2.1. Design of double C-shaped tape spring

The tape spring is a thin-walled open cylindrical structure that can achieve a deployable and folded state. To improve the stiffness and stability of the tape spring structure, two thin-walled open cylindrical structures are combined face-to-face to form a double C-shaped tape spring. They are tightly wrapped in a polyolefin tube by thermoforming, as shown in Fig. 1(a). This method could simplify the manufacturing process. Two open cylindrical structures are not fixed, reducing the creep effect of long-term accumulation. During the coiling process, the tight polyolefin tube can effectively reduce the tendency of the open cylindrical structure to buckle locally under pressure, improving the coiling efficiency. Fig. 1(b) shows the cross-section of the folded double C-shaped tape spring, which is flattened. The coiled double C-shaped tape spring is shown in Fig. 1(c), and the cross-sections of the deployable and folded segments correspond to Fig. 1(a) and (b), respectively. The double C-shaped tape spring structure can significantly improve the stiffness of the tape spring and the stability of the manipulator, which will be analyzed in Section 5.

Figure 1. Diagram of a double C-shaped tape spring. (a) Cross-section; (b) Cross-section after folding; (c) Coiled tape spring.

2.2. Mobile drive component

The tape spring has unique properties. Fig. 2(a) shows that the straight segments of the tape spring can be stiff enough to withstand reasonable compressive loads before buckling, and the folded segments have zero transverse curvature. As shown in Fig. 2(b), the folded segment can slide along the tape spring if it is subjected to an external torque. Inspired by the above characteristics, we place the MDC on the localized folded segment of the tape spring. The MDC has one driving pulley and three driven rollers, and the three driven rollers are paired under the driving pulley and located on the same arc, as shown in Fig. 2(c). The driving pulley and three driven rollers flatten the tape spring, reducing the folded segment’s resistant torsion. As the driving pulley rotates, the MDC imposes an external torque on the fold, and the fold and the MDC can move along the tape spring.

Figure 2. Design principle of the MDC. (a) Localized folded configuration; (b) Folded segment of the tape spring sliding; (c) Schematic diagram of MDC.

The detailed structure of the MDC is shown in Fig. 3. The MDC mainly consists of a U-shaped frame, driving shaft, driving pulley, driven shaft, driven rollers, bearings, and a straight pin. When the driving pulley is actuated by the servo motor, the MDC and the localized fold can move along the tape spring.

Figure 3. The detailed structure of the mobile drive component.

2.3. Fixed drive component

The fixed drive component is designed to store the coiled tape spring and extend or retract the tape spring. The main parts of the FDC are a servo motor, a shaft coupling, a hub, a braced frame, a fixed frame, and a constant tension mechanism, as shown in Fig. 4(a). The servo motor and braced frame are mounted on the fixed frame. One end of the tape spring is fixed on the hub. The hub is connected to the servo motor by shaft coupling, and the tape spring can extend or retract by rotating the hub.

Figure 4. Fixed drive component. (a) Structure; (b) Constant tension mechanism.

To flatten the tape spring and enhance motion reliability, a constant tension mechanism is designed to press the tape spring upon the hub. As shown in Fig. 4(b), the constant tension mechanism consists of a compression block, a rotating shaft, a tension spring, a mount pin, and a compression roller. While the tape spring is retracted or extended, the compression block presses the tape spring upon the hub, preventing it from expanding.

3.1. Structure and DOF analysis of a 2-DOF deployable parallel tape-spring manipulator

As shown in Fig. 5, the combined mobile drive component (CMDC) consists of two MDCs connected by a synchronous belt and a connecting rod. The connecting rod serves as the moving platform, and a gripper is mounted on it. Two MDCs can move synchronously by a servo motor. When the CMDC moves, the two MDCs may produce the same deviation angle with respect to the moving platform, but the moving platform does not produce the deviation angle relative to the horizontal axis. Therefore, the CMDC can solve the problem of the deviation angle of the MDC. Similarly, two FDCs are connected by a synchronous belt to form a combined fixed drive component.

Figure 5. Combined mobile drive component.

Herein, a novel 2-DOF deployable parallel tape-spring manipulator is proposed based on CMDC and CFDC. As shown in Fig. 6, the double V-shaped 2-DOF deployable parallel manipulator consists of two tape springs, a CMDC, a CFDC, and two revolute joints. The CFDC is fixed on the aluminum frame, and the ends of the tape springs are hinged with the frame through the revolute joints. The CMDC located on the fold can move along the tape springs. The two FDCs are driven by a synchronous belt that allows the two V-shaped tape springs to simultaneously shorten or extend the same length. By controlling the CMDC and CFDC, the moving platform can translate in the plane.

Figure 6. 2-DOF double V-shaped deployable parallel manipulator.

Fig. 7 illustrates the schematic diagram of the mechanism. The mobile drive component can be considered as a 2-DOF kinematic pair, encompassing one translational mobility and one rotational mobility, as depicted in Fig. 7(a). The fixed drive component can be regarded as a 1-DOF kinematic pair with one translational mobility, as presented in Fig. 7(b).

Figure 7. Schematic diagram of the mechanism. (a). Mobile drive component, (b). Fixed drive component, (c). Single V-shaped configuration, (d). Double V-shaped configuration.

To calculate the mobility of a mechanism, the Unified Modified G-K Criterion is given by relevant literature [Reference Gogu38–Reference Dai and Jones40]

(1) \begin{equation} M=d\!\left(n-g-1\right)+\sum _{i=1}^{g}f_{i}+v-\zeta \end{equation}

where $M$ denotes the mobility of a mechanism, $d$ is the order of a mechanism, $n$ is the number of links including the frame, $g$ is the number of kinematic joints, $f_{i}$ is the number of freedoms of the ith joint, $v$ is the number of redundant constraints except the number having been accounted in common constraints. $\zeta$ is the number of local freedoms.

The schematic diagram of the single V-shaped configuration is shown in Fig. 7(c). In this single V-shaped mechanism, $n$ is 4, $g$ is 4, $\sum _{i=1}^{4}f_{i}$ is 5. The mobility of this mechanism is

(2) \begin{equation} Ms=3\times \left(4-4-1\right)+5=2 \end{equation}

We connect two V-shaped mechanisms in parallel, as shown in Fig. 7(d). The mobility of the double V-shaped mechanism is

(3) \begin{equation} Md=3\times \left(8-10-1\right)+12=3 \end{equation}

By utilizing the Gr FC;bler’s formula, we can initially determine the DOF of the mechanism. However, details regarding the DOF of the mechanism and how to increase constraints need further investigation through the screw theory.

