Trajectory-Based Motion-Plane Modeling in Sports Biomechanics: A Comprehensive Review of Computational and Analytical Approaches

Featured Application

The motion-plane framework reviewed in this study can be applied to high-speed sport movements such as baseball and softball pitching, volleyball, badminton, and other sports that incorporate overhand striking and throwing. By quantifying planar orientation and deviation, this approach provides a practical way to characterize movement strategies and compare motion organization across sports.

Abstract

The purpose of this review was to evaluate the current literature using plane-based analyses to describe open-chain proximal-to-distal sport motions and to clarify how these approaches can extend to other activities to advance biomechanical assessment. Open-chain sport motions typically rely on a coordinated rotational axis that allows momentum to be transferred efficiently through the kinetic chain. Although this directional organization is central to performance, most biomechanical studies have relied on discrete, event-based variables rather than modeling the continuous trajectory structure of the movement. This review summarizes applications of motion-plane models in sports and discusses how their conceptual foundations can apply to other movements. Four primary approaches for deriving optimal-fit planes from three-dimensional trajectories are described: Principal Component Analysis (PCA), Singular Value Decomposition (SVD), Orthogonal Least Squares (OLS), and the Functional Swing Plane (FSP). These methods rely on different algebraic formulations to model kinematic trajectories. By comparing their mathematical foundations, strengths, and limitations, we highlight how plane-based models provide a meaningful perspective for examining movement efficiency, movement strategy, and potential injury risk across open-chain proximal-to-distal sports. Future research should apply these models across multiple sports to generate individualized trajectory planes, quantify plane deviation, and integrate measures of joint loading and performance, and may combine models to build motion planes.

1. Introduction

1.1. Background and Sports Applications

Open-chain rotational movements are central to performance in multiple sports including golf, tennis, baseball, softball, volleyball, handball, kicking, and javelin throwing. These sports require the rapid generation and efficient transfer of angular momentum via a coordinated proximal-to-distal sequence [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Larger, slower proximal segments initiate motion and transfer energy to smaller, distal segments through the kinematic chain. The kinematic chain is fundamental to achieving high distal segment velocities. For instance, distal segment velocities in baseball reach approximately 7000–9000 deg/s [17,18,19,20,21] at approximate ball release and in the golf swing at approximately 2000 deg/s [22,23]. Volleyball spikes, tennis serves, and handball throwing also achieve high distal segment velocities at approximately 1600 deg/s [24]. Soccer-kicking ankle velocity can reach approximately 1800 deg/s [25]. To achieve these high velocities repeatedly, athletes must move consistently and efficiently along a coordinated movement path. When adjacent segments of the kinematic chain rotate in a similar direction, the torque-producing lever arms are optimized, enabling angular momentum to transfer efficiently from proximal-to-distal segments to maximize the resultant velocity of the distal segment.

Multiple studies have shown that trunk motion predominantly involves rotation about its transverse-plane axes, while movements in the sagittal plane generate the kinetic energy needed to prepare for transfer to the distal segments [1,2,6,26]. These rotations not only align segmental momentum along a consistent orientational orbit but also produce the centripetal forces required to maintain curved trajectories of the segments [27]. This principle has been observed in both lower-limb striking, swinging, and throwing motions. The planar arc is defined by coordinating the pelvic, trunk, and torso motions (proximal segments) with distal segments (thighs → shanks → feet)/(shoulder → elbow → wrist). These energy flows suggest that the directional organization of open-chain rotational sports may result from the body’s dynamic redistribution of energy.

A key idea behind these orientation measures is the “motion-plane”, which represents the main geometric surface describing the movement’s overall direction. To achieve distal velocity, the motion must remain close to the optimal plane of motion, with each segment contributing in a coordinated manner to avoid energy dissipation during the motion. A fitted motion-plane provides quantitative descriptors such as slope (inclined) angles, direction angles, and deviation [28]. These variables can capture the differences in how athletes execute a movement and their ability to consistently repeat the sport-specific task. The slope angle of the plane often represents how an athlete organizes trunk and arm coordination. The direction angle represents the angle to the target. The deviation from the plane has been shown to affect mechanical efficiency, segmental sequencing, and the magnitude of the performance [9,22,29,30]. Moreover, to enhance the efficiency of transferring energy, controlling the radius during motion has been proven to be an important factor in golf swing study [31]. Consistent motion on the plane supports mechanical advantages and enhances the fluid propagations across segments.

Small deviations from an athlete’s preferred movement plane may lead to compensatory adjustments that can interfere with the smooth transfer of momentum. These effects are particularly noticeable in high-velocity tasks, where timing and segmental alignment become increasingly sensitive to small perturbations. Specifically in overhand throwing sports, a small deviation in trunk lateral tilt and shoulder abduction are particularly consequential because they directly influence the primary rotational axes responsible for transferring momentum from the trunk to the throwing arm. Greater trunk lateral tilt both in the coronal- and sagittal planes is associated with increased ball velocity but also elevated elbow and shoulder joint loading, and trunk posture, which interacts with the arm position to modulate joint stresses during pitching reliance on compensatory upper-body rotations [32,33,34,35]. Moreover, shoulder abduction angle describes how the upper arm is positioned relative to the trunk and serves as an important bridge between trunk momentum and upper-extremity acceleration. Excessive or misaligned shoulder abduction has been associated with reduce performance and increase shoulder and elbow loading, contributing to a reduction in performance and increased injury risks [18,22,36,37,38,39,40,41].

Overall, the construction of the orientation relies on customized anatomy and technique strategies, but movement close to the plane has been proven to be more efficient in channeling energy through the limbs. Radius from the plane view can be used to monitor performance and risk of injury. Recognizing and developing within the subjects’ plane may serve as a strategy for improving performance while decreasing injury risk in open-chain rotational sports. Until now, only the striking and kicking motion have been built up to discuss motion planes in biomechanics. Therefore, estimating an individualized motion plane may be useful for characterizing organization of movement in open-chain striking and kicking motions across multiple sports.

1.2. Plane-Related Variables in Different Sports and Research Gaps

Although baseball and softball biomechanics frequently reference orientational features that are conceptually related to a motion plane, these variables have not been formalized into a full geometric plane derived from the three-dimensional motion trajectories. In baseball pitching, the shoulder abduction angle is maintained between 85 and 95 degrees from the stride foot contact to the ball release (arm cocking phase and acceleration phase), which means creating a larger arm lever might be beneficial during these periods [19,33,42,43]. Additionally, “arm slot” is widely used to describe the angle of the arm at ball release (vector from the hand center of mass to the shoulder joint center from the frontal view). Arm slots are often broken down to further describe pitching styles, which have been associated with performance and joint loading [44,45]. Specifically, trunk lateral lean angle, a kinematic variable that helps make up arm slot, has been utilized as a surrogate for alignment to the throwing direction [24,26,46]. Trunk rotation occurs about this longitudinal axis, which, when in excess, has been associated with increased upper extremity loading [47]. Additionally, excessive shoulder abduction, another kinematic variable that makes up arm slot, has been linked to shoulder internal-rotation torque in baseball [48].

These plane-related descriptors are not limited to baseball throwing. In volleyball spiking, tennis serving, and handball throwing, researchers describe similar movement characteristics using arm elevation angles, trunk rotation patterns, and arm-path kinematic metrics rather than trajectory-based planes [37,49,50,51,52,53]. Kinematic studies of the volleyball spike consistently report substantial upper-arm elevation and trunk rotation, reflecting the same orientational demands as in other overhead sports [54,55,56]. Similar patterns have been observed in handball throwing, where sequential activation from trunk rotation followed by upper-arm elevation occurs during the acceleration phase. These findings show that essential kinematics, including upper-arm elevation and coordinated trunk rotation with lateral tilt, are highly similar across overhead sports.

Underarm movements, including windmill softball pitching, also demonstrate patterns that align with a specific movement plane. Pitchers demonstrating greater variability in the arm-circle path have shown higher shoulder proximal forces, suggesting that deviations from the primary path increase mechanical loading [41,57]. The arm-circle path represents a largely planar, circular trajectory in which the throwing arm rotates around the shoulder in a continuous, high-velocity loop. This motion produces a rotational pathway and could be characterized by a fitted movement plane. However, most of the existing softball literature, including analyses of trunk and pelvis rotation, arm-circle timing, shoulder kinematics, and segmental contributions, depends on isolated components such as maximum shoulder abduction, peak trunk lean, or specific angular positions at release [41,57,58]. These variables provide valuable insight into specific moments of the pitching cycle, but they do not capture how the arm maintains or deviates from its intended rotational pathway across the full arm-circle motion. As a result, the directional organization of the windmill pitch is described through discrete angle-based metrics rather than a continuous, trajectory-based representation of the movement.

