Using a spherical inverted pendulum and statokinesigram for modeling and evaluating quiet standing posture

This paper proposes a new approach to model and analyze erect posture, based on a spherical inverted pendulum which is used to mimic the body posture. The pendulum oscillates in two directions, θ and ϕ , from which the mathematical model was derived and two torque components in oscillation directions were introduced. They are estimated using stabilometric data acquired by a foot pressure mapping system. The model was quantitatively investigated using data from 19 participants, who were first were classified into three groups, according to the foot arch-index. Stabilometric data were then collected and fed into the model to estimate the torque’s components. The components were statistically processed, and the results revealed that the components in direction θ are able to reject intrinsic perturbation. The frequency spectrum of the components in direction ϕ was processed using fast Fourier transform, and the results showed the feasibility of the component in segregating foot deformities. In addition, high-arched foot cases tended to be more stable than other cases because the exerted torque is less. The torque profiles estimated by our model were compared with the profiles derived from a classical inverted pendulum. In most cases, our results showed a significant change (t-test p < 0.05).


Introduction
In humans, quiet standing is a complex function dominated by the central nervous system in which descending signals from visual, vestibular, and somatosensory systems are used to ensure equilibrium. This function is an involuntary physiological adjustment mechanism in which a deficit in the upcoming signals may cause a serious injury. Moreover, some neuromuscular disorders such as Parkinson's and cerebral palsy in children can impact balance.
Models of balance control can thus contribute in the assessment of the equilibrium condition, joint prostheses design, and clinical kinesiology research. 1,2 These models employ the trajectory of the body center of pressure (CoP) to keep the body center of mass (CoM) in the base of support. The CoP oscillates in a sagittal plane, anterior/posterior (A/P), and frontal plane, medial/lateral (M/L). According to, 1,3,4 A/P oscillation is mostly controlled via a torque exerted in the subtalar joint, which is generated by the plantar/dorsiflexor muscle arrangement, while M/L oscillation is produced due to the load/unload mechanism of the hip. [5][6][7] CoP oscillations can be recorded via foot pressure mapping and return a stabilometric diagram which is the sway path of the CoP. This diagram is useful when diagnosing balance disorders. [8][9][10] However, due to the rather chaotic shape of the diagram, it requires further processing to extract diagnosis features for use in the clinical assessment of the quiet stance.
In this work, we investigated a biomechanical model for estimating ankle joint torques using stabilometric data. The model is based on a spherical inverted pendulum where the motion of CoP is assumed to take place in two spherical coordinates, u and f. The u component of the motion occurs in the sagittal plane, while motion in direction f occurs in the frontal plane.
Our model differs from previous models, which use a simple inverted pendulum to identify the trajectory of the CoP as a single degree of freedom (DOF), and moves in sagittal or frontal planes. We assumed that the lateral motion occurs in the ankle joint, and is related to A/P motion in which the torque in direction f induces that motion.
To the best of our knowledge, our model is the first to combine A/P and L/M movements as a control model of the erect posture. We believe it will assist researchers in developing simplified control algorithms for quiet standing.
In the literature on modeling human postural sway, a single DOF inverted pendulum (IP) has frequently been proposed to evaluate quiet standing posture 6,7,11,12 and has been used as a control model for human postural regulation. [13][14][15] Other works, such as, 11 used motions in sagittal and coronal planes and fed them in the proposed control scheme for postural stability evaluation. Modeling of the quiet postural motion in the frontal plane was considered in, 3 where the authors propose, in a quiet-stance postural sway, to control the M/L motion by the inverted/everted subtalar joint. This assumption is in line with our proposed model. However, in, 7 frontal plane motion was modeled as three links, in which two of these links were parallel, while the third link demonstrated the body. The parallel links were connected with hinges and supported on the floor. A similar model was introduced in, 16 in which the proposed models were used to analyze human balance on a rolling board. Unlike our proposal, these models did not consider the relation between the motions occurring in sagittal and frontal planes, but dealt with each motion separately.
Static posturography is the study of body sway in an upright posture. This is usually carried out by placing the subject of a force-plate device that can sense tiny oscillations of the CoP of the body in both A/P and L/ M directions. [17][18][19] The device measures two time series of oscillations which can be post-processed to evaluate the status of the body sway. Oscillations of CoP can be combined in a statokinesigram graph, which is the trajectory of the CoP. The force pressure platform is an alternative device for obtaining this trajectory since it depends on the plantar pressure distributions of both feet and then estimates the precise location of body CoP. [20][21][22] We used a plantar pressure mapping platform to acquire the body CoP in both A/P and M/L directions.
Estimating postural ankle torque entails a quantitative approach, which is vital in the assessment of the quiet stance in humans. 23,24 This torque needs to be modulated frequently in order to preserve stability in response to internal and external perturbations. 25 According to, 25,26 the ankle torque consists of two components: intrinsic torque and active torque. The intrinsic torque is caused by joint mechanical resistance to its movements, while the second torque is produced due to changes in muscle activation.
Recent approaches to estimate the joint torque fall into two groups. The first is based on motion capture systems and pressure, or force platforms to measure the complete dynamics of the human body. It then uses the acquired data in kinematic chain models to estimate the intrinsic body forces and movements. The second approach employs joint kinematics and ground reaction forces. This latter approach abandons the inertial properties of the body segments in favor of symbolizing the subject as a single mass point. 24,27 In our work, in order to simplify and reduce the computation, we estimated the ankle joint torques following the second approach.
The rest of the paper is organized as follows. Section 2 presents the mathematical description of the proposed model. In addition, the instrument used, the subjects recruited, and the protocols tested are illustrated. Sections 3 and 4 present the results and discussion. Finally, the conclusions are drawn in Section 5.

