Reducing Complexity in Muscle‐Tendon Kinematics Parameterization Improves Convergence Speed in Musculoskeletal Simulations

ABSTRACT

Musculoskeletal simulations play a crucial role in rehabilitation, orthopedic implant design and athletic performance enhancement. A computational challenge within these simulations involves efficiently estimating muscle‐tendon lengths and moment arms, especially in multijoint, multidegree‐of‐freedom (DoF) systems. When modeling muscles spanning joints with six DoFs, the required number of terms can significantly increase, which could compromise computational speed. This study introduces a method that significantly reduces the polynomial coefficients needed for muscle‐tendon length and moment arm parametrization, ensuring computational efficiency without compromising accuracy. The approach was applied with two different error thresholds and was validated across four gait movements recorded from an elderly subject with a knee prosthesis, using data from the dataset of the Grand Challenge to predict in vivo contact forces. The results indicate that this reduction strategy decreased the required polynomial coefficients by approximately 50%, particularly for muscles spanning the knee and ankle joints, while preserving high accuracy in joint angle and knee contact force tracking. We demonstrate that our method decreases the computation time required for simulating full‐body dynamics by 15.6%, estimating knee contact pressures including a knee joint with all six DoFs. We observed minimal differences between the optimal solutions obtained using the full and reduced polynomials. This approach offers a simplified and computationally efficient method for muscle‐driven simulations, making it more practical for clinical applications like in physiotherapy, robotic‐assisted surgery, and athletic training. By increasing computational speed without losing accuracy, this method marks a notable advance in musculoskeletal modeling.

Reducing musculotendon kinematics parameterization decreases computational cost in musculoskeletal simulations while preserving their accuracy. The optimized polynomial model reduces 50% of terms and decreases the computation time by 16%. Joint angle and knee contact force tracking remain almost unaffected, ensuring reliable movement simulations.

1. Introduction

Musculoskeletal simulations provide valuable insights for developing rehabilitation protocols, designing orthopedic implants, and enhancing athletic performance [1, 2, 3, 4, 5, 6]. Their accuracy and efficiency are crucial, particularly when considering their potential application in scenarios where rapid computations are needed, such as in clinical settings (e.g., robotic‐assisted surgery or physiotherapy) or athlete training systems.

Muscle‐driven musculoskeletal simulations are typically obtained by solving an optimization problem that simultaneously estimate joint kinematics, muscle forces, and ground‐reaction forces. One of the main steps at each optimization iteration is the accurate evaluation of musculotendon kinematics, especially muscle‐tendon length ($l^{\mathit{MT}}$) and moment arms ($r$). These quantities are computed from the geometry of musculotendon paths [7], which in OpenSim are defined by wrapping objects (e.g., spheres, cylinders, ellipsoids, or toroidal surfaces) together with attachment and via points, some of which are active only over specific joint coordinate ranges (conditional path points). The muscle‐tendon lengths $l^{\mathit{MT}}$ and moment arms $r=\delta l^{\mathit{MT}}/\mathit{\delta q}$ follow from this path geometry. Solving for muscle paths with exact wrapping geometry is computationally expensive and often impractical within trajectory optimization. Nonetheless, accurate modeling of muscle lengths and moment arms is essential for computing muscle forces and joint loads. To improve computational efficiency, polynomial approximations [8, 9] and, more recently, neural networks [10] are used as surrogate models for musculotendon paths. Both provide smooth, differentiable, and computationally efficient representations that integrate well with direct‐collocation methods and gradient‐based optimizations. In practice, polynomial parameterizations remain the most widely used because of their simplicity when formulating optimal control problems (OCPs) [3, 11].

