2.1. Target scenarios in ADLs

The design of WELiBot is conceptualized based on three target scenarios in ADLs. Since the use of both hands is challenging for individuals with hemiplegia after a stroke, WELiBot is designed to assist the hemiplegic upper limb. The following three specific scenarios were considered in its conception.

• • Opening a bottle cap (Figure 2a): The hemiplegic limb stabilizes the bottle in a fixed position while the non-hemiplegic limb opens the cap.

• • Lifting a box (Figure 2b): The hemiplegic limb receives upward assistance, supporting the lifting force while the box is moved upward.

• • Cutting vegetables (Figure 2c): The hemiplegic limb is provided with downward assistance to hold the vegetables against the cutting board, while the non-hemiplegic limb operates the knife.

Figure 2. Three scenarios in ADLs. (a) Opening a bottle cap; (b) Lifting a box; (c) Cutting vegetables.

In these scenarios, the hemiplegic upper limb primarily functions as a stabilizing or supportive aid, assisting in fixing or steadying objects. Tasks such as twisting a cap or holding a knife to cut vegetables require precise coordination across multiple joints and controlled force application. Providing robotic assistance for these complex movements would necessitate multiple actuators and sensors, leading to increased weight and cost. Given these constraints, it is more practical and efficient for the non-hemiplegic upper limb to perform these complex tasks, while WELiBot supports the hemiplegic limb in a stabilizing role.

Based on the above considerations, the basic functional requirements for WELiBot are defined. First, it should provide vertical force assistance to support the hemiplegic limb. Second, it should constrain the motion of the hemiplegic limb within a predefined movement range to ensure stability and usability.

2.2. Constrained conditions

In rehabilitation therapy, two primary movements are distinguished in the evaluation and treatment process: the “reaching movement,” where the hand extends forward, and the “lifting movement,” where the hand is raised [Reference Suhaimi, Talha, Wan and Ariffin33]. In addition, movement around the trunk is important for adapting to different environments. Thus, in this study, the mechanism must be capable of producing three types of one-dimensional movements: reaching movement, lifting movement, and circumferential movement.

Previous studies on robotic systems that constrain the end-effector within a specific geometry include the work by Hou et al. [Reference Hou and Kiguchi34] and Zhang et al. [Reference Zhang, Guo and Sun35] on upper limb assistance and rehabilitation using a “virtual tunnel.” This approach employs impedance control to create a guided movement sensation, where the user’s hand follows a predefined path. The system minimizes resistance along the set path while increasing resistance perpendicular to the path to constrain motion. Higuchi et al [Reference Higuchi and Ogasawara36] developed the PAS-Arm, a human-compatible assistive arm for lifting tasks, which mechanically constrains the end-effector within a specific geometry. By using a continuously variable transmission, the robot’s joints synchronize to create a guiding plane. These robots rely on multi-joint serial mechanisms, where constraints are achieved by synchronizing joint control or mechanical design.

In this study, we propose that if a parallel mechanism can be partially decoupled with the three one-dimensional movements, a geometric constraint could be achieved by fixing one or two active joints without consuming energy. This approach eliminates the need for synchronized joint control, which is required in serial mechanisms, thereby simplifying the system’s control complexity.

2.3. Proposed mechanism

A 4R-5R parallel mechanism, shown in Figure 3, is proposed to fulfill the functional requirements under the constraints described above. An arc-shaped guide rail is attached around the human trunk, and the end-effector is connected to the hemiplegic forearm through a passive rotational arm cuff. Two sliders, Slider 1 and Slider 2, move along the arc-shaped guide rail, while the end-effector is positioned at the ends of the two chains originating from each slider.

Figure 3. Proposed mechanism of WELiBot.

The two chains connecting Slider 1 and Slider 2 are designated as Limb 1 and Limb 2, respectively. Since the sliders move along the arc-shaped guide rail, their motion is kinematically equivalent to a rotational joint at the center of the guide rail. The entire mechanism is represented as $\underline{\overline{R}}(\underline{\hat{R}}\hat{R}\hat{R})-\underline{\tilde{R}}\tilde{R}(\dot{R}\dot{R}\dot{R})$ , where the same superscript symbols indicate parallel joints, and underscored symbols denote active joints. Slider 1, Slider 2, and the rotational joint A₁ serve as active joints.

