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Abstract

In this study, a wrist human–machine interface with a single internal rotation center, which is efficient in terms of performance, size, and avoidance of kinematic interference, is proposed to realize the 3-degree-of-freedom rotational motion of the human wrist. For this purpose, an over-actuated coaxial spherical parallel mechanism is applied. The over-actuated mechanism is specially considered to improve the force feedback performance and to overcome the workspace limitations caused by the singularity. From a mechanical perspective, two links are coaxially connected to the base, and this rotation axis is designed to coincide with the wrist pronation–supination, which has the largest operating range in wrist motion. A prototype was fabricated and evaluated to validate the proposed mechanism. In addition, a usable design index was proposed for the design that prevents interference with the user in the workspace while minimizing the link inertia to optimize the device. This index is optimized with the general performance indices, that is, condition index and stiffness index. It was verified through simulation that the optimized wrist human–machine interface exhibited higher performance than other similar devices. In addition, range of motion is at least 104.3% than a daily living range of motion of the human wrist, guaranteeing all ranges of motion.

Keywords

Human–machine interface spherical parallel mechanism wrist exoskeleton

Introduction

Recently, human–machine interfaces (HMIs), which focus on physical force feedback, have been actively proposed in various fields.¹ To improve the performance of force feedback, it is necessary to achieve low inertia, back drivability, large workspace compared to device size, high stiffness, high accuracy, and wide bandwidth for the force response.² In particular, for an exoskeleton robot, which is one of the special-purpose HMI types, a match between the human joint rotation center and the device rotation center becomes an important element in design because it is related to safety and operation.³

The wrist has the characteristics of a spherical joint that operates with 3 degrees of freedom (DOF)—pronation–supination (PS), flexion–extension (FE), and abduction–adduction (AA)—for a single rotation center in the body. In general, orthogonal serial structures have been applied to the wrist exoskeleton HMI. They have been mainly proposed because they are easy to design and analyze; moreover, the range of motion (ROM) can be easily considered, and the rotation center can be placed close to the human wrist center.⁴ However, the small bandwidth for the force response and the machine impedance that is increased by the inertia of the actuator attached to the moving part have been identified as major drawbacks.⁵ As an alternative, parallel wrist human–machine interface (WHMI) using a prismatic drive has been developed.^6,7 In this instance, a parallel mechanism was applied to the FE and AA DOF. This reduced the inertia of the moving part by placing the actuator in the base of the parallel structure, and high stiffness and high force feedback performance were realized. The applied 2-DOF rotation and 1-DOF translation mechanism, however, can cause a mismatch problem between the human wrist center and the device rotation center.⁸ In addition, a large inertia is applied because the actuator must be additionally mounted on the base of the parallel structure for the PS operation. The application of the parallel mechanism including the above-mentioned proposed device has a serious problem in that the workspace is limited by the singularity. Therefore, a serial-parallel mixed type or an over-actuated mechanism is typically used to address this problem.^9–12

Gosselin and Hamel¹³ proposed a 3-RRR type spherical parallel mechanism with the advantage of a parallel structure for 3-DOF operation of a single rotation center. Due to the above-mentioned advantages of the mechanism, it was mainly developed as a non-exoskeleton-type desktop HMI.^14,15 In particular, SHaDe,¹⁵ a haptic device with 3-DOF, showed that the wrist-oriented design is more appropriate for joint movement with the human body than the palm-oriented design.

Despite its various advantages, the spherical parallel structure was difficult to apply to the wrist HMI in the previous studies for two major reasons. The first reason is that the human wrist joint ROM is very large. In the parallel structure, there is a limitation in the ROM that can perform the combined operation due to the influence of the singularity. The second reason is that the large ROM leads to the complex movements of the links in the spherical parallel structure. This can cause safety problems due to the collision between the user and the links. It is possible to increase the size of the links to avoid interference with the user, but this increases the link inertia, diluting the advantages of the spherical parallel structure mechanism.