In lower-limb striking sports such as soccer and rugby kicking, the movement pattern reflects a directional structure that can incorporate a specified movement plane. The kicking leg accelerates through a coordinated flow of energy, involving pelvic rotation, hip flexion, and rapid knee extension, allowing the foot to strike the intended object. Kinematic studies have shown that elite kickers maintain a relatively consistent swing-leg path, with deviations from this primary trajectory associated with reduced ball-contact precision and altered joint loading at the hip and knee. Similar to overhead motions, kicking is usually characterized by using discrete measures like thigh elevation, hip rotation, or kicking-foot orientation at ball contact [59,60], rather than by reconstructing the continuous three-dimensional trajectory of the foot. Although the kicking action is fundamentally directional and planar in nature, its underlying movement plane has not been formally modeled in the existing biomechanics literature.

Despite the close associations commonly used in biomechanical metrics and the motion-plane concepts, most biomechanical analyses are limited to discrete moments rather than capturing movement orientation across the entire motion. Direction-based indicators, such as arm slot or segment angles measured at specific time points, provide useful information but do not fully characterize the spatial organization of overhand movements or quantify when and how athletes deviate from their preferred motion path. As a result, important information may be missed during critical phases of the movement, as these measures offer only partial representations of movement direction.

Across the aforementioned sports, current plane-related metrics typically describe orientation at discrete instances and do not capture the continuous three-dimensional trajectory required to define a true motion plane.

The lack of trajectory-based plane models represents a pivotal gap in the biomechanics literature. Developing a more systematic, trajectory-based representation of the motion plane may enable more comprehensive characterization of directional mechanics and offer deeper insight into the structure of sport-specific movements.

2. Method

2.1. Study Design

This review summarizes research that either derives optimal-fit motion planes from kinematic trajectories or applies plane-related descriptors to characterize movement organization in sports. The conceptual foundations underlying these plane-based approaches are discussed, with emphasis on how they represent spatial organization and deviation relative to a reference plane. By comparing their mathematical assumptions, strengths, and limitations, this review highlights how plane-based analyses provide a useful framework for examining movement organization, efficiency, and strategy in open-chain, proximal-to-distal sports. The literature was identified and organized using a systematized approach informed by the PRISMA guidelines (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) [61].

2.2. Search Strategy

A literature search was conducted through PubMed, Embase, CINAHL, and SPORT Discus databases to identify studies applying plane-based analyses to three-dimensional kinematic trajectories in sports. The search was performed on 5 January 2026 (Figure 1). Search terms included combinations of keywords related to sport movements and motion-plane analysis, such as “motion plane”, “trajectory”, “plane fitting”, “kinematic trajectory”, “functional swing plane”, “arm slot”, “planar organization”, and sport-specific terms (e.g., “baseball”, “softball”, “golf”, “handball”, “volleyball”, “tennis”, “soccer”, “rugby”, “throwing”, “striking”, or “kicking”).

2.3. Study Screening and Selection

Titles and abstracts were screened by two authors independently to assess relevance to the aims of this review. Duplicate records were removed prior to screening. Full-text articles were then reviewed when eligibility could not be determined from the abstract alone. Studies were included if they (1) applied plane-based, trajectory-based analyses, or utilized plane-related spatial descriptors (e.g., arm slot, trunk lean relative to a motion axis) to three-dimensional kinematic data; (2) examined sport-related movement tasks involving open-chain, proximal-to-distal sequencing; and (3) provided sufficient methodological detail to describe motion-plane construction, principal component decomposition, or plane-related descriptors. Studies were excluded if they (1) focused exclusively on discrete joint angles (e.g., isolated peak flexion) or event-based variables that lacked a spatial reference to the movement’s global planar structure or trajectory organization; (2) did not report original kinematic analyses (e.g., review articles or purely kinetic modeling without kinematic data).

2.4. Scope and Task Selection

This review focused primarily on high-velocity, open-chain, proximal-to-distal sport movements—specifically throwing (e.g., baseball, softball, handball), striking (e.g., golf, tennis, volleyball), and kicking (e.g., soccer, rugby). These tasks were selected because they are characterized by coordinated rotational trajectories that frequently exhibit a dominant planar organization or a consistent spatial orientation (e.g., arm slot or swing plane).

While the review prioritizes tasks with clear trajectory-based planes (

Table 1. Summary of included studies that explicitly derived motion planes from three-dimensional kinematic trajectories.

Study	Sport	Plane-Related/Concept	Data Type	Key Outcome
Coleman & Rankin [62]	Golf	Projected planes	Trajectory point (left arm and clubhead shaft)	Arm and clubhead motion exhibit continuously varying planarity rather than a single fixed plane.
Coleman & Anderson [63]	Golf	OLS	Trajectory point (clubhead shaft)	RMS error > 8 cm swing only semi-planar.
Kwon et al. [28]	Golf	FSP	Trajectory point (clubhead and mid hand point)	Skilled golfers demonstrated a consistent functional swing plane with semi-planar downswings characterized by a transition phase followed by a stable planar execution phase.
Kwon et al. [64]	Golf	FSP	Trajectory point (clubhead and mid hand point)	X-factor parameters show no direct correlation with clubhead velocity.
Kwon et al. [30]	Golf	FSP	Trajectory point (clubhead)	Hand motion dictates swing patterns.
Han et al. [9]	Golf	FSP	Trajectory point (clubhead)	Downswing sequences are inconsistent among skilled golfers’ separation styles.
Cheng et al. [65]	Softball	SPP	Trajectory point (throwing wrist joint center)	Less RMS deviation predicts higher ball velocity.
McGuire et al. [15]	Golf	FSP	Trajectory point (club-head)	Short game patterns are unique and lack standard proximal-to-distal sequencing.
Madrid et al. [23]	Golf	FSP	Trajectory point (club-head)	Speed correlates with rotation velocity, not X-factor or wrist cock.
Willmott & Dapena [66]	Filed Hockey	OLS	Trajectory point (stick face)	High RMS indicated in inaccuracy group.
Bezodis et al. [67]	Rugby (kicking)	OLS	Trajectory point (kicking foot COM)	Swing plane and support foot dictate accuracy.
Navandar et al. [68]	Soccer	PCA	Trajectory point (kicking hip and knee joint center)	Kicking kinematics differ by sex at hip, not knee.

PCA: Principal component analysis;; OLS: Orthogonal least-squares; SPP: Softball pitching plane; FSP: Functional swing pane; COM: Center of mass; RMS: Root mean square.

), it also encompasses studies examining planar descriptors in multi-segmental sequences (

Table 2. Summary of studies using plane-related descriptors without explicit trajectory-based plane estimation.

Study	Sport	Plane-Related Descriptor(s)	Kinematic Representation	Key Finding
Seroyer et al. [19]	Baseball	Six phases of pitching	Kinetic chain of muscle recruitment	Pitching is a high-speed coordinated effort.
Wagner et al. [24]	Handball	Technique-specific movement patterns	Qualitative and quantitative determinants of performance	Handball performance is a holistic combination of physical intensity and tactical interaction.
Wagner et al. [26]	Handball, Tennis, and Volleyball	Upper-body joint motion; Acceleration phase angles	Trajectory points; kinematic sequencing	Shared proximal-to-distal sequencing despite sport-specific modifications.
Manzi et al. [42]	Baseball	Shoulder abduction; External rotation	Upper extremities kinematics and kinetics	Lowering abduction at release reduces shoulder force without losing velocity.
Fortenbaugh et al. [43]	Baseball	Shoulder abduction and trunk tilt	Kinematic and kinetic relationships	Specific joint alignments and timing dictate ball velocity and joint stress.
Oliver et al. [41]	Softball	360° arc motion	Path analysis of upper arm, forearm, and hand kinematics	Full kinetic chain coordination, not just shoulder, drives windmill velocity.
Van den Tillar et al. [37]	Handball	Upper-body joint motion	Upper-body kinematics	Sequence exists for movement initiation but not for peak velocities.
Manzi et al. [44]	Baseball	Lateral trunks lean and arm slot	Upper-body kinematics and kinematics	Arm slot position dictates joint torque and upper trunk rotation timing.
Escamilla et al. [45]	Baseball	Trunk angles and arm slot	Upper-body kinematics and kinetics	Sidearm slots favor transverse motion; overhand slots favor sagittal motion.
Wagner et al. [46]	Handball	Upper-body motion and acceleration	Upper-body kinematics	Skilled players compensate for high variability to maintain accuracy and speed.
Aguinaldo et al. [47]	Baseball	Trunk rotation	Truk kinematics and shoulder kinetics	Delayed trunk rotation reduces shoulder torque by conserving momentum.
Manzi et al. [48]	Baseball	Shoulder horizontal abduction/adduction	Upper-body kinematics and kinetics	Less horizontal adduction increases ball velocity and reduces shoulder force.
Van den Tillar et al. [37]	Handball	Upper-body joint motion	Upper-body kinematics	Sequence exists for movement initiation but not for peak velocities.
Coleman et al. [49]	Volleyball	Whole-body motion	Whole-body kinematics	Humerus angular velocity is the primary predictor of post-impact ball speed.
Bahamonde [50]	Tennis	Whole-body motion	15-segment angular momentum model	Momentum transfers from trunk to arm about parallel and towards axes.
Van den Tillar et al. [52]	Handball	Elbow extension and internal rotation; Pelvis timing	Trajectory points and angular velocity	Ball velocity depends on early pelvis rotation and shoulder/elbow speed.
Van den Tillar & Ettema [52]	Handball	Linear velocity of upper body	Whole-body kinematics	Performance goals change velocity and timing but not core technique.
Guo & Li [55]	Volleyball	Upper-body motion	Upper-body kinematics	Specific shoulder and elbow angles at impact are critical for spike velocity.
Reeser et al. [56]	Volleyball	Upper-body motion	Upper-body kinematics	High shoulder abduction at impact increases joint kinetics during spiking.
Werner et al. [57]	Softball	Upper-body motion	Upper-body kinematics and kinetics	Windmill shoulder distraction stress is comparable to professional baseball pitching.
Friesen et al. [58]	Softball	Upper-body motion	Upper-body kinematics and kinetics	Trunk position and elbow mechanics are primary drivers of shoulder distraction force.
Nunome et al. [59]	Soccer	Lower-body motion	Lower-body kinematics and kinetics	Hip external rotation torque rotates the thigh-shank plane to square the foot for impact.
Kellis & Katis [60]	Soccer	Lower-body motion	Lower-body kinematics	Sequential acceleration of thigh, shank, and foot creates multi-planar motion for ball speed.