The schematic diagram of the 2-DOF manipulator is shown in Fig. 8. $\mathit{D}_{1}$ and $\mathit{D}_{2}$ denote the FDCs, $\mathit{M}_{1}$ and $\mathit{M}_{1}$ denote the MDCs, and $\mathit{A}_{1}$ and $\mathit{A}_{2}$ denote the hinge joints which connect the end of the tape springs to the aluminum frame. $\mathit{D}_{1}$ and $\mathit{D}_{2}$ , $\mathit{M}_{1}$ and $\mathit{M}_{2}$ are connected by synchronous belts to realize the synchronous motion. The lengths of D ₁ D ₂, A ₁ A ₂, and M ₁ M ₂ are L ₁, and D ₁ A ₁ and D ₂ A ₂ are L ₂. The midpoint between D ₁ and A ₂ is set as the coordinate origin. The lengths of D ₁ M ₁, M ₁ A ₁, D ₂ M ₂ and M ₂ A ₂ are l ₁, l ₂, l ₃ and l ₄. Angles $\mathit{\theta }_{1},\mathit{\theta }_{2},\mathit{\theta }_{3},\mathit{\theta }_{4},\mathit{\theta }_{5},\mathit{\theta }_{6},\mathit{\theta }_{21},\text{ and }\mathit{\theta }_{41}$ are as shown in the figure. Therefore, the coordinates of each point are as follows: D ₁ $(x_{\mathrm{d}1},y_{\mathrm{d}1},0)$ , M ₁ $(x_{\mathrm{m}1},y_{\mathrm{m}1},0)$ , D ₂ $(x_{\mathrm{d}1}+L1,y_{\mathrm{d}1},0)$ , M ₂ $(x_{\mathrm{m}1}+L1,y_{\mathrm{m}1},0)$ , A ₁ $(x_{\mathrm{d}1}+L2,y_{\mathrm{d}1},0)$ , and A ₂ $(x_{\mathrm{d}1}+L1+L2,y_{\mathrm{d}1},0)$ .

Figure 8. Schematic diagram of the 2-DOF manipulator.

Then, the screw coordinates of the different joints can be determined. Limb D ₁ M ₁ is used as an example, while other limbs are not listed.

(4) \begin{equation} \begin{array}{l} \textit{\$} _{\mathrm{D}1}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 1 & y_{d1} & -x_{d1} & 0 \end{array}\right)\\[3pt] \textit{\$} _{\mathrm{D}1\mathrm{M}1}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & x_{m1}-x_{d1} & y_{m1}-y_{d1} & 0 \end{array}\right)\\[3pt] \textit{\$} _{\mathrm{M}1}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 1 & y_{m1} & -x_{m1} & 0 \end{array}\right) \end{array} \end{equation}

Specifically, selecting $\overset{\frown }{D_{1}M_{1}M_{2}D_{2}}, \overset{\frown }{D_{1}M_{1}A_{1}}$ , and $\overset{\frown }{D_{2}M_{2}A_{2}}$ , their screw equations are as follows:

(5) \begin{equation} \begin{array}{l} \textit{\$} _{\mathrm{D}1}\dot{\theta }_{1}+\textit{\$} _{\mathrm{D}1\mathrm{M}1}\dot{l}_{1}+\textit{\$} _{\mathrm{M}1}\dot{\theta }_{2}=\textit{\$} _{\mathrm{D}2}\dot{\theta }_{3}+\textit{\$} _{\mathrm{D}2\mathrm{M}2}\dot{l}_{3}+\textit{\$} _{\mathrm{M}2}\dot{\theta }_{4}\\[3pt] \textit{\$} _{\mathrm{D}1}\dot{\theta }_{1}+\textit{\$} _{\mathrm{D}1\mathrm{M}1}\dot{l}_{1}+\textit{\$} _{\mathrm{M}1}\dot{\theta }_{21}=\textit{\$} _{\mathrm{A}1}\dot{\theta }_{5}+\textit{\$} _{\mathrm{A}1\mathrm{M}1}\dot{l}_{2}\\[3pt] \textit{\$} _{\mathrm{D}2}\dot{\theta }_{3}+\textit{\$} _{\mathrm{D}2\mathrm{M}2}\dot{l}_{3}+\textit{\$} _{\mathrm{M}2}\dot{\theta }_{41}=\textit{\$} _{\mathrm{A}2}\dot{\theta }_{6}+\textit{\$} _{\mathrm{A}2\mathrm{M}2}\dot{l}_{4} \end{array} \end{equation}

These equations can be formulated into the matrix,

(6) \begin{equation} \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \textit{\$} _{\mathrm{D}1} & \textit{\$} _{\mathrm{M}1} & -\textit{\$} _{\mathrm{D}2} & -\textit{\$} _{\mathrm{M}2} & 0 & 0 & 0 & 0\\[3pt] \textit{\$} _{\mathrm{D}1} & 0 & 0 & 0 & -\textit{\$} _{\mathrm{A}1} & 0 & \textit{\$} _{\mathrm{M}1} & 0\\[3pt] 0 & 0 & \textit{\$} _{\mathrm{D}2} & 0 & 0 & -\textit{\$} _{\mathrm{A}2} & 0 & \textit{\$} _{\mathrm{M}2} \end{array} \begin{array}{c@{\quad}c} \textit{\$} _{\mathrm{D}1\mathrm{M}1} & 0\\[3pt] \textit{\$} _{\mathrm{D}1\mathrm{M}1} & -\textit{\$} _{\mathrm{A}1\mathrm{M}1}\\[3pt] 0 & 0 \end{array}\begin{array}{c@{\quad}c} -\textit{\$} _{\mathrm{D}2\mathrm{M}2} & 0\\[3pt] 0 & 0\\[3pt] \textit{\$} _{\mathrm{D}2\mathrm{M}2} & -\textit{\$} _{\mathrm{A}2\mathrm{M}2} \end{array}\right]\left[\begin{array}{l} \dot{\theta }_{1}\\[3pt] \dot{\theta }_{2}\\[3pt] \dot{\theta }_{3}\\[3pt] \dot{\theta }_{4}\\[3pt] \dot{\theta }_{5}\\[3pt] \dot{\theta }_{6}\\[3pt] \dot{\theta }_{21}\\[3pt] \begin{array}{l} \dot{\theta }_{41}\\[3pt] \begin{array}{l} \dot{l}_{1}\\[3pt] \begin{array}{l} \dot{l}_{2}\\[3pt] \dot{l}_{3}\\[3pt] \dot{l}_{4} \end{array} \end{array} \end{array} \end{array}\right]=0 \end{equation}

where the coefficient matrix is

(7) \begin{equation} Q=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \textit{\$} _{\mathrm{D}1} & \textit{\$} _{\mathrm{M}1} & -\textit{\$} _{\mathrm{D}2} & -\textit{\$} _{\mathrm{M}2} & 0 & 0 & 0 & 0 & \textit{\$} _{\mathrm{D}1\mathrm{M}1} & 0 & -\textit{\$} _{\mathrm{D}2\mathrm{M}2} & 0\\[3pt] \textit{\$} _{\mathrm{D}1} & 0 & 0 & 0 & -\textit{\$} _{\mathrm{A}1} & 0 & \textit{\$} _{\mathrm{M}1} & 0 & \textit{\$} _{\mathrm{D}1\mathrm{M}1} & -\textit{\$} _{\mathrm{A}1\mathrm{M}1}& 0 & 0\\[3pt] 0 & 0 & \textit{\$} _{\mathrm{D}2} & 0 & 0 & -\textit{\$} _{\mathrm{A}2} & 0 & \textit{\$} _{\mathrm{M}2} & 0 & 0 & \textit{\$} _{\mathrm{D}2\mathrm{M}2} & -\textit{\$} _{\mathrm{A}2\mathrm{M}2} \end{array} \right]\end{equation}