A computational challenge arises when modeling $l^{\mathit{MT}}$ and $r$ for muscles that span multiple joints with several degrees of freedom (DoFs). For instance, biarticular muscles such as the hamstrings spanning the hip (three DoFs) and the knee (six DoFs), introduce challenges for accurately computing $l^{\mathit{MT}}$ and $r$ [12, 13]. A cubic parametrization for $l^{\mathit{MT}}$ spanning nine DoFs would require up to 220 polynomial coefficients, complicating the optimization process and increasing computational cost [14].

To overcome these limitations, we introduce a method to reduce the number of polynomial coefficients required to approximate $l^{\mathit{MT}}$ and $r$, while preserving sufficient accuracy for musculoskeletal simulations. We evaluate the impact of this coefficient reduction on tracking simulation performance, including knee contact forces, joint kinematics, muscle forces, and ground reaction forces. Furthermore, we compare the resulting optimal solutions obtained using reduced versus full polynomial representations. By improving the efficiency of musculotendon modeling, this work aims to facilitate faster, more scalable, and clinically applicable musculoskeletal simulations.

2. Methods

2.1. Experimental Data

The experimental data used for this analysis were from the Knee Grand Challenge Competition to Predict In Vivo Knee Loads [15]. The data used were: marker trajectories, knee contact forces and ground reaction forces during gait cycles, for an 88‐year‐old subject (mass: 65 kg, height: 166 cm) wearing a knee prosthesis on the right leg. Four different movements were selected: normal gait, bouncy gait, forefoot strike gait, and treadmill gait performed using a subject's normal gait pattern at a fast walking speed of 1.4 m/s.

2.2. Musculoskeletal Model

A full‐body OpenSim skeletal model with 34 DoFs, including 20 lower‐limb DoFs actuated by 94 muscles, and 8 upper‐limb DoFs actuated by torques. The model was applied to compute $l^{\mathit{MT}}$ and $r$, and then within a musculoskeletal tracking simulation to assess the effectiveness of the proposed method. The right leg was constructed from subject‐specific information (from a CT scan), as in previous works [16, 17]. The left leg and upper‐body limbs were scaled from the model of [18]. The right knee joint had all six DoFs, while the left knee was kept with only the flexion‐extension DoF. The right knee contained the femoral and tibial prosthetic components, represented by geometry meshes, and a mesh‐based knee contact pressure model was considered between both elements as in [13].

2.3. Parameterization of Muscle‐Tendon Lengths and Moment Arms

Muscle‐tendon lengths and moment arms were parameterized with polynomials, as functions of joint angles. The OpenSim model was used to compute $l^{\mathit{MT}}$ and $r$ [19] for 10.000 random skeleton configurations within the expected ranges of motion during walking. We computed the polynomial coefficients of 50 muscles (those spanning the right lower‐limb and the lumbar joint) to best fit simultaneously the $l^{\mathit{MT}}$ and $r$ (using a least‐square method) [14, 20]. Two distinct approximation thresholds were applied in this study. Threshold 1 allowed a maximum root mean squared error (RMSE) of 1 mm for $l^{\mathit{MT}}$ and $r$ related to rotational DoFs and 0.01 for translational DoFs. Threshold 2 permitted a maximum RMSE of 2 mm for rotational DoFs and 0.02 for translational DoFs. These values are overall within 1 and 2% of the maximum $l^{\mathit{MT}}$ and $r$ values for rotational and translational DoFs. Cubic polynomial functions were overall sufficient to represent $l^{\mathit{MT}}$ and $r$ accurately, except for threshold 1, where three specific muscles required biquadratic polynomial functions to maintain RMSE within the acceptable limits. The total number of coefficients for each muscle polynomial can be calculated using the following expression:

$$n_{\mathit{\text{coef}}}=\frac{\left(n+k\right)!}{n!k!}$$ (1)

where $n$ is the polynomial's order, and $k$ is the number of DoFs spanned by the muscle. An illustrative example of a cubic polynomial formulation for $l^{\mathit{MT}}$ is shown in Equation (A1), and the reduced version is given in Equation (A2) in Appendix A.