), where movement efficiency is often defined by the alignment of distal-segment trajectories relative to the trunk or a functional axis. Closed-chain or highly multi-planar tasks (e.g., gymnastics or wrestling) were considered outside the primary scope when a stable or interpretable motion plane could not be meaningfully defined. This focus ensures conceptual consistency across the reviewed models while capturing the diversity of planar analysis in competitive sports.

3. Theoretical Framework of Trajectory-Based Motion Plane Modeling

It is important to distinguish trajectory-based motion-planes from traditional anatomical planes (sagittal, frontal, and transverse). Anatomical planes are fixed, body-centered reference frames used to describe joint rotations and segment orientations, whereas motion planes are task-specific constructs derived from the three-dimensional trajectory of a moving point or segment. A motion plane does not replace anatomical planes but instead provides a functional description of how movement is organized in space during a specific task. In some movements, the fitted motion plane may approximate an anatomical plane (e.g., predominantly sagittal motion), whereas in many sport skills it is obliquely oriented and reflects the combined contribution of rotations across multiple anatomical planes (e.g., baseball pitching). As such, motion-plane modeling complements traditional joint-based analyses by capturing the global spatial organization of movement rather than segment rotations relative to fixed anatomical axes.

3.1. Conceptual Definition of a Motion Plane

Many biomechanical studies have described movement organization using plane-related or “plane-like” descriptors derived from discrete events, such as arm slot angles, trunk lateral tilt at ball release, or segment orientation at impact [24,26,46]. While these variables provide useful snapshots of movement direction at specific instants, they do not define a motion plane in a geometric sense. In contrast, trajectory-based motion planes are estimated from the full three-dimensional path of a point or segment over a defined phase window. This approach captures the continuous spatial organization of the movement and allows deviation from the plane to be quantified across time. As a result, trajectory-based planes provide information about movement consistency and coordination that cannot be inferred from isolated, event-based angles alone.

A motion-plane in biomechanical analysis is not a fixed geometric plane but an exercise-dependent representation of motion organization. Rather than reflecting an immutable anatomical motion plane, a motion plane is customized to the dominant spatial structure of motion trajectory of each person over a specified portion of the sport [28,63,69,70]. Across the biomechanics literature, a motion plane is identified by three components: (1) trajectory of the points of interest, (2) the selected interval, and (3) the metric utilized to quantify planarity. Clarifying these components is fundamental before introducing models, as differences in any one of these elements can substantially alter the estimated plane and its biomechanical interpretation.

The first component of a motion plane is the trajectory of the points of interest used to define the motion plane. The previous literature usually applied distal points to constructing motion planes. In proximal-to-distal movements, distal points typically move at higher velocities and follow smoother, more organized trajectories, which increases the likelihood that their motion can be approximated by a plane with lower deviation [27,28,69]. Moreover, in many sports, performance is strongly linked to the ability to generate high velocity at the distal segment or joint, making distal-point trajectories particularly relevant for motion-plane analysis [28,69].

Following the points of interest, the selected interval is critical to make a motion plane. A plane may be estimated using the full motion, mechanically relevant sub-phases (e.g., downswing, acceleration phase), or event-aligned windows defined by key kinematic or kinetic events. Including the entire motion may obscure phase-specific organization, whereas restricting the analysis to functionally important intervals can yield a plane that better reflects the dominant mechanical strategy. The interval should be meaningful for specific movements. For example, for the softball, the downswing to ball release is related to ball velocity, and from the mid-downswing to mid follow-through is vital for golf-swing performance [28,69]. Therefore, the selected phase window plays a central role in determining both the geometry of the plane and its biomechanical meaning, and it must be chosen consistently with the study’s objectives.

In addition to defining the motion plane, metrics are required to quantify how closely the movement follows that plane. Commonly reported planarity metrics include root-mean-square (RMS) deviation, maximum deviation, and time-varying deviation. These metrics describe different aspects of deviation from the plane and are used to characterize movement organization. Orientation-related variables, such as direction angles relative to the target and slope angle relative to the ground, are also frequently reported to describe the global orientation of the motion plane.

3.2. Conceptual Consideration in Quantifying Planarity

3.2.1. Global vs. Local Measures of Planarity

Planarity can be quantified using global or local measures, which differ in how deviation from the motion plane is summarized over time. Global (phase-aggregated) measures, such as RMS deviation, summarize distances between trajectory points and the motion plane across the entire analyzed period and are commonly used to characterize overall planarity and movement consistency. In contrast, local (instantaneous) measures preserve frame-by-frame information, with maximum deviation capturing peak departures from the plane and time-varying deviation describing how off-plane motion evolves throughout the movement. Accordingly, global measures are the most appropriate for assessing overall consistency, whereas local measures are better suited for identifying peak deviations or phase-specific characteristics.

3.2.2. Effects of Motion Curvature and Noise

Although many high-speed sport movements can be reasonably approximated as planar, movement trajectories rarely lie within a perfectly flat plane. Changes in trajectory curvature or movement direction can influence deviation-based metrics, particularly when a single plane is used to represent the entire motion. In such cases, increased deviation may reflect factors such as fatigue, compensatory movement strategies, or the absence of an optimal movement pattern, rather than reduced motor control. In addition, measurement noise can amplify estimated off-plane deviations, especially in high-velocity segments where small spatial errors translate into larger deviations relative to the plane. Therefore, deviation-based planarity metrics should be interpreted with caution when motion curvature is pronounced or when data quality is limited, as elevated deviation values may arise from methodological factors rather than true biomechanical variability.

3.2.3. Phase Weighting and Mechanical Relevance

Not all phases of a movement have the same importance for performance or mechanical demand. In many sport movements, specific phases such as acceleration, impact, or release are of greater interest because they are more directly related to performance outcomes and joint loading. When planarity metrics are calculated without phase weighting, slower or preparatory phases may have a greater influence on the results, even though these phases may have limited mechanical relevance. Applying phase weighting allows greater emphasis on motion during critical phases while reducing the influence of less relevant portions of the movement. Therefore, phase relevance should be considered when selecting planarity metrics and motion-plane estimation methods.

3.3. Model of Principal Component Analysis

3.3.1. Principal Component Analysis (PCA)

The current use of principal component analysis (PCA) for identifying an optimal movement plane in human motion is rooted in more than a century of mathematical and statistical development. This technique originates from Karl Pearson’s foundational work “On lines and planes of closest fit to systems of points in space”, where he showed the best-fitting geometric subspace for a set of data points is obtained by minimizing the sum of squared orthogonal distances from each point to the candidate linear or plane [71]. Pearson’s formulation was mathematically identical to what is now recognized as PCA [71]. A couple of years later, Harold Hotelling transformed Pearson’s geometric insights into a complete statistical framework by formalizing eigenvalue–eigenvector decomposition, proving optimality conditions, and introducing PCA as a multivariate analysis method widely applicable across behavioral, biological, and physical sciences [72]. These two contributions established PCA as the canonical approach for dimension reduction and for extracting interpretable structure from high-dimensional motion data.