3.1. Mobility analysis

Figure 4 illustrate the coordinates and denotations for the joints and links of the mechanism. s _i,j represents the axis vector, while $\hat{\textbf{S}}_{i,j}$ represents the screw, where the first subscript denotes the joint number and the second subscript denotes the limb number. The x-axis is aligned with the direction of OA₁, the z-axis is aligned with the direction of s _1,1 = s _1,2, and the y-axis is determined based on the right-hand coordinate system. The movements of Slider 1 and Slider 2 along the guide rail are kinematically equivalent to rotational movements around the z-axis. The joints exhibit the following parallel relationship that s _1,1 || s _1,2 || s _2,2, s _2,1 || s _3,1 || s _4,1, and s _3,2 || s _4,2 || s _5,2. The angle between the x-axis and OA₂ is defined as $\phi$ . The angle between OA₂ and A₂C₂ is defined as $\psi$ .

Figure 4. Coordinates and denotations for the joints and links of the mechanism.

The velocity at point P on Limb 1 and Limb 2 can be expressed as:

(1) \begin{align} \hat{\textbf{S}}_{\textrm{P},1}&=\dot{\theta }_{1,1}\hat{\textbf{S}}_{1,1}+\dot{\theta }_{2,1}\hat{\textbf{S}}_{2,1}+\dot{\theta }_{3,1}\hat{\textbf{S}}_{3,1}+\dot{\theta }_{4,1}\hat{\textbf{S}}_{4,1} \end{align}

(2) \begin{align} \hat{\textbf{S}}_{\textrm{P},2}&=\dot{\theta }_{1,2}\hat{\textbf{S}}_{1,2}+\dot{\theta }_{2,2}\hat{\textbf{S}}_{2,2}+\dot{\theta }_{3,2}\hat{\textbf{S}}_{3,2}+\dot{\theta }_{4,2}\hat{\textbf{S}}_{4,2}+\dot{\theta }_{5,2}\hat{\textbf{S}}_{5,2} \end{align}

According to Figure 4, the screw representation for each joint on Limb 1 and Limb 2 is given by:

(3) \begin{align} \hat{\textbf{S}}_{1,1}&=\left[\begin{array}{l} {\boldsymbol{s}}_{1,1}\\ \overset{\rightharpoonup }{\textrm{OP}}\times {\boldsymbol{s}}_{1,1} \end{array}\right]=\quad \left[\begin{array}{llllll} 0 & 0 & 1 & y_{\textrm{P}}-y_{\textrm{O}} & -\left(x_{\textrm{P}}-x_{\textrm{O}}\right) & 0 \end{array}\right]^{\textrm{T}} \end{align}

(4) \begin{align} \hat{\textbf{S}}_{2,1}&=\left[\begin{array}{l} {\boldsymbol{s}}_{2,1}\\ \overset{\rightharpoonup }{\textrm{A}_{1}\textrm{P}}\times {\boldsymbol{s}}_{2,1} \end{array}\right]=\left[\begin{array}{llllll} 0 & 1 & 0 & -\left(z_{\textrm{P}}-z_{{\textrm{A}_{1}}}\right) & 0 & -\left(x_{\textrm{P}}-x_{{\textrm{A}_{1}}}\right) \end{array}\right]^{\textrm{T}} \end{align}

(5) \begin{align} \hat{\textbf{S}}_{3,1}&=\left[\begin{array}{l} {\boldsymbol{s}}_{3,1}\\ \overset{\rightharpoonup }{\textrm{B}_{1}\textrm{P}}\times {\boldsymbol{s}}_{3,1} \end{array}\right]=\left[\begin{array}{llllll} 0 & 1 & 0 & -\left(z_{\textrm{P}}-z_{{\textrm{B}_{1}}}\right) & 0 & -\left(x_{\textrm{P}}-x_{{\textrm{B}_{1}}}\right) \end{array}\right]^{\textrm{T}} \end{align}

(6) \begin{align} \hat{\textbf{S}}_{4,1}&=\left[\begin{array}{l} {\boldsymbol{s}}_{4,1}\\ \overset{\rightharpoonup }{\textrm{C}_{1}\textrm{P}}\times {\boldsymbol{s}}_{4,1} \end{array}\right]=\left[\begin{array}{llllll} 0 & 1 & 0 & 0 & 0 & 0 \end{array}\right]^{\textrm{T}} \end{align}

(7) \begin{align} \hat{\textbf{S}}_{1,2}&=\left[\begin{array}{l} {\boldsymbol{s}}_{1,2}\\ \overset{\rightharpoonup }{\textrm{OP}}\times {\boldsymbol{s}}_{1,2} \end{array}\right]=\left[\begin{array}{llllll} 0 & 0 & 1 & y_{\textrm{P}}-y_{\textrm{O}} & -\left(x_{\textrm{P}}-x_{\textrm{O}}\right) & 0 \end{array}\right]^{\textrm{T}} \end{align}