Determining the null space vectors of the coefficient matrix $Q$ ,

(8) \begin{equation} P=null\!\left(Q\right) \end{equation}

This yields a three-column vector, indicating that the mechanism has 3 DOFs namely two translations and one rotation. To design a 2-DOF translational mechanism, it is necessary to maintain the parallelogram configuration, that is, l ₁ = l ₃ and l ₂ = l ₄. To satisfy these conditions, we introduce synchronous pulley constraints between D ₁ and D ₂, as well as between M ₁ and M ₂. Then, $\dot{l}_{1} = \dot{l}_{3}$ and $\dot{l}_{2} = \dot{l}_{4}$ .

Therefore, the coefficient matrix is augmented by adding two rows for the constraints.

(9) \begin{equation} Q=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \textit{\$} _{\mathrm{D}1} & \textit{\$} _{\mathrm{M}1} & -\textit{\$} _{\mathrm{D}2} & -\textit{\$} _{\mathrm{M}2} & 0 & 0 & 0 & 0 & \textit{\$} _{\mathrm{D}1\mathrm{M}1} & 0 & -\textit{\$} _{\mathrm{D}2\mathrm{M}2} & 0\\[3pt] \textit{\$} _{\mathrm{D}1} & 0 & 0 & 0 & -\textit{\$} _{\mathrm{A}1} & 0 & \textit{\$} _{\mathrm{M}1} & 0 & \textit{\$} _{\mathrm{D}1\mathrm{M}1} & -\textit{\$} _{\mathrm{A}1\mathrm{M}1} & 0 & 0\\[3pt] 0 & 0 & \textit{\$} _{\mathrm{D}2} & 0 & 0 & -\textit{\$} _{\mathrm{A}2} & 0 & \textit{\$} _{\mathrm{M}2} & 0 & 0 & \textit{\$} _{\mathrm{D}2\mathrm{M}2} & -\textit{\$} _{\mathrm{A}2\mathrm{M}2}\\[3pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0\\[3pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 \end{array}\right] \end{equation}

The null space vector of the coefficient matrix $Q$ is two columns, indicating that the mechanism has 2 DOFs. We can select $\dot{l}_{1}$ and $\dot{l}_{2}$ as the two driving inputs. Substituting the null space vectors of the coefficient matrix $Q$ into the expression $\textit{\$} _{\mathrm{D}1}\dot{\theta }_{1}+\textit{\$} _{\mathrm{D}1\mathrm{M}1}\dot{l}_{1}+\textit{\$} _{\mathrm{M}1}\dot{\theta }_{21}$ , two motion screws for the moving platform B ₁ B ₂ can be obtained:

(10) \begin{equation} \textit{\$} _{\mathrm{M}1\mathrm{M}2}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right] \end{equation}

3.2. Structure and DOF analysis of a 3-DOF deployable parallel tape-spring manipulator

To further propose a 3-DOF deployable parallel tape-spring manipulator, we consider the 2-DOF manipulator as a sub-chain. As shown in Fig. 9, with the addition of a revolute joint based on the 2-DOF manipulator, the sub-chain can be equivalently represented by the blue lines.

Figure 9. Schematic diagram of the mechanism consisting of two 2-DOF manipulators.

The screw vectors for the endpoint of the equivalent sub-chain are:

(11) \begin{equation}\textit{\$}_{\mathrm{E}1}\left\{\begin{array}{l}\textit{\$}_{1} =\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & -y_{d} \end{array}\right)\\[6pt] \textit{\$} _{\mathrm{M}1\mathrm{M}2}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right)\\[14pt] \textit{\$} _{2}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & -y_{m} \end{array}\right) \end{array}\right. \end{equation}

The corresponding constraint screw is derived as follows:

(12) \begin{equation} \textit{\$} _{\mathrm{E}1}^{r}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 0 & 1 & 0\\[3pt] 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \end{equation}

When the above sub-chain is rotated by 90 degrees, the corresponding constraint screw for the endpoint of the sub-chain becomes:

(13) \begin{equation} \textit{\$} _{\mathrm{E}2}^{r}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \end{equation}

The constraint screw of the mechanism constructed by the two sub-chains is

(14) \begin{equation} \textit{\$} _{\mathrm{E}}^{r}=\left\{\begin{array}{l} \textit{\$} _{\mathrm{E}1}^{r}\\[3pt] \textit{\$} _{\mathrm{E}2}^{r} \end{array}\right. \end{equation}

Consequently, the DOF of the composite mechanism is

(15) \begin{equation} \textit{\$} _{\mathrm{E}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 0 & 0 & 1 & 0\\[3pt] 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

Therefore, the composite mechanism has three translational mobilities. It is evident that if the endpoints of the two sub-chains are rigidly connected, the moving platform of the parallel tape-spring mechanism exhibits three translational DOFs. In this context, each sub-chain contributes two input DOFs, denoted as $\dot{l}_{1}$ and $\dot{l}_{2}$ . Therefore, the composite parallel mechanism has four input DOFs and three output DOFs.

To eliminate redundancy in the composite mechanism, one potential solution is to retain only one input DOF in one of the sub-chains. However, for the tape-spring mechanism, retaining only one input cannot guarantee the expected motion. This is due to the lower stiffness of tape springs compared to traditional rigid links. When the tape spring is significantly extended, ensuring effective force transmission becomes challenging. For instance, if the tape spring is subjected to substantial pressure, it might undergo buckling, resulting in the failure of the motion transmission. Therefore, we choose to preserve four input DOFs, while the output DOF is 3. However, there is another issue that needs to be considered. To achieve coordinated motion, these four input DOFs must maintain interdependent relationships. The mismatched motion may result in mechanical jamming or deformation, posing significant challenges for control.