Then, the number of coefficients was reduced, while keeping a certain allowed accuracy. First, the significance of each coefficient was assessed following the methods of polynomial regression [21], which involved computing the p‐value associated with the t‐statistic value. A t‐test was performed for each coefficient to test the hypothesis that the polynomial coefficient equals zero (null hypothesis) or differs from zero (alternative hypothesis).

To identify the most effective polynomial coefficients, we applied a systematic selection method. Starting with the coefficient that had the smallest p‐value (most significant), each coefficient was added sequentially. A flowchart showing the coefficient reduction process is included in Appendix A Figures A1 and A2. The RMSE between the OpenSim data and model values was recalculated at each step. For rotational DoFs, the RMSE threshold was also set at either 1 or 2 mm, while for translational DoFs, it was 0.01 or 0.02, depending on the threshold in use. If the RMSE remained within these limits, the number of coefficients was considered valid. If the RMSE exceeded the threshold, the next coefficient in the ordered list was added, and the RMSE was recalculated, continuing this process until the target RMSE was achieved. This iterative method ensured an accurate fit across all muscles. Hereinafter, these polynomials are called the “reduced polynomials.”

2.4. Effectiveness Evaluation: Application to an Optimal Control Problem

To evaluate the impact of the proposed polynomial coefficient reduction, we tested the difference of using either the full or the reduced polynomials in a full‐body tracking simulation for four different movements. This musculoskeletal simulation was formulated using a direct collocation method (with 40 mesh intervals) and implicit dynamics formulation. The information of the equations of motion was obtained from the automatic differentiation version of OpenSim, described in [3]. The tracked experimental data consisted of full‐body kinematics (joint coordinates obtained from an inverse kinematics analysis considering the knee as a pin joint), ground reaction forces, and knee medial and lateral resultant contact forces, as in [13]. The range of motion of the secondary knee DoFs was minimized. Ground reaction forces were computed using a smooth foot‐ground contact model [22], and muscle forces using a smooth muscle force model [23]. The formulation closely follows the approaches described in [13, 24], with two key differences compared to the former: (1) our tracking simulations incorporate contact forces throughout the movement, (2) six DoFs are considered for the knee instead of a single DoF.

The variables $l^{\mathit{MT}}$ and $r$ at the optimal solutions obtained with both full and reduced polynomials were compared by computing the root mean square differences (RMSD). The tracking of knee contact forces was also assessed for both types of polynomials at both thresholds for all the studied movements by computing the RMSE between model and experimental values. The convergence performance was assessed comparing the computational time and the number of iterations taken to obtain the optimal solution. The computational time and hardware specifications used for the coefficient reduction process are reported in Appendix A. The implementation of the polynomial coefficient‐reduction method, together with scripts and examples to reproduce the manuscript's results, is available at https://github.com/gilserrancoli/ReducePoly_forMSK.

3. Results

The number of polynomial coefficients to estimate $l^{\mathit{MT}}$ and $r$ decreased significantly with the proposed method, approaching 50% in both cases studied, especially for muscles spanning the knee and ankle joints (Figure 1). For the 2 mm threshold, a total of 1451 out of 2886 coefficients (50.3%) with cubic polynomials were sufficient for all muscles (reduced polynomials), ensuring that the fit remained within the expected error threshold. Specifically, 297 out of 420 coefficients were selected for monoarticular hip muscles (70.7%), 714 out of 1540 for biarticular hip‐knee muscles (46.4%), 186 out of 336 for monoarticular knee muscles (55.4%), 117 out of 330 for biarticular knee‐ankle muscles (35.5%), and 137 out of 260 for monoarticular ankle muscles (52.7%) (Table A1). Increasing the accuracy, for the 1 mm threshold, 1694 out of 3153 coefficients were used (53%). In this case, a cubic polynomial order was sufficient for nearly all muscles except for three muscles (“bfsh,” “glmax3,” “vaslat”), which required a biquadratic polynomial order to meet the stricter fitting threshold. Specifically, 362 out of 435 coefficients were selected for monoarticular hip muscles (83.2%), 739 out of 1540 for biarticular hip‐knee muscles (48.0%), 291 out of 588 for monoarticular knee muscles (49.4%), 126 out of 330 for biarticular knee‐ankle muscles (38.2%), and 176 out of 260 for monoarticular ankle muscles (67.7%) (Table A1).