(8) \begin{align} \hat{\textbf{S}}_{2,2}&=\left[\begin{array}{l} {\boldsymbol{s}}_{2,2}\\ \overset{\rightharpoonup }{\textrm{A}_{2}\textrm{P}}\times {\boldsymbol{s}}_{2,2} \end{array}\right]=\left[\begin{array}{llllll} 0 & 0 & 1 & y_{\textrm{P}}-y_{{\textrm{A}_{2}}} & -\left(x_{\textrm{P}}-x_{{\textrm{A}_{2}}}\right) & 0 \end{array}\right]^{\textrm{T}} \end{align}

(9) \begin{align} \hat{\textbf{S}}_{3,2}&=\left[\begin{array}{l} {\boldsymbol{s}}_{3,2}\\ \overset{\rightharpoonup }{\textrm{A}_{2}\textrm{P}}\times {\boldsymbol{s}}_{3,2} \end{array}\right]=\left[\begin{array}{l} \cos \left(\varphi +\psi \right)\\ \sin \left(\varphi +\psi \right)\\ 0\\ -\left(z_{\textrm{P}}-z_{{\textrm{A}_{2}}}\right)\sin \left(\varphi +\psi \right)\\ \left(z_{\textrm{P}}-z_{{\textrm{A}_{2}}}\right)\cos \left(\varphi +\psi \right)\\ \sin \left(\varphi +\psi \right)\left(x_{\textrm{P}}-x_{{\textrm{A}_{2}}}\right)-\cos \left(\varphi +\psi \right)\left(y_{\textrm{P}}-y_{{\textrm{A}_{2}}}\right) \end{array}\right] \end{align}

(10) \begin{align} \hat{\textbf{S}}_{4,2}&=\left[\begin{array}{l} {\boldsymbol{s}}_{4,2}\\ \overset{\rightharpoonup }{\textrm{B}_{2}\textrm{P}}\times {\boldsymbol{s}}_{4,2} \end{array}\right]=\left[\begin{array}{l} \cos \left(\varphi +\psi \right)\\ \sin \left(\varphi +\psi \right)\\ 0\\ -\left(z_{\textrm{P}}-z_{{\textrm{B}_{2}}}\right)\sin \left(\varphi +\psi \right)\\ \left(z_{\textrm{P}}-z_{{\textrm{B}_{2}}}\right)\cos \left(\varphi +\psi \right)\\ \sin \left(\varphi +\psi \right)\left(x_{\textrm{P}}-x_{{\textrm{B}_{2}}}\right)-\cos \left(\varphi +\psi \right)\left(y_{\textrm{P}}-y_{{\textrm{B}_{2}}}\right) \end{array}\right] \end{align}

(11) \begin{align} \hat{\textbf{S}}_{5,2}&=\left[\begin{array}{l} {\boldsymbol{s}}_{5,2}\\ \overset{\rightharpoonup }{\textrm{C}_{2}\textrm{P}}\times {\boldsymbol{s}}_{5,2} \end{array}\right]=\left[\begin{array}{llllll} \cos \left(\varphi +\psi \right) & \sin \left(\varphi +\psi \right) & 0 & 0 & 0 & 0 \end{array}\right]^{\textrm{T}} \end{align}

The reciprocal screw can be obtained using the following equations:

(12) \begin{align} \left.\begin{array}{l} {\hat{\textbf{S}}^{r}}_{1,1}\circ \,\hat{\textbf{S}}_{1,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,1}\circ \,\hat{\textbf{S}}_{2,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,1}\circ \,\hat{\textbf{S}}_{3,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,1}\circ \,\hat{\textbf{S}}_{4,1}=0 \end{array}\right\} \end{align}

(13) \begin{align} \left.\begin{array}{l} {\hat{\textbf{S}}^{r}}_{2,1}\circ \,\hat{\textbf{S}}_{1,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{2,1}\circ \,\hat{\textbf{S}}_{2,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{2,1}\circ \,\hat{\textbf{S}}_{3,1}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{2,1}\circ \,\hat{\textbf{S}}_{4,1}=0 \end{array}\right\} \end{align}

(14) \begin{align} \left.\begin{array}{l} {\hat{\textbf{S}}^{r}}_{1,2}\circ \,\hat{\textbf{S}}_{1,2}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,2}\circ \,\hat{\textbf{S}}_{2,2}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,2}\circ \,\hat{\textbf{S}}_{3,2}=0\\[3pt] \begin{array}{l} {\hat{\textbf{S}}^{r}}_{1,2}\circ \,\hat{\textbf{S}}_{4,2}=0\\[3pt] {\hat{\textbf{S}}^{r}}_{1,2}\circ \,\hat{\textbf{S}}_{5,2}=0 \end{array} \end{array}\right\} \end{align}