To overcome such problems and develop an efficient exoskeleton-type WHMI, an over-actuated coaxial spherical parallel structure with a 2-RRR structure was proposed in this study. The over-actuated structure was proposed to extend ROM by overcoming the singularity and to improve the feedback performance. In addition, one joint of each link chain was coaxially connected to the base to fully guarantee 1-DOF.¹⁶ This allowed the proposed WHMI to independently and completely cover the PS axis with the widest DOF in the wrist. This design approach had the advantage that only 2-DOF movements had to be reflected in ROM selection or optimization considerations. Furthermore, the spherical mechanism with a single internal rotation center made it easy to locate the rotation center of the device close to the center of the human wrist. In the design optimization process, we proposed a usable design index for the design that can reduce the link inertia while preventing the collision between the moving links and the user. In the development of the proposed device, this index was considered together with the condition and stiffness indices, which are general performance indices, and applied to the WHMI optimization.

The kinematics of the proposed device are analyzed and reported in section “Mechanical analysis,” and its performance is verified through simulation in section “Prototype validation.” A simple prototype is fabricated to verify the validity of the exoskeleton-type HMI device. In section “Optimization,” the optimal design results are shown with optimization indices. In section “Discussion,” the derived optimal design verifies the excellence of the proposed mechanism through the simulation verification. Finally, the conclusions are provided in section “Conclusion.”

Mechanical analysis

Kinematics analysis

The proposed mechanism has a parallel structure in which the first joint rotation axis of the 2-RRR structure is coaxially connected, and its kinematic analysis is based on the spherical law of cosines¹⁷ like the general spherical parallel structures. Figure 1 shows the prototype of the proposed mechanism and Figure 2 shows the joint angles and design parameters. Subscripts $B$ and $M$ represent the base platform and mobile platform, respectively. For $i = 1, 2$ , when the rotation axis vector of the joint coaxially connected to the base platform is denoted by $u_{i}$ , the rotation axis vector of the joint connected to the mobile platform by $v_{i}$ , and the rotation vector of the middle joint of the two joints by $w_{i}$ , $u_{i} \cdot w_{i} = \cos (α_{1})$ , and $w_{i} \cdot v_{i} = \cos (α_{2})$ . $∠ M_{i} O O_{B} = β$ , and the angle between the plane $O_{B} M_{1} M_{2}$ and the straight line $\bar{O_{B} O_{M}}$ is denoted by $γ$ . $Y_{M}$ is the y-axis of the mobile platform coordinate system, and the angle between $Y_{M}$ and $\bar{O_{M} M_{i}}$ is denoted by $η_{2 i}$ . As the two link chains are symmetrical, $η_{21} = η_{22}$ . $v_{1} \cdot v_{2} = \cos (α_{3})$ , and $α_{3}$ can be obtained from the triangular pyramid $O_{B} O_{M} M_{1} M_{2}$ using the spherical law of cosine and the equation with $β$ and $γ$ . The over-actuated coaxial spherical parallel mechanism consists of four active joint axes and two passive joint axes, and the active joints correspond to $u_{i}$ and $w_{i}$ . $θ_{i}$ , $ϕ_{i}$ , and $ψ_{i}$ represent the joint angles obtained from the kinematic shape, as shown in Figure 2. The analysis in the existing analysis method is complex because $ϕ_{i}$ and $ψ_{i}$ must be derived. In this mechanism, however, the kinematics are analyzed using the method of deriving a passive angle $ψ_{i}$ because $ϕ_{i}$ corresponds to the angle of the active joint. For simple denotation, $c (ζ) = \cos (ζ)$ and $s (ζ) = \sin (ζ)$ are applied for the angle parameter $ζ$ . First, $μ_{BC}$ is defined as the angle of $∠ B O_{B} C$ and if $θ_{3} = θ_{1} + θ_{2} - π$ , $μ_{BC}$ is obtained using equation (1)

c (μ_{BC}) = c {(α_{1})}^{2} + s {(α_{2})}^{2} c (θ_{3})

(1)

Figure 1.

Prototypes of the over-actuated spherical parallel mechanism. (a) and (b) indicate the pre-optimized design and the optimized design, respectively.

Figure 2.

Denotation of the over-actuated mechanism: (a) the angles of the joints and (b) the design parameters. The active joints are shown in yellow, and the passive ones are in white.

