Biomechanical framework for the inverse dynamic analysis of swimming using hydrodynamic forces from swumsuit

Abstract This study aims to integrate an open-source software capable of estimating hydrodynamic forces solely from kinematic data with a full-body biomechanical model of the human body to enable inverse dynamic analyses of swimmers. To demonstrate the methodology, intersegmental forces and joint torques of the lower limbs were computed for a six-beat front crawl swimming motion, acquired at LABIOMEP-UP. The hydrodynamic forces obtained compare well with existing numerical literature. The intersegmental forces and joint torques obtained increase from distal to proximal joints. Overall, the results are consistent with the limited literature on swimming biomechanics, which provides confidence in the presented methodology.


Introduction
Human motion involves complex interactions, between the skeletal system and the surrounding environment, whose understanding is essential for many different fields, including orthopedics, medical rehabilitation, or sports science (Quental et al. 2016;Castro et al. 2019;Antunes et al. 2021). Fundamental quantities of interest in human motion research include intersegmental forces and torques acting at joints, whose in vivo or in vitro measurement is, when possible, extremely difficult. Computational biomechanical models based on multibody formulations are powerful and attractive tools that have been extensively applied to evaluate these quantities (Quental et al. 2018;Castro et al. 2019). However, most applications in human swimming analysis are limited due to technical difficulties related with experimental data acquisition. For instance, the airwater interface reduces marker visibility, and normal splashing produces reflections that make tracking of markers difficult. Another major difficulty is the estimation of external forces (Lauer et al. 2016), an essential requirement for an inverse dynamic analysis.
Despite advances in pressure measurement systems for the estimation of external forces, their application in human swimming analysis is still limited to specific body segments. Moreover, they do not reflect the instantaneous points of force application (Takagi et al. 2016). Computational Fluid Dynamics , done either with Smoothed Particle Hydrodynamics or Finite Element Methods, is a promising alternative in aquatic motion research for estimating these quantities. However, its computation time is usually unreasonably high for large-scale applications. Considering this limitation, Nakashima et al. (2007) developed the simulation software Swumsuit (SWimming hUman Model with Synthetic User Interface Tools), which estimates the fluid forces acting on the whole body without solving the flow field, thus speeding up the simulation. Fluid forces, here after called hydrodynamic forces, are determined from the local kinematics of each body at each instant of time and from experimentally measured force coefficients (Nakashima et al. 2007). The Swumsuit software is a promising tool to promote the musculoskeletal analysis of swimming movements, but it lacks import and export functions that allow its wide-spread application with other modeling tools (Langholz et al. 2016).
The aim of this study is to propose a systematic and reproducible biomechanical framework for swimming analysis, by integrating the Swumsuit software and a 3 D full-body biomechanical model, which enables the computation of internal forces consistent with the surrounding environment. A modeling interface is developed in Matlab (Mathworks, Natick, MA, USA) to generate automatically the input files for the Swumsuit software and to process the output files for the identification of the external forces to be applied to the biomechanical model considering the framework proposed by Quental et al. (2022). As an application case for the developed biomechanical framework, an inverse dynamic analysis of a six-beat front crawl swimming stroke, acquired at LABIOMEP-UP, is carried to evaluate the intersegmental joint forces and joint torques of the swimmer lower limbs.

Full-body biomechanical model for swimming analysis
Using Cartesian coordinates and Euler parameters to describe the body kinematics (Nikravesh 1988), a 3 D full-body model of the human body is developed in Matlab considering 16 rigid bodies: pelvis, torso, neck, head, 2 thighs, 2 legs, 2 feet, 2 arms, 2 forearms, and 2 hands (Silva and Ambr osio 2003). Body segment inertial parameters are based on Dumas et al. (2007), except for the head and neck, which are proposed by P amies-Vil a (2012). Fifteen joints constrain the motion of the body segments: eight spherical joints represent the cervical, lumbar, and right and left hip, shoulder, and ankle joints; four universal joints represent the right and left elbow and wrist joints; and three revolute joints represent the atlantooccipital and the right and left knee joints (Silva and Ambr osio 2003). The biomechanical model has 41 degrees of freedom.

