perpendicular to the slackline. As described before, the slackline can freely rotate around the *X*-Axis and therefore no torque can be applied around this axis.

#### 2.5.2. Subject Modeling and Dynamics Computations

The subject model consists of 17 segments. The length of each segment and the joint locations were estimated from the marker positions of the static trial recording following the Gait-IOR marker set guidelines. Figure 3 shows the static pose recording of the subject at the left and the specific rigid body model in the middle. Joint center positions are visualized in white. Virtual markers are placed on each segment of the model according to the static pose. These virtual markers are used to track the recorded marker data as shown at the right. For the dynamic properties of the model, such as segment mass, inertia or the relative segment CoM position, we refer to the measurements by de Leva [13] and linearly adjust them to the individual segment length. The degrees of freedom (DoF) and the kinematic structure is described in Table 1.

**Table 1.** Description of the kinematic structure of the model with all degrees of freedom.


**Figure 3.** Left: The static pose of the subject. Middle: The rigid body model in Puppeteer. Joint centers are visualized in white. Virtual markers were placed on the model following the static pose. Right: Inverse kinematic fit of the model to a slackline pose.

The dynamics of a rigid body system with *ndo f* DoF using generalized coordinates *q* are described by the following equation [16]:

$$H(q)\ddot{q} + \mathcal{C}(q, \dot{q}) = \tau \tag{4}$$

where the matrix *<sup>H</sup>***(***q***)** <sup>∈</sup> <sup>R</sup>*ndo f* <sup>×</sup>*ndo f* is the generalized inertia matrix that is constructed from the current configuration of the model and the inertia and CoM positions of each segment, *<sup>C</sup>***(***q***, ˙***q***)** <sup>∈</sup> <sup>R</sup>*ndo f* is the generalized bias force (e.g., gravity or the Coriolis force) and *<sup>τ</sup>* are the generalized forces applied at the joints. This equation holds when the subject is in the air and no external forces are applied. When the model is in contact with the slackline, a contact force *F* and a contact torque *M* are acting at the contact point *p* as derived in Section 2.5.1. We can compute the generalized forces *τ<sup>c</sup>* resulting from an external force using:

$$\pi\_{\varepsilon} = G(q\_{\prime}p)^T \ast \begin{bmatrix} M \\ F \end{bmatrix} \tag{5}$$

where *G***(***q***,** *p***)** is the 6D Jacobian for a point on a body that when multiplied with *q***˙** gives a 6-D vector that has the global angular velocity as the first three entries and the global linear velocity as the last three entries. With external contact forces Equation (4) becomes:

$$H(q)\ddot{q} + \mathcal{C}(q, \dot{q}) = \pi + \sum\_{\text{contracts}} \pi\_c \tag{6}$$

For known joint angles *q*, joint velocities *q***˙** and joint torques *τ* we are able to compute the joint acceleration *q***¨**. This is known as Forward dynamics and implemented in RBDL under the ForwardDynamics function. The contact Jacobean was computed using the CalcPointJacobian6D function. The exact recursive implementation is described in [16].

#### 2.5.3. Optimal Control Problem Formulation

We formulate the dynamic reconstruction of the recorded motion as an OCP. The general multi-phase OCP formulation is derived for example in [17]. We only describe the formulation that was used for the specific problem of motion reconstruction. Unlike our previous work [8,9] where we tracked joint trajectories that were computed beforehand, we now formulate an OCP that tracks the marker positions directly. In the following cost function we minimize the distance between the positions *m <sup>i</sup>* of the virtual markers on the model and the 45 recorded marker trajectories *m*-∗ *<sup>i</sup>* over the time *t* ∈ [0, *T*] of the motion:

$$\min\_{\mathbf{x}, \mathbf{u}} \int\_0^T \sum\_{i=0}^{45} ||\vec{m}\_i(q(t)) - \vec{m}\_i^\*(t)||^2 dt \tag{7}$$

with respect to the state vector *x* and control vector *u*. They are the same for all phases. The state vector consists of joint angles and joint velocities: **x**(*t*)=[**q**, **q˙**]. The control vector represents the torques of the actuated joints and all variables of the slackline contact model: **u**(*t*)=[*τ*(*t*), *λL*, *λR*] These are for each foot: three contact forces, two contact torques and the CoP position. All are given in the coordinate system of the slackline as it is shown in Figure 1.