Assuming the angle between the plane $B O_{B} A$ and the plane $B O_{B} C$ is $ν_{3}$ , and the angle between the plane $C O_{B} A$ and the plane $B O_{B} C$ is $ν_{4}$ , equation (2) can be derived

ν_{3}, ν_{4} = c^{- 1} (\frac{c (α_{1}) - c (μ_{BC}) c (α_{1})}{s (μ_{BC}) s (α_{1})})

(2)

Assuming the angle between the plane $M_{1} B O_{B}$ and the plane $CB O_{B}$ is $ν_{1}$ , and the angle between the plane $M_{2} C O_{B}$ and the plane $BC O_{B}$ is $ν_{2}$ , $ν_{1} = ϕ_{1} - ν_{3}$ and $ν_{2} = ϕ_{2} - ν_{4}$ can be obtained using $ν_{3}$ and $ν_{4}$ , respectively. $μ_{M_{1} C}$ , which is the angle of $∠ M_{1} O_{B} C$ , can be obtained using the triangular pyramid $O_{B} M_{1} BC$ , and $ψ_{2}$ can be derived using equation (3)

c (μ_{M_{1} C}) = c (μ_{BC}) c (α_{2}) + s (μ_{BC}) s (α_{2}) c (ν_{1})

(3)

ψ_{2} = \pm c^{- 1} (\frac{c (μ_{M_{1} C}) - c (α_{2}) c (α_{3})}{s (α_{2}) s (α_{3})})

(4)

In the same manner, $μ_{M_{2} B}$ , which is the angle of $∠ M_{2} O_{B} B$ , can be obtained and $ψ_{1}$ is also derived using the same procedure. In this instance, the sign of $ψ_{i}$ must be determined because it depends on the posture of the device. Using the triangular pyramids $O_{B} A M_{1} B$ and $O_{B} A M_{2} C$ , $μ_{A M_{1}}$ , that is, $∠ M_{1} O_{B} A$ , and $μ_{A M_{2}}$ , that is, $∠ M_{2} O_{B} A$ , can be obtained through equation (5)

\begin{matrix} μ_{A M_{2}} = c^{- 1} (c (α_{1}) c (α_{2}) + s (α_{1}) s (α_{2}) c (ϕ_{2})) \\ μ_{A M_{1}} = c^{- 1} (c (α_{1}) c (α_{2}) + s (α_{1}) s (α_{2}) c (ϕ_{1})) \end{matrix}

(5)

Assuming the angle between the plane $O_{B} BA$ and the plane $O_{B} B M_{2}$ is $ν_{1}'$ , and the angle between the plane $O_{B} CA$ and the plane $O_{B} C M_{1}$ is $ν_{2}'$ , $ν_{1}'$ and $ν_{2}'$ can be obtained through the spherical law of cosines, as shown in equation (6)

\begin{array}{l} {ν^{'}}_{1} = c^{- 1} (\frac{c (μ_{A M_{2}}) - c (μ_{M_{2} B}) c (α_{1})}{s (μ_{M_{2} B}) s (α_{1})}) \\ {ν^{'}}_{2} = c^{- 1} (\frac{c (μ_{A M_{1}}) - c (μ_{M_{1} C}) c (α_{1})}{s (μ_{M_{1} C}) s (α_{1})}) \end{array}

(6)

The sign of $ψ_{i}$ is determined by comparing the obtained $ν'_{i}$ and $ϕ_{i}$ . When $ν'_{1} > ϕ_{1}$ , $ψ_{2} > 0$ and when $ν'_{1} \leq ϕ_{1}$ , $ψ_{2} \leq 0$ . The sign of $ψ_{1}$ is determined in the same manner. Consequently, as all the joint angles of a link chain can be obtained, the rotation matrix $Q^{*}$ of the mobile platform for the base platform can be obtained through direct kinematics. The final rotation matrix is derived so that it can have the initial value $ϵ$ for the X-axis rotation, as shown in Figure 2. $Q$ in equation (8) is the rotation matrix of the end effector for the base obtained through the chain $i = 1$ , and $R_{n} (z)$ means a matrix rotated by $z$ angle in the n-axis direction

\begin{array}{l} Q^{*} = R_{z} (π - θ_{1}) R_{y} (α_{1}) R_{z} (π + ϕ_{1}) R_{y} (α_{2}) R_{z} \\ (π - ψ_{1}) R_{y} (\frac{α_{3}}{2}) R_{X} (γ) \end{array}

(7)

Q = Q^{*} R_{x} (ϵ)

(8)

The wrist rotation DOFs are represented in the PS-FE-AA order. PS corresponds to the Z-axis rotation, FE to the Y-axis rotation, and AA to the X-axis rotation. The final rotation angles will be represented by the Z-Y-X Euler angle denotation and denoted by $Φ$ (PS), $Θ$ (FE), and $Ψ$ (AA).

