Biomechanical Comparison of Human Walking Locomotion on Solid Ground and Sand

The biomechanical study of human walking locomotion has been a fundamental aspect of understanding human physiology and developing assistive devices and wearable robotic systems. However, the majority of gait analysis research and open-source datasets were performed on level, solid ground [1]. This focus has yielded significant insights into the basic biomechanics of walking gait but has not encapsulated the full spectrum of challenges and variables present during everyday ambulation. When it comes to real-world walking locomotion, navigating through granular terrains such as sand or small pebbles can be particularly challenging due to their constantly yielding and shifting nature [2]. Sand and other yielding terrain surfaces are prevalent in everyday life, including beaches, playgrounds, and off-road trails [3]. The distinct properties of sand, such as its constantly shifting and deformable nature, present unique challenges for maintaining stability and efficiency during walking locomotion [4].

Several research works have provided a comprehensive understanding of how different terrain conditions affect human walking locomotion. The metabolic cost of walking on yielding substrates was investigated in Ref. [5], providing a foundational understanding of the energy demands of walking on terrains such as sand. Similarly, the work of Ref. [4] examined the mechanical work and energetic cost associated with walking and running on sand. This study highlighted that walking on sand requires approximately 1.8 times more energy than walking on solid ground. The mechanical work and energetic cost during walking and running on sand and on a hard surface were compared in Ref. [6]. In Ref. [7], the authors quantitatively compared the speed, temporal segmentation, and variability of walking on solid ground and sand. Recent studies on resistive force models have explored the effects of foot shape and walking speed on energy expenditure during bipedal locomotion on granular substrates [8,9]. These studies underline the importance of terrain in influencing gait mechanics and energy efficiency.

The research work on human gait in natural settings [2] and on different surfaces such as forest terrains [10] provided a comprehensive picture of the interaction between terrain type, gait parameters, and energy expenditure. Specific adaptations to walking on uneven terrains such as unanticipated steps were observed, and corresponding lower-limb biomechanical response was analyzed in Ref. [11]. Another comprehensive overview of the terrain impacts on human sprint locomotion demonstrated the effects of different surfaces such as natural and artificial turf, and sand [12]. The study in Ref. [13] discussed the influence of deformation height on estimating the center of pressure (COP) during walking on level ground and sand, showing that the magnitude of COP changes in the anterior–posterior and medial–lateral directions differed in both level and cross-slope conditions.

Recent advancements in assistive technologies have significantly contributed to the understanding of human–terrain interaction during walking locomotion. Developments in exoskeleton control for uneven terrains in Ref. [14] and the integration of human-in-the-loop control in soft exosuits on different terrains in Ref. [15] have enhanced activity recognition and response adaptability. The use of inertial sensor-based algorithms for foot–ground contact detection [16,17] and terrain topography detection [18] illustrates the complexity of walking dynamics on different surfaces. These methods play a crucial role in real-time walking gait phase estimation, especially on uneven terrain [19]. Additionally, studies focusing on the asymmetry of ground contact during running on sand have extended the understanding of the biomechanical adaptations required for locomotion on sand [20]. Particularly, Jafarnezhadgero et al. [21,22] observed specific adaptations to increased surface instability, such as reduced anterior–posterior ground reaction forces (GRFs) and increased muscle activation. In Ref. [23], crucial insights into foot–substrate interactions and footprint formation were provided. Moreover, Grant [24] provided a holistic view of how human gait and energetics are altered when traversing substrates of varying compliance. Recent work in Ref. [25] has further advanced the understanding by analyzing how foot sinking depth influences gait patterns and energy cost on sand.

Understanding the biomechanics of human locomotion on sand is crucial for improving rehabilitation, robot control, and exoskeleton performance on different terrains [26]. All the above-mentioned research emphasizes the complex interplay between human biomechanics and different terrain types. For example, the work of Ref. [4] demonstrated that walking on sand requires more energy than on solid ground due to increased muscle activation for stability and propulsion. Research work has also demonstrated that yielding terrains lead to notable shifts in COP [13] and sinkage depth [25]. Despite these valuable insights, comprehensive joint-level biomechanics analysis on walking locomotion over such yielding terrain remains scant. Recognizing this deficit, this study provides a detailed analysis of human walking locomotion on solid ground and sand. The central hypothesis is that GRFs and joint moments during walking gaits differ significantly on solid ground and sand. This difference reflects the compensatory mechanisms humans employ to navigate on yielding terrains. To test this hypothesis, we collect synchronized motion capture and GRF data from 20 able-bodied adults in environments designed to replicate natural walking conditions on both solid ground and sand. The study captures both joint-level kinematic and kinetic aspects of walking locomotion along with a calibration approach to ensure accurate GRF measurements through sand.