To enhance the feasibility of the manipulator, when the moving platforms of two sub-chains are connected, a practical approach involves enabling relative vertical movement between these two moving platforms. This configuration translates the four inputs into four outputs, with one of the outputs representing the relative platform movement. In this way, even if there is a control deviation during motion, it may result in positional errors, but it will not lead to jamming or deformation. This design serves to enhance the practicality of the mechanism.

Overall, the 3-DOF parallel tape-spring manipulator is proposed as shown in Fig. 10. The distance between the center of the synchronous belt is L ₁, and the distance between the FDC and the revolute joint is L ₂. When L ₂ < L ₁, the double V-shaped parallel manipulator can evolve into the W-shaped parallel manipulator. Two W-shaped parallel manipulators are distributed vertically to form a 3-DOF translational manipulator. The 3-DOF manipulator is actuated by four servo motors, so it is a redundant manipulator. The 3-DOF manipulator mainly includes two 2-DOF W-shaped parallel manipulators, connections, rods, T-shaped connections, and moving rods. The manipulator is mounted on the aluminum frame. The T-shaped connection is fixed on the aluminum frame. Two bases are hinged to the T-shaped connections so that the moving platform is always parallel to the ground during the movement. The connections are hinged to the rods, and the moving rod is fixed on the upper connection, while the lower connection can translate along the moving rod. Relative motion can occur between two W-shaped parallel manipulators, reducing unstable movement caused by over-coupling. The 3-DOF tape-spring manipulator can realize space operation and is easy to control.

Figure 10. Detailed structure of the 3-DOF deployable tape-spring manipulator.

4.1. Kinematic analysis of a 2-DOF deployable parallel tape-spring manipulator

The kinematic model of the 2-DOF deployable parallel tape-spring manipulator is shown in Fig. 11. $\mathit{D}_{1}$ , $\mathit{D}_{2}$ , $\mathit{M}_{1}$ , and $\mathit{M}_{2}$ are connected by synchronous belts to realize synchronous motion. The lengths of D ₁ D ₂, A ₁ A ₂, and M ₁ M ₂ are L ₁, and D ₁ A ₁ and D ₂ A ₂ are L ₂. Points D, M, and A are the midpoints of D ₁ D ₂, M ₁ M ₂, and A ₁ A ₂, respectively. The midpoint between D ₁ and A ₂ is set as the coordinate origin. To simplify the kinematic model, the impact of the pulley radius on the positions of M ₁ and M ₂ is not considered. The initial lengths of the tape spring between D ₁ and M ₁, A ₁ and M ₁, D ₂ and M ₂, and A ₂ and M ₂ are all $\mathit{l}_{0}$ . We assume that $\mathit{s}_{1}$ is the change in the overall length of the tape spring caused by the FDC $\mathit{D}_{1}$ , that is, the length of the tape spring extending or shortening by rotating the hub. $\mathit{s}_{2}$ denotes the tape spring length variation between $\mathit{A}_{1}$ and $\mathit{M}_{1}$ caused by the MDC $\mathit{M}_{1}$ . When D ₁ rotates in a clockwise direction, s ₁ is defined as positive. Likewise, when M ₁ rotates in a clockwise direction, s ₂ is defined as positive. The lengths of D ₁ M ₁ and D ₂ M ₂ are both l ₁ = l ₀ - s ₁ + s ₂, and M ₁ A ₁ and M ₂ A ₂ are l ₂ = l ₀ - s ₂.

Figure 11. Kinematic model of the 2-DOF deployable parallel tape-spring manipulator.

According to geometrical relationships, the coordinates of midpoint M of $\mathit{M}_{1}\mathit{M}_{2}$ are

(16) \begin{equation} \left\{\begin{array}{l} x=\dfrac{\left(2\mathrm{s}_{2}-s_{1}\right)\left(2l_{0}-s_{1}\right)}{2L_{2}}\\[15pt] y=-\sqrt{\left(l_{0}+s_{2}-s_{1}\right)^{2}-\left(x_{0}+L_{2}/2\right)^{2}} \end{array}\right. \end{equation}

According to Eq. (16), s ₁ and s ₂ can be expressed by x and y, respectively,

(17) \begin{equation} \left\{\begin{array}{l} s_{1}=2l_{0}-\sqrt{\left(x+L_{2}/2\right)^{2}+y^{2}}-\sqrt{\left(x-L_{2}/2\right)^{2}+y^{2}}\\[7pt] s_{2}=l_{0}-\sqrt{\left(x-L_{2}/2\right)^{2}+y^{2}} \end{array}\right. \end{equation}

Equation. (16) and (17) are the forward and inverse kinematic equations, respectively. Furthermore, s ₁ and s ₂ are analyzed as follows.

As the layer of the coiled tape spring increases, the radius of $\mathit{D}_{1}$ will increase. Therefore, the radius of $\mathit{D}_{1}$ is not constant. Herein, the coiled tape spring can be regarded as the Archimedes spiral. The radius of $\mathit{D}_{1}$ can be expressed as

(18) \begin{equation} r_1=a+b\theta_1 \end{equation}

where a is the initial radius of the spiral, b is the rate of increase of the spiral, and $\theta _{1}$ is the rotation angle of $\mathit{D}_{1}$ . The arc length of the spiral is

(19) \begin{equation} L=\int \sqrt{r_1^{2}+\left(\frac{\mathrm{d}r_1}{\mathrm{d}\theta_1}\right)^{2}}\mathrm{d}\theta_1 \end{equation}

Therefore, we have

(20) \begin{equation} L\!\left(\theta_1\right)=\frac{b^{2}\theta_1+ab}{2b^{2}}\sqrt{a^{2}+b^{2}+2ab\theta_1+b^{2}\theta_1^{2}}+\frac{b}{2}\arcsin \mathrm{h}\!\left(\frac{b\theta_1+a}{b}\right)+C \end{equation}

Herein, we assume that the initial angle of D ₁ is 0. The tape spring length variations caused by the rotation angle of FDC $\theta _{1}$ are denoted as s ₁ and can be expressed as

(21) \begin{equation} s_{1}=L\!\left(\theta _{1}\right)-L\!\left(0\right) \end{equation}

Given the initial radius a = 19.5 mm and the rate of increase of the spiral b = 5/2π mm/rad, the relationship between s ₁ and the rotation angles of FDC $\theta _{1}$ is shown in Fig. 12. In addition, we also give the result when the radius r ₁ is assumed to be constant. When the FDC is rotated by 50 rad, s ₁ is approximately 2000 mm, while s ₁ is 1000 mm if the coiled tape spring is considered a constant radius. In other words, s ₁ is twice as large as the result if the coiled tape spring is considered a constant radius, and the error increases as the FDC rotates. Therefore, considering the coiled tape spring as an Archimedean spiral can significantly improve the accuracy of kinematic analysis and facilitate kinematic control.