Thus, the reciprocal screw for Limb 1 and Limb 2 is given by:

(15) \begin{equation} \left[\begin{array}{lll} {\hat{\textbf{S}}^{r}}_{1,1} &\quad {\hat{\textbf{S}}^{r}}_{2,1} &\quad {\hat{\textbf{S}}^{r}}_{1,2} \end{array}\right]=\left[\begin{array}{ccc} 0 &\quad 0 &\quad 0\\ 0 &\quad 1 &\quad 0\\ 0 &\quad 0 &\quad 0\\ 1 &\quad h_{1} &\quad -\sin \left(\varphi +\psi \right)\\ 0 &\quad 0 &\quad \cos \left(\varphi +\psi \right)\\ 0 &\quad x_{\textrm{P}}-x_{\textrm{O}} &\quad 0 \end{array}\right] \end{equation}

where h ₁ can be assigned an arbitrary value.

The constraint matrix of the mechanism is derived by combining the reciprocal screws of both limbs:

(16) \begin{equation} \textbf{J}^{\prime}_{\textbf{C}}=\left[\begin{array}{l} {\hat{\textbf{S}}^{r}_{1,1}}{}^{\textrm{T}}\\[3pt] {\hat{\textbf{S}}^{r}_{2,1}}{}^{\textrm{T}}\\[3pt] {\hat{\textbf{S}}^{r}_{1,2}}{}^{\textrm{T}} \end{array}\right]=\left[\begin{array}{llllll} 0 &\quad 0 &\quad 0 &\quad 1 &\quad 0 &\quad 0\\ 0 &\quad 1 &\quad 0 &\quad h_{1} &\quad 0 &\quad x_{\textrm{P}}-x_{\textrm{O}}\\ 0 &\quad 0 &\quad 0 &\quad -\sin \left(\varphi +\psi \right) &\quad \cos \left(\varphi +\psi \right) &\quad 0 \end{array}\right] \end{equation}

By rearranging columns and applying elementary row operations, the constraint matrix can be rewritten as:

(17) \begin{equation} \textbf{J}_{\textbf{C}}=\left[\begin{array}{llllll} 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad x_{\textrm{P}}-x_{\textrm{O}} &\quad 0 &\quad 1 &\quad 0\\ 0 &\quad 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \end{array}\right] \end{equation}

Since the constraint forces do not perform mechanical work on the mechanism, the motion constraints are obtained as follows:

(18) \begin{align}& \textbf{J}_{\textbf{C}}\left[\begin{array}{l} \textbf{w}\\ \textbf{v} \end{array}\right]=\textbf{J}_{\textbf{C}}\left[\begin{array}{l} \omega _{x}\\ \omega _{y}\\ \omega _{z}\\ v_{x}\\ v_{y}\\ v_{z} \end{array}\right]=0 \end{align}

(19) \begin{align}& \left.\begin{array}{l} \omega _{x}=0\\ \omega _{z}\cdot \left(x_{\textrm{P}}-x_{\textrm{O}}\right)+v_{y}=0\\ \omega _{y}=0 \end{array}\right\} \end{align}

It is evident that the output link exhibits three DoF. The rotations about the x-axis and y-axis are constrained, while the angular velocity about the z-axis and translational velocity in the y-direction have a proportional relationship with the coefficient x _P − x _O.

As shown in Figure 5, modifying the distance between the two sliders causes the end-effector to move approximately in the radial direction with a slight change in the height, corresponding to the “reaching movement.” Rotating the active rotational joint A₁ moves the end-effector in the vertical direction, corresponding to the “lifting movement.” Moving both sliders in the same direction while maintaining their relative distance results in the circumferential motion of the end-effector.

Figure 5. The motion of the mechanism in (a) Vertical direction, (b) Radial direction with a slight change in the height, and (c) Circumferential direction.

3.2. Displacement analysis

The coordinates, mechanism constants, and variables used for displacement analysis are defined in Figure 6. The O-XYZ coordinate system is a stationary coordinate system with its origin O at the center of the arc-shaped guide rail. The link lengths of Limb 1 are l _1,1, l _2,1, l _3,1, while those of Limb 2 are l _1,2, l _2,2, l _3,2. The angular displacement of Slider 1 and Slider 2 relative to the X-axis on the XY-plane is given by θ _1,1 and θ _2,1. The angular displacement of joint A₁ relative to the XY-plane is given by θ _2,1.