The main contributions of this work are twofold. First, this study provides a comprehensive analysis of GRFs and joint torque profiles during human walking on both solid ground and sand, complementing a knowledge gap in the biomechanics literature. While previous studies have examined isolated gait metrics on individual substrates, this work uniquely compares full-body adaptations across terrains within a controlled experimental framework. Key findings, such as differences in single-stance phase duration, knee flexion timing, and peak flexion magnitude, highlight the compensatory strategies humans employ to maintain stability and energy efficiency on yielding surfaces. Second, this study introduces a novel experimental dataset that integrates synchronized motion capture and GRF measurements with calibration references, offering essential data for advancing research into foot–terrain interactions and the development of adaptive wearable assistive devices.

In this study, 20 able-bodied healthy participants (14 males and 6 females, age: $24.8±4.0$ years old, height: $171.7±9.5$ cm, and weight: $74.5±16.1$ kg) participated in experiments and are labeled as S1 to S20. All participants are self-reported to be in a good health condition. An informed consent form was signed by each participant, and the experimental protocol was approved by the Institutional Review Board at Rutgers University.

Figure 1 shows the experimental setup in a laboratory with a three-segment walkway (7.5 m long and 0.2 m high). Segment 1 was built with reinforced plywood with one embedded force plate (model ACG-O from Advanced Mechanical Technology, Inc., Watertown, MA) to replicate a solid ground condition. Segments 2 and 3 were constructed to have width of 0.76 m, and segment 1 (with a width of 0.9 m) was designed to be slightly wider. Segment 2 was a sandbox filled with sand to represent a common granular terrain condition. A force plate (Bertec Corporation, Columbus, OH) was buried 14 cm beneath the sand surface to capture the GRFs. A calibration process for the GRF results is presented in Appendix  A. Segment 3, made of the same material as Segment 1, was connected to Segment 2 to ensure participants could comfortably leave the segment without changing their gaits. We conducted pilot tests to ensure force plate placement allowed natural stepping for subjects of varying heights. The sand used in this study was the natural, finely graded play sand with particle sizes ranged from approximately 0.1 to 1.5 mm in diameter. The sand materials were screened, washed, and dried to ensure consistency across trials. All experiments were conducted with dry sand in a controlled laboratory environment to prevent moisture variations that could affect sand properties and, consequently, GRF and gait measurements.

Starting at Segment 1 of the walkway, each participant was instructed to walk naturally and comfortably along the walkway without specifying which foot should make contact with the force plate. For the first five trials, the participant was required to walk back and forth on the walkway to get familiar with walking with the sensing devices and the terrain condition of the walkway. After that, another six to eight trials of one-directional walking experiments were conducted. The sand surface was flattened before each trial. The experimental setup is designed to study steady-state walking gait on both terrains. Unlike prior studies that focus on single-terrain, uniform-surface walking, the experimental protocol simulates real-world scenarios where participants transition naturally between different terrains. While the analysis focuses on steady-state walking rather than immediate gait changes during transitions, the setup allows us to observe the gait adaptations as humans encounter different terrains. We analyzed one representative stable step per trial, excluding initial transitional strides. This design captured how humans navigate on yielding surfaces like sand, and ensured the results primarily reflect steady-state gait rather than transitional adjustments [27].

For the walking motion, a motion capture system (ten Vantage cameras, Vicon Motion Systems Ltd.) was used to collect lower-limb motion information with the sampling frequency 100 Hz. A total of 18 reflective markers were firmly attached using adjustable straps over clothing to ensure participant comfort and consistency across trials. To prevent movement relative to the skin or clothing, straps were tightened appropriately, and marker positions were verified before each trial. Figure 2 shows the specific schematic of optical marker positions and their corresponding labels. Using the marker data, a customized lower-limb model was constructed within the Vicon nexus software for subsequent analysis. Meanwhile, the GRF measurements were synchronized via hardware and software setup with the motion data at 1000 Hz.

Motion and GRF data were combined to identify major gait events, such as heel-strike (HS) and toe-off (TO), and results were averaged across trials, assuming gait symmetry. The GRFs in the horizontal and vertical directions were smoothed by a moving average filter. The walking gait progression, namely, the joint angles and moments, was normalized by a defined phase variable (between 0 and 1) according to the strike events. Kinematic variables, including stride length/width, stance/swing time, vertical variation of the center of mass (COM), and average walking velocity, were analyzed to capture potential gait pattern variations between solid ground and sand. Spatiotemporal variables, such as stride length/width and the COM height, were calculated using three-dimensional kinematic data obtained from the motion capture system. Additionally, all spatiotemporal variables were normalized to each participant's height to account for interindividual variability and enable meaningful comparisons. The GRFs were normalized by the weight of each participant to enable a fair comparison across individuals. To standardize comparisons, the contact phase of the GRFs was normalized from heel-strike to toe-off for each participant.