Methodology
Our approach to modeling erect stance is based on using an inverted spherical pendulum to mimic human posture. It involves mass movements on a spherical surface. This spherical motion is described by spherical coordinates l, u, f ð Þ. Since the trajectory of the body CoP is described in two components, A/P and M/L movements, these components can be combined into a model that can be used to estimate ankle torques. 3,11,25 We used a statokinesigram graph to compute the model kinematics, u and f, according to (1) in which the displacements were defined in Cartesian coordinates: where l is the length between the ankle joint and body center of mass, u and f are spherical angular displacements as shown in Figure 1. The mathematical dynamic model of the spherical pendulums was derived in terms of u and f, where kinetic and potential energies of the setup were obtained using equations (2) and (3). We then implemented a Lagrange equation, equation (4), for each degree of freedom.
where m is body mass and g is gravity.
d dt where q i is the degree of freedom for the system, Q i is the applied load in each degree of freedom, and D: E is dissipation energy which we neglected due to the fact that it has little effect on the model and in order to simplify the computation. 25,28 The dynamic equations for our model are: where t i is exerted torque at the ankle joint.
To estimate the components of the ankle torque, kinematics data are fed into equations (5) and (6).
In the case of a simple inverted pendulum, and assuming that motion occurs in the sagittal plane or frontal plane separately, the dynamic equation is simplified as: The angular displacements u and f were obtained from the time series of the stabilometric diagram. This diagram is a projection vector of the point mass of the spherical inverted pendulum on the xy ð Þ plane. In mathematical terms, u and f are calculated by:

Instrumentation
Our approach requires stabilometric data to estimate ankle torque. A Tactilus high performance footplate, (USA), was used to acquire the data that were sampled at 100 Hz. Two time series for A/P and M/L oscillations were collected. These series were then used to compute the spherical angular displacement according to equations (8) and (9). The angular velocity and acceleration in both u and f directions are then estimated numerically. By identifying the total mass of the body and the location of the body center of mass, the two components of the ankle joint torque were evaluated using equations (5) and (6).

Participants and protocol
A quantitative approach, based on foot deformities, was used to assess the feasibility of the proposed model. Nineteen adults were recruited and Table 1 shows the demographic and anthropometric information of the subjects. The participants were categorized into three groups, according to the value of the foot arch index (AI): normal, flat and high arched foot. The value of the arch index was obtained by dividing the area of midfoot contact with the ground to the total area of foot contact with the ground. An AI of between 20 and 30%, was considered as normal. An AI less than 20%, was considered as a high arched foot. An AI higher than 30%, was considered as a flat foot. [29][30][31] The AI values were acquired using the Tactilus platform.
These groups helped verify the model's ability to assess foot abnormalities using the components of the joint torques. All participants were informed about the experimental protocol and signed a consent form in accordance with the Helsinki Agreement. The subjects were asked to step up to the foot pressure platform and to stand still for 20 s while the point data acquisition was initiated. The subjects were requested to keep their eyes open and to gaze on a fixed object. The platform then recorded the A/P and M/L oscillations.