$$
\lambda\_{\text{RJL}} = [F\_{\text{x}}, F\_{y}, F\_{\text{z}}, T\_{y}, T\_{\text{z}}, p] \tag{8}
$$

Controls are approximated to be piecewise linear continuous. To retrieve a feasible motion, the EoM must be satisfied and are therefore handled as constraints in the OCP. We reformulate Equation (4) and (6) as first order differential equations for the state vector:

$$\dot{\mathbf{x}}(t) = f\_j(t, \mathbf{q}(t), \mathbf{\tau}(t)) \tag{9}$$

where *<sup>j</sup>* <sup>∈</sup> <sup>N</sup> is the phase index. At this point we implemented two formulations: In the first implementation, we used the joint trajectories from the conventional inverse kinematics fit to determine the three different phases of the motion. Figure 4 shows the height of the feet above the ground plotted over time. The slackline is mounted at a rest height of 31 cm. Therefore, we can define a flight phase whenever both feet are above this height. This is indicated in red for the motion at hand. Hence, phases 1 and 3 are contact phases and subject to Equation (6), Phase 2 is a flight phase and subject to Equation (4).

**Figure 4.** Feet positions are plotted against time. The rest height of the slackline is 31 cm. We determine the times of contact and flight phases when both feet are higher than the slackline. The flight phase is indicated in red.

In the second implementation, we defined the whole motion as one phase subject to the contact dynamics (Equation (6)). Additionally, we formulated the following discontinuous path constraint throughout the motion:

$$r^{eq}(\mathbf{u}(0), \dots, \mathbf{u}(T)) = \begin{cases} \lambda\_{R\_{\prime}} & \text{if } \text{Right FOet Height} > 0.31 \text{ m} \\ \lambda\_{L\_{\prime}} & \text{if Left Foot Height} > 0.31 \text{ m} \\ 0, & \text{otherwise} \end{cases} \tag{10}$$

This allows the optimizer to determine the exact timing of contact which is fixed within the other implementation. Additionally, this formulation allows for different contact timings, such as jumping or landing with one foot after the other. This should enable for better marker positions tracking. On the other hand, such a formulation results in non-differentiabilities in the model which might cause numerical problems.

In both implementations, further boundary constraints *g*(·) are implemented for joint angles, velocities and torques and must be respected throughout the motion. The friction cone is implemented as an inequality constraint that requires the normal force to be larger than the horizontal forces. To determine reasonable upper limits for contact forces and torques, we performed a similar jumping motion on two force plates (Bertec, Columbus, OH, USA) and recorded the acting GRF. Limits were set to 1.5 times the measured maximum values. We solve both implementations of the OCP numerically using MUSCOD-II [17,18]. It was developed at the Interdisciplinary Center for Scientific Computing, IWR, Heidelberg University. The state variables are parameterized by the direct multiple-shooting method as it is derived in [19]. The control variables were discretized by piecewise linear continuous functions. On all multiple-shooting intervals, the dynamics of the system are computed in parallel. The same intervals are apply for states and controls. We ensure a continuous solution by imposing continuity constraints at the shooting interval transitions for all state variables. This way, a large but structured nonlinear programming problem (NLP) is obtained. It is solved by an adapted sequential

quadratic programming (SQP) method. Further detail can be found in [17]. We expect the first implementation to show better convergence, since discontinuities in the dynamics are supposed to be formulated as phase changes and constraints should be differentiable throughout one phase. However, in practical tests we observed that formulating phase changes as constraints also works in the present case. We did not use a regularization term, which is often used to account for possible redundancies in the contact forces during the double support phases.

#### 2.5.4. Validation

We validate our method by showing that important mechanical properties of the overall system are satisfied due to the fact that Newton's EoM have been formulated as constraints to the OCP. This includes:


With the change of momentum being equal to the CoM acceleration *c***¨** times the subject mass *m* and *L* being the angular momentum, we reformulate Newton's EoM:

$$
\vec{\mathbf{c}} \ast \boldsymbol{m} = \sum\_{\lambda\_{L/R}} \mathbf{F} \tag{11}
$$

$$\dot{L} = \sum\_{\lambda\_{L/R}} \left( \mathcal{M} + (\mathfrak{p}\_i - \mathfrak{c}) \times \mathcal{F} \right) \tag{12}$$

where *p* is the point where the force is acting. During all phases Equations (11) and (12) must hold. For the contact phases we can compute the left hand side of Equations (11) and (12) using the CalcCenterOfMass function of RBDL and the right hand side from result values of *λL*/*<sup>R</sup>* of the OCP result.