The measurements and calculations of interest are joint angles of hip, knee, and ankle in the sagittal plane of the human walking locomotion, namely, the XOZ plane shown in Fig. 1. Likewise, we only presented the longitudinal and vertical forces in the X- and Z-axis directions, respectively. Joint moments in the sagittal plane were also computed through an inverse dynamics model given the GRFs applied on the foot; see the detailed description in Appendix  B. A paired sample t-test was conducted to evaluate the impact of the terrains on the above-mentioned metrics, with a significance level of $α=0.05$⁠. The data processing and statistical analysis were conducted by using custom algorithms in matlab software (Mathworks, Inc., Natick, MA).

Figure 3 illustrates stride profiles among all the participants. Compared with that on solid ground, stride length was found to be significantly longer (i.e., 20%) on sand, with an accompanying increase (i.e., 17%) of in stride width. Statistical analysis revealed that walking on sand resulted in a significantly larger normalized stride length (⁠$0.74±0.08$⁠) compared with on solid ground (⁠$0.663±0.19$⁠) (⁠$p<0.05$⁠; $d = 0.53$⁠). Similar changes in normalized stride widths were noticed when walking on sand (⁠$0.026±0.021$⁠) compared with stable ground (⁠$0.023 ± 0.017$⁠) (⁠$p<0.05$⁠; $d = 0.15$⁠). Walking on sand showed a significant difference in both swing time and stance time compared to walking on solid ground. The swing time for walking on sand (⁠$0.46 ± 0.057$ s) was significantly shorter than that on solid ground (⁠$0.48 ± 0.056$ s) (⁠$p<0.05$⁠; $d = 0.26$⁠). Conversely, the stance time for walking on sand (⁠$0.74 ± 0.09$ s) was significantly longer than that on solid ground (⁠$0.67 ± 0.09$ s) (⁠$p<0.05$⁠; $d = 0.57$⁠). The gait phase percentages reveal slightly differences between walking on sand and solid ground, with the swing phase taking up 42% of the gait cycle on solid ground and 38% on sand. Despite these changes in swing and stance times, participants tended to walk faster on sand than on solid ground. Despite these changes in swing and stance times, participants tended to walk faster on sand than on solid ground. The normalized average walking speed on solid ground (⁠$0.60 ± 0.11$⁠) was approximately 94% of the speed on sand (⁠$0.64 ± 0.08$⁠) (⁠$p<0.05$⁠; $d=0.54$⁠). The mean COM vertical variation during each step on the sand was also observed to increase by 6%, although this difference was not statistically significant (⁠$p>0.05$⁠; $d=0.01$⁠).

Figure 4 shows the mean and standard deviation profiles for the hip, knee, and ankle joint angles in the sagittal plane across both legs throughout the gait cycle. Statistical analysis revealed significant differences in peak joint angles. The peak hip flexion angle was significantly greater on sand (⁠$29.36 ± 3.84 deg$⁠) than on solid ground (⁠$p<0.05$⁠; $d=0.11$⁠). Similarly, the peak knee flexion angle was significantly greater on sand (⁠$75.21 ± 7.96 deg$⁠) compared to solid ground (⁠$p<0.05$⁠; $d=0.37$⁠). The peak ankle dorsiflexion angle was also significantly greater on sand (⁠$18.29 ± 7.50 deg$⁠) than on solid ground (⁠$p<0.05$⁠; $d=0.16$⁠). In addition to peak joint angle values, we also conducted a statistical analysis across the entire gait cycle using one-dimensional statistical parametric mapping (1DSPM) to capture joint angle variations. The last row of Fig. 4 shows the 1DSPM statistical analysis between two terrain conditions across the gait cycle. For the hip joint, two regions of statistical significance were observed within the gait cycle. The knee joint exhibited a significant region of statistical significance (⁠$p<0.001$⁠) during the latter phase of the gait cycle, corresponding to the push-off period. For the ankle joint, there are three regions of statistical significance. A significant difference (⁠$p<0.01$⁠) was observed in the midstance phase. In addition, the push-off phase of the gait cycle exhibited highly significant differences (⁠$p<0.01$⁠), and another significant difference (⁠$p<0.05$⁠) was identified during the terminal swing phase.

