Optimal control simulations tracking wearable sensor signals provide comparable running gait kinematics to marker-based motion capture

Introduction

Inertial measurement units (IMUs) are becoming increasingly common for assessing human movement, including running gait (Blazey, Michie & Napier, 2021; Benson et al., 2022; Zeng et al., 2022; García-de-Villa et al., 2023). These small, lightweight, and inexpensive devices consist of an accelerometer, gyroscope, and in some cases, a magnetometer. They can be used outside of laboratory constraints, providing a more ecologically valid analysis. Despite their numerous benefits, IMU-based methods for evaluating joint kinematics also present challenges. Sensor drift and noise, integration errors, and device calibration issues pose significant challenges when deriving joint kinematics from sensor signals, particularly in dynamic movements.

Sensor fusion methods (Picerno, 2017; Weygers et al., 2020b) or machine learning techniques (Gholami et al., 2020; Rapp et al., 2021; Xiang et al., 2022) can be used to derive body segment and joint kinematics. To minimize error, researchers have applied sophisticated algorithms to filter and fuse sensor data and/or, constrain the outputs via computational biomechanical models. These models can be combined with inverse kinematic methods, minimizing the errors between the orientations of experimental IMUs and analogous IMU frames on the model, while being subject to physiologically feasible joint constraints. While kinematic outcomes have been validated with this approach, Ferrari et al. (2010), Zhang et al. (2013), Tagliapietra et al. (2018), Karatsidis et al. (2019), Cereatti et al. (2024) limiting an analysis to joint kinematics overlooks the interplay between energetics, and internal and external loads in relation to gait. Biomechanical models offer the ability to estimate other outcomes, including muscle or actuator activation dynamics, joint and ground reaction forces, and locomotive cost (Delp et al., 2007; Karatsidis et al., 2019; Slade et al., 2021; Lloyd, 2021). However, many applications of musculoskeletal modelling are based on optical motion capture methods in a laboratory setting (Dorn, Schache & Pandy, 2012; Apte, 2021).

Parameters, including kinematic measures and external and/or internal loads, are commonly computed in a consecutive manner using inverse kinematics, inverse dynamics, and static optimization or computed muscle control approaches (Delp et al., 2007; Karatsidis et al., 2019). Inverse methods depend critically on input data, with the potential for error accumulation across steps. The temporal independence of inverse kinematics may result in unrealistic joint angle fluctuations, as solving for each time step separately can lead to closely tracking noise or abrupt changes in the input data. Employing inverse dynamics to estimate joint moments and internal forces introduces dynamic inconsistencies, necessitating residual forces and moments in the solution (Faber, van Soest & Kistemaker, 2018). Additionally, inverse dynamics relies on directly measured external loads, limiting data acquisition to environments where the ground reaction force is measurable. Finally, static optimization proves suboptimal for modelling muscle forces during highly dynamic activities, failing to account for the tendon compliance that is pertinent in fast running (Lin et al., 2012).

One approach to reducing the dependence of the modelled kinematics on the accuracy of the experimental data and its associated error propagation is to use biomechanical models within an optimal control framework (Dembia et al., 2019). Optimal control methods solve for the control variables that minimize or maximize an objective function. In biomechanics, the objective can include minimizing errors between experimental and modelled data and/or locomotor objectives such as minimizing muscular effort, the cost of transport or mechanical loading (van den Bogert, Blana & Heinrich, 2011; Wang et al., 2012; Lin, Walter & Pandy, 2018). A more dynamically consistent simulation can be obtained in a single trajectory optimization, minimizing error propagation and the use of residual forces or moments (van den Bogert, Blana & Heinrich, 2011; Fluit et al., 2014; Lee & Umberger, 2016; De Groote et al., 2016). This method is highly suitable for solving problems with unstable dynamics as seen in running gait (Dorschky et al., 2019b; Nitschke et al., 2020; Haralabidis et al., 2021; Hosoi & Fay, 2024), while offering the ability to model other metrics not available with inverse kinematic methods (e.g., internal forces and torques, and energy cost, with ground contact models an option in lieu of externally measured loads).