Results
The proposed approach was quantitively implemented to estimate the components of the joint torque for the three groups of subjects: flat, normal, and high arch foot. Time series of the stabilometric data, A/P and M/ L, were collected for the 19 subjects according to the proposed protocol. The data acquired were then processed to numerically compute the first and second derivatives of the time series.
Using the dynamic model, and equations (5) and (6), the torque components exerted in the ankle joint were estimated. Figure 2 shows the estimated torques in both f and u directions. The raw torque profiles were then further processed to investigate the relation between the torque profiles and foot deformities. Figure 2 highlights that the torque values in direction u are higher than in direction f, where the vector direction of the torque in u twists the body, while the torque component in direction f effects the movement along the sagittal plane. Figure 3 shows boxplots for the torque profiles in direction u for the cases that were classified as normal, Figure 3(a), and abnormal, Figure 3(b) and (c). Figure  3 shows outliers in the subjects' plots 12, 16, and 17. Figure 3 shows mean values of the u component of the torque in the range of 1 to 2.3 NÁm. These variations are likely due to the slight range of oscillations in sagittal and frontal planes.
The mean values of component f of the torque were less than 0.001 NÁm for all subjects, see Figure 4. The torque values were not normally distributed as in component u, thus component f requires further investigation. Since fast Fourier transform (FFT) can be used to analyze the spectrum of frequencies in which the maximum frequency indicates a unique feature of the torque values, it was used to analyze the frequency spectrum of the torques in f direction, see Figure 5. Table 2 presents the maximum frequencies of the three groups. The maximum frequencies in high arched cases were lower than those for the normal and flat cases, and the flat case was higher than the normal case. Figure 6 shows the differences between the normal, flat, and high-arched foot for all subjects. The figure shows that the mean value of torque in direction u for   the flat foot cases is higher than for the normal and high-arched foot. In addition, the mean value of the torque is approximately equal to the value of the high arched foot cases. However, no significant difference (paired t-test p . 0.05), was found between the three groups. Figure 7 shows the maximum frequencies of the torque components in direction f for all subjects. This highlights that the range of frequencies in the flat foot cases was larger than the high arched cases. In addition, there was a significant change between the groups: p \ 0.001.