#### **3. Results and Discussion**

Both implementations converged. Due to the problem's complexity, we were numerically limited to ≈80 shooting nodes. We chose 40 multiple-shooting intervals per 1 s of motion and reconstructed around 2 s of motion. Other choices are possible, but we found that the computation time drastically increases with more shooting nodes and that the solver is not always able to find solutions for less shooting nodes per second of motion. As expected, the single-phase implementation resulted in a slightly lower tracking error. Therefore we present the results for the single-phase implementation. The resulting motion and the interaction forces (visualized as yellow arrows) are shown in Figure 5. Contact and flight phases are clearly distinguished. We see that the subject equally used both feet to initiate and land the jump and that contact torques were acting to build the necessary angular momentum for the rotation. The feet were aligned with the Slackline during the first contact phase and were turned perpendicular for landing.

We present the fitting error of our model to the measured marker data: Figure 6 shows the average marker residuum on the top and the frame by frame error on the bottom. Overall we achieved an average tracking error of 3 cm per marker. This accuracy is similar to the least squares kinematic fit of the conventional analysis that was used to initialize the OCP and to what is reported in the literature. The largest deviations occur for the two shoulder markers (L\_SAE and R\_SAE). One reason for this is that the arms were aligned to the upper body during the static pose on which the model is based. During slacklining, however, the arms are turned 90◦ compared to this pose and mainly parallel to the ground which results in skin and marker movement relative to the bone and shoulder joint. This offset is visible throughout the motion. Additionally, the shoulder is modeled

as a spherical joint with only three DoF. In reality this joint is much more complex and also has translational DoF. This currently limits the tracking accuracy; however, a more precise kinematic shoulder model could be used.

**Figure 5.** The result of this work: A fully dynamic reconstruction of a jumping motion with rotation. The contact forces and torques are visualized.

**Figure 6.** Top: Average marker error for all 45 markers. Largest deviations occur for the shoulder markers. This is due to the fact that the model is based on a N-Pose capture and the slackline motion mainly had a T-Pose like arm positioning. Bottom: Frame by frame marker error.

Figure 7 shows the CoM velocity and acceleration throughout the motion. The conventional motion analysis is plotted in green, the OCP result of the proposed method in purple. The most apparent differences occur during the flight phase when no forces are acting. As derived in Section 2.5.4, we expect constant accelerations and conservation of momentum in horizontal direction. Gravity should be the only force acting in vertical direction. This is indeed the case for the OCP result but is not given for the conventional analysis. Looking at the CoM acceleration we see values of up to 0.5 *<sup>m</sup> <sup>s</sup>*<sup>2</sup> in X direction and 0.7 *<sup>m</sup> <sup>s</sup>*<sup>2</sup> in Y direction during the flight phase. With the maximum values during the whole motion being 2.5 *<sup>m</sup> <sup>s</sup>*<sup>2</sup> and 3.5 *<sup>m</sup> <sup>s</sup>*<sup>2</sup> , respectively, this leads to an estimated relative error of up to about 20% for the conventional analysis.

**Figure 7.** Center of Mass velocity and acceleration in global coordinates during the motion. Results from the conventional method are plotted in green, the proposed method is plotted in purple. During the flight phase all horizontal accelerations are supposed to be zero and the according velocities should be constant. We see inconsistencies and changes in the range of 20% for the conventional method whereas the proposed method is physically feasible.

Angular momentum is plotted in Figure 8. It should also be conserved during the flight phase and the change should be equal to zero when no external torques are acting. Again, we observe the desired properties for the optimized motion. The conventional analysis shows high variability during the flight phase and errors in the range of 30% in the horizontal plane.

On the left of Figure 9 we plotted the contact forces for each foot stacked on top of each other and the CoM acceleration times the subject mass as a dashed line. We can see that they exactly match and that Equation (11) holds. On the right we have the same result for the contact torques. Again, Equation (12) is satisfied throughout the motion. This shows that the optimization result follows Newton's EoM also during the contact phases. The resulting forces are consistent with the CoM acceleration and the resulting contact torques and CoP position are consistent with the change of angular momentum. We have established that the dynamics of our optimization framework are physically feasible and can further analyze the motion itself.