FIGURE 1.

Mean and standard deviation of the polynomial coefficients taken in the reduced polynomials across muscle groups (in percentage with respect to the full polynomials). M and B stand for monoarticular and biarticular muscles, while T1 and T2 stand for 1 mm threshold and 2 mm threshold, respectively.

The average computational time (across 10,000 evaluations) to evaluate the function that computes $l^{\mathit{MT}}$ and r of right lower limb and the lumbar joint (50 muscles) with the reduced polynomials for the case of 1 mm was 0.20 ms per iteration, and the one using the full polynomials 0.23 ms (13% faster), while for the case of 2 mm, the average computational time with the reduced polynomials was 0.19 ms, and the one using the full polynomials 0.21 ms (9.5% faster).

This reduction in the number of coefficients parameterizing $l^{\mathit{MT}}$ and r had a significant impact in the computational time solving musculoskeletal simulations. When these coefficients were applied to the full tracking simulation formulated as an OCP, including the knee pressure contact model, the mean computational time across all movements was 3.1 h using the full polynomials and 2.7 h using the reduced ones with a 1 mm threshold (reduction of 12.2%) (Figure 2a). For the 2 mm threshold, the computational times were 2.4 and 2 h (reduction of 15.6%), respectively. The average number of iterations required for convergence was 1765 (full) and 1717 (reduced) at 1 mm, and 1502 (full) and 1236 (reduced) at 2 mm (Figure 2b).

FIGURE 2.

(a) Mean and standard deviation of the computational time required to achieve the optimal solution and (b) mean and standard deviation of the number of iterations for full and reduced polynomials at both thresholds.

The optimal solutions obtained using both the full and the reduced polynomials across all movement conditions were similar. The average RMSD of $l^{\mathit{MT}}$ was slightly lower with the 1 mm threshold polynomials than with the 2 mm threshold polynomials for monoarticular ankle and slightly higher for the biarticular knee‐ankle. In contrast, the average RMSD of r was nearly identical between the polynomial models. These results are presented across different muscle categories, as shown in Figure 3.

FIGURE 3.

Mean and one standard deviation of RMSD for (a) muscle‐tendon lengths ($l^{\mathit{MT}}$) across muscles and trials and (b) moment arms (r) across trials, muscles and joints. Rot and Trans stand for rotational and translational DoF. Note that the muscle force contributions to the knee forces are dimensionless.

These small differences arose from slight variations observed in the kinematics. The difference of mean RMSE for joint angles obtained using the full and reduced polynomials compared to the experimental data was ${0.02}^{°}$ at 1 mm threshold and ${0.05}^{\circ }$ at 2 mm threshold (Figure 4a). The capability of tracking knee contact forces was also similar between polynomial models. The tracking of medial knee contact forces (measured with RMSE) presented a difference (between the use of full and reduced polynomials) of 0.6 and 6.86 N using the thresholds 1 and 2, respectively compared to the experimental data. These differences for the tracking of lateral knee contact forces were 0.5 and 3.5 N for thresholds 1 and 2, respectively (Figure 4b). The curves of knee contact forces for all tracking simulations are shown in A.6, and the detailed tracking simulation results in Table A2.

FIGURE 4.

(a) RMSE comparison of joint angles for full versus reduced models (blue and pink, respectively) at threshold 1 (darker shades) and 2 (lighter shades). (b) RMSE of medial and lateral knee contact forces for full and reduced models using thresholds 1 and 2.