Figure 12. The relationship between s₁ and the rotation angles of FDC $\theta _{1}$ in two cases.

The radius of MDC is r ₂, and $\mathit{\theta }_{2}$ is the corresponding rotation angle. The tape spring length variation s ₂ due to the rotation angles of the MDC $\mathit{\theta }_{2}$ is

(22) \begin{equation} s_{2}=r_{2}\theta _{2} \end{equation}

Differentiating Eq. (21) and (22) with respect to time, we have

(23) \begin{equation} \begin{array}{l} \dot{s}_{1}=\dot{L}=\dot{\theta }_{1}\sqrt{\left(a+b\theta _{1}\right)^{2}+b^{2}}\\[3pt] \dot{s}_{2}=r_{2}\dot{\theta }_{2} \end{array} \end{equation}

Based on the above formula, the workspace of the 2-DOF deployable parallel tape-spring manipulator can be given. When s ₁ is invariable, the trajectory of M is an ellipse as s ₂ changes, with the foci located at $\mathit{D}({-}\mathit{L}_{2}/2,0)$ and A ( $\mathit{L}_{2}/2,0)$ . The focal length is c= L ₂/2, the length of the major axis is $a=(2l_{0}-s_{1})/2$ , and the length of the minor axis is $b=\sqrt{((2l_{0}-s_{1})/2)^{2}-(L2/2)^{2}}$ . Herein, assuming that the total length of the tape spring lies in the range $[\eta _{\min },\eta _{\max }]$ , the workspace can be regarded as the set of ellipses whose major axis is $a\in [\eta _{\min }/2,\eta _{\max }/2]$ . Given the length L ₂ = 500 mm, the maximum total length of the tape spring η _max = 2400 mm, and its minimum total length η _max = 600 mm, the workspace of the manipulator is shown in Fig. 13(a).

Figure 13. (a) Workspace of the 2-DOF manipulator; (b) Variation of r₁ in the workspace.

The boundary equations of the workspace $\psi _{1}$ and $\psi _{2}$ can be expressed as

(24) \begin{equation} \left\{\begin{array}{l} \psi _{1}\colon \dfrac{x^{2}}{\left(\eta _{\min }/2\right)^{2}}+\dfrac{y^{2}}{\left(\eta _{\min }/2\right)^{2}-\left(L_{2}/2\right)^{2}}=1\\[15pt] \psi _{2}\colon \dfrac{x^{2}}{\left(\eta _{\max }/2\right)^{2}}+\dfrac{y^{2}}{\left(\eta _{\max }/2\right)^{2}-\left(L_{2}/2\right)^{2}}=1 \end{array}\right. \end{equation}

As shown in Fig. 13(b), the radius of FDC r ₁ varies in the working space. The radius r ₁ lies in the range [19.5, 57] mm. The closer to the boundary $\psi _{1}$ , the larger the radius. Conversely, the closer to the boundary $\psi _{2}$ , the smaller the radius.

Differentiating Eq. (16) with respect to time, the velocity equations are

(25) \begin{equation} \left[\begin{array}{c@{\quad}c} \dfrac{x+L_{2}/2}{\sqrt{\left(x+L_{2}/2\right)^{2}+y^{2}}} & \dfrac{y}{\sqrt{\left(x+L_{2}/2\right)^{2}+y^{2}}}\\[15pt] \dfrac{x-L_{2}/2}{\sqrt{\left(x-L_{2}/2\right)^{2}+y^{2}}} & \dfrac{y}{\sqrt{\left(x-L_{2}/2\right)^{2}+y^{2}}} \end{array}\right]\left[\begin{array}{l} \dot{x}\\[3pt] \dot{y} \end{array}\right]=\left[\begin{array}{c@{\quad}c} -1 & 1\\[3pt] 0 & -1 \end{array}\right]\left[\begin{array}{l} \dot{s}_{1}\\[3pt] \dot{s}_{2} \end{array}\right] \end{equation}

Equation (25) can be rewritten as

(26) \begin{equation} J_{\mathrm{K}}\left[\begin{array}{l} \dot{x}\\[3pt] \dot{y} \end{array}\right]=\left[\begin{array}{l} \dot{s}_{1}\\[3pt] \dot{s}_{2} \end{array}\right] \end{equation}

Let $l_{1}=\sqrt{(x+L_{2}/2)^{2}+y^{2}},\text{ and }l_{2}=\sqrt{(x-L_{2}/2)^{2}+y^{2}}$ . The Jacobian matrix J _k is

(27) \begin{equation} J_{\mathrm{K}}=\left[\begin{array}{c@{\quad}c} \left(L_{2}/2-x\right)/l_{2}-\left(L_{2}/2+x\right)/l_{1} & -y/l_{1}-y/l_{2}\\[3pt] \left(L_{2}/2-x\right)/l_{2} & -y/l_{2} \end{array}\right] \end{equation}

The determinant of J _k is

(28) \begin{equation} \left| J_{\mathrm{K}}\right| =\frac{L_{2}y}{l_{1}l_{2}} \end{equation}

Equation (28) shows that the 2-DOF manipulator is singular when y = 0. Therefore, when the 2-DOF manipulator moves, the above condition should be avoided.

The condition numbers of Jacobian matrix J _k in the workspace are calculated to evaluate the kinematic performance of the manipulator, as shown in Fig. 14. The figure shows that the condition numbers become much larger as the moving platform approaches y = 0, indicating that the manipulator is singular when y = 0. In addition, we also give the result when r ₁ is assumed to be constant. The condition number when the coiled tape spring is considered an Archimedean spiral is larger than that when the radius of the coiled tape spring is considered constant, and the difference is obvious. Therefore, it is necessary to consider the coiled tape spring as the Archimedes spiral in the kinematic analysis.

Figure 14. Condition number of the Jacobian matrix under different conditions.

Figure 15. Kinematic model of the 3-DOF deployable tape-spring manipulator.

4.2. Kinematic analysis of the 3-DOF deployable tape-spring manipulator

The kinematic model of the 3-DOF deployable tape-spring manipulator is shown in Fig. 15. ¹ A ₁, ¹ A ₂, ² A ₁, and ² A ₂ denote revolute joints. ¹ D ₁, ¹ D ₂, ² D ₁, and ² D ₂ denote FDCs. ¹ M ₁, ¹ M ₂, ² M ₁, and ² M ₂ denote MDCs. E is the midpoint between ¹ M ₁ and ¹ M ₂, and F is the midpoint between ² M ₁ and ² M ₂. The coordinate origin O ₁ is set in the middle between ¹ D ₁ and ¹ A ₂. The coordinates of O _2, which is the midpoint between ² D ₁ and ² A _2, are (0, h ₀, 0). ¹ D ₁ ¹ A ₂ can rotate about the X-axis, and ² D ₁ ² A ₂ can rotate about the Z-axis. ¹ D ₁ and ¹ D ₂, ¹ M ₁ and ¹ M ₂, ² D ₁ and ² D ₂, and ² M ₁ and ² M ₂ are connected by synchronous belts. The center distance of the synchronous belt is L ₁, and the distance between the FDC and the revolute joint is L ₂.