Figure 6. The coordinates, mechanism constants, and variables for displacement analysis: (a) Mechanism in 3-Dimensions, (b) Projection of the mechanism onto the XY-plane, and (c) Projection of the mechanism onto the uZ- plane.

Figure 6b shows a projection of the mechanism onto the XY-plane, viewed from the positive Z-axis direction. The angle between OC₁ and A₂C₂ is denoted as $\alpha$ . The position of the end-effector P is defined as the intersection of the line passing through C₁, perpendicular to OC₁, and the line passing through C₂, perpendicular to A₂C₂. The distance between P to A₂C₂, as well as OC₁, is presented as l _e. Point Q is the intersection of the extended line of OC₁ and A₂C₂. A new u-axis is defined along OC₁ in Figure 6b, allowing Limb 1 to be described in the uZ-plane, as shown in Figure 6c. The projection of point P on the u-axis is designated as point R.

For simplicity, interference between the mechanism components, and interference between the mechanism and the human body are neglected in the following calculations.

3.2.1. Forward displacement analysis

In forward displacement analysis, the position of the end-effector P(X _P, Y _P, Z _P) is determined based on the displacements of the active joints θ _1,1, θ _2,1, and θ _1,2.

First, the XY-coordinates of point P are considered. The coordinates of point Q(X _Q, Y _Q) in Figure 6b can be calculated as follows:

(20) \begin{equation} \left.\begin{array}{l} X_{\textrm{Q}}=\dfrac{l_{1,2}\sin \theta _{1,2}-l_{1,2}\cos \theta _{1,2}\tan \left(\theta _{1,1}-\alpha \right)}{\tan \theta _{1,1}-\tan \left(\theta _{1,1}-\alpha \right)}\\ Y_{\textrm{Q}}=\tan \theta _{1,1}\cdot X_{\textrm{Q}} \end{array}\right\} \end{equation}

Using X _Q and Y _Q, the XY-coordinates of point P(X _P, Y _P) are expressed as:

(21) \begin{equation} \left.\begin{array}{l} X_{\textrm{P}}=X_{\textrm{Q}}-\dfrac{l_{\textrm{e}}}{\sin \frac{\alpha }{2}}\cos \left(\theta _{1,1}-\dfrac{\alpha }{2}\right)\\[5pt] Y_{\textrm{P}}=Y_{\textrm{Q}}-\dfrac{l_{\textrm{e}}}{\sin \frac{\alpha }{2}}\sin \left(\theta _{1,1}-\dfrac{\alpha }{2}\right) \end{array}\right\} \end{equation}

Next, the Z-coordinate of point P is determined. In Figure 6c, Z _P can be calculated as the intersection between a circle with a radius of l _3,1 centered at point B₁ and the line u = u _P:

(22) \begin{equation} Z_{\textrm{P}}=\sqrt{{l_{3,1}}^{2}-\left(u_{\textrm{P}}-u_{\textrm{B},1}\right)^{2}}+Z_{\textrm{B},1} \end{equation}

where

(23) \begin{equation} \left.\begin{array}{l} u_{\textrm{P}}=\textrm{OR}=\sqrt{{x_{\textrm{P}}}^{2}+{y_{\textrm{P}}}^{2}}\cos \left\{\tan ^{-1}\left(\dfrac{y_{\textrm{P}}}{x_{\textrm{P}}}\right)-\theta _{1,1}\right\}\\ u_{\textrm{B},1}=l_{1,1}+l_{2,1}\cos \theta _{2,1}\\ Z_{\textrm{B},1}=l_{2,1}\sin \theta _{2,1} \end{array}\right\} \end{equation}

3.2.2. Inverse displacement analysis

In inverse displacement analysis, the displacements of the active joints θ _1,1, θ _2,1, and θ _1,2 are determined based on the position of the end-effector (X _P, Y _P, Z _P).

The angular displacement θ _1,1 and θ _1,2 of Slider 1 and Slider 2 can be calculated using X _P and Y _P.

(24) \begin{equation} \theta _{1,1}=\tan ^{-1}\left(\frac{X_{\textrm{P}}}{Y_{\textrm{P}}}\right)-\sin ^{-1}\left(\frac{l_{\textrm{e}}}{\sqrt{{X_{P}}^{2}+{Y_{P}}^{2}}}\right) \end{equation}

Using the calculated θ _1,1, the coordinates of point Q(X _Q, Y _Q) can be expressed as follows:

(25) \begin{equation} \left.\begin{array}{l} X_{\textrm{Q}}=X_{\textrm{P}}+\dfrac{l_{\textrm{e}}}{\sin \frac{\alpha }{2}}\cos \left(\theta _{1,1}-\dfrac{\alpha }{2}\right)\\ Y_{\textrm{Q}}=X_{\textrm{P}}+\dfrac{l_{\textrm{e}}}{\sin \frac{\alpha }{2}}\sin \left(\theta _{1,1}-\dfrac{\alpha }{2}\right) \end{array}\right\} \end{equation}