The use of optimal control in biomechanics has burgeoned with advancements in computational power, available toolboxes, and methodological approaches (De Groote & Falisse, 2021; Febrer-Nafría et al., 2023; Hosoi & Fay, 2024). Tracking IMU signals with musculoskeletal models, combined with physiologically relevant cost functions, represents a potentially novel method to assess locomotion in a more holistic, dynamically consistent, and individualised manner (Dorschky et al., 2019a). Recent work has tracked acceleration and angular velocity signals with 2D musculoskeletal models in optimal control simulations. Minimizing the error between experimental IMUs and simulated signals from model IMUs was able to accurately simulate walking and running gait (Dorschky et al., 2019a). Whilst pioneering work, the use of a two-dimensional model limited the analysis to the sagittal plane, neglecting changes in some joint angles commonly assessed in running gait analysis, such as pelvis or hip rotations (Ceyssens et al., 2019; Vannatta, Heinert & Kernozek, 2020).

The aim of this study was to compare biomechanical simulations using two IMU-based methods with those from optical marker-based motion capture. Three-dimensional joint angles derived from optimal control simulations, which used raw IMU signals as tracking inputs, were compared with those obtained from optical marker-based motion capture. Additionally, the joint angles computed with IMU-based inverse kinematics were compared with motion capture angles. It was hypothesised that the optimal control method would better model joint kinematics compared with an inverse IMU-based method that is more contingent on IMU data accuracy, Borno et al. (2022) and that the level of agreement would be within the range of marker registration and scaling error associated with motion capture (Uchida & Seth, 2022). It was anticipated that employing an optimal control approach, while offering greater fidelity in the outcomes, would incur greater computational costs and time.

Methods

The University of Adelaide’s Human Research Ethics Committee approved the protocol for this observational study (#H-2022-120). All participants were fully informed of the experimental procedures and any associated risks and provided written informed consent prior to enrolment in the study. A graphical overview of the study method is shown in Fig. 1.

Experimental setup

Motion capture

A lower body and torso marker set consisting of 32 reflective markers on bony landmarks of the lower limb, pelvis and trunk (Cappozzo et al., 1995) was fitted to each participant. To minimize movement or detachment, markers were attached using superglue on the skin locations and double-sided adhesive tape secured with hyperfix tape over clothing or footwear. Marker trajectories were collected via a 10-camera Vicon Vantage V5 motion capture system (Vicon 4.2 Motion Systems, Oxford Metrics, Oxford, UK) at 100 Hz.

IMU sensors

Eight IMeasureU Blue Trident sensors (Vicon Motion Systems Ltd, Oxford, UK) were placed on the participants at recommended locations (Apte, 2021; Scalera et al., 2021). Sensors were placed on the sacrum and sternum, and the anterior-medial tibia, lateral mid-thigh, and proximal aspect of the shoe of both limbs (Fig. 2). Sensors were secured using hyper-fix and strapping tape to limit soft-tissue artefacts (Johnson et al., 2020). A separate custom-made housing for the pelvis sensor secured it to the participant’s waistband. Acceleration, magnetometer and angular velocity signals were collected at 1,125 Hz.

Optimal control simulations

IMU data tracking simulations of running gait were formulated as optimal control problems which were solved using direct collocation in OpenSim Moco (version 4.4.1) (Dembia et al., 2019) in Python 3.8 on a Phoenix High-Performance Computer (256 GB RAM, 72 CPU).

The models were identical to those used in the IMU-based inverse method, except that in the optimal control simulations, the model joints were driven by torque actuators with an activation time constant of 0.01 s. The optimal torques ranged from 50 to 500 Nm depending on the joint and degree of freedom, with a control activation limit of 5. Additional residual actuators were added to the pelvis rotational and vertical translation degrees of freedom with optimal values of 1 NM and 25 N respectively.