Although these foundational works established the theoretical basis, the modern mathematical formulation of PCA applied to human movement follows contemporary treatments in standard texts such as Jolliffe [69], Bishop [73], and Shlens [74]. Jolliffe’s monograph provides the classical statistical formulation of PCA, detailing the construction of the covariance matrix, the extraction of eigenvalues and eigenvectors, and the interpretation of variance maximization [69]. Bishop extends this into the machine-learning context, emphasizing orthogonal projections, least-squares minimization, and the geometric meaning of PCA as an optimal low-dimensional subspace [73]. Shlens offers a concise tutorial summarizing these results and showing how PCA emerges naturally from minimizing reconstruction error and maximizing captured variance [74]. Building from these modern sources, PCA operates on the premise that human movement, although recorded in three-dimensional space, often varies primarily along a limited set of coherent directions. Given a matrix of three-dimensional time-series coordinates from joint centers or end-effectors,

$$X=\left[\begin{array}{c}{X}_1^T\\ X_2^T\\ ⋮\\ X_n^T\end{array}\right]\in {\mathbb{R}}^{n\times 3}$$

(1)

where $X_i\in {\mathbb{R}}^3$ denotes the position vector at frame i, and n is the total number of frames. The data are mean-centered by subtracting the average position ($\overline{X}$) yielding the mean-centered vectors.

$$\widetilde{X_i}=X_i-\overline{X}$$

(2)

where $\widetilde{X_i}$ represents the position at frame i after mean centering, computed by subtracting the average position $\overline{X}$ from the original coordinates $X_i$.

Before centering:                 After centering:

            ●                                         ●

       ●        ●                             ●          ●

   ●                ●                  ●                       ●

       centroid ≠ 0                    centroid = (0,0,0)

The sample covariance matrix is then computed as

$$C={\displaystyle \frac{1}{n-1}}{\sum }_{n=1}^n\widetilde{X_i}{\stackrel{~}{{\widetilde{X_i}}_i}}^T$$

(3)

where C characterized the variance and linear coupling among the spatial dimensions of the motion.

       Principal spread direction →

         ●       ●       ●

      ●       ●       ●

   ●      ●      ●

Principal components are obtained by solving the symmetric eigenvalue problem:

$$C{v}_k= {\lambda }_k{v}_k$$

(4)

where ${\lambda }_k$ is the variance explained by the k-th principal direction, and $v_k$ is the eigenvalues are ordered

$${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$$

(5)

So that the dominant directions $v_1$ and $v_2$ span the primary motion-plane (Figure 2). This ordering follows directly from the Rayleigh quotient, which ensures that $v_1$ maximizes projected variance, while $v_2$ captures the greatest remaining orthogonal variance.

The optimality of PCA can also be formulated through the minimum reconstruction criterion:

$$\underset{V}{\min }{\sum }_{i=1}^n{\Vert \widetilde{x_i}-V{V}^T\widetilde{x_i}\Vert }^2$$

(6)

The overall objective seeks a two-dimensional subspace that minimizes the total reconstruction error across all n frame observations.

$$V=[v_1{v}_2 ]$$

(7)

where $v_1$ and $v_2$ are the first two principal directions spanning the dominant motion subspace. The normal vector of the PCA plane can be obtained either from the third principal direction $v_3$, which corresponds to the smallest variance, or equivalent from the normalized cross product.

$$v_3={\displaystyle \frac{v_1\times v_2 }{\Vert v_1\times v_2\Vert }}$$

(8)

The sign of v3 is arbitrary due to the inherent sign ambiguity of PCA eigenvectors. To project each mean-centered data point onto the PCA plane, the projection matrix is defined as

$$P=I-v_3{v}_3^T$$

(9)

The projected point at frame i is then computed as

$$P_i=P\widetilde{X_i}$$

(10)

This removes the out-of-plane component while preserving the in-plane motion structure. This framework is particularly relevant for rotational or ballistic movements in which mechanical efficiency and proximal-to-distal sequencing typically occur within a quasi-planar trajectory. The PCA plane thus provides an objective, data-driven representation of the functional movement plane underlying human motion (Figure 2). For the calculation of the off-plane deviation, following the previous studies, out of plane error:

$$d_i= \Vert \widetilde{X_i}- P_i\Vert$$

(11)

where $d_i$ is the off-plane deviation, $X_i$ is the raw data points at i frame, and the $p_i$ is the projected points at i frame. The root means squares (RMS) deviation:

$$RMS= \sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{d}_i^2}$$

(12)

where n is the total number of sampled frames, and $d_i$ is a scalar representing the orthogonal distance from the i-th kinematic point to the computed motion-plane. The root-mean-square (RMS) of the off-plane distances $d_i^2$ provides a quantitative measure of planarity, where n is the total number of sampled frames (Figure 3).

PCA is a strong mathematical method to make a motion plane. By using orthogonal directions with the greatest variance, it can identify the main direction of the movement and create a good low-dimensional representation. After mean-centering, PCA is invariant to rigid-body translations and rotations of the coordinate system, meaning that the estimated motion plane does not depend on the global laboratory orientation or on predefined anatomical axes. However, PCA is not invariant to scaling and is sensitive to outliers, as both can alter the covariance structure and disproportionately influence the estimated principal directions. Consequently, comparisons of PCA-derived planes across athletes or across sessions require consistent coordinate scaling, careful preprocessing, and cautious interpretation. In the present framework, PCA offers a robust data-driven representation of the dominant motion subspace, but its variance-based nature implies that all frames are weighted equally, making it less sensitive to phase-specific mechanical importance, such as acceleration or ball release. Additionally, PCA implicitly assumes a flat planar structure and may not fully capture curved- or phase-dependent movement trajectories.

3.3.2. Singular Value Decomposition (SVD)

Singular value decomposition (SVD) builds on a rich mathematical history that parallels, and eventually connects to, the development of PCA. It can be traced back to the early 20th century. The first announcement of the SVD method was from Eckart and Young studying optimal low-rank approximations [75]. It indicated that a general rectangular matrix can be approximated in the least-squares sense by truncating its singular value decomposition. Their result formalized the notion that the dominant singular vectors capture the most important structure in the data. By the mid-20th century, with further advances in linear algebra development, SVD had become a stable and widely used tool for decomposing large, noisy data matrices into orthogonal spatial and temporal components [70,76]. PCA emerges as a special case of SVD applied to a mean-centered data matrix, where the eigenvectors of the covariance matrix correspond to the right-singular vectors and the variances to the squared singular values. Thus, while PCA is often introduced from a statistical perspective, SVD provides the underlying geometric and computational machinery that is now standard in scientific computing, signal processing, and movement analysis.

Singular value decomposition (SVD) provides a straightforward way to identify the dominant directions of movement from a three-dimensional trajectory. Consider a marker or joint center recorded across $n$ time frames. Its coordinates can be arranged into a matrix:

$$X= \left[\begin{array}{ccc}{x}_1& y_1& z_1\\ ⋮& ⋮& ⋮\\ x_n& y_n& z_n\end{array}\right]\in {\mathbb{R}}^{n\times 3}$$

(13)

where each row represents the three-dimensional position at a given frame, and n denotes the total number of frames. To remove the overall offset, the trajectory is mean-centered column-wise by subtracting the average position across all frames:

$$\widetilde{X_i}= X_i- \overline{X}$$

(14)

where $\widetilde{X_i}$ is the mean-centered coordinates at i frame, $\overline{X}$ is the average position across all frames, and $X_i$ is the raw position at the i frame. After that the centered trajectory is decomposed using

$$\widetilde{X_i}=U\Sigma V^T$$

(15)

where $\mathbf{U}$ $\in {\mathbb{R}}^{n\times 3}$ contains an orthonormal temporal coefficient associated with each spatial mode, $\Sigma$ = diag(σ1,σ2,σ3) is a diagonal matrix of singular values ordered as σ1 ≥ σ2 ≥ σ3 ≥ 0. V contains orthonormal spatial directions. The first two right-singular vectors, $v_1$ and $v_2$, correspond to the directions of greatest variance in the mean-centered trajectory and the dominant two-dimensional motion subspace, which defines the best-fitting plane in a least-squares sense. The third right-singular vector, $v_3$, corresponds to the direction of minimum variance and is orthogonal to this subspace. Therefore, $v_3$ represents the normal vector of the motion plane (Figure 4).