**Figure 8.** Angular Momentum and Change of Angular Momentum around the Center of Mass. Similar to Figure 7 we see constant values during the flight phase and zero values of the derivative for the proposed method. Values computed with the conventional method are not respecting Newtons EoM.

**Figure 9.** Left: Sum of contact forces and Center of Mass acceleration times subject mass. Right: Sum of contact torques and torques produced by the contact forces plotted against the change of angular momentum. Equations (11) and (12) are satisfied throughout the motion.

We do so by looking at the sideways and vertical direction and forces and torques individually. As shown in Figure 1 the *X*-Axis is aligned with the slackline, the *Y*-Axis perpendicular and the *Z*-Axis in vertical direction. The Y direction is particularly interesting since the subject is very unstable due to the slackline and torques can only be applied controlling the contact force. Looking at the CoM

acceleration we see spring-like behavior before and after the jump. The subject is constantly swinging sideways. Amplitudes are small when the subject is in balance, larger after the landing and decreasing during the stabilization process. The oscillation frequency appears to be the same throughout the motion. The interaction forces of both feet point into the same direction, back towards the center of the anchor points of the slackline. We also observe spring spring-like properties before and after the jump in the vertical direction. The slackline model proposed by Paoletti and Mahadevan [1] models the force of the slackline also as a spring with the direction towards the resting position. However, they only considered sideways movement and did not take vertical movement of the subject into account. The lower left plot of Figure 9 shows the sum of contact torques and the sum of the torque generated from the contact forces. We see that the subject initiates the rotation applying contact torques when the feet are parallel during the first contact phase. After landing, the feet are perpendicular to the slackline and the contact forces are used to decrease the rotation.

In the future, we plan to apply the method to several other slackline tricks such as front or backflips, chest or pelvis bounces. For this task we plan to introduce other contact points for example at the hands, the chest or the pelvis. Another possibility would be a thorough gait analysis of slackline walking and the comparison to flat ground walking regarding stability parameters. We intend to reconstruct a full gait cycle and gain more insight in human balancing strategies. Furthermore, we want to analyze the resulting contact forces relative to the foot positions and develop a thorough spring model of the slackline. We showed that the current model by Paoletti and Mahadevan [1] is oversimplified as it does not take the whole range of motion of the slackline into account.

#### **4. Conclusions**

In this work we derived a general contact model for slackline balancing and demonstrated how it can be applied to analyze jumping on slackline. We employed the model inside an optimization framework to reconstruct the dynamics of a slackline jump with 180◦ rotation. We successfully implemented a multi-phase and a single-phase formulation leading to almost equivalent results. We found that the resulting motion has similar fitting error to the markers when compared to the inverse kinematics approach, but is physically feasible and time consistent. The advantage of our approach becomes apparent during the flight phase of the motion and when comparing the CoM and angular momentum dynamics. We found variations of up to 30% for the conventional method. Due to numeric complexity and the high amount of variables necessary to formulate the OCP, this method is limited to a few seconds of motion. This might limit the application to compare longer slackline motions of beginners and experts as initially intended. However, we can apply it to interesting parts of motion and analyze specific movements connected to balance recovery.

**Author Contributions:** Conceptualization, K.S. and K.M.; Formal analysis, K.S.; Funding acquisition, K.M.; Investigation, K.S.; Project administration,K.M.; Supervision, K.M.; Writing—original draft, K.S.; Writing—review & editing, K.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** Funding by the Carl Zeiss Foundation within the "Heidelberg Center for Motion Research" is gratefully acknowledged.

**Acknowledgments:** We want to thank the Simulation and Optimization research group of the IWR at Heidelberg University for giving us the possibility to work with MUSCOD-II. We also want to acknowledge financial support by the Baden-Württemberg Ministry of Science, Research and the Arts and by Ruprecht-Karls-Universität Heidelberg.

**Conflicts of Interest:** The authors declare no conflict of interest.

*Appl. Sci.* **2020**, *10*, 1094

#### **Abbreviations**

The following abbreviations are used in this manuscript:


ZMP Zero Moment Point

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