Contact between the foot and the ground was modelled using 11 ground-contact spheres on each foot, approximating a previously published Hunt-Crossley foot-ground contact model (Hunt & Crossley, 1975; Serrancolí et al., 2019). The spheres were placed evenly across the plantar surface in proportion to the foot size (Fig. 3). The static and dynamic friction and transition velocity properties were consistent with reference values of 0.8 and 0.5 respectively. The modulus (3.06 MPa) and damping coefficient (2.0 s/m) of the contact elements modelled the deformation and energy return of the heel region of a human foot in an athletic shoe (Aerts & De Clercq, 1993). Passive damping was added to the lower limb and lumbar joints, to model ligaments and other passive structures (Anderson & Pandy, 2001). To improve the modelled foot-ground contact mechanics (Falisse, Afschrift & Groote, 2021) the MTP joint was unlocked in these simulations, with a linear rotational spring force with a stiffness of 25 Nm/rad added to represent the MTP joint passive structures (Sasaki, Neptune & Kautz, 2009).

Initial guess

Optimization problems require an initial guess that serves as a starting point for the solver to search the solution space. A suitable initial guess is pertinent for optimal control modelling to avoid excessive computational effort or convergence on a local minimum that does not resemble the desired motion. As such, a guess was generated from a tracking simulation of generic kinematics from running at 14.4 km/h (Nitschke et al., 2020).

Objective equation

The optimal control problem can be formulated as follows: Minimize $J(x,u)$ subject to the following constraints:

(2) $$\begin{array}{c}\dot{x}=f(x(t),u(t))\hfill \\ x(0)=x_0\hfill \\ u(t)\in U,\mathrm{\forall }t\in [0,T].\hfill \end{array}$$

In this equation, the objective function $J(x,u)=J_{effort}+J_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{I}\mathrm{M}\mathrm{U}}+J_{trackIK}+J_{GRF}$ represents the cost to be minimized over the time period T. The primary objective of these simulations was to determine the model states $x$ and controls $u$ that minimize errors in tracking experimental data, subject to the system dynamics and constraints. The objective function consisted of the following terms each with individual weightings $w_{Ni}$

As the primary objective, the 3D orientations, accelerations and angular velocities of the input IMUs are tracked with the signals from the analogous virtual IMUs placed on the model (Eq. (3)).

(3) $$J_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{I}\mathrm{M}\mathrm{U}}={\int }_0^T\left[\frac{1}{N_1}{w}_{N1}\sum _{k=1}^{N_1}\sum _{i=x,y,z}{\left(\frac{a_{i,k}(t)-{\mu }_{a_{i,k}}(t)}{{\sigma }_{a_k}}\right)}^2+{\left(\frac{{\omega }_{i,k}(t)-{\mu }_{{\omega }_{i,k}}(t)}{{\sigma }_{{\omega }_k}}\right)}^2+{\left(\frac{{\theta }_{i,k}(t)-{\mu }_{{\theta }_{i,k}}(t)}{{\sigma }_{{\theta }_k}}\right)}^2\right]dt$$where $N_1$ is the number of IMUs (8) and $a$, $w$, and $\theta$ represent the accelerations and angular velocities of the IMU frame and orientations in the global frame in the $x$, $y$ and $z$ coordinates respectively. $\mu$ is the corresponding signal from the virtual sensor on the model. Differences were normalized to the measurement standard deviation $\sigma$.

A secondary aim was to minimize effort (Miller et al., 2012; Miller & Hamill, 2015; De Groote & Falisse, 2021; Veerkamp et al., 2021) (Eq. (4)).

(4) $$J_{effort}=\frac{1}{D}\underset{0}{\overset{T}{\int }}\left[w_{N3}\sum _{i=1}^{N_3}{\mu }_i^3(t)\right]dt$$where D is the distance travelled by the model’s centre of mass, ${\mu }_i(t)$ is the value of the control actuator at time $t$ and $N_3$ is the number of actuators. A larger weighting $w_3$ was applied to the model’s residual actuators in the effort goal term to penalize their use.

An objective tracking generalized joint kinematics from a dataset of fast running kinematics was added to act as a regularisation term, to assist with convergence and noise reduction (Nitschke et al., 2020) (Eq. (5)).