Figures 5(a) and 5(b) show the longitudinal and vertical GRFs normalized by the participant's weight, respectively. The longitudinal and vertical GRFs showed significant differences between sand and solid ground, with distinct magnitudes and variability observed across the two terrains (⁠$p<0.05$ for both forces and $d=0.58$ and 0.36, respectively, for $Fx$ and $Fz$⁠). It is observed that the normalized magnitude of the maximum longitudinal force $Fx$ on sand is relatively smaller than that on solid ground in both forward (0.0464$±$0.03 versus 0.15$±$0.035) and backward directions (−0.076$±$0.024 versus −0.1$±$0.047). However, regarding the vertical force $Fz$⁠, sandy terrain might provide more supporting force during the heel-strike (1.22$±$0.22 versus 1.03$±$0.084) compared with the solid ground. It is also noticeable that there is a slight difference with the double-hump pattern of the vertical force $Fz$ throughout the contact phase. The magnitudes of two humps on the solid ground appear almost equivalent. Nevertheless, for the sandy terrain, the first hump of the force is slightly higher than the second one in the late contact phase.

The inverse dynamics model and calculation of joint moments were presented in Appendix  B. Figure 6 shows the normalized joint moments in the sagittal plane for the hip, knee, and ankle joints during walking on sand and solid ground. During the stance phase, the hip joint moments on sand show increase peak values compared to solid ground, particularly in late stance (50–60% of the gait cycle). Similarly, knee joint moments are slightly higher on sand from early-stance to midstance (10–50% of the gait cycle). At the ankle joint, dorsiflexion moments are notably higher on sand during late stance (50–60% of the gait cycle). In the swing phase, joint moments for all three joints exhibit smaller differences between sand and solid ground. From the comparison results, we observe that for able-bodied locomotion, the joint moments kept consistent bilaterally both on sandy and solid terrain.

Figure 7 shows the comparison of the knee stiffness profiles on sandy and solid terrain. It is clear that, compared with that on solid ground, the knee moment contour of walking on sand shifts toward the large angle amplitude and lower moment amplitude directions. The characteristic feature such as knee stiffness defined in the stance phase [28], i.e., shown as the trajectory R-HS$→$L-TO$→$L-HS in the figure, did not change significantly. The mean knee flexion stiffness values during the R-HS$→$L-TO phase are 0.15 N·m/(deg kg), while the average knee extension stiffness values (⁠$Ke$⁠) during the L-TO$→$L-HS phase are 0.17 N·m/(deg kg) on both solid ground and sand. However, for the swing phase (R-TO$→$R-HS), the knee stiffness values on sand shift due to the increased knee angles, although the average magnitudes remain similar at 0.011 N·m/(deg kg) on both terrains.

To further explore the interdependencies between biomechanical variables on both terrains, we conducted a multiple linear regression (MLR) analysis. The MLR analysis investigated the relationship between joint angles and moments across sand and solid ground for knee, hip, and ankle joints. For the knee joint, the model shows 6.8% of the variance (⁠$R2=0.068$⁠) with no significant predictors (⁠$p>0.05$⁠). Interaction terms for knee moment and angle did not improve the model (⁠$R2=0.0822$⁠, $p=0.426$⁠). For the hip joint, the $R2$ was 0.0117, and no significant effects were observed for condition or peak hip angle (⁠$p>0.05$⁠). In contrast, the ankle joint model demonstrated moderate explanatory power (⁠$R2=0.255$⁠), with condition (sand) as a significant predictor (⁠$p=0.0025$⁠), although peak ankle angle was not significant (⁠$p=0.398$⁠). Test set $R2$ values for knee, hip, and ankle joints were $−0.060$⁠, 0.006, and $−0.279$⁠, respectively, and these results confirm poor generalizability of the models.

In this study, we examined human walking locomotion across two different terrain conditions, with an emphasis on a granular sandy condition. We collected walking data and extracted detailed kinematic and kinetic measurements and calculations. Moreover, we encompassed comparisons of joint angles, joint moments, and GRFs throughout the walking gait profile.

Different from the experiment setup designed in Ref. [3], the participants in this study were asked to walk across two different terrains sequentially from the solid ground to the sandbox, allowing each participant to keep the preferred comfortable walking pace. Our findings suggest that walking strategies exhibit terrain-specific adaptations during steady-state walking on solid ground and sand. The extended stride length and width observed on the sand were likely compensatory mechanisms to maintain balance on the yielding surface. Participants demonstrated a higher average walking speed on sand compared to solid ground; see the normalized velocity result in Fig. 3. This result is consistent with Ref. [7], which reported that the self-selected speed on a 60 m distance increased when walking on sand with respect to solid ground, with the highest speed on wet sand. This observation suggests that individuals compensate for the challenging terrain by adopting a faster pace, potentially to minimize the time spent on an unstable surface. However, Grant et al. [25] observed a decreased walking speed and increased stride length on sand compared with solid ground, while Svenningsen et al. [3] reported reductions in both walking speed and stride length on sand. The discrepancies between the findings in this work and those studies are likely due to differences in experimental conditions and walking tasks. This study allows participants to walk sequentially across solid ground and sand at their preferred pace. In contrast, the studies in Refs. [3] and [25] used controlled indoor environments on single terrains and consistent starting conditions for all trials, thereby minimizing variability from self-selected walking strategies specific to each terrain. These findings underscore the importance of replicating real-world conditions when studying gait adaptations on yielding terrains.

