https://orcid.org/0009-0004-7326-3968
Abstract
Exoskeleton robots mimic the structure of the human body and are physically connected to the wearer through attachment parts such as cuffs or straps. Nevertheless, misalignment between the human body and the robot can occur due to improper wearing of the exoskeleton, the elasticity of human skin, and the geometric complexity of human joint movements. Such misalignment increases unnecessary physical interaction forces, causing discomfort and pain to the wearer. Therefore, these interaction forces should be considered when designing exoskeleton robots to ensure wearability. In this study, we propose a method for estimating human-exoskeleton interaction forces through posture prediction. The human-robot connection is modeled as an elastic element, and posture is predicted using an energy optimization algorithm. The predicted posture is then used to calculate the interaction forces. Since this method considers only the physical characteristics of the exoskeleton robot and wearer, it enables objective evaluation of the robot without the need to manufacture actual prototypes. We performed quantitative experiments using prototypes of a sensor-equipped dummy and an exoskeleton to confirm the effectiveness of the modeling method. Consequently, the proposed method is expected to reduce the time and costs associated with developing exoskeleton robots and obviate the need for human subject testing.
We propose a method to predict interaction forces between a human and an exoskeleton robot.
Generalization of the human–robot model and a simulation involving a numerical optimization process were performed.
The simulation considers parameters like stiffness, length, weight, and wearing position of body segments.
The proposed method can be used to test a robot's wearability without human subject tests.
1. Introduction
Various forms of robots have been researched to assist humans in repetitive tasks, thereby reducing fatigue and ensuring safety. Among such robots, exoskeletons that are worn directly have garnered particular attention. These robots are typically designed in anthropomorphic form, resembling the anatomical structures of humans (Gopura et al., 2015; Gull et al., 2020). Thus, exoskeleton robots can effectively facilitate motion, alleviating physical burden and enhancing physical ability (De Looze et al., 2015; Yin et al., 2021; Schmalz et al., 2019). These robots have been researched to augment the muscle strength of workers in manufacturing tasks and to improve the combat capabilities of military personnel. More recently, they have been researched for use in rehabilitation therapy for the elderly, traffic accident victims, and stroke patients (Chen et al., 2016; Young & Ferris, 2017; Bao et al., 2019; Hussain et al., 2021; Calafiore et al., 2022).
An exoskeleton robot has cuffs that are physically attached to a part of the human body to maintain the alignment between the human body and the robot structure (Yandell et al., 2017). Nevertheless, misalignment can occur for the following reasons:
Initial wearing error. Owing to differences in physical characteristics among individuals, perfectly adjusting the length of an exoskeleton robot's links and the attachment position of the cuff is impossible, even with the help of a therapist. Moreover, in most cases, the alignment between the human joint and the corresponding robot joint is distorted because the user wears the exoskeleton based on their subjective perception of comfort or past experiences with similar devices.
Elastic properties of human soft tissues. Human skin exhibits elastic properties. Therefore, even if the robot and the human structures are perfectly aligned when the exoskeleton is initially worn, misalignment may occur due to skin deformation at the contact positions (Pons, 2008).
Complexity of human joints. Unlike the joints of exoskeleton robots, human joints have a complex mechanism that allows simultaneous sliding and rotational motion. This mechanism results in continuous variation in the position of the rotation centre (Hirschmann & Müller, 2015). In several studies, self-alignment mechanisms have been applied to robotic joints to address issues arising from the complexity of human joint movement (Celebi et al., 2013; Sarkisian et al., 2021).
These factors unavoidably result in misalignment between the human body and robot, producing additional interaction forces at the contact positions (Zanotto et al., 2015; Mallat et al., 2019; Bessler-Etten et al., 2022). Unfortunately, these forces can cause discomfort and pain for the wearer. If the wearer uses the robot for prolonged periods under a continuous load, skin damage may occur in the form of pressure ulcers. Therefore, the physical human–robot interaction force is considered a critical parameter in terms of the wearability and safety of exoskeleton robots (Massardi et al., 2022; Wang et al., 2023).
Since the physical interaction forces between the human body and robot are closely related to wearability, several studies have been conducted to quantify these forces (Bartenbach et al., 2015; Akiyama et al., 2016; D'Elia et al., 2017; Georgarakis et al., 2018; Suh & Choi, 2019; Yousaf et al., 2021). Most of these studies involved direct-wearing experiments using exoskeleton robots equipped with force-sensitive resistors or load cells to measure the interaction forces. However, such human subject tests can be inconsistent due to differences in the physical characteristics among the wearers, resulting in a lack of objectivity in the experimental results. Additionally, such experiments raise ethical concerns regarding potential risks to the wearer. Therefore, additional time is required for planning clinical tests to ensure the wearer's safety and to allow the wearer to undergo approval tests. To avoid clinical tests, some studies have employed dummies with human-like joints and skin to measure the interaction forces (Akiyama et al., 2012; Aso et al., 2013). However, this approach incurs additional costs and requires additional time for designing and fabricating the dummy. Moreover, with both of the aforementioned methods, it is difficult to consider the various physical characteristics of the wearer, such as body length and weight.
The limitations of existing studies thus necessitate a wearability assessment method that can be used for human subjects with diverse physical characteristics, without the need for clinical tests. Therefore, we propose a novel method for modelling human–exoskeleton interaction to predict the wearer's posture and the resultant physical interaction forces. We validated the proposed model using a dummy that mimics a human arm and an exoskeleton robot equipped with multi-axis force sensors. The remainder of this paper is structured as follows. Section 2 presents a detailed kinematic model of the human–exoskeleton interaction. Section 3 explains the optimization algorithm and describes the simulation process for estimating the posture and interaction forces. Subsequently, Section 4 presents the validation experiment with the dummy and exoskeleton prototypes, followed by a comparison between the experimental measurements and the predicted values from the simulation. Finally, Section 5 discusses the findings, and Section 6 concludes the paper.
2. Human–Exoskeleton Interaction Model
In this section, we describe the proposed interaction model between an exoskeleton robot and a human body in planar space. Additionally, we explain the contact points and the kinematic models for the human–exoskeleton interaction and outline the relationships between the variables of the model and the potential energy. By restricting the model to planar space, all joint motions were assumed to occur within a 2D plane.
2.1 Contact models
Due to its elasticity, human skin deforms under load and regains its original state when the load is removed (Pons, 2008). This deformation occurs in three directions in a planar space. Figure 1 illustrates an example of the elastic deformation of skin on a human forearm, demonstrating how the position of the exoskeleton robot's link can be shifted in three directions. These deformations can be modelled with a spring connecting the initial position of the link when the exoskeleton is worn and its final position after the deformation, as depicted in Figure 2. Here, the initial position (marked with a red dot) and the final position (marked with a blue dot) represent human–robot contact points before and after deformation, respectively. Since an actual exoskeleton robot link can have more than one cuff, the contact point is set as the centre position of the cuff. Additionally, although the spring is represented as a single element, it encompasses deformations in all three directions including torsion. For simplicity, the interaction between the human skin and the attaching cuff is assumed to result solely from skin deformation without any slipping. Additionally, the human joints are modelled as simple pin joints to simplify the kinematic analysis.