The value of θ _1,2 is determined as follows:

(26) \begin{equation} \theta _{1,2}=\pm \cos ^{-1}\left\{\frac{Y_{\textrm{Q}}\cos \left(\theta _{1,1}-\alpha \right)-X_{\textrm{Q}}\sin \left(\theta _{1,1}-\alpha \right)}{l_{1,2}}\right\}+\left(\theta _{1,1}-\alpha +\frac{\pi }{2}\right) \end{equation}

From the sign of the first term on the right-hand side, it is evident that the inverse kinematic solution for $\theta\!$ _1,2 has two possible values. The positive solution results in link A₂B₂ being positioned inside the arc-shaped guide rail, which is unsuitable. Therefore, the negative solution is selected for the subsequent analysis. The value of θ _2,1 can be calculated as follows, based on Figure 6c.

(27) \begin{equation} \theta _{2,1}=\angle \textrm{PA}_{1}\textrm{R}-\angle \textrm{PA}_{1}\textrm{B}_{1}=\tan ^{-1}\left(\frac{Z_{\textrm{P}}}{u_{\textrm{AR}}}\right)-\cos ^{-1}\left(\frac{{u_{\textrm{AR}}}^{2}+{Z_{\textrm{P}}}^{2}+{l_{2,1}}^{2}-{l_{3,1}}^{2}}{2l_{2,1}\sqrt{{u_{\textrm{AR}}}^{2}+{Z_{\textrm{P}}}^{2}}}\right) \end{equation}

where

(28) \begin{equation} u_{\textrm{AR}}=\textrm{A}_{1}\textrm{R}=\sqrt{{X_{\textrm{P}}}^{2}+{Y_{\textrm{P}}}^{2}}\cos \left\{\tan ^{-1}\left(\frac{Y_{P}}{X_{P}}\right)-\theta _{1,1}\right\}-l_{1,1} \end{equation}

3.3. Workspace analysis

Similar to the displacement analysis, interference between the mechanism’s joints and links, as well as interference between the mechanism and the human body, are neglected in this study. In addition, the constraint $\theta\!$ _1,1 < $\theta\!$ _1,2 is applied to the sliders since the two sliders cannot pass each other. In practice, two sliders contact each other at a certain angular difference. For this analysis, however, the sliders are theoretically treated as points.

Since the XY-coordinates (X _P, Y _P) of the end-effector are determined solely by the slider positions $\theta\!$ _1,1 and $\theta\!$ _1,2, the workspace of the mechanism is analyzed in the XY-plane and Z-coordinate, respectively. Let r _P represent the distance between the origin point O and the end-effector P, with its maximum and minimum values denoted as r _P,max, r _P,min, respectively.

This can be analytically derived based on the results of the displacement analysis. In Figure 6b, the shape of the quadrilateral PC₁QC₂ remains constant, regardless of $\theta\!$ _1,1 and $\theta\!$ _1,2. Therefore, when the distance r _Q between the origin point O and point Q reaches its maximum or minimum value, the distance r _P also takes its respective maximum or minimum value. Moreover, since the mechanism is symmetric about the Z-axis, when evaluating the maximum and minimum values of r _Q, $\theta\!$ _1,1 can be fixed at 0, and only $\theta\!$ _1,2 is considered as a variable. By substituting $\theta\!$ _1,1 = 0 into Eq. (20), we obtain the Eq. (29):

(29) \begin{equation} \left.\begin{array}{l} X_{\textrm{Q}}=\dfrac{l_{1,2}}{\sin \alpha }\left(\cos \alpha \sin \theta _{1,2}+\sin \alpha \cos \theta _{1,2}\right)=\dfrac{l_{1,2}}{\sin \alpha }\cdot \sin \left(\theta _{1,2}+\alpha \right)\\ Y_{\textrm{Q}}=0 \end{array}\right\} \end{equation}

so that,

(30) \begin{equation} r_{\textrm{Q}}=\sqrt{{X_{\textrm{Q}}}^{2}+{Y_{\textrm{Q}}}^{2}}=\frac{l_{1,2}}{\sin \alpha }\cdot \sin \left(\theta _{1,2}+\alpha \right) \end{equation}

Therefore, r _Q reaches its maximum value when the term sin( $\theta\!$ _1,2+ $\alpha$ ) on the right-hand side reaches its maximum value, satisfying OA₂⊥A₂C₂:

(31) \begin{equation} \theta _{1,2}=\frac{\pi }{2}-\alpha \end{equation}

Similarly, r _Q reaches its minimum value when sin( $\theta\!$ _1,2+ $\alpha$ ) takes its minimum value. Since sin( $\theta\!$ _1,2+ $\alpha$ ) increases monotonically within the range 0 ≤ $\theta\!$ _1,2 ≤ π/2- $\alpha$ , its minimum value occurs when $\theta\!$ _1,2 = 0. Therefore, r _P = r _P,min occurs when $\theta\!$ _1,1 = $\theta\!$ _1,2.