Estimation of hydrodynamic forces
Hydrodynamic forces are estimated in this study using the Swumsuit software (Nakashima et al. 2007). To validate Swumsuit, experimental and simulated swimming velocities and hydrodynamic forces have been compared for different swimming strokes, showing errors of 1%-7.5% in swimming velocity and of 10% in hydrodynamic forces (Takagi et al. 2021), which is acceptable considering the complexity and inaccuracy associated with the experimental assessment of hydrodynamic forces in swimming. As input, Swumsuit requires data regarding body properties, Figure 1. Correspondence between body segments of the full-body biomechanical model developed in this study and that available in Swumsuit. joint motion, analysis settings, and the absolute motion of the body made of three linear and three angular velocities of the whole-body center of mass. Due to differences in model topology and formulation between Swumsuit and the biomechanical model in this study, a modeling interface is developed in Matlab to integrate Swumsuit and the developed fullbody biomechanical model considering the framework proposed by Quental et al. (2022).
The biomechanical model included in Swumsuit, which cannot be customized, is composed of 21 body segments, each of them represented by a series of elliptic plates. Correspondence between the body segments of the models used here and in Swumsuit is defined to allow a direct relationship between them ( Figure 1). Geometric and anthropometric data are scaled from the dataset provided by Nakashima et al. (2007), based on a Japanese male (20-to 29-year-old male; height: 1.71 m; body mass: 64.9 kg), using the subject's height and body mass.
Swumsuit describes the position of each body segment with respect to the previous body segment in the kinematic chain through sequential rotations about their body-fixed reference frames. In Swumsuit, the human body is described by two kinematic chains, as illustrated in Figure 2, and all joints are assumed spherical joints, with 3 degrees of freedom. In the modeling interface, the kinematic data input to Swumsuit result from a kinematic analysis of the biomechanical model guided by the acquired kinematic data that ensures consistency of the motion under analysis with the biomechanical model. Since the biomechanical model developed considers a different type of coordinates than that in Swumsuit, kinematic data output from the kinematic analysis are processed according to Swumsuit requirements. For all time steps, and for each parent-child pair of body segments, Tait-Bryan angles are computed. An extrinsic sequence of rotations ZYX is used here, although any possible sequence may be considered. For each child segment, Swumsuit requires the definition of the Tait-Bryan angles for the parent and child body segments.
Once the Swumsuit simulation is defined, the force data output are processed to be consistent with the biomechanical model. Swumsuit force data include the total hydrodynamic forces acting at the center of mass of each body segment and the hydrodynamic forces, excluding buoyancy, acting at the center of each elliptic plate. Note that each body segment is composed of 10 elliptic plates that discretize the body shape and enable the identification of the hydrodynamic forces distribution in the surface of each body segment. Since in the multibody formulation used here external forces must describe a force-couple system acting at the center of mass of each body segment (Nikravesh 1988), the forces output at the center of each elliptic plate are processed to compute equivalent force-couple systems for each body. By summing all forces acting on all elliptic plates of a body and comparing the resultant force with the hydrodynamic forces including buoyancy, output from Swumsuit, the contribution of buoyancy alone is computed. The equivalent force-couple systems due to buoyancy are computed for each body assuming that the buoyancy forces are distributed equally over all its elliptic plates. For further detail on the data processing between the formulations of Swumsuit and the biomechanical model applied here, see Quental et al. (2022).

Inverse dynamics
An inverse dynamic analysis is performed for a sixbeat front crawl swimming stroke to evaluate the intersegmental joint forces and joint torques of the swimmer lower limbs. Experimental data were collected at LABIOMEP-UP from a healthy regional level swimmer (25-year-old male; height: 1.80 m; body mass: 70.3 kg), with no history of injuries in the two months before the experiment, using a Qualisys Track Manager system (Qualisys, Gothenburg, Sweden) at 100 Hz. Twelve above water cameras (3x Oqus 310þ, and 9x Oqus 400) and 10 underwater cameras (4x Oqus 300 þ u, 4x Oqus 700 þ u, and 2x Oqus 310 þ u) were used in a 1.90 m deep, 25 m long and 6 lanes indoor swimming pool, with a 27 C water temperature. To synchronize the above water cameras, the underwater cameras, and the air-water interface, a three-step calibration was performed (Andersen et al. 2021), leading to a calibration volume of 28 m 3 (7 m in length, 2 m in width, and 2 m in height), located 9 m from the start wall, and centered relatively to the pool width. Sixty-six reflective markers, described in detail in the supplementary material, were placed on anatomical bony landmarks to track motion. After a standardized personal warm-up routine (1000 m at submaximal pace) (Silva et al. 2019), the swimmer performed 4 times 25 m bouts of the six-beat frontcrawl technique, at the same submaximal and selfcontrolled speed, with 3 min rest in between. Each 25 m bout began from a push start, and the swimmer was instructed not to breathe inside the calibration volume (marked on the bottom of the pool) to avoid effects of breathing on swimming technique (McCabe et al. 2011). This study was conducted in accordance with the Declaration of Helsinki and followed a protocol approved by the Ethics Committee of the Faculty of Sport of University of Porto (CEFADE 24 2020, 11 November 2020) (Fernandes et al. 2022).
From the 4 swimming trials collected, the one with the least noise was selected for analysis in this study. After proper filtering of the data, through a residual analysis, body-fixed reference frames were defined for all body segments according to the recommendations of the International Society of Biomechanics (Wu et al. 2002(Wu et al. , 2005. The elbow, wrist, knee, and ankle joint centers were estimated as the midpoint between two reflective markers (Dumas et al. 2007). The atlantooccipital (P amies-Vil a 2012), cervical (Reed et al. 1999), shoulder (P amies-Vil a 2012), lumbar (Murphy et al. 2011), and hip (Hara et al. 2016) joint centers were estimated using regression equations.
The inverse dynamics problem is solved in Matlab considering a fully determined problem, in which the degrees of freedom of the biomechanical model are driven by joint actuators. The external forces, i.e., the hydrodynamic forces, are applied according to the methodology described in Section 2.2. Through the solution of the equations of motion, given by intersegmental forces and joint torques acting on the human body are computed. In Equation (1), M€ q are the inertial forces, in which M is the mass matrix and € q is the vector of generalized accelerations of the biomechanical system; U T q k are the internal forces, in which U q is the Jacobian matrix of the kinematic constraints and k is the corresponding vector of Lagrange multipliers; and g is the vector of external forces.