Planar motion of an attached exoskeleton link due to skin deformation: (a) shear movement, (b) normal movement, and (c) torsional movement.
Contact point model.
2.2 Kinematic model
As an example, Figure 3 presents the kinematic model of the interaction between an upper-limb exoskeleton robot and a human body. Here, |${H_i},{\rm{\ }}{R_i},$| and G represent the human joint, robot joint, and base, respectively, where the y-direction of coordinate G is aligned opposite to gravity. Additionally, |$H{C_i}$| and |$R{C_i}$| correspond to the contact points described in Figure 2, where |$H{C_i}$| indicates the initial position of the contact point (marked with a red dot) and |$R{C_i}$| represents the final position of the contact point after deformation (marked with a blue dot). Moreover, the curly brackets enclosing these symbols denote the coordinate system. The human body contains one more link than the robot, including the base link. Additionally, a one-degree-of-freedom (1-DOF) joint is placed between the human body and the robot. Considering these details, a more generalized kinematic model of the human body and robot can be established, as demonstrated in Figure 4. In the figure, |${L_{{\rm{h}}i}}$| and |${L_{{\rm{r}}i}}$| represent the lengths of the links, while |${L_{{\rm{hc}}i}}$| and |${L_{{\rm{rc}}i}}$| denote the distances between the joints and contact points. Here, the subscript i indicates the link number and ranges from 1 to n. In addition, |${\theta _{{\rm{h}}0}}$| represents the angle between |$\{ G \}$| and |$\{ {{H_1}} \}$|. Moreover, |${\theta _{{\rm{h}}i}}$| and |${\theta _{{\rm{r}}i}}$| represent the angles between |$\{ {{H_i}} \}$| and |$\{ {{H_{i + 1}}} \}$| and between |$\{ {{R_i}} \}$| and |$\{ {{R_{i + 1}}} \}$|, respectively. Based on the kinematic model, the kinematic information of the model can be summarized as shown in Table 1. As the contact point has 3 DOFs, it does not affect the kinematic model's DOF. Consequently, the model has |$( {2n + 1} )$| links and |$( {2n - 1} )$| joints, resulting in (|$2n + 2$|) DOFs, according to Kutzbach's criterion.
Kinematic model of a human body and a 2-DOF upper-limb exoskeleton robot.
Generalized human–exoskeleton kinematic model.
Summary of the number of links, joints, and contact points in the generalized kinematic model.
| Human | Robot | |
|---|---|---|
| Number of links | |$n + 1$| | n |
| Number of joints | n | |$n - 1$| |
| Number of contact points | n | |
| Human | Robot | |
|---|---|---|
| Number of links | |$n + 1$| | n |
| Number of joints | n | |$n - 1$| |
| Number of contact points | n | |
Summary of the number of links, joints, and contact points in the generalized kinematic model.
| Human | Robot | |
|---|---|---|
| Number of links | |$n + 1$| | n |
| Number of joints | n | |$n - 1$| |
| Number of contact points | n | |
| Human | Robot | |
|---|---|---|
| Number of links | |$n + 1$| | n |
| Number of joints | n | |$n - 1$| |
| Number of contact points | n | |
Referring to Figure 1, the shear, normal, and torsional deformations at the contact point can be represented as X- and Y-axes, and counterclockwise directional deformations, respectively. Therefore, the magnitude and direction of the spring deformation at the contact point can be determined through the transformation matrix between the i-th contact point on the robotic arm and the corresponding contact point on the human arm as below:
Here, |${l_{{\rm{s}}i}},\ {l_{{\rm{n}}i}},$| and |${\psi _i}$| indicate the shear-directional, normal-directional, and the torsional deformation, respectively. Additionally, according to the vector loop equation, the relationship among the coordinate systems of all the contact points can be expressed as follows:
Based on Equation (2), all |${}_{R{C_i}}^{H{C_i}}T$| terms can be represented by |${}_{R{C_1}}^{H{C_1}}T$|. Therefore, the positions of all the coordinates based on the global coordinate system can be expressed with the following |$( {2n + 2} )$| variables: |${\theta _{{\rm{h}}0}}, \cdots ,{\theta _{{\rm{h}}( {n - 1} )}},{\theta _{{\rm{r}}1}}, \cdots ,{\theta _{{\rm{r}}( {n - 1} )}},{\rm{\ }}{l_{{\rm{s}}1}},{l_{{\rm{n}}1}},$| and |${\psi _1}$|.
The robot joints must align perfectly with the human joints to ensure ideal wearing of an exoskeleton robot. However, since perfect alignment is not realistically possible, the wearing error that occurs during initial wearing must be considered. Initial wearing errors arise from improper adjustment of the cuff. Therefore, additional parameters are needed to represent the wearing errors caused by improper cuff adjustment. Wearing errors can be represented by the transformation matrix that determines the contact position. The |${}_{R{C_i}}^{{R_i}}{T_{{\rm{perfectly}} - {\rm{aligned}}}}$| term in Equation (3) represents the transformation matrix in the perfectly aligned state. By adding specific parameters to account for misalignment, |${}_{R{C_i}}^{{R_i}}{T_{{\rm{perfectly}} - {\rm{aligned}}}}$| can be modified to |${}_{R{C_i}}^{{R_i}}T$|, as shown in Equation (4):
The additional parameters, |${L_{{\rm{so}}i}},{L_{{\rm{no}}i}},$| and |${\theta _{{\rm{to}}i}}$|, denote the degrees of the initial wearing offset in the shear, normal, and torsional directions, respectively. The concept of wearing offset is illustrated in Figure 5. In this figure, the robot's coordinate system is shown in black, while the human's coordinate system is represented in yellow. Figure 5a depicts the perfectly aligned state, free from wearing errors, where the joints of the human body and robot align perfectly. In contrast, Figures 5b–d demonstrate a misaligned state, where the robot's forearm cuff deviates from the ideal position. In these cases, the joints of the two structures are already misaligned from the beginning. Such wearing offsets result in increased interaction forces on the human body.
Concept of the initial wearing offset: (a) perfectly aligned, (b) misaligned by |${{{L}}_{{{\it so}}{{i}}}}$|, (c) misaligned by |${{{L}}_{{{\it no}}{{i}}}}$|, and (d) misaligned by |${{{L}}_{{{\it toi}}}}$|.