Based on this, r _P,max and r _P,min can be determined as follows:

(32) \begin{equation} \begin{array}{l} \left.\begin{array}{l} r_{\textrm{P},\max }=\sqrt{\left(\dfrac{l_{1,2}}{\sin \alpha }-\dfrac{l_{e}}{\tan \left(\alpha /2\right)}\right)^{2}+{l_{e}}^{2}}\left(when\,\theta _{1,2}=\theta _{1,1}+\dfrac{\pi }{2}-\alpha \right)\\ r_{\textrm{P},\min }=\sqrt{\left(l_{1,2}-\dfrac{l_{e}}{\tan \left(\alpha /2\right)}\right)^{2}+{l_{e}}^{2}}\left(when\,\theta _{1,1}=\theta _{1,2}\right) \end{array}\right\}\\ \end{array} \end{equation}

The workspace in the XY-plane is represented as a doughnut-shaped region, indicated by the gray area in Figure 7a.

Figure 7. The workspace of the mechanism in (a) XY-plane and (b) Z-axis direction.

Next, the maximum value Z _P,max and minimum value Z _P,min are determined, given values of X _P and Y _P. First, consider only Limb 1. Figure 7b represents Limb 1 in the uZ-plane. Z _P reaches its maximum value Z _P,max,1 or minimum value Z _P,min, when points A₁, B₁, and C₁ are aligned in a straight line. Therefore, Z _P,max,1 and Z _P,min,1 can be expressed as follows:

(33) \begin{equation} \begin{array}{l} Z_{\textrm{P},\max ,1}=\sqrt{\left(l_{2,1}+l_{3,1}\right)^{2}-{u_{\textrm{AR}}}^{2}}\\ Z_{P,\min ,1}=-Z_{\textrm{P},\max ,1}=-\sqrt{\left(l_{2,1}+l_{3,1}\right)^{2}-{u_{\textrm{AR}}}^{2}} \end{array} \end{equation}

The value of u _AR can be determined using (28).

Similarly, considering only Limb 2, Figure 7c represents Limb 2 in the vZ ₂-plane, where the Z ₂-axis is defined as the line parallel to the Z-axis, passing through point A₂, and the v-axis corresponds to the A₂C₂ direction as shown in Figure 6b. For Limb 2, Z _P reaches its maximum value Z _P,max,2 or minimum value Z _P,min,2 when points A₂, B₂, and C₂ are aligned in a straight line. Therefore, Z _P,max,2 and Z _P,min,2 can be expressed as follows:

(34) \begin{equation} \begin{array}{l} \left.\begin{array}{l} Z_{\textrm{P},\max ,2}=\sqrt{\left(l_{2,2}+l_{3,2}\right)^{2}-{v_{\textrm{AS}}}^{2}}\\ Z_{\textrm{P},\min ,2}=-Z_{\textrm{P},\max ,2}=-\sqrt{\left(l_{2,2}+l_{3,2}\right)^{2}-{v_{\textrm{AS}}}^{2}} \end{array}\right\}\\ \end{array} \end{equation}

where

(35) \begin{equation} v_{\textrm{AS}}=\sqrt{\left\{x_{\textrm{P}}-l_{e}\sin \left(\theta _{1,1}-\alpha \right)-l_{1,2}\cos \theta _{1,2}\right\}^{2}+\left\{y_{\textrm{P}}+l_{e}\cos \left(\theta _{1,1}-\alpha \right)-l_{1,2}\sin \theta _{1,2}\right\}^{2}} \end{equation}

Based on this, considering both Limb 1 and Limb 2, Z _P,max is constrained by the smaller value between Z _P,max,1 and Z _P,max,2, while Z _P,min is constrained by the larger value between Z_P,min,1 and Z_P,min,2. Therefore, Z _P,max and Z _P,min can be expressed as follows:

(36) \begin{equation} \left.\begin{array}{l} Z_{\textrm{P},\max }=\min \left(Z_{\textrm{P},\max ,1},\,Z_{\textrm{P},\max ,2}\right)\\ Z_{\textrm{P},\min }=\max \left(Z_{\textrm{P},\min ,1},\,Z_{\textrm{P},\min ,2}\right) \end{array}\right\} \end{equation}

In addition, the end-effector cannot reach within a circular region centered at A₁ of Limb1 and A₂ of Limb2, with radii r = ∣l _2,1−l _3,1∣ and r = ∣ l _2,2−l _3,2∣, respectively. Therefore, these regions must be excluded from the workspace.