4. Discussion

This study aimed to reduce the number of polynomial coefficients used in the parameterization of $l^{\mathit{MT}}$ and r without compromising accuracy. The findings show a significant reduction in the number of coefficients, specifically 53.0% and 50.3% for threshold 1 and 2, respectively (allowing 1 and 2 mm of error reconstructing $l^{\mathit{MT}}$). The most significant reduction occurred in muscles spanning the knee and ankle joints (Table A1). These reductions are likely due to the fact that polynomials of $l^{\mathit{MT}}$ and r muscles spanning a significant number of DoF (like bi‐articular muscles spanning the knee) do not need all terms of the polynomial (with all combinations of coordinates) to obtain a high accuracy (Table A1). This reduction leads to considerable decrease in computational time, with an average reduction of 15.6% and 12.2%, respectively, for the reduced compared to the full polynomials across all the movements to achieve the optimal solution. In terms of iterations, the mean number of iterations required for convergence decreased by 2.69% and 17.66% for the 1 and 2 mm thresholds, respectively (Figure 2).

The coefficients multiplying the cubic terms experienced the most significant decrease (see Table A1). For monoarticular ankle muscles, coefficients associated with less critical joint DoFs, such as dorsiflexion‐plantarflexion, were unnecessary to maintain simulation accuracy at both thresholds. However, this trend varied across muscle groups. For instance, monoarticular hip muscles retained most of their coefficients, suggesting that simple standard linearization methods would not apply to all muscles. This observation indicates that hip muscles, which may involve more complex moment‐arm behavior and multi‐DoF coupling, require more polynomial terms to maintain simulation accuracy. The cubic terms associated with the biarticular muscles spanning both knee and ankle joints were the ones experiencing the highest reduction. Coefficients of the gastrocnemius lateralis were reduced to 37 and 30 coefficients (out of 120) using 1 and 2 mm thresholds, respectively. This represents a reduction of 69.2% and 75.0%, respectively. For the case of the gastrocnemius medialis, the number of cubic terms was reduced to 28 (76.7% reduction) for both thresholds. The coefficients associated with ankle DoFs in this muscle group experienced the most considerable decrease.

The results were consistent with the muscle functions. The nonsignificant terms were overall associated to DoFs where the muscle was not playing a crucial role. For instance, the polynomial for rectus femoris was reduced to 84 coefficients (out of 219) for both thresholds. This muscle is mainly responsible for the hip flexion and knee extension DoFs [25]. Of the remaining coefficients, 33 multiply terms affecting the knee flexion, while only 20 and 12 affect knee abduction‐adduction and internal‐external rotation. For biceps femoris [26], the number of polynomial coefficients was reduced to 94 terms for 1 mm threshold, of which 27 contribute to hip flexion, whereas only 14 and 13 contribute to the hip abduction‐adduction and hip internal‐external rotation. This approach allowed for targeted reductions in coefficients, maintaining terms crucial to primary muscle actions while eliminating irrelevant terms, resulting in a more efficient computation without compromising model accuracy.

Musculoskeletal simulations produced remarkably similar optimal solutions when using either type of polynomial functions (reduced and full). The mean RMSD of $l^{\mathit{MT}}$ was low across muscle groups, as shown in Figure 3a, with the largest difference observed in the biarticular knee‐ankle muscles between both thresholds, reaching $7\times {10}^{-3}$ mm. For monoarticular muscles, such as those at the hip and ankle, the RMSD values remained very low. The difference of RMSD values of r for rotational DoFs and for translational DoFs were negligible (Figure 3b).