Inverse kinematics analysis

Assuming the rotation matrix of the mobile platform for the base platform is $Q$ and the $v_{i}$ joint expressed for the mobile platform coordinate system is $v_{i}^{*}$ , $v_{i} = Q v_{i}^{*}$ . If $z_{0} = [0, 0, 1]^{T}$ , the $w_{i}$ rotation axis can be expressed as shown in equation (9)

w_{i} = R_{z} (π - θ_{1}) R_{y} (α_{1}) z_{0}

(9)

In this instance, as $α_{1}$ , $α_{2}$ , and $Q$ are known, inverse kinematics can be derived by expanding $w_{i} \cdot v_{i} = \cos (α_{2})$ .

Jacobian analysis

As only rotation movements are generated in the spherical parallel structure, the Jacobian of the device is defined as a rotation Jacobian. The rotation angular velocity $ω_{M}$ of the mobile platform can be expressed as a vector sum using the rotation speed of each joint of the ith chain, as shown in equation (10)

u_{i} {\overset{\cdot}{θ}}_{i} + w_{i} {\overset{\cdot}{ϕ}}_{i} + v_{i} {\overset{\cdot}{ψ}}_{i} = ω_{M}

(10)

When the velocity vector of the active joint is defined as $\overset{\cdot}{Θ} = [{\overset{\cdot}{θ}}_{1}, - {\overset{\cdot}{θ}}_{2}, {\overset{\cdot}{ϕ}}_{1}, - {\overset{\cdot}{ϕ}}_{2}]^{T}$ considering the rotation direction of the joint then $J = A^{- 1} B$ is obtained using $A$ and $B$ expressed in as follows

A = [\begin{matrix} {(w_{1} \times v_{1})}^{T} \\ {(w_{2} \times v_{2})}^{T} \\ {(u_{1} \times v_{1})}^{T} \\ {(u_{1} \times v_{2})}^{T} \end{matrix}], B = diag ([\begin{matrix} (w_{1} \times v_{1}) \cdot u_{1} \\ (w_{2} \times v_{2}) \cdot u_{2} \\ (u_{1} \times v_{1}) \cdot w_{1} \\ (u_{2} \times v_{2}) \cdot w_{2} \end{matrix}])

(11)

A ω_{M} = B \overset{\cdot}{Θ}

(12)

where $diag (M)$ represents the diagonal form of the $M$ matrix.

Prototype validation

To validate the kinematics of the proposed device, simple parameters were applied and numerical inverse Jacobian position control simulation was performed. Table 1 lists the parameters applied to the simulation. The position control equation that used inverse Jacobian was $J_{A}^{- 1} (Ω d - Ω c) = \overset{\cdot}{Θ}$ . $Ω_{d}$ represents the target Euler angle, $Ω_{c}$ the current end Euler angle, and $J_{A}$ the analytical Jacobian for the Euler angle. Figure 3 shows a simple block diagram of the inverse Jacobian position control. In Figure 4, the graphs showing that the results of the FE and AA position control for the target operation are illustrated. The red lines represent $Ω_{d}$ and the blue lines $Ω_{c}$ . Simulation results showed that the position control was performed properly, and the mechanism analysis was verified. Based on this, a prototype device with simple parameters was fabricated, and its validity as an exoskeleton device was evaluated through simple user tests. Figure 5 shows the scenes of the user with the prototype and movements were made to each axis direction. It was confirmed that the device can operate independently in the PS axis, create point-oriented motion, and place the rotation center close to the wrist center.

Table 1.

Parameters of the prototype.

Parameters	$α_{1}$	$α_{2}$	$β$	$η_{21}$	∈
Value (°)	65	65	20	30	−60

Figure 3.

Simple block diagram of the inverse Jacobian position control.

Figure 4.

Results of the inverse Jacobian position control simulation. Subscripts “d” and “c” represent the desired and current values, respectively. The blue lines represent the input angles and the red lines the output angles through the control results. Θ denotes the angle corresponding to FE and Ψ denotes the angle corresponding to AA.

Figure 5.

A subject with the pre-optimized prototype.

Optimization

In this section, the wrist device ROM is selected and optimized to ensure the activities of daily living (ADL) ROM.^18,19 Condition, stiffness, and usable design indices are introduced as optimization indices, and the validity of the results is shown. As the ROM of PS is guaranteed independently, optimization is performed only for FE and AA. Therefore, only 2-DOF is considered for optimization, making it possible to lower the complexity of the optimization process. The ROMs of FE and AA were set as 120° and 60°, respectively, to guarantee higher values than the ADL ROM (Table 2).

