Simulating Finger-Tip Force Using Two Common Contact Models: Hunt-Crossley and Elastic Foundation

Abstract

Musculoskeletal models of the hand rarely include fingerpad contact mechanics, thereby limiting our ability to simulate and examine hand-object interactions. The objective of this study was to evaluate whether two common contact models (Hunt-Crossley and Elastic Foundation) can accurately represent the fingerpad. Two musculoskeletal models of the index finger were created by adding fingerpad contact geometry using either the Hunt-Crossley or Elastic Foundation contact models. Key contact parameters (target force, contact area, and stiffness) were then systematically varied through 432 forward dynamic simulations to examine how these parameters influenced estimation of finger-tip forces. Across all simulations, variation in target force, contact area, and stiffness parameters impacted the computation time required to complete the simulations and the accuracy of the predicted finger-tip force. Computation time was over three times longer in simulations with high versus low values of contact area and stiffness in both contact models. For both contact models, larger contact area and stiffness values resulted in simulations that more closely predicted target force. However, across all simulations, the Hunt-Crossley model produced a greater proportion of accurate finger-tip force simulations than the Elastic Foundation model, suggesting that the Hunt-Crossley contact model may be preferable for modeling the fingerpad. Overall, our study demonstrates how the Hunt-Crossley and Elastic Foundation contact models behave in low-force biomechanical scenarios, such as those experienced during hand-object manipulation, and provides a foundation for incorporating contact mechanics into musculoskeletal models of the hand.

Introduction

Our ability to use our hands as end-effectors is facilitated by the fingerpad – the patch of skin on the distal phalanx involved in finger-object contact. The fingerpad provides sensory feedback during both fine and gross movements. We can precisely manipulate objects and differentially apply forces of varying magnitudes because of the fingerpad.

Experimental studies examining the fingerpad’s biomechanical properties and its role in finger-object interaction provide foundational knowledge regarding hand contact forces. For example, experiments investigating fingerpad properties have provided insight into its friction properties, stiffness, and size (Dzidek et al., 2017; Liu et al., 2018; Moran et al., 1985; Pérez-González et al., 2013; Tomlinson et al., 2007; Tomlinson et al., 2009). This work importantly describes the fingerpad’s function and provides an empirical basis for modeling finger-object contact. Hand contact forces required for a variety of activities have also been experimentally measured (Hussain et al., 2018; Kargov et al., 2004; Smaby et al., 2004; Vergara et al., 2014). These studies quantify the magnitude and frequency of fingerpad contact forces experienced in everyday life.

Despite the fingerpad’s importance and the variety of experimental studies characterizing it, most musculoskeletal models do not include fingerpads. In fact, although models have been extensively used to examine external forces in the hand (An et al., 1985; Chao et al., 1976; Fok and Chou, 2010; Sancho-Bru et al., 2003a; Sancho-Bru et al., 2003b; Weightman and Amis, 1982), these models typically include either a contact model representing soft tissues (e.g., Cooney and Chao, 1977) or muscle-tendon actuators (e.g., Goislard de Monsabert et al., 2014), not both. This is problematic because without a model that incorporates physiologically accurate representations of both the fingerpad and muscle-tendon actuators, we are limited in how finger-object interactions can be examined through musculoskeletal simulations. Contact modeling offers a potential method for modeling the fingerpad. Hunt-Crossley and Elastic Foundation contact models have been used to model foot-floor contact (e.g., Febrer Nafría et al., 2018; Haralabidis et al., 2019; Mansouri et al., 2015; Serrancolí et al., 2018) and joint analyses (e.g., Chan and Walker, 2018; Hast and Piazza, 2013; Zargham et al., 2019). However, neither contact model is regularly used to model fingerpad forces. Consequently, how these contact models perform in low force biomechanical scenarios, such as those experienced during hand-object interaction, is not well understood.

This study aimed to examine how two common contact models (Hunt-Crossley and Elastic Foundation) can be used to represent contact mechanics of the fingerpad. We specifically model the index finger given its role in daily activities and relative kinematic simplicity compared to the thumb. We hypothesized that varying contact model parameters – target force, contact area, and stiffness – would impact the accuracy of simulated finger-tip forces in both contact models.

Methods

To evaluate how the selected contact model and its underlying parameters influence simulated finger-tip forces, two musculoskeletal models were created in OpenSim (v. 3.3.; Delp et al., 2007). Each index finger model incorporated a spherical representation of the fingerpad using either the Hunt-Crossley or Elastic Foundation contact models, which are native to OpenSim (Seth et al., 2018). Through forward dynamic simulations, how contact model (Hunt-Crossley vs. Elastic Foundation) and contact model parameters (target force, contact area, and stiffness) influenced finger-tip forces was examined.