Results
Hydrodynamic force profiles, containing added mass fluid force, normal drag force, tangential drag force, and buoyancy, are presented in Figure 3 for the right thigh, leg, and foot. For the sake of comparison, force profiles obtained using the datafiles of Nakashima et al. (2007) for a similar motion are also shown. The stroke cycle under analysis begins with the downbeat of the left lower limb. Considering all the presented body segments, the most relevant force components are along the anterior-posterior direction (F z ), perpendicular to the water surface. The force components along the medial-lateral (F x ) and inferior-superior (F y ) directions are generally larger for the foot than for the thigh and leg segments.
The intersegmental forces estimated through inverse dynamics are shown in Figure 4 for the right and left hip, knee, and ankle joints. The maximum forces increase from distal to proximal body segments: 200 N for the hip joint, 120 N for the knee joint, and 80 N for the ankle joint. Despite intra cycle variations among joints, intersegmental force magnitudes are similar for the left and right lower limbs. Figure 5 presents the flexion-extension joint torques for the right and left hip, knee, and ankle joints. As for the intersegmental forces, the flexionextension joint torques increase in magnitude from distal to proximal body segments: 55 Nm for the hip joint, 22 Nm for the knee joint, and 5 Nm for the ankle joint. Forces are presented in a global reference frame in which the x-, y-, and z-axes represent, approximately, the medial-lateral direction, inferior-superior direction, and anterior-posterior direction. The x-axis represents the stroke cycle in percentage. For the sake of comparison, results obtained using the datafiles of Nakashima et al. (2007) are also displayed.