2.3 Static analysis
Assuming that the human joints do not move rapidly, the interaction force between the human body and the robot can be calculated through static analysis of the previously described kinematic model. Static analysis method is commonly used in conditions with slow movements, as it simplifies the model and calculations while maintaining accuracy, making it a widely adopted approach in various studies (Corke, 2023). The potential energy of the model equals the sum of the gravitational potential energy that results from the weight of the links and the elastic potential energy stored in the springs. The posture of the kinematic model is determined where potential energy is minimized. First, to calculate the gravitational potential energy, the heights of the centres of mass of the ith human limb and the ith exoskeleton limb, denoted by |${h_{{\rm{h}}i}}$| and |${h_{{\rm{r}}i}}$|, is computed as follows:
In these equations, |${y_{{\rm{h}}i}}$| and |${y_{{\rm{r}}i}}$| represent the y-coordinate of each link in the global coordinate system. In addition, |${G_{{\rm{h}}i}}$| and |${G_{{\rm{r}}i}}$| signify the ratio of the centre of mass position, defined as the distance from the start of the link to its centre of mass divided by the total length of the link. Based on Equations (5) and (6), |${U_{{\rm{gravity}}}}$| is calculated as follows:
Here, |${m_{{\rm{h}}i}}$| and |${m_{{\rm{r}}i}}$| represent the mass of each link, and g signifies the gravitational acceleration.
The elastic potential energy |${U_{{\rm{spring}}}}$| can be computed using the transformation matrix |${}_{R{C_i}}^{H{C_i}}T$| as follows.
From the above equation, |${K_{{\rm{s}}i}},{K_{{\rm{n}}i}},$| and |${K_{{\rm{t}}i}}$| denote the shear-directional, normal-directional, and torsional stiffness, respectively. Finally, the total potential energy U is calculated as follows, using Equations (7) and (8):
3. Posture-Prediction Algorithms and Simulation
In this section, the optimization algorithm for predicting the posture of the interaction model is explained. The relationship between the variables and the potential energy of the model is used for optimization. Additionally, the simulation that calculates the interaction forces between the human body and the robot based on the predicted posture is described.
3.1 Posture-prediction algorithm
In a robot system where a human leads the movement and an exoskeleton provides the assistive force, the posture of the kinematic model based on the human joint angle can be represented with |$( {n + 2} )$| variables: |${\theta _{{\rm{r}}1}}, \cdots ,{\theta _{{\rm{r}}( {n - 1} )}},{l_{{\rm{s}}1}},{l_{{\rm{n}}1}}$|, and |${\psi _1}$|. Using these variables and the processes described in Equations (5)–(9), the potential energy of the robot system can be calculated. According to physical laws, the posture of the model is determined at the position where the potential energy is minimized. To identify this minimized position, the partial derivatives of the potential energy with respect to each variable are calculated. These derivative values represent the rate of energy change for those variables. Therefore, the values of the variables at which the rate of energy change becomes zero determine the minimum potential energy and the system's posture. By finding these variables, the proposed model's posture can be accurately predicted.
Figure 6 presents an optimization algorithm that predicts the postures of an exoskeleton robot and a human body. Based on the aforementioned principle, potential energy U was set as the objective function, and |${\theta _{{\rm{r}}1}}, \cdots ,{\theta _{{\rm{r}}( {n - 1} )}},{\rm{\ }}{l_{{\rm{s}}1}},{l_{{\rm{n}}1}}$|, and |${\psi _1}$| were defined as design variables. Additionally, the Newton–Raphson method, which is well-suited for multivariable function optimization, was applied for the optimization process. In the DEFINE FUNCTION stage, the first partial derivative of U with respect to |${\theta _{{\rm{r}}1}}, \cdots ,{\theta _{{\rm{r}}( {n - 1} )}},{\rm{\ }}{l_{{\rm{s}}1}},{l_{{\rm{n}}1}}$|, and |${\psi _1}$| is defined as |$F( X )$|, and the second partial derivative of U is defined as |$J( X )$|. During the INPUT stage, the initial values of the human motion, wearing offset, model variables, and tolerable error are entered. At this point, considering the fact that the spring deformation is small, the initial values of variables |${l_{{\rm{s}}1}},{l_{{\rm{n}}1}}$|, and |${\psi _1}$| are set to zero, while those of |${\theta _{{\rm{r}}1}}, \cdots ,$| and |${\theta _{{\rm{r}}( {n - 1} )}}$| are set to be equal to the corresponding human joint angles. In the DO stage, the optimal variables are found numerically using the Newton–Raphson method, and their values are printed.
Pseudocode of optimization algorithm for minimizing potential energy.
3.2 Simulation
3.2.1 Interaction force calculation
Using the human joint angles as input and the robot joint angles as output, all coordinate positions in the global coordinate system can be determined, which enables the prediction of both the human and robot postures. Moreover, by substituting |${l_{{\rm{s}}1}},{l_{{\rm{n}}1}}$|, and |${\psi _1}$| into Equation (2), the deformations of all the springs can be calculated. If the spring stiffness is known, the force generated at each contact point can be calculated as follows, according to Hooke's law:
Here, |${F_{{\rm{s}}i}},{F_{{\rm{n}}i}}$|, and |${M_i}$| represent the forces exerted on the shear-directional, normal-directional, and torsion springs of the ith link, respectively. According to the contact point model, the interaction forces between the human body and the robot can be estimated based on these forces.
3.2.2 Simulation condition
A simulation was conducted to analyse the interaction between a 2-DOF upper-limb exoskeleton robot and a human body during overhead drilling tasks. This simulation focused on the most common type of wearing offset. The constants used in the simulation are presented in Table 2. The constants related to the robot links were arbitrarily determined, and the initial contact position was assumed to be at the centre of each human link. Additionally, the length, mass, and mass centre ratio of the human links were derived from information on a male participant (height: 1741 mm and weight: 73 kg) in an anatomical study (De Leva, 1996).
Constants related to each link used in the simulation.
| Human link1 | Human link2 | Human link3 | Robot link1 | Robot link2 | Robot link3 | |
|---|---|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 531.9 | 281.7 | 268.9 | 300 | 281.7 | 268.9 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 381.9 | 140.9 | 134.5 | 150 | 140.9 | 134.5 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [kg] | 31.73 | 1.978 | 1.183 | 6 | 2.817 | 2.689 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 55.14 | 57.72 | 45.74 | 0.5 | 0.5 | 0.5 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1700 | 1700 | 1700 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5200 | 5200 | 5200 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 18.72 | 18.72 | 18.72 | |$\times $| | |$\times $| | |$\times $| |
| Human link1 | Human link2 | Human link3 | Robot link1 | Robot link2 | Robot link3 | |
|---|---|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 531.9 | 281.7 | 268.9 | 300 | 281.7 | 268.9 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 381.9 | 140.9 | 134.5 | 150 | 140.9 | 134.5 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [kg] | 31.73 | 1.978 | 1.183 | 6 | 2.817 | 2.689 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 55.14 | 57.72 | 45.74 | 0.5 | 0.5 | 0.5 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1700 | 1700 | 1700 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5200 | 5200 | 5200 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 18.72 | 18.72 | 18.72 | |$\times $| | |$\times $| | |$\times $| |
Constants related to each link used in the simulation.