Musculoskeletal Model

A previously described index finger model (Figure 1a) was used (Binder-Markey and Murray, 2017). Briefly, this model included four extrinsic muscles, one unconstrained degree-of-freedom [metacarpophalangeal (MCP) flexion-extension], and two constrained degrees-of-freedom [proximal interphalangeal (PIP) and distal interphalangeal (DIP) flexion-extension were held constant]. The index finger position was initialized at 70° MCP flexion, 20° PIP flexion, and 10° DIP flexion. A massless, spherical representation of the fingerpad was added and defined with either a Hunt-Crossley or Elastic Foundation contact model (see Supplemental Material for details).

Figure 1.

Model and simulation framework. (a) Musculoskeletal model of the hand with four extrinsic muscles of the index finger and contact geometry at the distal phalanx representing the fingerpad, (b) Simulation framework showing inputs (contact model, contact area, and stiffness paired with muscle activations for each contact model and target force combination) and outputs (finger-tip force) for each forward dynamic simulation, (c) Generic forward dynamics simulation profile showing finger-tip force versus time.

Simulation Framework

To evaluate differences between the contact models across a range of input parameters, forward dynamic simulations were run. During each simulation, the index finger rotated in flexion about the MCP joint until the contact sphere representing the fingerpad pressed against the planar contact surface (Figure 1a). The inputs were muscle activations to generate the target force (see Supplemental Material for details). The output was the force vector between the simulated fingerpad and planar contact surface (Figure 1b). Each simulation included pre-contact movement and post-contact steady-state force generation during which negligible joint movement (less than 0.1°) occurred (Figure 1c). Normal forces were averaged over 0.4 to 0.5 seconds and reported; this time interval corresponds to the force immediately following contact. Off-axis forces were negligible (less than 10^-3N).

Pilot testing indicated that variation in dissipation and friction parameters between 0.1 and 0.9 did not alter generated finger-tip forces, while variation in contact area and stiffness strongly influenced generated finger-tip forces. Thus, parameters of interest were limited to target force, contact area, and stiffness. Parameterizations of target force (Drost et al., 2019; Smaby et al., 2004; Vocelle et al., 2020), contact area (Liu et al., 2018), and stiffness (Hast et al., 2019) were adopted from literature and wide ranges were chosen to account for human variability. A total of 432 simulations were performed varying contact model (Hunt-Crossley, Elastic Foundation), target force (5, 12, or 20 N), contact area (40-80mm², increments of 5mm²), and stiffness (10⁶-10¹³ N/m, multiples of 10).

Simulation Analysis

To test our hypothesis, we compared to what extent the Hunt-Crossley versus Elastic Foundation contact models generated simulated finger-tip forces that accurately represented the target forces. Specifically, three accuracy brackets (5%, 10%, and 15%) were created to quantify the number of simulations for which the predicted finger-tip forces were within a given percent error of the target force. Accuracy brackets were separately defined to examine contact model only, contact model and target force, contact model and contact area, and contact model and stiffness. Additional supplemental analyses examined computation time and the effects of parameter variance (see Supplemental Materials).

Results

Across all simulations, variation in target force, contact area, and stiffness parameters impacted the accuracy of predicted finger-tip forces. Variance of predicted forces ranged from 0 to 0.42N².

Target Force

For the Hunt-Crossley model, accuracy of finger-tip forces increased with decreasing target force. In contrast, in the Elastic Foundation model, there was no clear relationship between finger-tip force accuracy and target force. Specifically, compared to simulations with 20N target force, the Hunt-Crossley model generated finger-tip forces within a 10% accuracy bracket for 29.8 and 15.9 times more simulations parameterized with 5N and 12N target forces, respectively (c.f., Figure 2, 3% compared to 83% and 44%). The Elastic Foundation model generated finger-tip forces within a 10% accuracy bracket for 9.4 times more simulations parameterized with 12N target force compared to those with 20N (c.f., Figure 2, 26% compared to 3%,). Interestingly, the overall proportion of accurate finger-tip forces increased for the Elastic Foundation model in the 15% accuracy bracket when target force was increased from 12N to 20N, but the inverse was true for simulations generated by the Hunt-Crossley model (Figure 2).

Figure 2.

Percentage of accurately simulated finger-tip forces by contact model, accuracy bracket, and target force.

Figure 3.

Percentage of accurately simulated finger-tip forces by contact model, accuracy bracket, and contact area.

Contact Area

For both contact models, accuracy of finger-tip forces increased as contact area increased. However, the Hunt-Crossley model produced more simulations with finger-tip forces similar in magnitude to the target force for each contact area value tested compared to the Elastic Foundation model (Supplementary Figure 1). Specifically, the Hunt-Crossley model generated finger-tip forces within a 10% accuracy bracket for 2.5 and 3.4 times more simulations parameterized with 55-65mm² and 70-80mm² contact area compared to those with 40-50mm² (c.f., Figure 3, 47% and 64% compared to 19%). Similarly, the Elastic Foundation model generated finger-tip forces within a 10% accuracy bracket for 6.0 and 22.0 times more simulations parameterized with 55-65mm² and 70-80mm² contact area compared to those with 40-50mm² (c.f., Figure 3, 6% and 22% compared to 1%). Similar trends were observed for 5% and 15% accuracy brackets in both models (Figure 3).