(5) $$J_{trackIK}=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left[\frac{1}{N_2}{w}_{N2}\sum _{i=1}^{N_2}{\left(x_i(t)-{\mu }_i(t)\right)}^2\right]dt.$$

Here, $N_2$ is the number of joint kinematics and ${\mu }_i(t)$ is the value of the input state variable at time $t$. An additional tracking goal was added to track the knee and hip flexion angles from the IMU inverse kinematics solved for in OpenSense, as this was found to improve joint angle agreement in the sagittal plane in pilot testing.

A generalized ground reaction force trajectory in the horizontal and vertical directions from the same dataset was tracked (Nitschke et al., 2020) to improve the model foot contact with the ground. The error between the force trajectory scaled in proportion to the participant’s bodyweight and ground reaction forces modelled from the contact sphere, were minimized in the vertical and horizontal directions (Eq. (6)).

(6) $$J_{GRF}=\frac{1}{W}\underset{0}{\overset{T}{\int }}\left[w_{N4}\sum _{j=x,z}\left(GR{F}_j(t)-GR{F}_{j,\mathrm{r}\mathrm{e}\mathrm{f}}(t)\right)\right]dt$$where GRF is the ground reaction force from the modelled contact spheres at time $t$, and $GR{F}_{ref}$ is the reference force. W is the model’s body weight.

The optimal control formulation weights $w$ for each of these objectives were determined heuristically. The values were chosen to minimize the sum of Pearson correlations when comparing model kinematics with the analogous experimentally tracked/measured data on a subset of five strides from each participant. The weighting factors for the acceleration, angular velocity and orientations of the IMUs were 0.002, 0.01 and 3, respectively. The joint kinematics tracking, effort minimization, and ground reaction force goal weighting factors were 1, 0.5 and 1,000, respectively. Note that due to the different magnitudes and units of the goal inputs, these weighting factors do not represent the relative contribution to the overall simulation objective. Instead, these weightings were chosen with the aim of having approximately 90% of the objective comprised of the tracking of each of the three IMU signals, with the remaining contribution from the auxiliary goals. An example optimal control setup file is available at https://doi.org/10.5281/zenodo.14796986.

Bounds on the time taken to complete a full gait cycle, along with constraints on problem states and controls, were applied to limit the solution space and ensure that the solution was within a feasible range for human running locomotion. The state bounds were set to to the joint ranges of the optical motion capture reference data $\pm$5–10°, depending on the variable. A CasADi solver (Andersson et al., 2019) within OpenSim Moco was used to solve the predictive simulations. Convergence and constraint tolerances were set to 0.001 and 0.005 respectively.

Results

The participant characteristics and their personal best times are outlined in Table 1

Out of the 270 total simulations, 266 successfully converged on a solution. Among them, three were identified as outliers, involving Participant 3 running at medium and fast speeds. The mean simulation time across all successful optimal control simulations was 46 $\pm$ 60 min. The longest time was 8 h and 46 min, again for Participant 3 (Fig. 4A). IMU-based simulation and marker-based inverse kinematics had much shorter and more consistent computational times, with average solve times of 19.3 $\pm$ 3.7 and 3.7 $\pm$ 0.4 seconds per stride respectively (Fig. 4B).

Compared with marker-based inverse kinematics, the mean ( $\pm$ SD) RMSE across all joint angles was 7° $\pm$ 1° in the optimal control simulations, ranging from 4°–12°. The errors ranged from 4°–26°, with a mean of 10° $\pm$ 1° using IMU-based inverse kinematics (Table 2A, Fig. 5). Errors in the optimal control simulations did not differ in magnitude with faster speeds (Fig. 6A). This finding was also seen in the IMU-based inverse kinematics (Fig. 6B), excluding knee flexion where errors were greater at slower speeds (Fig. 5). When normalising differences to the joint range of motion, lumbar extension still exhibited the greatest differences comparing the optimal control approach with motion capture. The normalized errors for the IMU-based kinematics were greatest for pelvic tilt (Table 2B).