Additionally, knee contact forces for both medial and lateral compartments tracked experimental data accurately in both models across all the studied movements, with similar accuracy levels as the ones reported in other studies in the literature [16, 27, 28, 29]. The maximum RMSE difference between the use of full and reduced polynomials for medial $\mathit{KCF}$ was 0.63 N and for lateral $\mathit{KCF}$ was 0.47 N at the 1 mm threshold, while the difference at the 2 mm threshold was 6.55 N for the medial $\mathit{KCF}$ and 3.44 N for the lateral $\mathit{KCF}$ (Figure 4). Furthermore, RMSE values for kinematics tracking between both models were almost identical, with differences of 0.02° and 0.05° at 1 and 2 mm, respectively (Figure 4). These results highlight the balance achieved between computational efficiency and accuracy, particularly for muscles with complex paths, such as those requiring extensive wrapping surface, which typically required high‐order polynomial representations due to their intricate path geometry.

The new method requires fewer polynomial coefficients compared to prior studies introducing polynomials to parameterize $l^{\mathit{MT}}$ and r [9], or using higher‐order polynomials and B‐splines [8, 30], which already improved the computational speed of musculoskeletal simulations. Our method represents a simple workflow to further reduce the number of computations compared to the mentioned studies, particularly relevant when applied within an OCP. We envisage that the computational difference would be even higher when using polynomials with higher degrees and for models including more joints with multiple DoF, like in shoulder, neck or spine muscles [31, 32]. This method could also be applied to complex ligament models, such as the multiscale phenomenological approach [33], to enhance computational efficiency while maintaining biomechanical accuracy.

In conclusion, the reduction of polynomial coefficients proposed in this study offers a promising approach to enhance computational efficiency in musculoskeletal simulations. By reducing the computational time without compromising accuracy, this method facilitates the application of these simulations in clinical and robotic contexts. The ability to maintain high accuracy with reduced computational time supports broader adoption of musculoskeletal modeling, particularly in clinical applications that require rapid, patient‐specific adjustments to model parameters and conditions.

Author Contributions

Mohanad Harba: conceptualization, methodology, investigation, writing – original draft. Joan Badia: investigation, validation, writing – review. Gil Serrancolí: resources, validation, visualization, project administration, writing – review and editing.

Funding

This work was supported by Agència de Gestió d'Ajuts Universitaris i de Recerca, FISDUR 00154. Agencia Estatal de Investigación, PID2023‐150189NA‐I00.

Ethics Statement

The authors have nothing to report.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. Additional Data

The process of reducing the polynomials is sequential. It optimizes the polynomial representation of ($l^{\mathit{MT}}$) and moment arms ($r$) by reducing the number of polynomial coefficients while maintaining the desired accuracy (see the flowchart in Figures A1 and A2). The flowchart includes the following steps:

1. Input data: the ($l^{\mathit{MT}}$) and moment arms ($r$) are provided as inputs.

2. Initial Polynomial Fit: a full polynomial is fit to the input data using a least‐square method, generating an initial set of coefficients for each muscle.

3. Statistical significance of coefficients: each coefficient is subjected to a statistical test (t‐test) to evaluate its significance. The resulting p‐values determine the relative importance of each coefficient.

4. Sort coefficients by p‐value: coefficients are ranked in ascending order of their p‐values, starting with the most significant.

5. Check RMSE: the root mean squared error (RMSE) is calculated between the polynomial approximation and the original data. This step checks whether the approximation meets the predefined accuracy threshold.

6. Iterative inclusion: if the RMSE exceeds the threshold, the next most significant coefficient is added to the polynomial. The RMSE is recalculated, and this step is repeated until the RMSE is within the acceptable limit.

7. Keep coefficients: once the RMSE satisfies the threshold, the current set of coefficients is finalized.

8. Output reduced polynomial coefficients: the reduced set of coefficients is output for use in musculoskeletal simulations.

FIGURE A1.

Flowchart illustrating the sequential coefficient reduction process used in this study.

FIGURE A2.

Knee contact force curves (medial and lateral) for the four tracking simulations (arranged in rows) using both full and reduced polynomial parameterizations at both thresholds (T1 and T2).