Each original point $\widetilde{X_i}$ can be projected onto the plane as

$$p_i=X_i+v_1\left({v_1}^T\left(X_i-\overline{X}\right)\right)+v_2{(v_2}^T(X_i-\overline{X}))$$

(16)

where $X_i$ are the original points, $\overline{X}$ is the mean position of the trajectory, and $p_i$ is the projection of each point onto the motion plane. The terms $v_1$ and $v_2$ are the first two right-singular vectors that defined the plane direction, and the inner products ${v_k}^T\left(X_i- \overline{X}\right)$ represent the scalar coefficient of the point along component $k$, with $k=\mathrm{1,2}$.

This gives a smooth, planar version of the trajectory. Using the formular as the off-plane deviation:

$$d_i= \widetilde{X_i}- \left[v_1\left({v_1}^T\widetilde{X_i}\right)+ v_2\left({v_2}^T\widetilde{X_i}\right)\right]$$

(17)

where $d_i$ is the off-plane deviation. Because the fitted plane is spanned by $v_1$ and $v_2$, and ${v_3}^T$ is orthogonal to this plane, the residual vector $d_i$ lies entirely along the ${v_3}^T$ direction. And the RMS deviation:

$$\Vert d_i\Vert = \left|{v_3}^T\widetilde{X_i}\right|$$

(18)

$$RMS=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\Vert d_i\Vert }^2}$$

(19)

RMS of the off-plane distance, ∥$d_i$∥ provides a quantitative measure of planarity, $v_3$ is the plane normal, its inner product gives the orthogonal distance, and n is the total. Lower RMS values indicate that the trajectory remains close to the fitted motion-plane, whereas larger values reflect greater out-of-plane motion.

Compared to PCA, singular value decomposition (SVD) is often favored as a computational implementation due to its numerical stability and direct linear-algebraic formulation. While PCA explicitly constructs and decomposes a covariance matrix, SVD operates directly on the mean-centered data matrix, which can reduce numerical error accumulation when handling large datasets or poorly conditioned trajectories.

Importantly, when applied to the same mean-centered trajectory data without additional weighting or scaling, PCA and SVD yield equivalent principal directions and therefore define the same motion plane. The distinction between the two approaches lies primarily in computational strategy rather than in the geometric interpretation of the resulting plane.

Similar to PCA, SVD treats each frame equally and does not incorporate biomechanical weighting for mechanically important phases of the movement. In addition, because SVD decomposes raw data matrices, careful preprocessing (e.g., consistent marker definitions, scaling, and normalization) is required to avoid bias in the estimated singular vectors. As a result, SVD is best viewed as a numerically robust implementation for variance-based plane estimation rather than a fundamentally distinct modeling approach.

3.3.3. Model of Orthogonal Least-Squares (OLS)

Orthogonal least-squares (OLS) provides a geometric method for estimating a best-fit plane by minimizing the perpendicular distance of all points to the plane. Unlike algebraic least-squares, which minimize residuals only in a single coordinate direction, OLS accounts for error in all three spatial dimensions, making it appropriate for representing movement trajectories where deviations occur in the full 3D space. This method has been widely applied in geometric modeling and motion analysis [77,78]. OLS has been used to estimate the best-fit plane in softball pitching, called the softball pitching plane (SPP), that represents the dominant spatial orientation of a 3D movement trajectory [65]. Given a sequence of $\mathit{n}$ three-dimensional points, the centroid of the trajectory was first computed, and each point was mean-centered to remove translational effects:

$$\widetilde{X_i}= X_i- \overline{X}$$

(20)

where $x_i$ represents the original 3D coordinates at frame $i$, $\overline{X}$ is the average position across time, and $\widetilde{x_i}$ is the centered coordinate used for plane fitting. Because different phases of the motion may contribute unequally to defining the functional plane, each point was optionally assigned a weight ${\omega }_i$(e.g., instantaneous hand or segment velocity). A weighted scatter matrix was constructed as

$$S={\sum }_{i=1}^n{\omega }_i\widetilde{X_i}{\widetilde{X_i}}^T$$

(21)

where ${\omega }_i$ represents the weighting factor, S $\in {\mathbb{R}}^{3\times 3}$ is the weighted scatter matrix summarizing spatial variance, and $\widetilde{X_i}{\widetilde{X_i}}^T$ is the outer product representing the spatial contribution of frame i. Based on prior work in softball windmill pitching, instantaneous hand or segment velocity was used as the weighting factor (ωᵢ) to emphasize late acceleration phases that are critical for momentum transfer and performance [69]. Because distal segment velocities peak during these high-intensity phases in open-chain proximal-to-distal movements, velocity-based weighting allows the fitted plane to reflect the spatial organization of biomechanically meaningful portions of the trajectory. OLS identifies the plane normal vector $n$ by minimizing the weighted sum of squared orthogonal distances:

$$\underset{n}{\min }{\sum }_{i=1}^n{\omega }_i{\left(n^T\widetilde{X_i}\right)}^{2}subject to \Vert n\Vert =1$$

(22)

This constrained minimization yields the eigenvalue problem:

$$S_n= {\lambda }_{min}n$$

(23)

where n is the unit vector of the best-fit plane, and ${\lambda }_{min}$ is the smallest eigenvalue of S. The eigenvector associated with ${\lambda }_{min}$ corresponds to the direction of minimum variance, and therefore the normal to the fitted plane. Because the fitted plane passes through the centroid, its equation can be expressed as

$$n^{T}X+d=0, d=-n^T\overline{X}$$

(24)

where d is the scalar offset that allows direct computation of point-to-plane distances, each original point was orthogonal projected onto the plane using

$$P_i=X_i-\left(n^T\right(X_i-\overline{X}\left)\right)n$$

(25)

where $P_i$ represents the projected point on the plane and $n^T(X_i- \overline{X}$) is the signed perpendicular distance of point $X_i$ to the plane, and $n$ is the spatial unit vector of the plane. Off-plane deviation for each frame was quantified as

$$d_i=n^T(X_i-\overline{X})$$

(26)

where $d_i$ is the deviation distance off the plane by using the raw position $X_i$ subtract the average $\overline{X}$ from the plane perspective $n^T$ The local coordinate system was defined using the global vertical axis $y_{lab}$. The n unit vector was assigned to ($i_{SSP}$). Two orientation angles were then computed to describe the plane relative to the global reference frame. The slope angle ($\theta$) quantified the inclination of the plane relative to the global horizontal plane and was computed as the angle between the global medial–lateral axis and $k_{SSP}$. The direction angle ($\mathrm{\varnothing })$ quantified the in-plane direction of the plane relative to the target direction and was defined as the angle between $j_{SSP}$ and the global target axis (Figure 5).

And overall movement consistency relative to the plane was summarized using a weight RMS metric:

$$RMS= \sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\omega }_i{d}_i}{{\sum }_{i=1}^n{\omega }_i}}}$$

(27)

In this formulation, $d_i$ is the orthogonal distance of frame $i$ from the plane, ${\omega }_i$ is the resultant linear velocities of $i$ frame, and $RMS$ reflects the global degree of planarity; lower RMS values indicate more coordinated, planar movement.

It is important to distinguish the geometric constraints and functional assumptions underlying different plane estimation approaches. In OLS-based plane fitting, the estimated plane is purely data-driven and is constrained to pass through the centroid of the trajectory after mean-centering. No additional task-specific or directional constraints are imposed. As a result, the OLS plane represents the geometrically optimal best-fit plane that minimizes the weighted sum of orthogonal distances across all frames.

This centroid-based formulation differs fundamentally from approaches in which the plane orientation is constrained by functional or task-related assumptions. For example, functional swing plane (FSP) models define the plane using a point whose position vector is normal to the plane and explicitly incorporate assumptions about swing mechanics and momentum transfer. Similarly, target-constrained planes impose alignment with an external reference direction (e.g., intended target line), prioritizing task relevance over global geometric optimality.

Accordingly, while OLS yields a mathematically optimal plane in a least-squares sense, it does not guarantee alignment with anatomical reference planes (sagittal, frontal, or transverse) or with task-specific directions unless such alignment emerges naturally from the trajectory data. This distinction highlights that differences among plane estimation methods arise not only from optimization criteria, but also from the presence or absence of functional constraints, which in turn influence the biomechanical interpretation of the resulting plane.

Unlike PCA and SVD, which minimize reconstruction errors in a reduced subspace, OLS directly minimizes the three-dimensional orthogonal distance between each point and the plane. As a result, it often produces a more geometrically faithful representation of the dominant orientation of the movement trajectory. Unlike algebraic least-squares, it is fully coordinate-independent and does not privilege any specific axis. Additionally, OLS can incorporate biomechanical weight (e.g., resultant linear velocity, joint loading), allowing the plane to identify the important periods that contribute most meaningfully. OLS also provides a straightforward mechanical interpretation of the smallest eigenvalue identifies the orientation of least variance (i.e., plane normal). However, OLS is sensitive to small fluctuations in the low-variance direction, meaning the experimental errors noise can disproportionately influence the plane normally. If the trajectory is inherently not planar, forcing a single plane may cover important curvature or the characteristics of the segment’s motions. Although compared to PCA and SVD, weight selection introduces subjectivity, applying the inappropriate weight might distort the estimated plane. Finally, OLS lacks the variance–structure output of PCA, reducing its diagnostic utility when comparing planarity across movements.