Qualitative comparisons between optimal control simulations and marker-based inverse kinematics revealed regions of notable difference for lumbar extension over the gait cycle, and periods of hip and knee flexion (40–80% gait cycle), and ankle flexion (0–20% gait cycle) (Fig. 7). When comparing OpenSense IMU inverse kinematics with marker-based inverse kinematics, these errors tended to be larger in magnitude, with regions of greater error in the lumbar and pelvis joints seen outside the sagittal plane.

Discussion

This study compared two IMU-based modelling approaches for reconstructing joint motion: optimal control simulations seeking to minimize differences between experimental and simulated IMU signals, and IMU-based inverse kinematics. A 3D, subject-specific biomechanical model was used in both methods to model running gait across various speeds. Optical marker-based motion capture was employed as the comparative gold standard to assess the accuracy of the modelled joint kinematics.

This study confirmed the hypothesis that an optimal control modelling approach more closely resembles the joint kinematics obtained from optical motion capture than traditional inverse-based methods using IMUs. The average RMSE was found to be 7° and 10° for each of the methods, with a consistent level of agreement observed across all speeds, where the mean errors within participants did not differ by more than 2°. The lowest errors with marker-based inverse kinematics in both methods were in joint angles outside the sagittal plane (Fig. 5) IMU-based inverse kinematics had greater errors than optimal control simulations in 10 out of 11 joint angles, with magnitudes up to 26° seen in knee flexion (Table 2). Conversely, when the joint range of motion was considered, normalised errors were smallest in knee flexion for the optimal control simulations compared with motion capture (Table 2B). These results underscore the ability of an optimal control approach to effectively capture three-dimensional motion at moderate to fast running speeds.

While optimal control generally showed better agreement with motion capture than did IMU-based inverse kinematics, this was not observed for the lumbar flexion degree of freedom. A fixed offset between joint angle outcomes, with the optimal control simulations having a more upright torso, was not observed between marker and IMU-based inverse-kinematic approaches. Optimal control simulations must also satisfy system dynamics to arrive at a solution. With the torso being a large and heavy model body segment, reducing lumbar flexion would minimize the need for high actuator forces to maintain any forward lean. The necessity of maintaining an upright posture may therefore come at the expense of tracking lumbar flexion. In the IMU-based kinematics the greatest difference was observed in the pelvic tilt degree of freedom. This fixed offset observed in pelvic tilt could be attributed to differences in static postures when placing the IMUs on participants. Given the lack of consistent offset observed across participants, method uniformity was prioritized over individual correction factors.

A similar offset pattern emerged for the ankle flexion angle during the stance phase. The difference here may reflect the simulation’s attempts to employ appropriate contact mechanics to appropriately balance forces and facilitate forward running propulsion, rather than adhering to tracking foot IMU signals. In contrast, the inverse simulations did not consider contact tracking with the ground; hence ankle angles were dictated only by kinematic constraints and the tracking of sensor orientations, allowing the foot to penetrate the ground. This observation highlights a trade-off between realistic contact modelling and the representation of true joint angles without considering external loads. Although better results may have been obtained in the optimal control simulations by tracking measured ground reaction forces rather than generic forces scaled with body weight, the requirement for force plates in conjunction with IMUs is less translatable to real-world data collection.

Past research comparing IMU kinematics with motion capture methods in running gait is limited when considering an inverse kinematics or optimal control approach. For inverse kinematics, the RMSE in two studies ranged from 5–8° (Lin et al., 2023) and 18–28°, reduced to 5–8° with an offset correction (Nüesch et al., 2017). In this study, no offset correction was applied, and comparable error magnitudes (4–26°) were observed in IMU-based inverse kinematics. In the case of optimal control, errors ranging from 5–9° have been reported (Dorschky et al., 2019b), typically increasing when fewer sensors are used (Dorschky et al., 2024) and consistent with the errors observed in this study (4–12°). The aforementioned study revealed increasing accuracy in more distal joints, unlike the findings of this study. The improved ankle flexion accuracy could be a consequence of the use of variable temporal weighting factors. Periods with large artefacts in the signal, such as during foot strike, would ideally not be tracked as closely in the optimal control simulations, however, a temporal change in weighting factors was not a feature currently available in the optimal control software.