Table 2.

ADL properties and joint limit of the wrist.

DOF	ADL ROM (°)	ADL torque (N m)
PS	150	0.06
FE	114	0.35
AA	57	0.35

Global condition index

The global condition index (GCI) is generally used as an index for representing force isotropy or dexterity performance. GCI is the total workspace average of the local condition index (LCI) that represents the condition under a certain state.²⁰ LCI ranges from 0 to 1, and the value closer to 1 means better force isotropy. Equation (13) represents the LCI derived from the nth posture and $‖ J ‖$ is the Euclidean norm of a Jacobian. Equation (14) represents GCI

LCI (n) = \frac{1}{‖ J {(Θ (n))}^{- 1} ‖ ‖ J (Θ (n)) ‖}

(13)

GCI = \frac{1}{n} \sum_{i = 1}^{n} LCI (n)

(14)

Global stiffness index

Stiffness means displacement compared to the force applied to the end in a certain posture, and a higher stiffness can express a virtual hard wall.²¹ The stiffness index can be expressed using the local stiffness index (LSI) and global stiffness index (GSI), like the condition index. The stiffness index is derived through the relationship $J^{T} τ = f$ between the moment force $f$ applied to the end and the actuator torque $τ$

L S I (n) = \frac{1}{‖ {(J {(Θ (n))}^{T} J (Θ (n)))}^{- 1} ‖ ‖ (J {(Θ (n))}^{T} J (Θ (n))) ‖}

(15)

GSI = \frac{1}{n} \sum_{i = 1}^{n} LSI (n)

(16)

Usable design index

Kinematic interference between the user and the moving links in the exoskeleton-type device can affect safety and operation. In general, it is possible to increase the size of the links to avoid interference with the user, but this causes a problem of increasing the inertia of the moving parts. Therefore, the design that reduces the inertia of the links while preventing interference is an important design consideration in HMI. In this study, the distance between the Z-axis which is the wrist–forearm-direction axis and the simplified links moving along the spherical surface centered at the wrist rotation point, is proposed as the usable design index. Let the distance between the Z-axis and the simplified moving links on the ROM be denoted by $D$ , and $\min (D)$ denote the closest distance. If $\min (D)$ is larger than the user forearm radius, it is possible to control the moving link size for reducing the link inertia while avoiding interference. Therefore, it is necessary to derive the design parameters that will increase by $\min (D)$ for better performance and avoidance of kinematic interference. As there is no interference between the link connecting AB and the link connecting AC due to the coaxial structure, the distance between the straight line connecting $B$ and $M_{1}$ or $C$ and $M_{2}$ and the PS axis on the plane projected in the z-axis direction can be modeled as $D$ . Taking the link chain of $i = 1$ as an example, when the points which the $B$ and $M_{1}$ joints projected on the XY-plane are denoted by $B (x_{1}, y_{1})$ and $M_{1} (x_{2}, y_{2})$ , respectively, and a straight line is modeled between the points, the minimum distance between the line and the Z-axis can be expressed as

D = \frac{| y_{2} - \frac{Δ y}{Δ x} x_{1} |}{\sqrt{{(\frac{Δ y}{Δ x})}^{2} + 1}} r

(17)

where $Δ x = x_{2} - x_{1}$ and $Δ y = y_{2} - y_{1}$ . For the optimization, the radius of the spherical surface to which the link belonged was normalized to one, and $\min (D)$ derived for all the link chains in all the ROMs was used as the usable design index.

Optimization result

To optimize the three indices—GCI, GSI, and usable design index—proposed in this study, the non-dominated sorting genetic algorithm II (NSGA-II), a multi-purpose function optimization algorithm, was used.²² The initial settings were crossover probability 0.9, mutation probability 0.2, and distribution index 20. The optimization of population 50 and generation 100 was performed 10 times. Equation (18) represents the objective function of the optimization, and Table 3 represents the upper and lower bounds of the parameters. The red dots in Figure 6 represent the final optimized results

\begin{matrix} f_{1} = maximize (GCI) \\ f_{2} = maximize (GSI) \\ f_{3} = maximize (\min (D)) \end{matrix}

(18)

Table 3.