Additionally, the increased COM vertical variation might indicate an adaptive response to optimize energy efficiency and maintain stability on the yielding terrain (see Fig. 3), which was also found for the midstance phase (21–60% gait cycle) in Ref. [3]. Previous studies used the inverted pendulum model to examine the kinetic and potential energies of the COM during walking on sand and solid ground [25]. While total energy exchange recovery was slightly higher on sand, no significant differences in pendular energy recovery were observed between the two terrains, and only modest changes in muscle activations were observed [25]. These adaptations are pivotal for navigating granular substrates, where maintaining stability and efficient locomotion poses greater challenges compared to solid ground. The increased stride lengths and widths indicate that human tend to adapt broader stance and greater joint flexion for stability, likely as a strategy to reduce mechanical work and metabolic costs associated with redirecting the COM during step transitions. These findings suggest that participants balance the tradeoff between minimizing mechanical work and maintaining lateral stability, adjusting their gait to the energy and stability demands of the sand.

The kinematic analysis of the lower limbs aligns with existing studies in Refs. [24,25], and [29]. The joint angles, particularly in the lower extremities, demonstrate distinct variations and adjustments in walking strategies on solid ground and sand terrains. Walking on the sandy terrain results in greater dorsiflexion of the ankle in the early stance phase but a similar peak amplitude of the ankle plantarflexion near toe-off. This is because the heel would experience an intrusion into the sand and then the forefoot was prevented from plantarflexion at the beginning of the stance phase; see the small portion (0–5%) of ankle plantarflexion on the solid ground shown as the dashed line in Fig. 4. This might reduce the energy storage potential in the ankle plantar flexors, thereby limiting the energy available for recovery and propulsion during push-off and resulting in greater mechanical work. Furthermore, a relatively larger variability of the ankle dorsi/plantarflexion on the sand was found compared with the hip flexion/extension and knee flex/extension.

The large flexion observed in the hip and knee joints during the toe-off and swing phases on sand can be interpreted as a compensatory mechanism for the loss of momentum during the stance phase, when the foot sinks and the sand underneath the step is displaced. The deformable nature of sandy terrain reduces the efficiency of energy transfer, necessitating additional joint flexion to facilitate foot clearance and forward progression. These kinematic findings underscore the adaptability and fine-tuned control required for efficient and stable locomotion on deformable surfaces.

The kinetic analysis revealed critical insights into the biomechanical demands of walking across solid ground and sand. In contrast to Ref. [13], which assumes that a separated sandbox experimental design minimizes force dissipation within the sand, the calibration results presented in Appendix  A reveal a notable discrepancy between the forces applied at the sand surface and the GRFs recorded by the embedded force plate. Despite requiring force calibration, the longitudinal and vertical GRFs in this work are in agreement in pattern to those reported by Xu et al. [13]. The differences in GRFs between sand and solid ground highlight the significant impact of terrain type on the forces exerted by the foot during walking. The normalized magnitudes and variability of the longitudinal (⁠$Fx$⁠) and vertical (⁠$Fz$⁠) GRFs across the gait cycle are illustrated in Figs. 5(a) and 5(b), respectively. The small normalized magnitude of the maximum longitudinal force on sand suggests that the yielding nature of sand reduce the ability to generate propulsive and braking forces, possibly due to the dissipation of energy as the foot interacts with the deformable surface. In contrast, the vertical force on sandy terrain exhibits a higher peak during the heel-strike phase and a different double-hump pattern compared to solid ground, indicating that the deformable surface significantly alters the loading pattern during the stance phase. The magnitudes of two humps on the solid ground appear almost equivalent. Nevertheless, for the sandy terrain, the first hump of the force is slightly higher than the second one in the late contact phase. The increased variability and different magnitudes of GRFs on sand also suggest that walking on this surface requires adapted biomechanical strategies to maintain stability and forward progression.

Compared with on solid ground, walking locomotion on sandy terrain required less hip extensor action until the midstance to control hip flexion and the forward rotation of the upper body. The joint moments shown in Fig. 6 have a small backward component (approximately 5%) at the hip joint which requires small hip extensors consequently. However, compared to solid ground, walking on sand demanded greater hip extension torques to pull the thigh forward and upward at the latter half of the stance phase (including the double stance) and get ready for the knee extension and swing. The knee joint also exhibits large flexion torques during the mid- and late stance phase to prepare for the swing phase. These findings may indicate large energy consumption to compensate for momentum loss at the stance leg on sand, as suggested by previous studies that have reported high metabolic cost on yielding terrains [5]. The MLR analysis also highlights notable differences in joint mechanics between sand and solid ground, particularly for the ankle joint, where condition significantly influenced peak ankle moment. This suggests that the ankle play a critical role in adapting to yielding terrains like sand. The relatively low percentages of variance observed in the knee and hip joints likely result from additional contributing factors that were not directly measured in this study. Potential contributors include muscle activation patterns, joint load distribution, and neuromuscular control strategies, all of which are critical to understanding gait adaptations. These factors might interact with joint mechanics and terrain conditions to influence overall movement patterns.