Machine learning methods are increasingly popular alternatives to both inverse kinematics and optimal control simulations for deriving biomechanical outcomes from IMU sensor data without relying on explicitly defined biomechanical models (Xiang et al., 2022). These methods have demonstrated accurate kinematic predictions across various movement types, often requiring fewer sensors and showing robustness to noise (Dorschky et al., 2020; Hernandez et al., 2021; Rapp et al., 2021; Xiang et al., 2022; García-de-Villa et al., 2023; Gholami, Napier & Menon, 2020). However, machine learning approaches also face challenges, including the need for large, high-quality datasets for training, the potential for over-fitting, and models trained only specific to a particular motion, population, or individual (Gurchiek, Cheney & McGinnis, 2019). Unlike a musculoskeletal modelling approach, these methods typically only evaluate a limited range of outcomes in one domain, such as kinematics, ground reaction forces, or joint forces (Xiang et al., 2022). While machine learning can eliminate the need for detailed input parameters, it typically lacks the direct insight that biomechanical models can provide, limiting its interpretability for understanding underlying movement strategies (Gurchiek, Cheney & McGinnis, 2019; Rodu et al., 2024).

Marker-based motion capture is considered the reference standard; however, it is pertinent to acknowledge the inherent errors associated with this method, which could impact the accuracy of the study findings when compared to true joint angles. Motion capture systems can have errors of 1–5 mm, Chiari et al. (2005) compounded by soft tissue artefacts of up to 10 mm (Leardini et al., 2005). The marker registration error further contributes to this uncertainty; a 2 cm difference resulted in a variation in peak ankle, knee, and hip sagittal angles of up to 15° (Uchida & Seth, 2022). The finding that the resulting joint angles were within the bounds of other experimental data provides further confidence in the study findings in light of this uncertainty. Ultimately, the modelling relevance of this study lies in the ability to measure gait changes within and between individuals, which were observed across participants at different running speeds.

A key strength of this modelling approach lies in the incorporation of a three-dimensional model in both methods. Modelling outside the sagittal plane could capture more intricate aspects of human locomotion, such as hip adduction and foot pronation, which can be particularly pertinent in running gait analysis. The ability of the model to capture these out-of-plane movements in optimal control simulations was demonstrated, with no region having an RMSE greater than 10°. Moreover, the multi-segment foot model used in the optimal control simulations allowed the MTP joint’s role during the push-off phase of running gait to be accounted for, enhancing the realism of the simulations and ground contact modelling (Falisse, Afschrift & Groote, 2021).

A limitation lies in the use of a torque-driven model in the optimal control simulations. These models lump synergistic muscle forces together and assume that the maximal torque exerted at a joint is a function of the kinematics of that joint alone. Torque-driven simulations typically have fewer unknown parameters compared to individual muscle-model simulations. It is feasible that this approach could be extended to incorporate muscle-driven simulations, where the optimization process includes muscle dynamics and activation patterns to more accurately represent the underlying physiology. This was avoided, however, due to the significant increase in computational cost and simulation complexity involved in simulating muscle dynamics, and remains an area for future improvement.

A further limitation lies in the potential generalizability of the approach. The small sample size in this study restricts the applicability of the results to broader populations, particularly those with varying anthropometric characteristics or gait patterns. Moreover, while the approach has been validated for running gait, its application to other dynamic activities or non-linear movement patterns remains unexplored. Additionally, the controlled laboratory environment may not fully capture the variability present in real-world conditions. Although the method appears translatable to field settings, it has not yet been tested in such environments.

The sensor setup employed in this study consisted of eight sensors. The IMU tracking strategy using optimal control could be adapted to situations with fewer IMUs, which offers advantages in cost and convenience (Dorschky et al., 2024). In such cases, emphasizing the minimization of effort and tracking generic running kinematics is likely needed to compensate for body segments lacking IMUs. Fewer sensors could thus make the motion more generalised, limiting the ability to model unique gait patterns. Given that the goal of this study was to achieve optimal agreement with subject-specific motion capture kinematics, all available sensors were utilized in the simulations. The IMU-based inverse kinematics approach, which is solely dependent on the sensor orientations to estimate joint angles, is unlikely to be adaptable to involving fewer sensors for gait analysis.