Discussion
Considering the integration of a 3 D full-body biomechanical model and the Swumsuit software, the present study presents a framework for the inverse dynamic analysis of swimming motions including estimated hydrodynamic forces. As an application case, hydrodynamic forces, intersegmental forces, and joint torques of the human lower limbs are evaluated for a six-beat front crawl swimming motion.
Overall, the hydrodynamic forces obtained here compare well with those estimated using the datafiles of Nakashima et al. (2007), but the magnitudes are usually larger in the present study, which is likely due to differences in swimming motions evaluated. All body segments of the lower limb are acted upon by hydrodynamic forces throughout the entire stroke cycle. Although it highly depends on the swimmer's ability and technique, keeping the feet submerged is expected to contribute to body propulsion and to decrease wave drag (Keys 2010). The hydrodynamic forces along the anterior-posterior direction for the right foot show three force peaks, approximately at 28%, 60%, and 99% of the stroke cycle, which are consistent with the three downbeats of the body segment during the stroke cycle. The peak at 28% is masked by an oscillation between 28% and 50%, which can be due to the different swimmers' techniques. The correlation of these force peaks with the ankle joint motion agrees with the literature assumption that the maximum propulsive forces in the lower limbs occur during the downbeat, when the ankle is in maximum plantarflexion (Wei et al. 2014;Sanders et al. 2018).  The intersegmental forces estimated through inverse dynamics increase from distal to proximal body segments of the lower limb. Their pattern is not consistent with that of the hydrodynamic forces, i.e., proximal joints of body segments with the highest hydrodynamic forces do not always present the highest intersegmental forces. The leg kick is described as a wave like motion that begins with the extension/ flexion of the thigh, and follows through the leg and foot, i.e., from proximal to distal joints (Keys 2010). As the first element of the kinematic chain of the human lower limbs, the hip concentrates larger intersegmental forces, compared to the remaining joints of the lower limb. For the ankle joint, the intersegmental forces show high and low forces peaks, which correspond, respectively, to three downbeats and three upbeats of the lower limb. During the upbeat, extension of the thigh occurs at the hip, followed by the upward motion of the leg and foot (Keys 2010). As these bodies approach the water surface, the joint force decreases. A similar result was found by Harrison et al. (2014) when analyzing the entry of the hand into the water. The large intersegmental force peaks during the downbeat of the lower limb are consistent with the higher hydrodynamic forces generated during this phase by the surrounding water, which propels the body forward.
The flexion-extension joint torques estimated in this study behaved like the intersegmental forces, i.e., they increase from distal to proximal joints. While studying the joint torques of the upper limbs, Harrison et al. (2014) also observed higher joint torques for the proximal joints compared to the distal joints. Throughout the stroke cycle, the hip, knee, and ankle joint torques vary according to the downbeat and upbeat sequences of the front crawl swimming motion. To the authors' knowledge, only Nakashima et al. (2015) presented hip joint torques for a front crawl swimming motion. In their study, they investigated the effect of knee joint motion on the swimming performance of an experienced swimmer with an acquired one-sided transfemoral prosthesis in the right lower limb with different knee joint stiffnesses. For the sake of comparison, our results are compared with those of the healthy (left) lower limb, as shown in Figure 6. Although significant differences exist between the results obtained here and those of Nakashima et al. (2015), similar general trends are identified in both. In addition to the swimmers' conditions and swimming techniques, differences in modeling conditions considered may justify these results. One assumption regarding the definition of the hip joint characteristics is especially relevant. Unlike the present study, which estimated the hip joint center using regression equations, Nakashima et al. (2015) assumed the hip joint center as the position of the greater trochanter, which certainly affected the accuracy and interpretation of the hip joint torques (Hara et al. 2016). The peak-to-peak joint torque amplitude obtained in this study (90 Nm) compares well with that obtained by Nakashima et al. (2015) (123 Nm for the Type A condition and 114 Nm for the Type C condition).
Despite presenting a framework for inverse dynamic analyses of swimming motions and providing further data on the lower limb biomechanics of a front crawl swimming motion, the present study is not without limitations, which must be considered. For the estimation of internal forces, hydrodynamic forces were computed using the software Swumsuit. Naturally, Swumsuit limitations are also limitations of the present study (Quental et al. 2022). One noteworthy limitation is that Swumsuit does not expand the calculation of the complex flow field, which could potentially lead to calculation errors when different flows, or a flow and an object, interact, or when swimming occurs near a wall. Still, validation studies have shown acceptable deviations between experimental measurements and Swumsuit estimates of swimming velocity and hydrodynamic forces (Takagi et al. 2021), especially if considering that the computational Figure 6. Flexion-extension hip joint torques estimated in this study and the study of Nakashima et al. (2015). The study of Nakashima et al. (2015) considered soft (Type A) and intermediate (Type C) knee joint conditions regarding the stiffness of a transfemoral prosthesis. The x-axis represents the stroke cycle in percentage.
complexity of more advanced computational methods preclude their wide application in large-scale analysis. The biomechanical model developed did not include the muscular system. Hence, the intersegmental forces presented do not represent actual joint contact forces since these are a result of muscle action. Since this study focused not only on the biomechanical analysis of swimming, but also on the biomechanical modeling aspects, only one swimming motion was evaluated, which limits its generality. Future studies focusing on the biomechanics of swimming should consider a larger number of motion acquisitions. Finally, due to the limitations of experimental data available in the literature regarding swimming, no further validation of results was possible.

Conclusion
By integrating a 3 D full-body model of the human body with the Swumsuit software, this work proposes a consistent procedure for the complete inverse dynamic analysis of the biomechanics of swimming that, by its overall reach, is novel. The results regarding hydrodynamic forces, intersegmental forces, and joint torques, computed from an inverse dynamic analysis of a six-beat front-crawl swimming motion, were shown to be qualitatively consistent with the limited literature available on swimming biomechanics, which provides confidence on the developed framework. The maximum intersegmental forces and flexion-extension joint torques increased from distal to proximal body segments: 200 N and 55 Nm for the hip joint; 120 N and 22 Nm for the knee joint; and 80 N and 5 Nm for the ankle joint, respectively.

Disclosure statement
There is not any financial or personal relationship with other people or organizations that could inappropriately influence this work.

Funding
This work was supported by the Portuguese Foundation for Science and Technology (FCT), through IDMEC, under LAETA, project UIDB/50022/2020.