Herein, we assume that ¹ s ₁ and ² s ₁ are tape spring length variations caused by FDCs, and ¹ s ₂ and ² s ₂ are tape spring length variations caused by MDCs. The initial lengths of the tape spring between ¹ D ₁ and ¹ M ₁, ¹ A ₁ and ¹ M ₁, ¹ D ₂ and ¹ M ₂, ¹ A ₂, ² D ₁ and ² M ₁, ² A ₁ and ² M ₁, ² D ₂ and ² M ₂, and ² A ₂ and ² M ₂ are all $\mathit{l}_{0}$ . The lengths of each tape spring segment can be expressed as

(29) \begin{equation} \left\{\begin{array}{l} l_{{}^{1}{M}{_{1}^{}}{}^{1}{A}{_{1}^{}}}=l_{{}^{1}{M}{_{2}^{}}{}^{1}{A}{_{2}^{}}}=l_{0}+{}^{1}{s}{_{2}^{}}\\[5pt] l_{{}^{1}{D}{_{1}^{}}{}^{1}{M}{_{1}^{}}}=l_{{}^{1}{D}{_{2}^{}}{}^{1}{M}{_{2}^{}}}=l_{0}-{}^{1}{s}{_{2}^{}}-{}^{1}{s}{_{1}^{}}\\[5pt] l_{{}^{2}{M}{_{1}^{}}{}^{2}{A}{_{1}^{}}}=l_{{}^{2}{M}{_{2}^{}}{}^{2}{A}{_{2}^{}}}=l_{0}+{}^{2}{s}{_{2}^{}}\\[5pt] l_{{}^{2}{D}{_{1}^{}}{}^{2}{M}{_{1}^{}}}=l_{{}^{2}{D}{_{2}^{}}{}^{2}{M}{_{2}^{}}}=l_{0}-{}^{2}{s}{_{2}^{}}-{}^{2}{s}{_{1}^{}} \end{array}\right. \end{equation}

We assume that the coordinate of the moving platform is E $(x,y,z)$ . The lengths of ${}^{1}{M}{_{1}^{}}{}^{1}{A}{_{1}^{}}$ and ${}^{1}{M}{_{2}^{}}{}^{1}{A}{_{2}^{}}$ are $\mathit{l}_{1}$ . The lengths of ${}^{1}{D}{_{1}^{}}{}^{1}{M}{_{1}^{}}$ and ${}^{1}{D}{_{2}^{}}{}^{1}{M}{_{2}^{}}$ are $\mathit{l}_{2}$ . The lengths of ${}^{2}{M}{_{1}^{}}{}^{2}{A}{_{1}^{}}$ and ${}^{2}{M}{_{2}^{}}{}^{2}{A}{_{2}^{}}$ are $\mathit{l}_{3}$ . The lengths of ${}^{2}{D}{_{1}^{}}{}^{2}{M}{_{1}^{}}$ and ${}^{2}{D}{_{2}^{}}{}^{2}{M}{_{2}^{}}$ are $\mathit{l}_{4}$ . The following expressions are established from the geometric relation:

(30) \begin{equation} \left\{\begin{array}{l} \sqrt{\left(x-\dfrac{L_{2}}{2}\right)^{2}+y^{2}+z^{2}}=l_{0}+{}^{1}{s}{_{2}^{}}=l_{1}\\[15pt] \sqrt{\left(x+\dfrac{L_{2}}{2}\right)^{2}+y^{2}+z^{2}}=l_{0}-{}^{1}{s}{_{2}^{}}-{}^{1}{s}{_{1}^{}}=l_{2}\\[15pt] \sqrt{x^{2}+\left(y+h_{t}-h_{0}\right)^{2}+\left(z+\dfrac{L_{2}}{2}\right)^{2}}=l_{0}+{}^{2}{s}{_{2}^{}}=l_{3}\\[15pt] \sqrt{x^{2}+\left(y+h_{t}-h_{0}\right)^{2}+\left(z-\dfrac{L_{2}}{2}\right)^{2}}=l_{0}-{}^{2}{s}{_{2}^{}}-{}^{2}{s}{_{1}^{}}=l_{4} \end{array}\right. \end{equation}

Differentiating Eq. (30) with respect to time, the velocity equations are

(31) \begin{equation} \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\left(x-L_{2}/2\right)}{l_{1}} & y/l_{1} & z/l_{1} & 0\\[15pt] \dfrac{\left(x+L_{2}/2\right)}{l_{2}} & y/l_{2} & z/l_{2} & 0\\[15pt] x/l_{3} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}} & \dfrac{\left(z+L_{2}/2\right)}{l_{3}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}}\\[15pt] x/l_{4} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} & \dfrac{\left(z-L_{2}/2\right)}{l_{4}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} \end{array}\right]\left[\begin{array}{l} \dot{x}\\[3pt] \dot{y}\\[3pt] \dot{z}\\[3pt] \dot{h}_{t} \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0\\[3pt] -1 & -1 & 0 & 0\\[3pt] 0 & 0 & 1 & 0\\[3pt] 0 & 0 & -1 & -1 \end{array}\right]\left[\begin{array}{l} {}^{1}{\dot{s}}{_{2}^{}}\\[3pt] {}^{1}{\dot{s}}{_{1}^{}}\\[3pt] {}^{2}{\dot{s}}{_{2}^{}}\\[3pt] {}^{2}{\dot{s}}{}{1}^{}{}{} \end{array}\right] \end{equation}

The Jacobian matrix $J$ is

(32) \begin{equation} J=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0\\[3pt] -1 & -1 & 0 & 0\\[3pt] 0 & 0 & 1 & 0\\[3pt] 0 & 0 & -1 & -1 \end{array}\right]^{-1}\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\left(x-L_{2}/2\right)}{l_{1}} & y/l_{1} & z/l_{1} & 0\\[13pt] \dfrac{\left(x+L_{2}/2\right)}{l_{2}} & y/l_{2} & z/l_{2} & 0\\[10pt] x/l_{3} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}} & \dfrac{\left(z+L_{2}/2\right)}{l_{3}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}}\\[10pt] x/l_{4} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} & \dfrac{\left(z-L_{2}/2\right)}{l_{4}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} \end{array}\right] \end{equation}

The determinant of the Jacobian matrix $J$ is

(33) \begin{equation} \left| J\right| =\frac{{L_{2}}^{2}y\!\left(h_{t}-h_{0}+y\right)}{l_{1}l_{2}l_{3}l_{4}} \end{equation}

Equation (33) shows that the 3-DOF manipulator is singular when y = 0 or $y=h_{0}-h_{t}$ . Therefore, when the 3-DOF manipulator moves, the above condition should be avoided.