| Human link1 | Human link2 | Human link3 | Robot link1 | Robot link2 | Robot link3 | |
|---|---|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 531.9 | 281.7 | 268.9 | 300 | 281.7 | 268.9 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 381.9 | 140.9 | 134.5 | 150 | 140.9 | 134.5 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [kg] | 31.73 | 1.978 | 1.183 | 6 | 2.817 | 2.689 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 55.14 | 57.72 | 45.74 | 0.5 | 0.5 | 0.5 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1700 | 1700 | 1700 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5200 | 5200 | 5200 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 18.72 | 18.72 | 18.72 | |$\times $| | |$\times $| | |$\times $| |
| Human link1 | Human link2 | Human link3 | Robot link1 | Robot link2 | Robot link3 | |
|---|---|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 531.9 | 281.7 | 268.9 | 300 | 281.7 | 268.9 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 381.9 | 140.9 | 134.5 | 150 | 140.9 | 134.5 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [kg] | 31.73 | 1.978 | 1.183 | 6 | 2.817 | 2.689 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 55.14 | 57.72 | 45.74 | 0.5 | 0.5 | 0.5 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1700 | 1700 | 1700 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5200 | 5200 | 5200 | |$\times $| | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 18.72 | 18.72 | 18.72 | |$\times $| | |$\times $| | |$\times $| |
Shear and normal stiffness values used in the simulation were determined based on references from existing literature. Several studies have measured and analysed the stiffness of human skin through experiments. According to Aso's study, the shear-directional stiffness of the thigh ranges from 700–2800 N/m, while Clark's study indicates that the normal-directional stiffness of the forearm skin ranges from 4200–7500 N/m (Clark et al., 1996; Aso et al., 2013). Furthermore, in Akiyama's study, experimental and regression analyses estimated the stiffness of the thigh to be 700–2730 N/m for shear-directional stiffness and 2800–7600 N/m for normal-directional stiffness (Akiyama et al., 2015). Meanwhile, torsional stiffness varies depending on the distance between cuffs, so a method based on normal stiffness was used for its estimation. Figure 7 illustrates a scenario where torsion occurs in a robot link composed of two cuffs due to the force F applied to one cuff. In this figure, |${L_{\rm{c}}}$| represents the distance between the cuffs, and |$\theta $| denotes the torsional angle. The torsional stiffness |${K_{\rm{t}}}$| can be expressed as follows:
Case of torsional deformation occurring in human forearm: (a) before deformation and (b) after deformation.
Since torsion between the human body and the robot link is typically minimal, the torsional angle can be approximated using the normal-directional stiffness |${K_n}$| as shown below:
Therefore, using Equations (13) and (15), torsional stiffness is calculated as follows:
In this simulation, |${L_{\rm{c}}}$| was set to 120 mm, considering the typical structure of robot cuffs. Therefore, all of the stiffness values were determined based on the existing literature and the torsional stiffness estimation method. The constants used in the simulation are presented in Table 2.
3.2.3 Simulation result
Figure 8 depicts the simulation results in a perfectly aligned state, without any wearing offset, while Figures 9 and 10 show the results of simulations considering shear- and normal-directional wearing offsets. In these figures, Cycle |$0\% $| represents the starting phase of the overhead drilling task, where the hand is at its lowest position, while Cycle |$100\% $| indicates the final phase, where the hand reaches its highest position. Comparing the predicted postures of the human body and robot in Figures 8a, 9a, and 10a, the mismatch becomes more pronounced when wearing offsets occur. Additionally, Figures 8b, 9b, and 10b show the interaction forces acting on the human body, revealing that wearing offsets increase the interaction forces. Specifically, in cases where wearing offsets occur, the moment increases significantly as the posture approaches Cycle |$100\% $|. Moreover, Figures 8c, 9c, and 10c illustrate the angular differences between the human and robot joints. In the perfectly aligned state, the angular differences were less than 4°. However, in the shear-directional wearing offset state, the angular difference exceeded 12° at the shoulder, while in the normal-directional wearing offset state, it exceeded 12° at the elbow. In this manner, simulations using the proposed optimization algorithm enable the quantitative prediction of interaction forces generated in each direction and the extent of angular differences between the human and robot joints.
Simulation results in perfectly aligned state: (a) posture, (b) interaction forces, and (c) angular difference.
Simulation results in misaligned state (|${{{L}}_{{{ so}}1}} = + 20\ {{\rm mm}},{\boldsymbol{\ }}{{{L}}_{{{so}}2}} = - 20\ {{\rm mm}},{\boldsymbol{\ }}{{\rm and}}\ {{{L}}_{{{ so}}3}} = - 20\ {{\rm mm}}$|): (a) posture, (b) interaction forces, and (c) angular difference.
Simulation results in misaligned state (|${{{L}}_{{{ no}}1}} = - 20\ {{\rm mm}},{\boldsymbol{\ }}{{{L}}_{{{ no}}2}} = + 20\ {{\rm mm}},{\boldsymbol{\ }}{{\rm and}}\ {{{L}}_{{{ no}}3}} = + 20\ {{\rm mm}}$|): (a) posture, (b) interaction forces, and (c) angular difference.
4. Experimental Validation
In this section, we describe the dummy and exoskeleton prototypes designed to validate the proposed interaction model. Additionally, we explain the experimental setup and method that can consider the wearing offset. Finally, we compare the results of the simulation and the experiment.
4.1 Setup
Experimental validation through clinical tests is challenging in terms of accurately distinguishing between the forces generated by human movement and those produced by human–exoskeleton interactions. Since the experiment in this study was intended to measure the interaction forces for validating the proposed interaction model, a dummy prototype for a 1-DOF exoskeleton robot was designed to replace the human arm, as shown in Figure 11a. The dummy consisted of an aluminum frame, encoders, silicone blocks, and silicone holders to fix the blocks. A silicone material with a durometer of 10 was used for the silicone blocks, and an AN25 encoder (Dream Solution) with a resolution of 1024 P/R was attached to the dummy's elbow joint.
Design of dummy and exoskeleton prototypes: (a) dummy and (b) exoskeleton.
The upper-limb exoskeleton prototype designed for the experiment is demonstrated in Figure 11b. This robot comprised an aluminum frame, motors, multi-axis force/torque (F/T) sensors, and 3D-printed cuffs. Each silicone cap on the cuff had grooves to allow the silicone blocks to be fixed. A Dynamixel XH540-W270-R motor equipped with an encoder was used for actuating the robot joint and measuring its angle. In addition, an ATI Mini58 multi-axis F/T sensor was placed between robot link1 and the cuff to measure the interaction forces between the exoskeleton and the dummy prototypes. Since the experimental device is interconnected, the interaction forces occurring at link2 also affect link1. Therefore, the F/T sensor placed in link1 is sufficient to validate the proposed method.