The upper and lower bounds of the parameters.

Bounds	$α_{1}$	$α_{2}$	$β$	$η_{21}$	∈
Upper (°)	90	90	90	60	−60
Lower (°)	60	60	20	10	−30

Figure 6.

A NSGA-II genetic algorithm results. The red dots represent the parameters derived from the final evolution in each generation. The distance (D) in the graphs represents $\min (D)$ , the usable design index.

The optimization results showed that GCI and GSI are proportional and D is inversely proportional. In other words, there is a trade-off relation between the performance and interference avoidance of the device, and appropriate parameter selection is required accordingly. Three types of IDs were derived as examples of the optimization results. ID1 has sufficient GCI and GSI performances and a large $\min (D)$ , ID3 has higher GCI and GSI performances and a smaller $\min (D)$ than ID1, and ID2’s performance lies between those of ID3 and ID1. Table 4 shows the parameters of each ID and their performances. In this study, simple parameters were selected based on the results of ID 3, which exhibited the best performances. Figures 7 and 8 represent the LCI and LSI for the FE and AA postures.

Table 4.

The upper and lower bounds of the parameters.

	$α_{1}$	$α_{2}$	$β$	$η_{21}$	∈	$GCI$	$GSI$	$\min (D)$
ID 1	82.87	64.73	68.75	31.56	−57.67	0.8597	0.6078	0.6628
ID 2	89.99	82.70	54.55	56.34	−53.32	0.9084	0.7096	0.5771
ID 3	80.16	90	54.72	57.67	−46.06	0.9114	0.7215	0.5027
Selected	80	90	55	60	−45	0.9092	0.7190	0.5029

Figure 7.

Graph of the local condition index map.

Figure 8.

Graph of the local stiffness index map.

Discussion

Table 5 compares the finally derived performance indices with those of the existing similar exoskeleton-type wrist HMI mechanisms. The performance comparison was conducted based on the GCIs, LCIs, and ROMs with similar studies. The parallel type was compared with SHaDe, which used a spherical parallel structure, and Rice wrist-A, which had a 3-RRR structure. In the case of Rice wrist-A, the GCI was not mentioned, but the maximum LCI was 0.22 and the minimum LCI was 0.08, and thus, the GCI, meaning the overall average LCI, could be predicted as being between 0.08 and 0.22. In the case of Gimbal, the mechanism was an orthogonal series type, and the kinematics was analyzed in the same PS-FE-AA order as the analysis of the proposed structure. In this case, for the series structure, the performance indices were analyzed for FE 120° and AA 60° in the same manner as the ROM of the proposed device.

Table 5.

Performance comparison with similar devices.

	Parallel type			Serial type
Mechanism	SHaDe¹⁵	Rice wrist-A⁷	Proposed mechanism	Gimbal
GCI	0.795	0.08 < GCI < 0.22	0.9092	0.8418
Min(LCI)	0.101	≈0.08	0.7830	0.5774
LCI > 0.5	95%	−	100%	100%
Pro/Sup (ADL: 150)	180	180	180	180
Flex/Ext (ADL: 115)	90	84	120	opt. 120
Abd/Add (ADL: 70)	90	Add > 33Abd > 19	60	opt. 60

Min(LCI) means the minimum LCI in the entire ROM. The proposed mechanism exhibited very high force isotropy and dexterity performance on average in the workspace with the GCI of 0.9092. The minimum LCI was 0.7830, which was higher than that of all other mechanisms in the ROMs. This means that WHMI is capable of showing a high performance in the ROM regardless of the posture. For GCI, the performance was 14.3% and 8% higher than those of SHaDe and the serial type. For the minimum LCI, the performance was 675.2% and 35.6% higher than those of SHaDe and the serial type. The ROM of the device was 56.5% wider for the PS axis, 4.3% for the FE axis, and 5.2% for the AA axis compared to the ADL ROM. As the finally derived parameters were optimized with the usable design index, they had the minimum link radius without interference, that is, the minimum link inertia, even with the same performance. The usable design index was 0.5029, which corresponded to $r = 99.42 mm$ when the radius of the wrist–forearm was assumed to be $D = 50 mm$ . Figure 9 shows the final model designed considering the driving part and the electric field using the derived optimal parameters. The optimal design form differs by 180° in the PS-axis direction from the two prototypes to facilitate the design of the handle. This does not affect the performance and the ROM analysis because it is a rotation on an independently guaranteed axis.