The distinct patterns in GRFs and joint moments suggest that walking on sand involve terrain-specific adjustments in joint mechanics to maintain stability and locomotion efficiency. However, these results do not necessarily indicate a qualitative shift in movement strategies, as the joint moments follow similar patterns across both terrains; see Fig. 7. Complementing our measurements, the work in Ref. [25] studied muscle activation patterns during walking on various sand substrates with different sinkage depths. The authors reported slightly higher muscle activation values on sand than those on solid ground and significant differences limited to only specific muscles (e.g., rectus femoris, vastus lateralis, vastus medialis, and lateral gastrocnemius). These findings highlight the complex interplay between joint mechanics and muscle activation, suggesting that terrain-specific adaptations involve subtle adjustments rather than fundamental shifts in muscle coordination. When walking on sand, the rectus femoris and vastus lateralis work hard to maintain knee extension and stabilize the leg during the stance phase, while the lateral gastrocnemius generates additional force during push-off to overcome the resistance and energy loss caused by sand deformation. Additionally, increased foot sinkage demands great muscular effort to lift the leg and prepare it for the next swing phase, further contributing to high activation levels. It was also found in Ref. [21] that walking on sand requires heightened stabilizing muscle activity, particularly in the tibialis anterior, to counterbalance the unstable surface. The findings in this study align with these observations, as evidenced by increased knee and ankle flexion moments during stance on sand. These findings indicate a complex interplay between maintaining stability, managing energy efficiency, and adapting to the terrain's physical characteristics.

This work focused on knee stiffness because of its linear relationship between joint moment and joint angle during certain gait phases. This would enable reliable stiffness estimation and can be further used for assistive device design [30]. In contrast, hip and ankle joints exhibit complex, nonlinear stiffness profiles throughout the gait due to their biomechanical roles: the hip contributes to stability and propulsion, while the ankle is involved in surface interaction and energy absorption. Nonlinearities at these joints complicate stiffness generalization for a complete gait cycle. The knee biomechanical profile potentially provides a tool for assistive device controllers to adjust strategies accordingly. By leveraging joint torque and angle data, assistive controllers can provide phase-specific assistance that aligns with the natural gait cycle, enhancing movement synergy. For example, using the knee exoskeleton in Refs. [26] and [31], we can further incorporate the active knee biomechanics profile with respect to knee flexion/extension. Adaptive real-time biomechanical feedback control can dynamically adjust assistance on varying terrains and, therefore, improve stability and energy efficiency for the user. This data-driven approach promises to refine the integration of exoskeletons with human locomotion and potentially enhance assistive device functionality.

The methodology in this study captures steady-state walking gaits on both solid ground and sand, providing insights into the human motor system's compensatory mechanisms for maintaining stability on less stable surfaces. Although relatively short, the 7.5-m walkway enables the capture of at least one complete gait cycle per terrain, offering valuable insights into immediate biomechanical adaptations during surface transitions. The findings in this work are consistent with the existing literature on gait adaptations to sand. For example, Grant et al. [25] highlighted foot–sand interaction and energy costs, whereas this study expands and examines GRFs and joint-level adaptations, emphasizing how changes in joint torques support stability and energy conservation. While D'Août et al. [23] focused on foot print depth and pressure distribution, we captured dynamic GRF variations throughout the gait cycle using a calibrated foot–terrain interactions. Additionally, the results in this study generalize findings from Refs. [21] and [22] that reported altered GRFs and increased muscle activation in pronated-foot populations, through detailed joint-level analyses. Together, these results not only enhance our understanding of human gait mechanics on yielding terrains but also have implications for the design of adaptive assistive technologies such as exoskeletons that can dynamically respond to deformable terrains.