The kinematic model used in the simulation was modified to match the experimental device, as depicted in Figure 12. The base link was moved to the robot, and both the dummy and robot were configured to have two links each. The constants related to the links are listed in Table 3. The link lengths, weights, and stiffness values were measured experimentally, while the centre of gravity ratios were calculated using a 3D modelling tool.
Kinematic model of experimental setup.
Constants related to each link used in the experiment.
| Dummy link1 | Dummy link2 | Robot link1 | Robot link2 | |
|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 310 | 310 | 310 | 310 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 90 | 220 | 90 | 220 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [g] | 273.5 | 187.5 | 758.5 | 189.5 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 68.33 | 54.44 | 50.53 | 56.60 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1702 | 1702 | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5739 | 5739 | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 21.78 | 21.78 | |$\times $| | |$\times $| |
| Dummy link1 | Dummy link2 | Robot link1 | Robot link2 | |
|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 310 | 310 | 310 | 310 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 90 | 220 | 90 | 220 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [g] | 273.5 | 187.5 | 758.5 | 189.5 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 68.33 | 54.44 | 50.53 | 56.60 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1702 | 1702 | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5739 | 5739 | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 21.78 | 21.78 | |$\times $| | |$\times $| |
Constants related to each link used in the experiment.
| Dummy link1 | Dummy link2 | Robot link1 | Robot link2 | |
|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 310 | 310 | 310 | 310 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 90 | 220 | 90 | 220 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [g] | 273.5 | 187.5 | 758.5 | 189.5 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 68.33 | 54.44 | 50.53 | 56.60 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1702 | 1702 | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5739 | 5739 | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 21.78 | 21.78 | |$\times $| | |$\times $| |
| Dummy link1 | Dummy link2 | Robot link1 | Robot link2 | |
|---|---|---|---|---|
| |${L_{{\rm{h}}i}},{L_{{\rm{r}}i}}$| [mm] | 310 | 310 | 310 | 310 |
| |${L_{{\rm{hc}}i}},{L_{{\rm{rc}}i}}$| [mm] | 90 | 220 | 90 | 220 |
| |${m_{{\rm{h}}i}},{m_{{\rm{r}}i}}$| [g] | 273.5 | 187.5 | 758.5 | 189.5 |
| |${G_{{\rm{h}}i}},{G_{{\rm{r}}i}}$| [%] | 68.33 | 54.44 | 50.53 | 56.60 |
| |${K_{{\rm{s}}i}}$| [N/m] | 1702 | 1702 | |$\times $| | |$\times $| |
| |${K_{{\rm{n}}i}}$| [N/m] | 5739 | 5739 | |$\times $| | |$\times $| |
| |${K_{{\rm{t}}i}}$| [Nm/rad] | 21.78 | 21.78 | |$\times $| | |$\times $| |
4.2 Experimental method
Figure 13 shows the assembly comprising the dummy prototype and robot prototype in a perfectly aligned state, illustrating two different postures used in the experiments. The arrows in the figure indicate the measurement directions of the F/T sensor. Figure 13a depicts the posture with |${\theta _{{\rm{r}}0}} = 180^\circ $|, where robot link1 is fixed in a horizontal position relative to the ground. Figure 13b illustrates the posture with |${\theta _{{\rm{r}}0}} = 270^\circ $|, where robot link1 is perpendicular to the ground, corresponding to a neutral posture considered comfortable for humans (Apostolico et al., 2013). These two postures were selected as experimental conditions because they are subject to different gravitational effects, making them suitable for validating the interaction model that utilizes the energy optimization method. In validation experiments, the motor located at the robot's elbow was repeatedly actuated five times in the range of |${\theta _{{\rm{r}}1}} = 225^\circ -360^\circ $| at a low speed of |$18{\rm{\ }}^\circ /{\rm{s}}$|, considering the range of motion of the human elbow and the possibility of collisions with the experimental setup components (Oosterwijk et al., 2018). Additionally, the forces along the X- and Y-axes and the moment along the Z-axis were measured by the F/T sensor, and the angles of the dummy's and robot's elbow joints were measured using their respective encoders.
Two different postures in perfectly aligned state: (a) |${{{\theta }}_{{{ r}}0}} = 180^\circ $| and (b) |${{{\theta }}_{{{ r}}0}} = 270^\circ $|.
Subsequently, the experiment was conducted considering three directions of wearing offset, and the wearing offset was set to |$+ 10$| mm, |$- 10$| mm, |$+ 5^\circ $|, and |$- 5^\circ $|. These values were determined based on typical wearing offsets and the physical limitations of the experimental equipment (Zanotto et al., 2015). Figure 14a shows the case where a −10 mm shear-directional wearing offset was applied, and the distance between |$\{ {{R_2}} \}$| and |$\{ {R{C_2}} \}$| was set to 210 mm. Figure 14b depicts the case where a normal-directional wearing offset was applied, with the robot's cuff shifted −10 mm along the Y-axis from its original position. Figure 14c illustrates the case where a torsional wearing offset was applied, with the robot's cuff rotated |$- 5^\circ \ $|from its original position. A comparison between Figures 13a and 14 reveals that the silicone blocks in the configuration with the wearing offsets were deformed more than those in the configuration without a wearing offset.
Experimental setup with wearing offsets applied: (a) |${{{L}}_{{{ so}}2}} = - 10\ {{\rm mm}}$|, (b) |${{{L}}_{{{ no}}2}} = - 10\ {{\rm mm}}$|, and (c) |${{{L}}_{{{ to}}2}} = - 5^\circ $|.
4.3 Experimental validation
Figures 15 and 16 compare the results of the simulation and the experiment for the two postures. The experimentally measured forces and angular differences were similar to the corresponding simulation results. In addition, the trends within the experimental results were observed to be similar, despite the different postures. This outcome implies that the effect of the wearing errors on the interaction forces was dominant over that of the posture. When a wearing offset is present, the interaction forces and angular differences exhibit varying patterns depending on the direction and sign of the offset. In contrast, in perfectly aligned cases, the interaction forces and angular differences were close to zero, as shown in Figures 15a and 16a. Moreover, due to the weight of the dummy link1, the |${F_{{\rm{s}}1}}$| curves in Figure 16 are approximately 3 N lower than those in Figure 15 in all cases. Overall, these findings indicate that human–exoskeleton interactions are accurately predicted by the proposed model. To validate the interaction model quantitatively, the root mean square errors (RMSEs) between the simulation and experimental results were calculated, and the values are listed in Tables 4 and 5. Additionally, in the captions of Figures 15 and 16, and Tables 4 and 5, all unspecified wearing offset values are set to 0.