Based on the above analysis, the workspace in three-dimensional space can be determined by obtaining the boundary of the workspace from the XY-plane and the Z-axis direction while excluding the unreachable regions. Since the mechanism is symmetric about the Z-axis, the three-dimensional workspace can be obtained by rotating the XZ plane workspace 360° around the Z-axis.

6.1. Experimental setup

In both the assistance experiment and the motion measurement, the arm cuff of the robot was moved along a straight trajectory with PD control, following a trapezoidal acceleration and deceleration profile with an acceleration of a = 0.5 m/s² and a maximum velocity of v _max = 0.3 m/s. The total motion time was about 1.67 s. Muscle activity measurements and the robot’s motion were synchronized using a trigger switch (Biometrics Ltd, UK). The block diagram of the controller is shown in Figure 12. In the block diagram, ${\boldsymbol{x}}_{d}$ is the target trajectory of the end-effector, and $\boldsymbol{\theta }_{d}$ and $\boldsymbol{\theta }_{a}$ indicate the target angle and measured angle of each active joint, respectively. $\dot{{\boldsymbol{x}}}_{d}$ and $\dot{\boldsymbol{\theta }}_{d}$ are the corresponding velocity and angular velocity. ${\boldsymbol{V}}$ is the driving voltage. ${\boldsymbol{F}}_{\textit{assist}}$ is the assistive force. ${\boldsymbol{K}}_{p}$ and ${\boldsymbol{K}}_{d}$ are the proportional gain and derivative gain, respectively. ${\boldsymbol{s}}$ is the Laplace variable.

Figure 12. The block diagram of the PD controller.

In the condition without robot assistance, participants were instructed to move their arms along the same trajectory as in the robot-assisted condition, maintaining the same start and end positions. The motion was guided by a metronome, set at a beat interval of 60/72 s. The specific motion was performed over two beats, ensuring consistency with the robot-assistance condition.

6.2. Results of muscle activity

In the assistance experiment, the specified motion involved a straight-line movement from the position at (0.24, 0.32, 0) to (0.1, 0.37, 0.30) (m) in the coordinate shown in Figure 9a. This motion simulates the daily task of picking up an object from a table and holding it in a static position. Figure 9b shows an example of the experimental setup with robot assistance.

The EMG voltage V _EMG(t) at each moment was computed from the raw EMG data e(t) using the following formula, with a window length T = 300 ms:

(45) \begin{equation} V_{\textrm{EMG}}\left(t\right)=\sqrt{\frac{1}{T}\int _{-T/2}^{T/2}e^{2}\left(t+\tau \right)d\tau } \end{equation}

This process was performed for each of the five trials under the same conditions, and the average V _EMG(t) across the trials is shown in Figure 13. Moreover, the mean absolute value (MAV) of V _EMG(t) was calculated over the entire duration of 1.67 s, representing the overall muscle activity for each trial. An independent two-tailed t-test has been conducted to compare the difference in conditions with and without robot assistance. The results shown in Figure 14 indicate that the muscle activities of the infraspinatus, anterior deltoid, and biceps brachii were reduced in the robot-assistance condition, confirming the effectiveness of assistance. In particular, for all four subjects, the biceps brachii muscle activity was significantly decreased.

Figure 13. RMS from four subjects under the condition with and without robot assistance.

Figure 14. Comparison of EMG data from four subjects.

6.3. Results of measured end-effector displacement

The measured coordinates of the starting and endpoint for each motion are summarized in Table III.

The standard deviations of relative displacement along the x, y, and z axes (σ _x , σ _y , σ _z ) were computed, and a 3D repeatability radius was defined as:

(46) \begin{equation} R=\sqrt{{\sigma _{x}}^{2}+{\sigma _{y}}^{2}+{\sigma _{z}}^{2}} \end{equation}

Table IV presents the mean displacement, standard deviations, and repeatability results for the four motions.

The repeatability was 0.43 mm for circumferential motion, 0.66 mm for rising motion, 0.46 mm for reaching motion, and 0.40 mm for combined motion. These results indicate that WELiBot consistently reproduced similar motion displacements across repeated motion tasks, demonstrating stable and reliable actuation performance.