Participants wore standardized swimming boots to provide uniform foot protection and minimize discomfort while maintaining barefoot-like conditions without significantly altering gait mechanics. This footwear choice ensured consistent optical marker and inertial measurement unit placements, reducing variability that could arise from barefoot walking or personal footwear. While the minimalist footwear might introduce slight variations in GRFs and gait parameters compared to true barefoot walking [32], this tradeoff balanced participant comfort with biomechanical consistency across trials, enabling accurate and reliable data collection. In experiments, the force plate was embedded at the 14 cm sand layer depth to obtain the GRFs, and this was inspired by previous studies and verified by numerical simulation and pilot testing. For example, Lejeune et al. [6] used a 7.5 cm sand layer to place the force plate in walking and running experiments, which was effective for capturing force data in relatively shallow substrate conditions. Grant et al. [25] studied foot sinking depth and its influence on gait patterns using walkways with 10 cm depth of sand. Additionally, Jafarnezhadgero et al. [21] employed a sand depth of 20 cm that facilitated stable GRF measurements. Therefore, we chose an intermediate sand depth according to the existing studies. While the work in Ref. [13] examined the influence of deformation height on the COP, it did not account for the dissipation of GRFs within the deformable substrate, leaving a critical aspect of force transmission unexplored. To account for this, the GRF measurements collected within Segment 2 of the experimental platform were calibrated using a correlation based on the sand layer thickness. Details regarding the sand depth selection, the calibration process, and the resulting correlation curves are provided in Appendix  A.

This study represents a significant advanced step in analyzing biomechanics during human locomotion across varying terrain types, specifically solid ground and sand. It is the first study that reports and analyzes the calibrated GRFs and joint moments for humans walking on yielding terrains such as sand. The locomotion dataset provided by this study also generates new knowledge and enables using wearable devices to assist human locomotion on yielding terrains. Understanding the gait adaptations and incorporating additional biomechanical variables, such as electromyography data and joint load analysis, can inform the design of assistive devices, prosthetics, and rehabilitation protocols, as well as guide the development of robotic systems. The study's design permitted participants to walk at their preferred pace, leading to variability in walking speeds, diverging from other studies such as Ref. [1] focused on locomotion velocity effects. This variability might influence biomechanical data consistency. In Ref. [13], the results showed the magnitude of COP changes on level ground in the anterior–posterior direction is within 4 mm when the sand deformation is 20 mm. Therefore, for the convenience of calculation, we take the COP position from the GRF data. The research scope in this study was limited to straight walking on level sand and solid ground, excluding complex movements like ascending, descending, and turning on granular terrains. Such limitations might restrict a comprehensive understanding of biomechanical adaptations in gait activities. The dataset lacks other lower limb biomechanics, such as electromyography signals, important for analyzing muscle activation patterns. Future research work should include different locomotive modes and participant demographics, using an array of sensors for gait analysis on additional terrain types to further enhance our understanding of human locomotion on granular terrains.

This paper presented a biomechanical analysis of human walking locomotion on sandy terrain, offering significant insights into gait adaptations on challenging granular surfaces. Using comprehensive biomechanical data collected from 20 able-bodied adults, the findings highlighted the intricate adjustments in stride patterns, joint mechanics, and ground reaction forces required to maintain stability and efficiency on yielding substrates. These insights offered valuable implications for the development of advanced assistive devices and responsive robotic systems. We also provided and shared open-source kinematic and kinetic dataset and a comparison of human walking on solid and sand surfaces. Despite certain limitations such as the variability in walking speeds and the focus on able-bodied individuals, this research paved the way for future studies to explore a broader range of locomotion conditions and participant demographics and, therefore, enhance the applicability of wearable sensing and assistive technologies in different environmental settings.

The authors thank Mr. Aditya Anikode of Rutgers University for his help in constructing the experimental setup in this study.

• U.S. National Science Foundation (NSF) (Award No. CMMI-2222880; Funder ID: 10.13039/100000001).

The authors attest that all data for this study are included in the paper.

The force plate and sandbox configurations in the experiment were similar to that in Ref. [13]. However, the dissipation of the force under this design cannot be negligible. To verify the choice of placing the force plate underneath the 14 cm sand layer and negligible sandbox boundary effect, we conducted foot–sand interactions simulation using the material point method-based nondilatant plasticity model [33]. The simulation examined stress and strain rate distributions in both sagittal and frontal planes. The stress distributions at each time interval show the dissipation of forces within the sand layer, while strain rate distributions illustrate the rate of deformation within the sand medium. The intruder size in the simulation was set to be $27×9$ cm to approximate the contact area of an adult foot, while the sandbox size and sand properties matched the dimensions and sand used in our experiments.

Figures 8 and 9 show the stress and strain rate distributions in the sagittal and frontal planes, respectively, at three time intervals with an intruding velocity of 1 m/s. The simulation results exhibit rapid dissipation with increasing depth and distance from the contact area. As shown from the figures, the intruding distance is within 0.04 m, and by 10 ms, the force transmits beyond 0.14 m. By 40 ms after foot impact on sand surface, the majority of the force is concentrated directly beneath and around the intruder, with only minor stress propagation reaching the edges of the sandbox in either sagittal or frontal plane. Meanwhile, Figs. 8(a) and 9(a) show a clear difference between the stress distribution at the contact surface and under the sand layer. This discrepancy highlights the necessity of a calibration process for accurately measuring GRFs using the force plate embedded beneath the sand. The 14 cm thick sand layer was therefore selected to provide a tradeoff between walking stability on platform and the accurate GRF measurements. This depth allows for accurate GRF capture with sufficient substrate depth for realistic foot–terrain interaction without excessive dissipation of force signals.