Comparison of simulation and experimental results for posture with |${{{\theta }}_{{{r}}0}} = 180^\circ $|: (a) perfectly aligned, (b) |${{{L}}_{{{ so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{ so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{ no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{ no}}2}} = - 10\ {{\rm mm}}$|, (f) |${{{\theta }}_{{{ to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{ to}}2}} = - 5^\circ $|.
Comparison of simulation and experimental results for posture with |${{{\theta }}_{{{\it r}}0}} = 270^\circ $|: (a) perfectly aligned, (b) |${{{L}}_{{{\it so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{ so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{\it no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{\it no}}2}} = - 10\ {{ \rm mm}}$|, (f) |${{{\theta }}_{{{\it to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{\it to}}2}} = - 5^\circ $|.
RMSEs between simulation and experimental results (|${{{\theta }}_{{{\it r}}0}} = 180^\circ $|): (a) perfectly aligned, (b) |${{{L}}_{{{\it so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{\it so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{\it no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{\it no}}2}} = - 10\ {{\rm mm}}$|, (f) |${{{\theta }}_{{{\it to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{\it to}}2}} = - 5^\circ .$|
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.1073 | 0.2809 | 0.0418 | 0.1458 | 0.1750 | 0.2118 | 0.1887 |
| |${F_{{\rm{s}}1}}$| [N] | 1.3587 | 2.0372 | 1.0644 | 1.6701 | 1.6865 | 1.9038 | 1.7764 |
| |${F_{{\rm{n}}1}}$| [N] | 0.7263 | 1.1954 | 0.4629 | 0.7251 | 0.8638 | 1.0168 | 1.1102 |
| Ang. diff. [|$^\circ $|] | 0.6799 | 0.9710 | 0.6019 | 0.7962 | 0.6304 | 0.4329 | 0.9993 |
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.1073 | 0.2809 | 0.0418 | 0.1458 | 0.1750 | 0.2118 | 0.1887 |
| |${F_{{\rm{s}}1}}$| [N] | 1.3587 | 2.0372 | 1.0644 | 1.6701 | 1.6865 | 1.9038 | 1.7764 |
| |${F_{{\rm{n}}1}}$| [N] | 0.7263 | 1.1954 | 0.4629 | 0.7251 | 0.8638 | 1.0168 | 1.1102 |
| Ang. diff. [|$^\circ $|] | 0.6799 | 0.9710 | 0.6019 | 0.7962 | 0.6304 | 0.4329 | 0.9993 |
RMSEs between simulation and experimental results (|${{{\theta }}_{{{\it r}}0}} = 180^\circ $|): (a) perfectly aligned, (b) |${{{L}}_{{{\it so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{\it so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{\it no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{\it no}}2}} = - 10\ {{\rm mm}}$|, (f) |${{{\theta }}_{{{\it to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{\it to}}2}} = - 5^\circ .$|
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.1073 | 0.2809 | 0.0418 | 0.1458 | 0.1750 | 0.2118 | 0.1887 |
| |${F_{{\rm{s}}1}}$| [N] | 1.3587 | 2.0372 | 1.0644 | 1.6701 | 1.6865 | 1.9038 | 1.7764 |
| |${F_{{\rm{n}}1}}$| [N] | 0.7263 | 1.1954 | 0.4629 | 0.7251 | 0.8638 | 1.0168 | 1.1102 |
| Ang. diff. [|$^\circ $|] | 0.6799 | 0.9710 | 0.6019 | 0.7962 | 0.6304 | 0.4329 | 0.9993 |
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.1073 | 0.2809 | 0.0418 | 0.1458 | 0.1750 | 0.2118 | 0.1887 |
| |${F_{{\rm{s}}1}}$| [N] | 1.3587 | 2.0372 | 1.0644 | 1.6701 | 1.6865 | 1.9038 | 1.7764 |
| |${F_{{\rm{n}}1}}$| [N] | 0.7263 | 1.1954 | 0.4629 | 0.7251 | 0.8638 | 1.0168 | 1.1102 |
| Ang. diff. [|$^\circ $|] | 0.6799 | 0.9710 | 0.6019 | 0.7962 | 0.6304 | 0.4329 | 0.9993 |
RMSEs between simulation and experimental results (|${{{\theta }}_{{{\it r}}0}} = 270^\circ $|): (a) perfectly aligned, (b) |${{{L}}_{{{\it so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{\it so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{\it no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{\it no}}2}} = - 10\ {{\rm mm}}$|, (f) |${{{\theta }}_{{{\it to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{\it to}}2}} = - 5^\circ .$|
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.0955 | 0.2152 | 0.1033 | 0.1916 | 0.1184 | 0.1607 | 0.1722 |
| |${F_{{\rm{s}}1}}$| [N] | 0.5830 | 0.9895 | 0.6366 | 1.0342 | 0.6556 | 0.7538 | 1.0384 |
| |${F_{{\rm{n}}1}}$| [N] | 0.6461 | 1.0622 | 0.9198 | 1.6462 | 0.8671 | 0.6412 | 0.8544 |
| Ang. diff. [|$^\circ $|] | 0.2953 | 0.4206 | 0.4402 | 0.3062 | 0.2703 | 0.5641 | 0.5586 |
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.0955 | 0.2152 | 0.1033 | 0.1916 | 0.1184 | 0.1607 | 0.1722 |
| |${F_{{\rm{s}}1}}$| [N] | 0.5830 | 0.9895 | 0.6366 | 1.0342 | 0.6556 | 0.7538 | 1.0384 |
| |${F_{{\rm{n}}1}}$| [N] | 0.6461 | 1.0622 | 0.9198 | 1.6462 | 0.8671 | 0.6412 | 0.8544 |
| Ang. diff. [|$^\circ $|] | 0.2953 | 0.4206 | 0.4402 | 0.3062 | 0.2703 | 0.5641 | 0.5586 |
RMSEs between simulation and experimental results (|${{{\theta }}_{{{\it r}}0}} = 270^\circ $|): (a) perfectly aligned, (b) |${{{L}}_{{{\it so}}2}} = + 10\ {{\rm mm}}$|, (c) |${{{L}}_{{{\it so}}2}} = - 10\ {{\rm mm}}$|, (d) |${{{L}}_{{{\it no}}2}} = + 10\ {{\rm mm}}$|, (e) |${{{L}}_{{{\it no}}2}} = - 10\ {{\rm mm}}$|, (f) |${{{\theta }}_{{{\it to}}2}} = + 5^\circ $|, and (g) |${{{\theta }}_{{{\it to}}2}} = - 5^\circ .$|
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.0955 | 0.2152 | 0.1033 | 0.1916 | 0.1184 | 0.1607 | 0.1722 |
| |${F_{{\rm{s}}1}}$| [N] | 0.5830 | 0.9895 | 0.6366 | 1.0342 | 0.6556 | 0.7538 | 1.0384 |
| |${F_{{\rm{n}}1}}$| [N] | 0.6461 | 1.0622 | 0.9198 | 1.6462 | 0.8671 | 0.6412 | 0.8544 |
| Ang. diff. [|$^\circ $|] | 0.2953 | 0.4206 | 0.4402 | 0.3062 | 0.2703 | 0.5641 | 0.5586 |
| (a) | (b) | (c) | (d) | (e) | (f) | (g) | |
|---|---|---|---|---|---|---|---|
| |${M_1}$| [Nm] | 0.0955 | 0.2152 | 0.1033 | 0.1916 | 0.1184 | 0.1607 | 0.1722 |
| |${F_{{\rm{s}}1}}$| [N] | 0.5830 | 0.9895 | 0.6366 | 1.0342 | 0.6556 | 0.7538 | 1.0384 |
| |${F_{{\rm{n}}1}}$| [N] | 0.6461 | 1.0622 | 0.9198 | 1.6462 | 0.8671 | 0.6412 | 0.8544 |
| Ang. diff. [|$^\circ $|] | 0.2953 | 0.4206 | 0.4402 | 0.3062 | 0.2703 | 0.5641 | 0.5586 |
5. Discussion
The RMSEs in Tables 4 and 5 indicate some discrepancies between the results from the experiments and simulations. These errors are presumed to be that although the interaction model assumed a 2D plane, the links of the dummy were not positioned on the same plane due to design limitations. A slight width between dummy link1 and dummy link2 due to the thickness of the aluminum frame might have caused deformation along the Z-axis during the experiment. Furthermore, the gap between the cuff and the silicone blocks and the simplification of the dummy's stiffness as a linear function may have contributed to the errors. In spite of these factors, the comparison between the graphs predicted by the simulation and those measured in the experiment shows similar trends, so it suggests the proposed interaction model is effectively applied.