Stiffness

For both contact models, accuracy of finger-tip forces increased with stiffness. The Hunt-Crossley model generated finger-tip forces within a 10% accuracy bracket for 1.5 times more simulations parameterized with 10⁸-10⁹ N/m, 10¹⁰-10¹¹ N/m, or 10¹²-10¹³ N/m stiffness compared to those with 10⁶-10⁷ N/m (c.f., Figure 4, 35%, 35%, and 36% compared to 24%). Similarly, the Elastic Foundation model generated finger-tip forces within a 10% accuracy bracket for 7%, 11%, and 11% of simulations parameterized with 10⁸-10⁹ N/m, 10¹⁰-10¹¹ N/m, or 10¹²-10¹³ N/m stiffness compared to 0% for those with 10⁶-10⁷ N/m (Figure 4). Similar trends were observed for 5% and 15% accuracy brackets in the Elastic Foundation model, but only the 15% accuracy bracket in the Hunt-Crossley model (Figure 4).

Figure 4.

Percentage of accurately simulated finger-tip forces by contact model, accuracy bracket, and stiffness.

Discussion

In this study, how two common contact models (Hunt-Crossley and Elastic Foundation) represent fingerpad contact mechanics was examined by evaluating how key model parameters influence simulated finger-tip forces. Our results confirm our hypothesis and highlight differences in the accuracy of finger-tip forces generated by each contact model for different values of target force, contact area, and stiffness. Interestingly, stiffness had the largest effect on finger-tip forces as it represents both material properties and geometry in calculations of contact forces during object collision. Both contact models can be used to represent the fingerpad, however, analysis of our results led us to conclude the Hunt-Crossley model is generally preferable for representing the reported experimental properties of the fingerpad.

Our analysis of the Hunt-Crossley and Elastic Foundation models in a low-force scenario representative of hand contact biomechanics is a novel contribution because both contact models were previously implemented in primarily high-force scenarios. For example, these models have been used to represent contact mechanics of the knee’s tibiofemoral joint (Chan and Walker, 2018; Hast and Piazza, 2013; Zargham et al., 2019) and foot-floor contact (Febrer Nafría et al., 2018; Haralabidis et al., 2019; Mansouri et al., 2015; Serrancolí et al., 2018). However, contact force estimates in these scenarios range between 2.2 to 6 (Costigan et al., 2002; Kutzner et al., 2010; Taylor et al., 2004) and 0.8 to 1.2 (Fluit et al., 2014) times bodyweight for the tibiofemoral joint and foot-floor contact, respectively. In comparison, typical contact forces on the fingerpad are less than 35N (Kargov et al., 2004; Smaby et al., 2004); assuming a 70 kg individual, this is only 0.05 times bodyweight.

The preferred contact model for representing the fingerpad varies based on simulation parameters. The Hunt-Crossley model is most appropriate when simulating scenarios that involve low values of contact area (≤ 60mm²), stiffness (≤ 10⁹ N/m), and target force (5N or 12N). In contrast, the Elastic Foundation model may be advantageous when simulating scenarios that involve high values of target force (≥ 20N). There are a range of parameter values where both models are adequate [c.f., Supplementary Figures 1–3, contact area (65–80mm²), stiffness (10¹⁰-10¹³ N/m), and 5N or 12N target forces], and other considerations such as computation time and model complexity should be considered. However, neither contact model is suitable for representing the fingerpad with very low values of contact area (≤ 40mm²) and stiffness (≤ 10⁶ N/m). It is important to consider that these conclusions were drawn in a low-force scenario, and therefore may not apply to high-force scenarios, such as foot-floor contact.

This study has several limitations. The target forces tested do not include those observed in the hand during upper limb weight-bearing activities (above 20N) or delicate finger-object manipulations (below 5N). However, prior studies indicate hand contact forces predominately fall within the range tested (5N to 20N) (Vergara et al., 2014). Prior studies also report ranges of fingerpad contact area (Dzidek et al., 2017; Liu et al., 2018; Tomlinson et al., 2009) and stiffness (Pérez-González et al., 2013) beyond those tested herein. Another limitation is that our usage of sphere-plane contact to represent finger-object geometry may not be representative of some activities. Finally, while we assumed static, dynamic, and viscous friction to be negligible, this may not be true for finger-object interactions involving complex kinematics or geometry. Despite these limitations, our study provides a foundation for accurately characterizing the fingerpad, thereby enabling broader incorporation of contact mechanics into musculoskeletal models of the hand.