The greater resolution of optimal control simulations comes at the expense of computational time and user input. On average, the simulations took nearly 150 times the duration of IMU-based inverse kinematics (Fig. 4). Not accounted for is the additional time spent iteratively testing the simulation parameters to ensure convergence on a feasible solution. Unlike inverse methods, optimal control presents limitless inputs, including the initial guess, weighting factors, and constraint combinations. While the ability of these simulations to more closely replicate gold-standard kinematics underscores the potential of optimal control in biomechanical modelling, it is important to acknowledge that this setup remains somewhat subjective, without a universal method of application. The consistent application of weighting factors and constraints across participants and speeds demonstrates the robustness of the approach used in this study. However, critical questions regarding the integrity of the control problem and the influence of input parameters on outcomes remain unanswered. As such, the intricate nature of optimal control may not align with the practicality and ease of use required for everyday individuals to deploy IMU sensors effectively in field settings, posing barriers to widespread adoption and implementation of this approach without further development.

On the other hand, inverse kinematics, both IMU-based and marker-based, is computationally simpler to implement and solve than optimal control methods, with significantly shorter computational times and fewer input parameters. These methods assume that joint motion can be accurately determined solely based on endpoint positions, neglecting the underlying dynamics of the system. This simplification may lead to less accurate representations of joint motion and its associated drivers, particularly in dynamic tasks or when dealing with complex movement patterns. The choice between these approaches depends on the specific research questions, available time and resources, and desired level of detail and accuracy in the modelling process.

The findings of this study have important implications for the field of biomechanics and the assessment of human movement. Optimal control simulations not only outperformed IMU-based inverse kinematics in accuracy in this study, but also offer distinct advantages in biomechanical modeling over both inverse and machine learning approaches. These simulations enable the simultaneous estimation of outcomes such as ground reaction forces, muscle activations, joint loads, and kinematics through a single trajectory optimization that adheres to physiological and physical principles. Beyond data tracking, optimal control methods allow a wide range of objectives to be considered, including optimality principles underlying gait (Miller et al., 2012; De Groote & Falisse, 2021). In addition to the tracking of IMU signals, the cost function in this study included goals related to minimizing effort, enforcing joint angle symmetry, and, to a lesser extent, tracking generic kinematics and ground reaction force trajectories. Aligning the modelling approach with the fundamental objectives of locomotion reduces the limitations associated with traditional IMU-based methods. Reducing the dependence on potentially error-prone experimental IMU data mitigates inaccuracies in the step-wise computation of outcome variables. In addition, by optimizing control inputs (muscles or torque actuators), these simulations provide insights into how activation patterns and strategies achieve specific movements, making them particularly effective at capturing the complex interactions of forces and dynamics in human movement. Future research should continue to explore the potential of optimal control methods with inertial sensors, encompassing diverse locomotion forms, populations, and clinical and sporting applications. Efforts to refine the objective function and associated weighting factors used in optimal control simulations could further enhance model accuracy and validity.

Conclusion

This study demonstrated the ability of two modelling methods using IMUs to reproduce joint kinematics, with closer agreement to optical marker-based motion capture observed with optimal control modelling. The use of a three-dimensional, torque-driven biomechanical model and a custom model-scaling method allowed for a precise and comprehensive representation of gait in a single dynamic optimal control simulation. An optimal control paradigm enhances the robustness of IMU data to noise and errors, presenting a promising solution to mitigate limitations seen in traditional inverse-based approaches that rely heavily on and are more susceptible to errors in experimental data. Inverse kinematics provides a more straightforward and computationally efficient solution, making it more suitable for real-time applications and situations where computational resources are limited. Optimal control simulations also integrate dynamic constraints, providing a more comprehensive understanding of the underlying forces and activations driving the observed movements. This approach, when integrated with IMUs, opens new possibilities for advanced biomechanical modelling methods, such as modelling metabolic cost or muscle activation dynamics.