In this study, a static analysis method was employed, and experiments were conducted at a slow speed of 18|$^\circ /s$| to validate the model. However, actual robots can operate at higher speeds than those used in the experiment, which can lead to different interaction forces. Therefore, to evaluate the limitations of static analysis, additional experiments were conducted under conditions with a + 10 mm shear-directional wearing offset. Joint speeds were set to three different conditions: |$18^\circ /{\rm{s}}$|, |$36^\circ /{\rm{s}}$|, and |$54^\circ /{\rm{s}}$|. Figure 17 presents the results for each speed condition, while Table 6 provides the corresponding RMSE values. Despite variations in motor speed, the experimental results demonstrated consistent trends across all speed conditions, indicating that the estimated values align well with the experiment data. Although the hysteresis of the angular difference increased significantly with motor speed, resulting in RMSE exceeding 40, the RMSE for the interaction forces exhibited only minimal changes. These findings suggest that the proposed model effectively estimates the overall trends of interaction forces and maintains its validity across different motor speeds. Therefore, the model can be reliably applied to scenarios involving joint speeds of up to |$54^\circ /{\rm{s}}$|. However, its applicability may face limitations in cases of extreme accelerations, where dynamic effects could significantly influence the interaction forces.
Experimental results at varying motor speed.
RMSEs between simulation and experimental results.
| Motor speed | |||
|---|---|---|---|
| 18|$^\circ /{\rm{s}}$| | 36|$^\circ /{\rm{s}}$| | 54|$^\circ /{\rm{s}}$| | |
| |${M_1}$| [Nm] | 0.2809 | 0.4448 | 0.4977 |
| |${F_{{\rm{s}}1}}$| [N] | 2.0372 | 2.7241 | 2.9752 |
| |${F_{{\rm{n}}1}}$| [N] | 1.1954 | 3.2895 | 4.1738 |
| Ang. diff. [|$^\circ $|] | 0.9710 | 34.9710 | 45.4963 |
| Motor speed | |||
|---|---|---|---|
| 18|$^\circ /{\rm{s}}$| | 36|$^\circ /{\rm{s}}$| | 54|$^\circ /{\rm{s}}$| | |
| |${M_1}$| [Nm] | 0.2809 | 0.4448 | 0.4977 |
| |${F_{{\rm{s}}1}}$| [N] | 2.0372 | 2.7241 | 2.9752 |
| |${F_{{\rm{n}}1}}$| [N] | 1.1954 | 3.2895 | 4.1738 |
| Ang. diff. [|$^\circ $|] | 0.9710 | 34.9710 | 45.4963 |
RMSEs between simulation and experimental results.
| Motor speed | |||
|---|---|---|---|
| 18|$^\circ /{\rm{s}}$| | 36|$^\circ /{\rm{s}}$| | 54|$^\circ /{\rm{s}}$| | |
| |${M_1}$| [Nm] | 0.2809 | 0.4448 | 0.4977 |
| |${F_{{\rm{s}}1}}$| [N] | 2.0372 | 2.7241 | 2.9752 |
| |${F_{{\rm{n}}1}}$| [N] | 1.1954 | 3.2895 | 4.1738 |
| Ang. diff. [|$^\circ $|] | 0.9710 | 34.9710 | 45.4963 |
| Motor speed | |||
|---|---|---|---|
| 18|$^\circ /{\rm{s}}$| | 36|$^\circ /{\rm{s}}$| | 54|$^\circ /{\rm{s}}$| | |
| |${M_1}$| [Nm] | 0.2809 | 0.4448 | 0.4977 |
| |${F_{{\rm{s}}1}}$| [N] | 2.0372 | 2.7241 | 2.9752 |
| |${F_{{\rm{n}}1}}$| [N] | 1.1954 | 3.2895 | 4.1738 |
| Ang. diff. [|$^\circ $|] | 0.9710 | 34.9710 | 45.4963 |
The stiffness of human skin was assumed to be linearized in this study to simplify the algorithm and enhance computational efficiency. In practice, human skin exhibits linear behaviour under small deformations, making the linearization of skin elasticity a validated approach widely used in previous studies (Jayabal et al., 2019; Malhotra et al., 2018; Yazdi et al., 2018). However, under large deformations, human skin demonstrates non-linear elastic behaviour due to its multilayered structure and viscoelastic properties, which cause its elasticity to change over time. Therefore, linear models have limitations in analysing large deformations. Various studies have proposed methods to model the mechanical behaviour of human skin more accurately. For instance, Delalleau used an inverse method to capture non-linear characteristics, while Jachowicz and Yazdi applied viscoelastic models such as the Kelvin–Voigt and Maxwell models (Delalleau et al., 2007; Jachowicz et al., 2007; Yazdi et al., 2018). While these approaches enhance the realism of human skin modelling, they require significant computational power and detailed parameter information, posing practical challenges. Moreover, the stiffness between the human and the robot is not determined only by the properties of human skin. Accurate stiffness estimation must consider various factors, including skin properties, the elasticity of the cuff, and the structure of the robot. Therefore, instead of focusing on upgrading human skin modelling, it is essential to develop effective methods to estimate the overall stiffness between the human and the robot.