Figure 10 shows the calibration setup for GRF measurements. We designed and built a sandbox with a force plate positioned beneath the sand layer. The calibration setup consists of a lever mechanism, featuring a force/torque load cell (model mini45 from ATI Industrial Automation, Apex, NC) at one end and a vertically adjustable displacement lift platform at the other end. A metal plate with a similar human foot ratio was attached to the load cell to reduce the variation of the calibration setup with actual human walking case. By raising the platform and leveraging the mechanical advantage of the bearing structure, the compression plate can firmly generate a controlled vertical force onto the sand's surface. The magnitude of the applied force was recorded by the load cell and was synchronized with the force plate readings. This setup allowed us to establish a correlation between the force exerted at the surface and the force recorded at the force plate that was embedded at varying sand thickness conditions.

The calibration was performed by applying static known forces (obtained from the load cell and denoted by $Fzs$⁠) and recording the corresponding force plate measurements (denoted by $Fzb$⁠) at different sand depths. The forces were exerted from 0 to 800 N with a 100 N increment and then lowered to 0 with the same decrement. We increased the sand depth from 0 to 14 cm by the increment of 1 cm. The sand surface was paved flat before each compression calibration. These measurements were then used to generate a force ratio curve (⁠$ζ=Fzb/Fzs$⁠) that served as a calibration reference for interpreting the GRFs during the walking trials. This curve was essential for compensating for the deformation of sands and energy dissipation at various depths, and this would allow for a precise adjustment in our biomechanical analysis. Moreover, it was confirmed that the location had no significant impact on the force plate measurements for the area directly above the force plate.

Figure 11 shows the force calibration results. The force measured beneath the sand was indeed smaller than that at the surface. Overall, the force ratio $ζ$ decreases as the sand layer becomes thicker. A significant ratio drop is observed when the sand thickness rises from 6 to 7 cm. This correction force ratio was used to calculate the actual GRFs from the force plate measurements. For the participant walking experiments, we used the ratio $ζ=0.81$ for the 14-cm sand depth, and therefore, the real vertical ground reaction force $Fzs=Fzb/ζ$⁠, where $Fzb$ is the force plate measurements in the vertical direction.

In this section, we discuss the calculation of the joint moments in the sagittal plane based on inverse dynamics. The formulations for ankle, knee, and hip joint moments are derived and articulated using Newtonian–Euler formulation specific to the foot, shank, and thigh segments, respectively. Similar derivations can be found in the support materials from the work of Ref. [28].

As shown in Fig. 12, the leg consists of three segments, namely, the thigh, shank, and foot segment. Each segment has two joints: proximal and distal joints. To obtain the joint moments of each segment, the Newtonian–Euler method is used. Since the procedure is consistent for each segment, we only present the derivation of the proximal and distal joint moments for the foot segment as an example. For the foot segment, the proximal joint is the ankle joint and the distal one is the toe, and the ground reaction force vector $FG$ and moment vector $MG$ are applied at the COP point.

where $MPF$ is the proximal joint moment, namely, the ankle moment. $lf$ and $lpf$ are the length of the foot segment and distance from the ankle to the center of mass of the foot segment, respectively. $ef$ is the unit vector of the foot segment direction (shown in Fig. 12) that represents the orientation of the foot.

where $es$ (⁠$et$⁠) is the unit vectors of the shank (thigh) segment. $ls$ (⁠$lt$⁠) is the length of the shank (thigh) segment. $lps$ (⁠$lpt$⁠) is the distance from the knee (hip) to the center of mass of the shank (thigh) segment. $Is$ (⁠$It$⁠) is the moment inertial about the center of mass of the shank (thigh) segment. The angular acceleration of the shank (thigh) segment is denoted as $ω˙s$ (⁠$ω˙t$⁠).

The formulations (B1)–(B3) are consistent for both the stance leg and swing leg. For the swing leg, the ground reaction forces and moments are zero, i.e., $FG=0$ and $MG=0$⁠. Translation accelerations and angular accelerations of the leg segments are extracted from the optical marker measurements. The corresponding anthropometry information of the participant such as segment length, mass, and location of COM can be found in Chap. 4 of Ref. [34]. We used estimate ratios for the joint moment calculation in this study.

Supplementary dataset to this article can be found online at this link.³