Discussion
In this work we have presented what we believe is the first ever model for human posture based on a spherical inverted pendulum. Our model reveals that a sway motion occurs in two directions, u and f, and that the   ankle joint acts as a spherical joint with two degrees of freedom. The model was inspired by the sway motion that occurs in A/P and M/L oscillations. This motion can be obtained by a joint that moves in two directions.
Since the subtalar and ankle joints are synovial joints, our model is in line with the joint kinesiology. 32 The dynamic equations of the model were derived, and two torques components were introduced in u and f directions. The point of effect of these components was located in the center of the ankle joint and we believe that these torque components are the cause for human postural balance. Both components were derived from the sway motion and anthropometric data of the subject. With reference to the direction components, we can infer that t f is a major contributor to keeping the center of mass within the base of support.
We found that component t u affects the rotational oscillation along the longitudinal axis of the body. This component works as a regulator of the external perturbations due to the strong impact of the inertial and gravitational forces.
In order to investigate the visibility of the proposed model by comparing it with the classical inverted single degree model, 19 subjects participated in the assessment procedure and were grouped into three categories according to the arch index. The foot pressure platform provides different levels of the arch index in order to accurately identify the type of foot deformity. Since a clear segregation between the three adopted groups is required, we used the following ranges of the arch index: 0.21-0.28 for a normal foot, 0.07-0.14 for cavus, and 35-0.42 for planus cases.
We found that the values of t u were higher than the torque component in the f direction. This variation was included in the parametric model proposed, where the inertial forces in equation (5) have a greater effect on the torque component than on the torque component in equation (6). Although there is a disparity between the torque components, especially in the f direction, and given the quantification analysis of t f , the torque in the f direction is nevertheless a potential feature for classifying foot deformities. By comparing our model with single degree IP presented in previous works, we estimated the torque using IP model using equation (7). We then compared the torque profile obtained with the torque profiles in our model in both directions. In most cases, a paired t-test showed a significant change, (p \ 0.05), between the torque profile generated based on the IP model and the proposed model. This confirms the uniqueness of our  approach for modeling human posture. Two cases showed insignificant changes -subjects 3 and 5 -which can be attributed to the age, location of CoM, and type of foot deformities. The trajectory of the CoP was used to estimate the torque's components in which noisy profiles were obtained. In general, the results reveal that the torque components in the u direction are distributed normally.
The mean values of the torque are in the range of 1 to 2.2 NÁm for normal cases, while the torque range lies between 1.6 to 2 NÁm in flat foot cases. The cavus cases reported torque values higher than 1 NÁm, which in several cases reached 3 NÁm. These values are consistent with previous knowledge of the high-arched foot and the topological nature of the plantar fascia compared with flat and normal cases. 29 Using FFT to analyze the frequency spectrum of the torque components in f direction clearly identified different types of foot deformities. Flat foot cases had a higher frequency range than high-arched cases. This would seem to indicate the high stability of a high arched foot compared with a flat foot, where the torque applied in the high-arched foot is less than that exerted in flat and normal cases. 33 Our model has some limitations. Estimating the CoM by assuming the body as a single segment is possibly not the optimal solution. 34 In fact, we plan to extend the current work to identify the CoM using an optical motion capture system where multi-body segments should improve the accuracy of the computation. Since the ankle joint is a synovial joint, a viscoelastic model should be added to the current model. This would help in understanding the change in intrinsic stiffness of the joint and its relation to the sway.
The potential relationship between the foot arch index and its influence on sagittal and frontal balance is not fully understood; however, we can hypothesize some possible causes based on existing literature. One possible reason for the alteration of sagittal and frontal balance with changes in the foot arch index could be the effect of foot arch adjustment on the moment arm of key muscles involved in balance control, such as the gastrocnemius and tibialis anterior. The gastrocnemius, as a major plantarflexor, contributes to sagittal plane control, while the tibialis anterior, as a dorsiflexor, plays a role in frontal plane control. Variations in the foot arch index could potentially impact the control of sagittal and frontal balance by affecting the moment arms of these muscles.
While direct evidence linking the foot arch index to sagittal and frontal balance is limited, some studies have explored the relationship between foot arch characteristics and balance control. For instance, 35 demonstrated that the anatomical structure of the foot is essential in assessing balance conditions, particularly in the elderly population. It was observed that elderly individuals exhibit lower hallux mobility and increased forefoot width compared to younger individuals, where the foot tends to be flatter and more pronated. This finding suggests a potential inverse relationship between the foot arch index and balance.
Further research is required to investigate the underlying mechanisms linking the foot arch index to sagittal and frontal balance control. Future studies could employ advanced biomechanical analysis techniques, such as motion capture and electromyography, to explore muscle activation patterns and joint kinematics associated with different foot arch profiles.
Further investigations are needed to reveal the true significance of this torque component. In fact, we plan to extend the group of participants to include those with balance disorders. This would then give better insights into how to exploit t f when estimating balance disorders.
We intend to validate the obtained torques by conducting force plate measurements. This validation process will furnish empirical evidence supporting the accuracy and precision of our model's predictions. Furthermore, we have set our sights on expanding our investigation by assembling an extensive dataset of force plate measurements encompassing a diverse range of postures. This expanded dataset will facilitate the refinement and optimization of our model, bolstering its effectiveness in applications related to human balance.
The proposed model has a limitation concerning the intrinsic forces responsible for maintaining frontal balance, which are primarily related to the hip joint. In order to address this limitation and provide a more comprehensive representation of the biomechanical factors influencing frontal balance, we recommend further improving the current model by incorporating the hip joint as a second spherical joint. This updated model would consist of two spherical joints in series and be linked with a lumped mass representing the center of body mass. By introducing an intrinsic torque exerted at the hip joint, the modified model would capture the contributions of both the ankle and hip joints to frontal balance. This expanded model would feature four degrees of freedom, accommodating two degrees of freedom for each joint. To effectively analyze and estimate the torque values of both the ankle and hip joints, additional measurements using motion capture systems and either force plate or foot pressure mapping systems would be required. This proposed modification and its subsequent analysis are necessary for future work, and the enhanced model would offer a more comprehensive understanding of the interaction between ankle and hip joint torques in frontal balance control

Conclusions
We have presented a new approach for modeling quiet standing posture. The approach is based on modeling the posture as a spherical inverted pendulum with two DOF. The model was used to estimate torque components in u and f directions where stabilometric data, A/P and M/L motions, were acquired using a foot pressure mapping system and then fed into the model to estimate the joint torques. These components were quantitatively analyzed to differentiate between three types of foot deformities, normal, high-arched, and flat foot. In addition, we found that the joint torque estimation is viable for designing wearable assistive devices such as exoskeletons.
Our findings revealed that the profile of the torque component in the f direction is a prominent feature that can be used to classify different types of foot deformities, while the u component of the torque can reduce the effect of the perturbations on the human body. Estimating these components may help in assessing patients and providing a proper rehabilitation plan. We have provided further evidence that a high arched foot is more stable than flat foot. A potential implementation of our model could be its use in a feedback control loop to control human balance in rehabilitation devices.
Using a quantitative approach to process torque components is essential since their profiles tend to be chaotic, especially in the f direction. Our testing protocol was used to evaluate the proposed model on 19 subjects with no history of neuromuscular disorder. However, given the small number of participants, caution is needed in drawing any firm conclusions.
Further studies, using the viscoelastic model of the subtalar and ankle joints, are needed. We are currently investigating how to integrate the viscoelastic model of the subtalar and ankle joints into our model, and we are setting up testing protocols with the assistance of surface-electromyography.

Declaration of conflicting interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author received no financial support for the research, authorship, and/or publication of this article.