This stepwise process ensures that only the most significant coefficients are retained, allowing efficient and accurate polynomial representations. The time required to perform the polynomial coefficient reduction was 338.78 s for threshold 1, and 280.8 s for threshold 2. This reduction was performed on a laptop equipped with a 1.70 GHz Intel i7‐1225U CPU and 16 GB of memory.

To assess the performance and selection criteria in our lower‐limb muscle simulations related to polynomial coefficient selection and simulation performance across various gait trails, we present data in Tables A1 and A2. Table A1 outlines the number of coefficients chosen for each muscle, categorized as monoarticular or biarticular, for two thresholds 1 mm and 2 mm. Table A2 provides simulation metrics, including computation time, iteration count, and RMSE values for $\mathit{KCF}$ and joint angles across different gait patterns, comparing full and reduced models at each threshold. As an illustrative example, the following cubic polynomial function was used to approximate the $l_{\mathrm{gem}}^{\mathit{MT}}$ of the gemellus muscle, which spans three rotational DoFs: hip flexion ($q_1$), hip adduction ($q_2$), and hip rotation ($q_3$).

$$\begin{array}{c}{l}_{\mathrm{gem}}^{\mathit{MT}}=c_0+c_1{q}_1+c_2{q}_2+c_3{q}_3+c_4{q}_1{q}_2+c_5{q}_1{q}_3+c_6{q}_2{q}_3+c_7{q}_1^2+c_8{q}_2^2+c_9{q}_3^2\\ +c_{10}{q}_1{q}_2{q}_3+c_{11}{q}_1{q}_2^2+c_{12}{q}_1{q}_3^2+c_{13}{q}_1^2{q}_2+c_{14}{q}_1^2{q}_3+c_{15}{q}_1^3\\ +c_{16}{q}_2{q}_3^2+c_{17}{q}_2^2{q}_3+c_{18}{q}_2^3+c_{19}{q}_3^3\end{array}$$ (A1)

As shown in Table A1, one quadratic term and five cubic terms can be excluded while still satisfying the accuracy threshold. The reduced polynomial will be as follows:

$$\begin{array}{c}{l}_{\mathrm{gem}}^{\mathit{MT}}=c_0+c_1{q}_1+c_2{q}_2+c_3{q}_3+c_4{q}_1{q}_2+c_5{q}_1{q}_3+c_6{q}_2{q}_3+c_7{q}_1^2+c_8{q}_2^2\\ +c_{13}{q}_1^2{q}_2+c_{14}{q}_1^2{q}_3+c_{15}{q}_1^3+c_{18}{q}_2^3+c_{19}{q}_3^3\end{array}$$ (A2)

Note that this is merely an illustrative example for a monoarticular muscle, and as mentioned, the reduction is more significant in biarticular muscles.

Knee contact force curves across all tracking simulations

TABLE A1.

Number of polynomial coefficients selected for all lower‐limb muscles used to parameterize the $l^{\mathit{MT}}$, according to their spanning joints for both thresholds (1 and 2).