3.3.4. Model of Newton–Raphson Method–Functional Swing Plane (FSP)

The functional swing plane was built by Kwon and his colleagues for the golf swing [28]. This method is generated by the convergence point of the three-dimensional data according to the Newton–Raphson method. This equation is widely applied in golf biomechanics studies [9,15,23,28,29]. The trajectory of a point of interest can be fit to a plane defined by a point whose position vector ($r_0$) is normal to the plane. $r_0$ is defined as the position vector from the global origin to a point on the plane such that $r_0$ is perpendicular to the plane:

$$r_o=\left[x_0 y_0 z_0\right], \mathbf{n}= {\displaystyle \frac{r_o}{\left|r_o\right|}}$$

(28)

where $\mathbf{n}$ denotes the unit normal vector of the fitted plane. Accordingly, the deviation of any trajectory point from this plane can be expressed directly in terms of its projection onto $n$:

$${\epsilon }_i=(r_i- r_o)·\mathbf{n}$$

(29)

where $r_i$ is the position in $Q_i$, $r_0$ is a point which is normal to the plane, ${\epsilon }_i$ is the deviation from the plane in each frame, $n$ is the spatial unit vector perpendicular on the plane (Figure 6).

Three trajectory points or more provide a sufficiently determined nonlinear system of equations:

$$F=\left[\begin{array}{c}{\omega }_1{\epsilon }_1\\ ⋮\\ {\omega }_i{\epsilon }_i\\ ⋮\\ {\omega }_m{\epsilon }_m\end{array}\right] \approx 0$$

(30)

where ${\omega }_i$ is the scalar weight and m is the point count (≥3); m is a function of the sampling frequency and the phase of swing from which the trajectory is extracted. From the equation above, the Jacobian matrix can be derived:

$$J= \left[\begin{array}{ccc}{\omega }_1{\displaystyle \frac{\partial {\epsilon }_1}{\partial x_0}}& {\omega }_1{\displaystyle \frac{\partial {\epsilon }_1}{\partial y_0}}& {\omega }_1{\displaystyle \frac{\partial {\epsilon }_1}{\partial z_0}}\\ ⋮& ⋮& ⋮\\ {\omega }_i{\displaystyle \frac{\partial {\epsilon }_i}{\partial x_0}}& {\omega }_i{\displaystyle \frac{\partial {\epsilon }_i}{\partial y_0}}& {\omega }_i{\displaystyle \frac{\partial {\epsilon }_i}{\partial z_0}}\\ ⋮& ⋮& ⋮\\ {\omega }_m{\displaystyle \frac{\partial {\epsilon }_m}{\partial x_0}}& {\omega }_m{\displaystyle \frac{\partial {\epsilon }_m}{\partial y_0}}& {\omega }_m{\displaystyle \frac{\partial {\epsilon }_m}{\partial z_0}}\end{array}\right]$$

(31)

where the partial derivatives assume the following form:

$$\left[\begin{array}{ccc}{\displaystyle \frac{\partial {\epsilon }_i}{\partial x_0}}& {\displaystyle \frac{\partial {\epsilon }_i}{\partial y_0}}& {\displaystyle \frac{\partial {\epsilon }_i}{\partial z_0}}\end{array}\right]= {\displaystyle \frac{1}{{\left(r_i·r_0\right)}^{\frac{1}{2}}}}\left(r_i-2r_0\right)-{\displaystyle \frac{\left(r_i-r_0\right)·r_0}{{\left(r_0·r_0\right)}^{\frac{1}{2}}}}{r}_0$$

(32)

The system of nonlinear equations was solved using the Newton–Raphson method of optimization [79]. Starting from an initial estimate of $r_0$, the solution was iteratively updated using the Jacobian matrix to minimize the weighted residual vector ${\epsilon }_i$, which represents the deviation of trajectory points from the fitted plane. Iterations continued until convergence, defined as when changes in r₀ between successive iterations fell below a predefined tolerance.

Because trajectory points were sampled at a constant temporal frequency, their spatial density varied with movement speed. To account for this effect, instantaneous segment or endpoint velocity was used as a weighting factor, so that faster, mechanically relevant phases of the motion contributed more strongly to the plane estimation. In practice, the optimization was applied to task-specific motion windows (e.g., mid downswing to mid follow-through in golf swing), as plane estimates are sensitive to the selected phase when trajectories exhibit substantial curvature [73]. This trajectory-plane fitting method enabled researchers to overcome the major limitation of the regression-based approach, which only fits the vertical coordinates of the trajectory points [62,63]. The planarity of the trajectory, meaning how closely the motion followed the fitted plane, was evaluated using the RMS and maximum fitting errors:

$$\epsilon RMS= \sqrt{{\displaystyle \frac{1}{{\sum }_{i=1}^m{\omega }_i}}{\sum }_{i=1}^m{\omega }_i{[\left(r_i·r_0\right)·n]}^2}$$

(33)

$${\epsilon }_{max}=\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|\left(r_i·r_0\right)·n\right|\right)$$

(34)

where m is the total number of sampled trajectory points used for constructing the functional swing plane, $r_i$ is the raw positions of the i-th frame, $r_o$ is the covert point of the plane, n is the unit vector of $r_o$, and the ${\omega }_i$ is the resultant linear velocity of each frame. The orientation of the FSP can be computed as

$${\varnothing }_{plane}={cos}^{-1}\left(n_{plane}·k\right),$$

(35)

$$d_{plane}={\displaystyle \frac{k\times n_{plane}}{\left|k\times n_{plane}\right|}},$$

(36)

$${\theta }_{plane}=sign(i·d_{plane})·{cos}^{-1}(j·d_{plane})$$

(37)

where i, j, and k are the X, Y, and Z axes unit vectors of the global frame, respectively. The slope of the plane (${\varnothing }_{plane}$) was calculated using the angle between the global ground and the plane, whereas the angle between the intersection line created by the plane with the ground ($d_{plane})$ and the global y-axis (direction of the target) was used as the direction ground angle $\left({\theta }_{plane}\right)$ (Figure 7).

The FSP method provides a nonlinear, iteratively optimized plane definition that is highly suited to movements dominated by circular or curved trajectories, such as striking and the softball windmill pitching. Unlike PCA, SVD, or OLSFSP does not rely on linear algebraic decompositions. FSP defines the plane through the point whose position vector is perpendicular to the plane and iteratively solves for this point using Newton–Raphson optimization. This approach supports more accurate modeling when the trajectory contains sections with uneven temporal spacing. Also, FSP is weighted by the velocity components for variations in sampling density. According to this characteristic, FSP strongly reflects the movement’s dynamic execution rather than its geometric variance features. However, these advantages come with several practical considerations. The method is more computationally demanding and requires an initial guess and close monitoring of convergence. Selecting the appropriate portion of the movement to initialize the FSP solution is essential. If the analyst is not familiar with the mechanics of the specific sport, it becomes easy to select an inappropriate phase and unintentionally distort the resulting plane. FSP also lacks closed-form solutions, making theoretical interpretation less straightforward. Furthermore, because FSP constructs the plane through a perpendicular reference point rather than a variance-based solution, cross-individual or cross-sport comparisons are less straightforward unless consistent procedures are applied.

3.4. Summary and Comparison of Motion-Plane Models

In conclusion, the four models (PCA, SVD, OLS, and FSP) represent different mathematical strategies for identifying the dominant motion-plane. All approaches aim to define a representative plane.

Although all approaches aim to define a representative plane, each model is grounded in different theoretical assumptions and emphasizes distinct characteristics of the trajectory. PCA and SVD rely on variance structure to estimate the principal direction and are efficient for highly planar and evenly sampled across time. OLS concentrates on minimizing the true perpendicular distance from the plane and allowing incorporating meaningful weighting based on the task demands. The FSP approach applied the nonlinear formulation that is better suited for circular or unevenly spaced trajectories. Because each method captures different aspects of movement organization, no single approach is universally optimal, and method selection should be guided by the characteristics of the motion and the specific biomechanical questions being addressed.