Figure 9.

Snapshots of the optimized WHMI.

Figure 10 shows the kinematic analysis results of the final proposed model. It shows the $θ_{1}$ , $θ_{2}$ , $ϕ_{1}$ , and $ϕ_{2}$ angles, which correspond to the active joints, when the end is moved with the sine waveform having the maximum ROM magnitude of each DOF. Figure 10(a) shows the graph of the operation from −80° to 80° for the PS DOF. The angles $θ_{1}$ and $θ_{2}$ of the active joints coaxially connected to the PS axis operated at the same magnitude; however, $ϕ_{1}$ and $ϕ_{2}$ did not exhibit a similar behavior. As the two active joints $θ_{1}$ and $θ_{2}$ were identically involved in the PS DOF, the load on the PS axis was distributed through them. In actual designs, as the PS-axis joint is the closest to the base and is affected by the inertia of all links, a relatively high torque performance is required. As the proposed model distributes the load in the corresponding axial direction, however, it can reduce the load on a single actuator. Figure 10(b) and (c) shows the graphs of the movements of the active joints generated by AA DOF motion from −30° to 30° and generated by FE DOF motion from −60° to 60°, respectively. The results confirmed that multiple active joints were involved in making motion possible in each axial direction. This allows the proposed mechanism to distribute the force load to the entire actuator, thereby realizing efficient torque feedback performance.

Figure 10.

Angles of the active joints according to the motion of the end. (a), (b), and (c) show the results of the PS, AA, and FE motion, respectively, when the end is moved with the sin waveform having the maximum ROM magnitude $θ_{1}$ is in red (circle marker), $θ_{2}$ in green (triangle marker), $ϕ_{1}$ in magenta (cross marker), and $ϕ_{2}$ in cyan (square marker).

Figure 11 shows the torque of the actuator with the largest load when 1 N m torque was applied axially to the wrist center for all the postures in the ROM. This can be confirmed by the relation $J^{T} F = τ$ when the torque at the end is $F$ and the actuator torque is $τ$ . Figure 11(a)–(c) shows the graphs of the simulation that applied torque in the AA, FE, and PS directions, respectively, for all the postures. The results show that the torque of the actuator, which was subject to the largest load when a 1 N m torque was applied to the end of the proposed mechanism, had a maximum value of 0.7939 N m and a minimum value of 0.3666 N m. This means that the maximum 2.727-fold and minimum 1.259-fold torque output compared to the actuator are possible if the axial torque is applied to the wrist center in all the postures. These results show that the proposed mechanism is capable of high-performance force feedback by force distribution through the parallel structure, unlike the serial structure, in which the torque applied in a certain axial direction directly limits the performance of the actuator in that direction.

Figure 11.

Torque of the actuator with the largest load when 1 N m torque is applied axially to the end for all the postures in the ROM. (a), (b), and (c) show the results when the torque was applied in the AA, FE, and PS direction, respectively. The numbers on each graph represent the maximum and minimum values.

Conclusion

In this study, a new 3-DOF exoskeleton-type WHMI mechanism based on the over-actuated coaxial spherical parallel mechanism was proposed. Due to the characteristics of the proposed wrist HMI mechanism, it is easy to closely place the rotation centers of the human wrist and the device, and the same DOFs are guaranteed. To achieve this, a coaxial structure was applied to guarantee the independent operation in the PS axis, which has the widest ROM, and the over-actuated structure was considered to overcome the ROM limit of the parallel structure and improve performance. First, the proposed mechanism was validated by kinematic analysis and simulation. A prototype was fabricated for experimental verification, and the applicability of the exoskeleton-type WHMI was confirmed. The usable design index that estimates the minimum link size to avoid interference with the user was additionally proposed for the extended research. This index was applied together with GCI and GSI to optimize the proposed mechanism. The excellence of the optimized mechanism as a HMI was confirmed through performance index comparisons with existing similar devices. The proposed mechanism will be experimentally verified in a follow-up study and will be completed after user evaluation in application areas, including rehabilitation.

Footnotes

Handling Editor: Yong Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Basic Science Research Program through the National Research 1 0 of Korea (NRF) funded by the Ministry of Education (2017R1A2B3010336) and by the Research Grant of Kwangwoon University in 2017.

ORCID iD

Jaeyong Lee

Woosung Yang

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