Muscle	Linear	Quad	Cubic	Bi‐Quad
MoA hip	3	6	10	15
addbrev	3 | 3	6 | 5	7 | 5	—
addlong	3 | 3	6 | 5	10 | 5	—
addmagProx	3 | 3	6 | 6	7 | 7	—
addmagMid	3 | 3	5 | 5	6 | 6	—
addmagDis	3 | 3	6 | 6	7 | 4	—
addmagIsch	3 | 3	6 | 6	7 | 4	—
gem	3 | 3	5 | 5	5 | 5	—
glmaxSup	3 | 3	6 | 5	4 | 2	—
glmaxMed	3 | 3	5 | 5	10 | 7	—
glmaxAnt	3 | 3	6 | 6	6 | 7	10 | —
glmedAnt	3 | 3	6 | 6	8 | 6	—
glmedMed	3 | 3	6 | 6	9 | 7	—
glmedPos	3 | 3	6 | 6	7 | 6	—
glminAnt	3 | 3	6 | 5	9 | 5	—
glminMed	3 | 3	5 | 5	5 | 5	—
glminPos	3 | 3	6 | 5	8 | 5	—
iliacus	3 | 3	6 | 3	5 | 2	—
pect	3 | 3	6 | 5	6 | 4	—
piri	3 | 3	5 | 2	6 | 6	—
psoas	3 | 3	5 | 2	7 | 1	—
quadfem	3 | 3	6 | 6	9 | 6	—
BiA Knee‐A	8	36	120	—
gaslat	8 | 8	22 | 20	37 | 30	—
gasmed	8 | 8	21 | 21	28 | 28	—
BiA Hip‐Knee	9	45	165	—
bflh	9 | 8	26 | 26	63 | 59	—
grac	8 | 8	33 | 33	82 | 82	—
recfem	9 | 9	27 | 27	47 | 47	—
sart	9 | 9	24 | 24	54 | 51	—
semimem	8 | 8	32 | 32	84 | 77	—
semiten	9 | 9	30 | 30	93 | 83	—
tfl	8 | 8	27 | 25	50 | 48	—
MoA Knee	6	21	56	126
bfsh	6 | 6	12 | 10	30 | 26	56 | —
vasint	6 | 6	19 | 19	26 | 25	—
vaslat	6 | 6	16 | 16	28 | 29	36 | —
vasmed	6 | 6	18 | 15	22 | 18	—
MoA ankle	2	3	4	—
edl	2 | 2	2 | 1	1 | 1	—
ehl	2 | 2	3 | 1	2 | 1	—
fdl	2 | 2	3 | 1	2 | 1	—
fhl	2 | 2	3 | 1	2 | 1	—
perbrev	2 | 2	1 | 1	1 | 1	—
perlong	2 | 2	2 | 1	1 | 1	—
pertert	2 | 2	2 | 1	1 | 1	—
soleus	2 | 2	3 | 3	2 | 2	—
tibant	2 | 2	3 | 1	2 | 1	—
tibpost	2 | 2	1 | 1	1 | 1	—

Note: MoA stands for monoarticular and BiA for biarticular. The numbers in bold are the total number of possible coefficients for each muscle category. Values are shown in light grey for threshold 1 and dark grey for threshold 2. The abbreviations of the muscles are the same as in generic OpenSim models, and also mentioned in [16].

TABLE A2.

Simulation results for different gait patterns, comparing full and reduced models at both thresholds 1 and 2 mm with respect to the experimental data.

Gait	Model	Threshold [mm]	Time [h]	Iterations	RMSE
					KCF [N]		Joint angles
					Lateral	Medial	[°]
Normal	Full	1	2.54	1508	83.31	49.91	±1.31
		2	2.93	1881	85.63	54.64	±1.34
	Reduced	1	2.56	1513	82.20	48.61	±2.59
		2	2.28	1360	80.75	49.00	±1.34
Bouncy	Full	1	2.99	1643	83.61	49.91	±1.12
		2	1.94	1122	77.99	54.64	±1.15
	Reduced	1	2.24	1281	81.61	55.44	±2.09
		2	1.51	908	79.86	56.59	±1.14
Treadmill	Full	1	4.99	2388	89.80	60.55	±2.36
		2	2.74	1538	90.80	63.02	±2.27
	Reduced	1	3.79	2442	86.01	63.29	±2.32
		2	2.43	1331	81.33	43.89	±2.62
Forefoot strike	Full	1	2.05	1522	83.67	54.78	±1.15
		2	1.96	1466	86.96	54.87	±1.17
	Reduced	1	2.19	1635	92.43	57.74	±1.14
		2	1.77	1347	85.68	51.89	±1.08

Note: For each gait, the table provides computation time, number of iterations, and RMSE values for medial and lateral knee contact forces as well as joint angles.

Data Availability Statement

The code for reducing the polynomial parameterization is included in an open GitHub repository referenced in the manuscript.