To improve comparability and reproducibility across studies, consistent reporting practices are needed in motion-plane analyses. At a minimum, studies should clearly report the object used to define the trajectory (e.g., joint center, segment, or end-effector), the coordinate frame of reference, the analyzed phase window, and any filtering or preprocessing procedures. In addition, the plane estimation method, the use of weighting (if applicable), and the deviation metric (e.g., RMS, maximum deviation, or time-varying measures) should be explicitly specified. Reporting these elements in a consistent manner would allow more meaningful cross-study comparisons and support the application of motion-plane modeling across different sports and movement tasks.

Phase window selection and deviation metrics are highly task- and sport-dependent and therefore are not listed as fixed attributes in

Table 3. Summary of motion-plane modeling method used in sports biomechanics.

Model	Core Concept	Key Characteristics	Advantages	Limitations	Typical Applications
PCA	Covariance eigen-decomposition	Variance-based; orthogonal axes	Fast; stable; easy to implement	No weighting; variance-driven	Highly planar motions; large datasets
SVD	Matrix factorization	Direct decomposition; scale-dependent	Robust to noise; no covariance step	No weighting; scaling sensitivity	Noisy or near-planar trajectories
OLS SPP	Orthogonal error minimization	Geometric best-fit plane; weightable	Physically interpretable; flexible	Sensitive to outliers; planar bias	Motions with a clear functional plane
FSP	Nonlinear optimization	Velocity- and phase-weighted plane	Captures curved motion structure	Initialization-dependent; computationally costly	Windmill pitching; striking motions

PCA: Principal component analysis; SVD: Singular value decomposition; OLS: Orthogonal least-squares; SPP: Softball pitching plane; FSP: Functional swing plane.

. These elements should be determined based on the mechanical relevance of the movement and the specific research question (

Table 4. Recommended reporting items for motion-plane analyses across sports.

Reporting Item	Description	Rationale
Trajectory definition	Point, segment, or end-effector used to define the trajectory	Different definitions lead to different plane orientations
Coordinate frame	Global or local reference frame and axis definitions	Plane orientation is frame-dependent
Phase window	Full motion or task-relevant phase (e.g., acceleration, release)	Mechanical relevance varies across phases
Filtering Preprocessing	Filtering method, cutoff frequency, normalization	Affects trajectory smoothness and planarity
Plane estimation method	PCA, SVD, OLS, SPP or FSP	Methods emphasize different geometric or statistical properties
Weighting strategy	None or task-based weighting (e.g., velocity)	Influences functional relevance of the fitted plane
Deviation metric	RMS, maximum deviation, or time-varying deviation	Captures different aspects of planarity
Reporting units	Units used for deviation and angles	Ensures interpretability and reproducibility

PCA: Principal component analysis; SVD: Singular value decomposition; OLS: Orthogonal least-squares; zeSPP: Softball pitching plane; FSP: Functional swing plane.

).

3.5. Cross-Sport Applications for Motion-Plane Models

3.5.1. Application Motion Plane Across Sports

Despite the long history of trajectory-based planes, their application to sport biomechanics remains limited. Among all sports, golf biomechanics provides the most comprehensive application of motion-plane modeling, integrating analyses of the club’s trajectory plane and hand segmental planes. Coleman & Rankin were among the first to challenge the traditional assumptions that the golf swing could be a single plane [62]. They constructed the left-arm to demonstrate the inclination and direction angles changing continuously via the entire downswing phases [62]. They also quantified the vertical distance of the clubhead deviation near the impact point. They found that neither the arm nor the clubhead travels in a fixed lane, contradicting simplified double-pendulum planar models. Their anatomical planes thus generated the first empirical evidence that planarity is not a realistic global assumption, which motivated the need for more trajectory-based plane identifications. Following this study, Coleman & Anderson built the regression-based best-fit plane to the club-shaft trajectory by applying the OLS formula [63]. Their results reported large RMS fitting errors (>8 cm), demonstrating that the club-shaft path is only semi-planar and deviates substantially from any single plane. Considering these observations, Kwon introduced a more function-oriented definition of the swing plane as Functional Swing Plane [28]. Rather than using simple projections or OLS, the FSP method employs an iterative Newton–Raphson optimization to fit a plane through the clubhead and midpoint-of-hands trajectory, achieving a plane with minimal orthogonal error. This motion-plane model allowed the FSP to reflect the portion from the mid of downswing to the follow through where the club’s movements are most coordinated and mechanically significant. Nesbit and colleagues projected the golf-club-head trajectories to a two-dimensional view (side view) and linked the radius to the golfer’s kinematic and kinetic variables [31]. Finally, Matsumoto applied SVD to examine the structure of the golf swing [80]. They decomposed the golf-swing data into principal movement patterns. Their results showed that the leading mode closely followed the main arc of the swing, effectively capturing the underlying functional swing plane. This study illustrates that although the golf swing is a three-dimensional motion, most phases of the swing can be described by low-dimensional patterns (i.e., a dominant swing plane). Their findings emphasize how plane-related features remain useful for understanding coordination in the golf swing.

Beyond golf, other striking sports have developed similar trajectory-plane concepts for swing movements. For instance, OLS methods have been adopted to identify the swing plane of the field-hockey stick during the swing. In the study of Willmott and Dapena [66], the stick-face trajectory was fitted with a OLS-derived plane, and the planar deviation was quantified based on the orthogonal residual between the trajectory and the plane during the downswing interval. Their finding indicated that the stick and forearms moved with lower variability relative to the plane, whereas the upper arm displayed greater deviations.

Kicking motion has also been analyzed using motion-plane concepts in the biomechanics literature. In a study by Bezodis and colleagues [67], they developed the OLS formula to create the rugby kicking motion-plane during the final portion of the downswing of the kicking-foot resampled at equal spatial intervals to remove the bias caused by increasing foot speed to identify the kicking motion-plane. The findings demonstrated the kicking-foot trajectory was highly planar, with small deviation (RMS) approximately 2 mm, confirming that the distal segment follows a very consistent direction. Importantly, the directional angle of the kicking plane distinguished accurate and inaccurate kickers. Navandar et al. [68] formed PCA to identify the dominant movement patterns of the soccer instep kick. By decomposing hip and knee flexion extension patterns into four principal components, the kicking motion exhibits a clear low-dimensional structure, meaning that a large portion of the movement can be described by a small number of coordinated “movement planes”. Cheng and colleagues demonstrated that the higher deviation off the plane the lower ball velocity in the adolescent softball pitchers [65]. A summary of the suitability of different motion-plane modeling approaches across sports tasks is presented in (

Table 5. Scheme summarizing the appropriateness of motion-plane modeling methods for different movement phases and trajectory characteristics.

Sport/Task	Trajectory Characteristic/Phase	Method	Modeled Trajectory	Key Outcome	Implication for Plane Modeling
Golf swing	Multi-phase, non-planar	OLS	Shaft	Large RMS error	Fixed plane insufficient
Golf swing	Quasi-planar distal	FSP	Clubhead	Minimal error	Functional plane appropriate
Soccer kick	Low-dimensional joint patterns	PCA	Hip and knee	Few PCs explain variance	PCA suitable for structure extraction
Softball pitching	Ballistic distal motion	OLS (SPP)	Wrist	Functional plane	Task-specific plane required

PCA: Principal component analysis; PCs: Principal components; OLS: Orthogonal least-squares; FSP: Functional swing plane; RMS: Root mean square; SPP: Softball pitching plane.

).

3.5.2. Performance and Joint Loading

Recent trajectory-plane research across golf, hockey, soccer, softball windmill pitching, and rugby show that motion-plane characteristics are not just geometric descriptors, but also meaningful contributors of the motion quality and mechanical styles. In the rugby kicking motion, the differences between accurate and inaccurate performance have been quantified based on their swing-plane orientation, with more accurate kickers demonstrating a shallower and more target-aligned plane [67]. These findings suggested that plane direction and inclination can be linked to accuracy-related performance metrics, although they do not constitute a unified predictive model.

In softball windmill pitching, trajectory-based analyses of the softball pitching plane have demonstrated high reliability, and greater RMS deviation has been quantitatively associated with lower ball velocity at both within- and between-pitcher levels [65]. These relationships are typically expressed using correlation coefficients or regression slopes, rather than standardized performance impact coefficients.

In golf, FSP further suggest that the angular organization of the downswing influences clubhead velocity, and analyses have revealed that plane orientation, and on-plane variables are associated with clubhead velocity and impact consistency variables that are strongly related to shot outcome. While these studies do not directly quantify scoring performance, they provided quantitative links between plane-based metrics and surrogate performance indicators such as ball speed, launch direction, and contact precision [28].