It is assumed that the length changes of each straight segment caused by the driving component are dl ₁, dl ₂, dl ₃, and dl ₄, and the position changes of the moving platform are dx, dy, dz, and dh _t. The position of the moving platform and the variation in the length of the straight section of the tape spring can be expressed as follows:

(34) \begin{equation} \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\left(x-L_{2}/2\right)}{l_{1}} & y/l_{1} & z/l_{1} & 0\\[13pt] \dfrac{\left(x+L_{2}/2\right)}{l_{2}} & y/l_{2} & z/l_{2} & 0\\[10pt] x/l_{3} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}} & \dfrac{\left(z+L_{2}/2\right)}{l_{3}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}}\\[10pt] x/l_{4} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} & \dfrac{\left(z-L_{2}/2\right)}{l_{4}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} \end{array}\right]\left[\begin{array}{l} dx\\[3pt] dy\\[3pt] dz\\[3pt] dh_{t} \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0\\[3pt] 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 1 & 0\\[3pt] 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} dl_{1}\\[3pt] dl_{2}\\[3pt] dl_{3}\\[3pt] dl_{4} \end{array}\right] \end{equation}

Substituting the length $L_{2}=350$ mm, $h_{0}=h_{t}=100$ mm, the maximum total length of the tape spring η _max = 2400 mm, and its minimum total length η _min = 460 mm into Eq. (34), the workspace of the 3-DOF manipulator is shown in Fig. 16.

Figure 16. Workspace of the 3-DOF deployable tape-spring manipulator.

5.1. Second moment of the area of the double C-shaped tape spring

To solve the stability workspace of the manipulator, it is necessary to analyze the buckling of the double C-shaped tape spring. The cross-sectional geometry of the double C-shaped tape spring is shown in Fig. 17, where r is the radius, t is the wall thickness, α is the angle of the arc edge to the vertical Y-axis, and $h\in [0,t]$ is the integration variable. Two thin-walled open cylindrical structures are tightly wrapped in a polyolefin tube, and the effect of the polyolefin tube is neglected. The second moment of the area of the double C-shaped tape spring is the sum of the two independent tape springs about the centroid O.

Figure 17. Cross-sectional geometry of the double C-shaped tape spring.

The cross-section of the double C-shaped tape spring is centrosymmetric, so the second moment of area I _x about the X-axis is

(35) \begin{equation} I_{x}=4I_{x/4}=\left(\left(r+t\right)^{4}-r^{4}\right)\left(\frac{\sin 2\alpha }{4}+\frac{\alpha }{2}\right)-\frac{4\!\left(r+t\right)^{3}r-4r^{4}}{3}\sin 2\alpha -\left(4r^{3}t+2r^{2}t^{2}\right)\alpha \cos ^{2}\alpha \end{equation}

Similarly, the second moment of area I _y about the Y-axis can be expressed as

(36) \begin{equation} I_{y}=4I_{y/4}=\left(\left(r+t\right)^{4}-r^{4}\right)\left(\frac{\alpha }{2}-\frac{\sin 2\alpha }{4}\right) \end{equation}

The second moment of the area can be considered the upper limit for the bending stiffness and ultimate load. As shown in Fig. 18, the second moment of the area corresponding to different α values is investigated under a specific flattening width and wall thickness (w = 25 mm, t = 0.1 mm). With increasing α, $\mathit{I}_{\mathit{x}}$ increases and $\mathit{I}_{\mathit{y}}$ decreases gradually. The cross-section of the double C-shaped tape spring can be regarded as circular when α = 90°, so $\mathit{I}_{\mathit{x}}$ and $\mathit{I}_{\mathit{y}}$ are equal. At this point, the double C-shaped tape spring has the best stability performance. In addition, the second moments of the area of the single tape spring about the X- and Y-axes are $I_{x}^{'}$ and $I_{y}^{'}$ . Under the same conditions, the second moment of the area of the double C-shaped tape spring is much larger than that of the single tape spring. Therefore, the manipulator we proposed adopts the double C-shaped tape spring, which can greatly improve the stability of the manipulator.

Figure 18. The second moment of the area corresponding to different α.

5.2. Simulation analysis

Tape springs with lengths of 490 mm, 540 mm, 590 mm, and 640 mm are simulated by ANSYS. The elasticity and Poisson’s ratio of the tape spring are set to $\mathit{E}=1.31\times 10^{5}\mathrm{N}/\mathrm{mm}^{2}$ and 0.3, the elasticity and Poisson’s ratio of the polyolefin tube are set to $\mathit{E}=1070\mathrm{N}/\mathrm{mm}^{2}$ and 0.41, and the actual load is set to 1. Fig. 19 shows the deformation diagram for the linear buckling of the tape springs. The simulation results of the tape springs with lengths of 490 mm, 540 mm, 590 mm, and 640 mm are 347 N, 303 N, 257 N, and 213 N, respectively.

Figure 19. Deformation diagram of a double C-shaped tape spring. (a) l = 490 mm; (b) l = 540 mm; (c) l = 590 mm; (d) l = 640 mm.

5.3. Buckling experiment

The experimental configuration is shown in Fig. 20, mainly including a mechanical testing machine, fixture, and tape spring. The mechanical testing machine can obtain the load F and displacement of the rod end $\mathit{\lambda }$ in real time. In the experiment, the moving speed of the loading mechanism is set as 5 mm/min. Buckling experiments are carried out on a tape spring with lengths of 490 mm, 540 mm, 590 mm, and 640 mm. Experiments of each length are performed three times. F- $\lambda$ curves of tape springs with different lengths are shown in Fig. 21.

Figure 20. Configuration of the buckling experiment.

Figure 21. F- $\lambda$ curves of double C-shaped tape springs with different lengths: (a) l = 490 mm; (b) l = 540 mm; (c) l = 590 mm; (d) l = 640 mm.

For clarity, the experimental critical buckling loads of tape springs with different lengths are shown in Table I. For each length of tape spring, three experimental results and average values are given. As the length of the tape spring increases, the critical buckling loads gradually decrease.

Table I. Critical buckling loads of the tape springs.