Measuring stiffness through experiments is possible, but determining it for each wearer is a time-consuming process. An alternative approach is to provide generalized stiffness values based on influential factors such as BMI, gender, age, and cuff design (Van Kuilenburg et al., 2012). This approach uses easily available information to provide stiffness values, thereby reducing the burden of stiffness measurement experiments. However, this method has limitations in terms of accuracy, as generalized models may not provide values tailored to individuals. To provide stiffness values tailored to individual wearers without conducting experiments, it is necessary to develop methods for estimating the stiffness between the human and the robot. One of the approach methods is to use a deep learning model trained on simulation data to inversely estimate stiffness values from experimental results. By applying this method, it will be possible to improve the accuracy of the simulation and applicability by estimating precise stiffness values.
As the Newton–Raphson method is an iterative approach to finding approximate values of a solution numerically, selecting improper initial values could negatively affect the algorithm's convergence and computational efficiency. Therefore, two methods for selecting appropriate initial values were devised. First method, as shown in Figure 6, leverages the fact that postures change continuously. It employs the results from the previous posture as the initial values to calculate the solution for the next posture. However, this method has a drawback in that if it diverges once, it cannot compute the next posture. Second method involves selecting initial values of |$0,0,0,{\theta _{{\rm{h}}1}}, \cdots ,$| and |${\theta _{{\rm{h}}( {n - 1} )}}$| for each posture. Unlike the first method, the second method does not provide appropriate initial values; thus, it requires more iterations for optimization and is more prone to divergence as well. However, in this method, the previous result does not influence the next calculation; therefore, it is suitable for analysing human motion involving singularity postures and could even enable parallel computing for analysing multiple postures simultaneously. Consequently, the first method is suitable for analysing various postures without divergence, whereas the second method is more appropriate for analysing human motions that are executed over a long duration or involve singularity postures.
In this study, the interaction model was simplified to two dimensions, focusing on planar movements, as the primary objective was to validate the proposed method. However, to reflect the movement of all joints, the proposed model must be expanded to three dimensions. To extend the model to 3D, Equation (1) can be revised as follows:
Here, |${\alpha _i},{\rm{\ }}{\beta _i},$| and |${\gamma _i}$| represent the torsional angles along the X-, Y-, and Z-axes, respectively, while |${l_{{\rm{xi}}}},{\rm{\ }}{l_{{\rm{y}}i}}$|, and |${l_{{\rm{z}}i}}$| represent the translational displacements along the X-, Y-, and Z-axes. In 3D modelling, additional constants and parameters related to the Z-axis are added, necessitating greater computational power for optimization. Particularly, the Newton method requires second-order derivatives, and as the number of variables increases, the size of the |$J( X )\ $| matrix grows significantly, leading to a substantial increase in computational complexity. Therefore, improving algorithm performance is essential for effectively implementing 3D modelling. For instance, applying the quasi-Newton method can significantly reduce the computational burden compared to the Newton method, as it only requires the gradient of the objective function instead of second-order derivatives (Dennis & Moré, 1977).
Typically, the wearing of an exoskeleton robot occurs in a posture where the arm and torso are aligned, as seen in Figure 8 at Cycle |$0\% $|. However, during the wearing process, the exoskeleton robot may shift laterally or vertically, resulting in the wearing offsets depicted in Figures 9 and 10. A comparison of Figures 8–10 shows that at Cycle |$0\% $|, corresponding to the initial wearing posture, all interaction forces are minimal, and the angular differences between the joints are nearly |$0^\circ $|. In contrast, at Cycle |$100\% $|, where the shoulder and elbow joints are flexed, the presence of wearing offsets leads to increased moments and angular differences exceeding |$10^\circ $|. These findings suggest that even if the wearer experiences no discomfort during the initial wearing of the exoskeleton, significant forces and angular differences may arise during motion. This could potentially cause user discomfort or injury and hinder the proper control of the robot. Therefore, the results highlight the importance of not only ensuring user comfort during the initial wearing but also verifying that the robot's joint positions are precisely aligned with the human joints.
The optimization algorithm proposed in this study is built upon the well-established principle of energy minimization, which has been extensively employed in previous research. For instance, energy minimization has been utilized in path planning to reduce energy consumption during motion, in predicting human postures to optimize ergonomic movements, and in the optimal design of mechanical systems to enhance energy efficiency and performance (Yang et al., 2010; Carabin et al., 2017; Gu et al., 2023). Meanwhile, this study extends the utility of energy minimization by customizing it for human–robot interaction modelling. Unlike prior research, which typically focuses on either human kinematics or robotic actuation, the proposed method offers a holistic approach to modelling human–robot interactions. This enables the simulation of complex kinematics between an exoskeleton robot and the human body, allowing for the estimation of both posture and interaction forces.
The proposed simulation method, based on kinematic analysis, allows for more detailed predictions when a human model is provided. A pin joint can be generalized as a rolling joint for modelling more complicated motions (Suh et al., 2015; Suh & Kim, 2018; Suh & Choi, 2023). Additionally, this generalization can be applied to human joints where rolling and sliding occur simultaneously. Therefore, if kinematic information about human joints is provided, the relationship between the ith and (|$i + 1$|)th links of the human body can be represented by a transformation matrix. In this regard, several studies have analysed the kinematics of the human shoulder joint using motion-capture systems (Zhang et al., 2020; Liang et al., 2021). Such research can refine the relationship between the human joints and enhance the accuracy of the proposed interaction model.
6. Conclusions
In this study, we modelled human–exoskeleton interaction and developed an energy-optimization algorithm to simulate the postures of and the physical interaction forces between the human body and the exoskeleton robot. Additionally, we conducted experiments with a dummy equipped with sensors and validated the interaction model by comparing the simulation results with the experimental measurements. For simulation, the proposed method only requires information on the physical characteristics (such as arm length, weight, skin elasticity, and contact position) of the exoskeleton robot and the wearer. Therefore, unlike existing techniques, this method can quantitatively analyse human–exoskeleton interactions without requiring a human participant to wear the robot. Moreover, this method can be used to analyse how the characteristics of the robot and human impact wearability. Consequently, the proposed modelling and optimization technique is expected to address ethical issues and the lack of objectivity in clinical tests, and reduce the time required for the development process of exoskeleton robots.
Conflicts of Interest
To the best of our knowledge, the named authors have no conflict of interest, financial or otherwise.
Author Contributions
Seungbum Lim (Conceptualization, Software, Methodology, Writing – original draft), Woojin Kim (Investigation, Writing – review & editing), and Jungwook Suh (Supervision, Methodology, Writing – review & editing)
Acknowledgments
This work was supported in part by the National Research Foundation of Korea Grant and in part by the Institute of Information and Communications Technology Planning and Evaluation grants funded by the Korea government under Grants NRF-2020-R1C1C1008707, 2022–0–00501, and RS-2023–00224546.