Importantly, plane-based metrics have frequently been used to characterize and classify movement strategies rather than to directly predict performance magnitude. Across multiple studies, analyses incorporating functional swing plane (FSP) orientation and on-plane variables have reported no significant differences in clubhead speed across variations in upper-extremity strategies, including wrist release timing, torsional styles, or hip–shoulder separation [9,15,29,79]. These findings suggest that plane-related descriptors capture stable features of individual movement organization rather than serving as universal predictors of performance output. Selected studies indicate that organizing motion relative to the swing plane may facilitate more effective proximal-to-distal sequencing. For example, McGuire and colleagues demonstrated that when upper-extremity motion was more closely aligned with the swing plane, the resulting kinetic sequence (pelvis → shoulder → girdle → arms → club) was associated with greater ball distance [15]. These findings suggest that FSP metrics are most appropriately interpreted as descriptors of movement strategy and coordination, providing a biomechanical context for interpreting kinematic variables, rather than as direct predictors of performance magnitude.

With respect to joint loading and risk of injury, Willmott and Dapena observed highly planar motion of the stick and forearm but substantially less planar upper-arm behavior, implying increased off-plane acceleration and deceleration demands at proximal joints in field hockey [76]. Although these plane-based analyses do not directly quantify injury outcomes or establish injury-risk coefficients, their mechanical implications are consistent with broader biomechanical evidence linking off-axis segmental motion to increased shoulder and elbow loading. Therefore, current evidence supports quantitative associations between plane deviation and mechanical demand, while injury risk reduction remains an open area for future research.

Importantly, most existing studies do not explicitly control for potential confounding variables such as anthropometrics, strength, technique variability, or workload exposure. As a result, current findings support associative rather than causal interpretations between motion-plane deviation and joint loading. Future studies incorporating multivariable regression models or matched control designs will be necessary to clarify independent effects and dose–response relationships.

Overall, the existing literature provides quantitative relationships between motion-plane characteristics and performance-related or mechanical outcomes, primarily through correlations, group comparisons, and regression-based analyses. However, standardized performance impact coefficients or unified predictive models have not yet been established. Motion-plane modeling thus offers a quantitative framework for describing movement quality and mechanical organization, while highlighting the need for future studies to formalize dose–response relationships between plane deviation, performance outcomes, and joint loading.

4. Conclusions, Limitations, and Future Directions

4.1. Conclusion

Recent advances in sports biomechanics have introduced a variety of computational approaches for analyzing complex movement patterns. However, the present review focuses specifically on trajectory-based motion-plane modeling, which addresses different research questions related to spatial organization and deviation from a reference plane. This comprehensive review demonstrated four motion-plane models computed by the trajectory of the position during the movements. The motion plane opens a new window to organize complex three-dimensional motions to project to a two-dimensional perspective for more practical applications. Open-chain, proximal-to-distal rotational sports are ideal candidates for constructing motion planes, as the goal is to achieve maximum velocity in the distal segment. This characterization facilitates coordinated distal segment motion that is aligned with the intended direction of movement. Based on the energy-transfer theory, athletes who can effectively transmit momentum through the kinetic chain can produce greater distal segment velocity [1,10,14,81,82]. The four mathematical approaches were introduced to make motion planes: Principal Component Analysis (PCA), Singular Value Decomposition (SVD), Orthogonal Least-Squares (OLS), and Functional Swing Plane (FSP). Those models have revealed simplified, two-dimensional patterns across different types of sports. These approaches explain how athletes generate and transfer momentum through coordinated trajectories, giving researchers different perspectives for describing the strategies across sports by applying the inclination and directional orientation of the motion plane. Since studies have demonstrated that the deviation from the motion plane influences performance, it has become a standard variable for evaluating motion stability and consistency.

Although plane-based measures do not always predict performance directly, they capture stable features of individual techniques and may help explain how athletes coordinate multi-segment motions. While deviations from a preferred trajectory may reflect individual coordination strategies or compensatory adaptations, their direct association with specific injury mechanisms requires further empirical validation through longitudinal data. Deviations from a preferred trajectory can indicate compensatory strategies and potential inefficiencies. Overall, motion-plane modeling offers a promising framework for quantifying movement organization in high-speed sports. Despite meaningful progress in recent years, several important gaps remain, particularly in determining how planar variability correlates with long-term mechanical loading and musculoskeletal health.

4.2. Future Study

Despite the strengths of motion-plane modeling, the applicability of these findings is not universal. The approaches reviewed here are most suitable for open-chain, proximal-to-distal movements in which trajectories exhibit a dominant planar organization. For movements characterized by large curvature, frequent changes in direction, or substantial out-of-plane components, a single representative plane may oversimplify the underlying coordination. In such cases, increased deviation from a fitted plane may reflect functional adaptations, fatigue-related changes, compensatory strategies, or the absence of a stable optimal solution rather than reduced motor control. Therefore, the interpretation of motion-plane features should consider the specific task demands, movement phase, and mechanical context. Accordingly, the findings summarized in this review should be interpreted as task-specific rather than universally applicable across all movement types.

4.2.1. From Plane-like Descriptors to True Trajectory-Based Motion Planes

In many sports, researchers have revealed plane-related variables such as arm slot, bat path, trunk lateral tilt, shoulder abduction angle, and kick-foot directions, but most of those sports have not computed an optimal-fit motion plane. As a result, the structure of the motion is often inferred indirectly rather than quantified mathematically. Future studies should concentrate on identifying true trajectory-based planes. Implementing a motion plane rather than depending on discrete variables or plane-like descriptors enables more accurate comparisons of mechanical strategies. This enables comparisons between strategies and supports the monitoring of performance and consistency throughout a season or across multiple years.

4.2.2. Linking Plane Features with Performance Outcomes and Mechanical Demands

Current studies primarily use orientation characteristics to categorize athletes, with only a limited number of papers addressing the relationship between off-plane deviation and performance or injury risks in striking motions. There is still a need for research that directly examines how plane-based features influence outcomes such as performance, accuracy, motion efficiency, and potential injury risks. One classic example of baseball pitching is shoulder abduction, which has been linked to the performance and the elbow and shoulder loading in the different studies [33,43,83,84,85]. Understanding whether plane deviation and radius predict performance success, compensatory behavior, or increased mechanical loading remains an unexplored opportunity, especially in high-risk upper-extremities injury sports (e.g., baseball). Additionally, future research can apply these models alongside outcomes measurements including ball flight, contact precision, joint kinetics, or joint loading. Doing so would clarify the practical value of planar organization and help identify whether specific plane characteristics reflect efficient or high-risk movement strategies. Based on these gaps,

Table 6. Proposed motion-plane modeling applications by movement phase and trajectory type that remain to be experimentally validated.

Model	Trajectory Type	Movement Phase	Example Sports	Rationale
PCA/SVD	Moderately curved	Arm cocking Acceleration Downswing	Baseball pitching (arm path), golf swing	Captures dominant variance
OLS/FSP	Highly curved	Acceleration	Softball windmill, tennis serves, hockey, Volleyball, Handball	Phase-weighted relevance
Hybrid approaches	Multi-planar	Arm cocking Acceleration Transition phases	All open-chain sports	May reduce model-specific bias

PCA: Principal component analysis; SVD: Singular value decomposition; OLS: Orthogonal least-squares; FSP: Functional swing plane.

summarizes proposed applications of motion-plane models across different movement phases and trajectory types that remain to be experimentally validated.

This scheme is intended as theoretical guidance based on trajectory characteristics and movement phase, rather than a summary of experimentally validated model performance.

4.2.3. Integrating Multiple Modeling Approaches to Develop Robust, Task-Relevant Plane Estimates

There is a gap in comparing the performance of all existing models across different sports. As each formula emphasizes a different aspect of planarity, future studies may benefit from combining geometric, statistical, and functional approaches. Hybrid models may help overcome method-specific limitations, provide more accurate estimates of plane orientation and deviation, and establish standardized benchmarks for comparing plane-fitting accuracy across movement types, datasets, and sports.

4.2.4. Empirical Validation and Robustness Benchmarking

Although the modeling approaches reviewed here differ in their theoretical sensitivity to noise, outliers, and trajectory curvature, direct empirical comparisons under controlled conditions remain limited. Most existing studies apply a single plane-fitting method within a specific sport or task, making it difficult to quantitatively evaluate relative robustness across models. Future research should conduct systematic benchmarking studies using shared datasets or simulated trajectories with controlled noise levels, outlier contamination, and curvature profiles. Such comparisons would help clarify the practical trade-offs among PCA-, SVD-, OLS-, and FSP-based approaches and provide evidence-based guidance for method selection in applied biomechanics.

Overall, future research should aim to expand the use of motion-plane modeling across a wider range of sports, connect plane characteristics to performance and loading outcomes, and develop improved estimation methods that combine the strengths of existing mathematical approaches. These steps will help advance motion-plane analysis from a descriptive tool toward a broader framework for understanding coordination, efficiency, and mechanical demand in high-speed athletic movements.