5.4. Comparison of critical buckling load results obtained by the three methods

The buckling in axial compression of the tape spring is quite complex. In this section, we assume that the critical buckling case is Euler column buckling. The buckling load F is given by the Euler formula as

(37) \begin{equation} F=\frac{\pi ^{2}EI}{\left(Kl\right)^{2}} \end{equation}

where I is the second moment of the area of the tape spring, K is the column effective length factor, l is the length of the tape spring, and F is Euler’s critical load. In the experiments and simulations, one end of the tape spring is fixed, and the other end is hinged. Substituting K = 0.7, $\mathit{w}=25\mathrm{mm}$ , $\mathit{t}=0.1\mathrm{mm}$ , α = 34°, and $\mathit{E}=1.31\times 10^{5}\mathrm{N}/\mathrm{mm}^{2}$ into Eq. (37), the theoretical critical load is obtained.

The critical loads obtained from the theoretical calculation, simulation, and experiments are shown in Table II.

Table II. Comparison of critical loads for the tape spring obtained by three methods.

Table II shows that the theoretical, simulation, and experimental results are similar, and the theoretical and simulation values are higher than the experimental values. However, for practical application, the predicted value of the critical load of the tape spring should be below the experimental value to ensure stability and safety in the actual working process. To make the structure more reliable, the Euler formula is multiplied by a safety factor $\Phi$ to calculate the critical loads of the tape spring. Herein, according to the relevant literature [Reference Zureick and Scott41–Reference Lee, Lee, Lee and Jang43] and experimental data, the safety factor is set as $\Phi$ = 0.85. The modified Euler formula is

(38) \begin{equation} F=\frac{\zeta }{l^{2}} \end{equation}

where

(39) \begin{equation} \zeta =\Phi \frac{\pi ^{2}EI}{K^{2}} \end{equation}

To verify the accuracy of the modified Euler formula proposed in this paper, the calculated results of the modified Euler formula are plotted against the experimental results in Fig. 22. The theoretical calculation results using the modified Euler formula are smaller than the experimental values with small errors. This ensures the calculation accuracy as well as the stability and safety in the actual working process.

Figure 22. Comparison between the modified Euler formula and experimental values.

5.5. Stability spaces of 2-DOF and 3-DOF deployable parallel manipulators

The double V-shaped tape-spring manipulator can be simplified to a single V-shaped tape-spring manipulator DMA. The lengths of D ₁ M ₁ , D ₂ M ₂ , and DM are l ₁, and the lengths of M ₁ A ₁, M ₂ A ₂, and MA are l ₂. l ₁ and l ₂ can be expressed as

(40) \begin{equation} \left\{\begin{array}{l} l_{1}=l_{0}-s_{1}+s_{2}\\[3pt] l_{2}=l_{0}-s_{2} \end{array}\right. \end{equation}

The x- and y-component forces upon the end-effector are denoted as f _x and f _y. Then, the compressions (tensions) f _s1 and f _s2 in DM and MA are

(41) \begin{equation} \left[\begin{array}{l} f_{s1}\\[3pt] f_{s2} \end{array}\right]=\left(J_{\mathrm{Kl}}^{\mathrm{T}}\right)^{-1}\left[\begin{array}{l} f_{x}\\[3pt] f_{y} \end{array}\right] \end{equation}

Herein, we assume that f _y is always along the negative Y-axis. The Jacobian matrix J _K can be expressed as

(42) \begin{equation} J_{\mathrm{K}}=\left[\begin{array}{c@{\quad}c} \left(x+L_{2}/2\right)/l_{1} & y/l_{1}\\[5pt] \left(x-L_{2}/2\right)/l_{2} & y/l_{2} \end{array}\right] \end{equation}

Then,

(43) \begin{equation} \left(J_{\mathrm{K}}^{\mathrm{T}}\right)^{-1}=\left[\begin{array}{c@{\quad}c} l_{1}/L_{2} & \left(l_{1}\left(L_{2}-2x\right)\right)/\left(2L_{2}y\right)\\[5pt] -l_{2}/L_{2} & \left(l_{2}\left(L_{2}+2x\right)\right)/\left(2L_{2}y\right) \end{array}\right] \end{equation}

Substituting Eq. (43) into Eq. (41), the following equations can be obtained:

(44) \begin{equation} \left\{\begin{array}{l} f_{s1}=f_{x}l_{1}/L_{2}+f_{y}\!\left(l_{1}\!\left(L_{2}-2x\right)\right)/\left(2L_{2}y\right)\\[5pt] f_{s2}=-f_{x}l_{2}/L_{2}+f_{y}\!\left(l_{2}\!\left(L_{2}+2x\right)\right)/\left(2L_{2}y\right) \end{array}\right. \end{equation}

To maintain the stability of the manipulator, we require

(45) \begin{equation} \left\{\begin{array}{l} -f_{s1}\leq \dfrac{2\zeta }{l_{1}^{2}}\left(f_{s1}\lt 0\right)\\[13pt] -f_{s2}\leq \dfrac{2\zeta }{l_{2}^{2}}\left(f_{s2}\lt 0\right) \end{array}\right. \end{equation}

Γ_s1 and Γ_s2 denote the first and second boundaries of the stability space, respectively, which come from the first and second expressions in Eq. (45).

When calculating the stability space of the 2-DOF manipulator, only a vertical downward force on the end-effector is considered, that is, f _y = −10 N. The manipulator parameters are those given in Section 4. The corresponding stability space of the 2-DOF manipulator is shown in Fig. 23.

Figure 23. Stability space of the 2-DOF manipulator.

For the 3-DOF manipulator, the Jacobian matrix can be written as follows according to Eq. (34).

(46) \begin{equation} J=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \dfrac{\left(x-L_{2}/2\right)}{l_{1}} & y/l_{1} & z/l_{1} & 0\\[12pt] \dfrac{\left(x+L_{2}/2\right)}{l_{2}} & y/l_{2} & z/l_{2} & 0\\[12pt] x/l_{3} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}} & \dfrac{\left(z+L_{2}/2\right)}{l_{3}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{3}}\\[12pt] x/l_{4} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} & \dfrac{\left(z-L_{2}/2\right)}{l_{4}} & \dfrac{\left(y+h_{t}-h_{0}\right)}{l_{4}} \end{array}\right]^{-1}\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0\\[3pt] 0 & 1 & 0 & 0\\[3pt] 0 & 0 & 1 & 0\\[3pt] 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

Assuming that the force on the end-effector of the 3-DOF manipulator is F, the relationship between F and the axial force N on the straight segment of each tape spring can be written as follows:

(47) \begin{equation} N=J^{T}F \end{equation}

Similarly, when calculating the stability space of the 3-DOF manipulator, only a vertical downward force on the end-effector is considered, that is, F _y = −10 N. When the axial force is less than the critical buckling load of the tape spring, it is regarded as the stability space of the manipulator, as shown in Fig. 24.

Figure 24. Stability space of the 3-DOF manipulator.