2.1. Analysis of the kinematics

The CDPR structure model is shown in Fig. 1, in which the ends of the upper three cables are linked to a point on the moving platform and the other ends are connected to the pulleys fixed on the upper part of the CDPR frame. The lower six cables are organized in groups of two cables, with each cable in a group running parallel to the other. One end of each cable is connected to the moving platform, and the other end is attached to a pulley below the CDPR frame. There are six actuators within the CDPR structure, of which actuators 1, 2, and 3 are connected to the lower three groups of cable and actuators 4, 5, and 6 are connected to the upper three cables. All the actuators are connected to lead screws and powered by motors. In the subsequent calculation, we consider each pair of parallel cables below as an equivalent cable located at the axis of symmetry between the parallel cables.

Figure 1. Structure of the CDPR platform.

The global coordinate system O-XYZ connecting the base and the local coordinate system P-xyz fixed on the moving platform are established, and the position of the local coordinate system origin point in the global coordinate system is P . If i represents the i-th cable, the global coordinates of the connection points between cables and moving platform are $\boldsymbol{A}_{\boldsymbol{i}}$ , and the local coordinates are $\boldsymbol{a}_{\boldsymbol{i}}$ . The global coordinates of the connection points between the cables and pulleys are $\boldsymbol{B}_{\boldsymbol{i}}$ . The relationship between Global Coordinates $\boldsymbol{A}_{\boldsymbol{i}}$ and Local Coordinates $\boldsymbol{a}_{\boldsymbol{i}}$ can be expressed as

(1) \begin{equation} \boldsymbol{A}_{\boldsymbol{i}} = \boldsymbol{R}\boldsymbol{a}_{\boldsymbol{i}} + \boldsymbol{P} \end{equation}

where R is the rotation matrix from the local coordinate system to the global coordinate system, which can be described as

(2) \begin{align} \boldsymbol{R} & =rot\!\left(z,\gamma \right)rot\!\left(y,\beta \right)rot\!\left(x,\alpha \right)\nonumber\\ & =\left[\begin{array}{l@{\quad}l@{\quad}l} c\beta c\gamma & \mathit{c}\mathit{\gamma }s\alpha s\beta -c\alpha s\gamma & s\alpha s\gamma +\mathit{c}\mathit{\alpha }c\lambda s\beta \\ c\beta s\gamma & c\alpha s\gamma +\mathit{s}\mathit{\alpha }s\beta s\gamma & \mathit{c}\mathit{\alpha }s\beta s\gamma -c\gamma s\alpha \\ -s\beta & c\beta s\alpha & c\alpha c\beta \end{array}\right] \end{align}

In Matrix R, c and s represent cos and sin, respectively. $\boldsymbol{l}_{i}$ is denoted as the cable length vector, which can be written as

(3) \begin{equation} \boldsymbol{l}_{i}=\boldsymbol{B}_{i}-\boldsymbol{A}_{i}=\boldsymbol{B}_{i}-\boldsymbol{R}_{i}\boldsymbol{a}_{i}-\boldsymbol{P} \end{equation}

Moreover, the cable length can be obtained as

(4) \begin{equation} l_{i}=\left\| \boldsymbol{B}_{i}-\boldsymbol{R}_{i}\boldsymbol{a}_{i}-\boldsymbol{P}\right\| \end{equation}

The unit direction vectors of the cables can be computed from

(5) \begin{equation} \boldsymbol{u}_{i}=\boldsymbol{l}_{i}/l_{i} \end{equation}

Generally, the angular velocity of a rigid body cannot be directly obtained by derivation of the angular rotation. According to the relationship between the angular velocity of the rigid body and the derivative of the attitude coordinates, the angular velocity of the rigid body is

(6) \begin{equation} \boldsymbol{\omega }=\boldsymbol{E}\dot{\boldsymbol{\theta }} \end{equation}

where the angular velocity and the first derivative of the angular rotation $\boldsymbol{\theta }$ with respect to time of the moving platform are denoted as $\boldsymbol{\omega }$ and $\dot{\boldsymbol{\theta }}$ , respectively. E is the transformation matrix between the angular velocity vectors and angular rotation of the moving platform:

(7) \begin{equation} \boldsymbol{E}=\left[\begin{array}{l@{\quad}l@{\quad}l} c\beta c\gamma & -s\gamma & 0\\ c\beta s\gamma & c\gamma & 0\\ -s\beta & 0 & 1 \end{array}\right] \end{equation}

The velocity vector of the moving platform can be written as

(8) \begin{equation} \left[\begin{array}{l} \boldsymbol{v}\\ \boldsymbol{\omega } \end{array}\right]=\left[\begin{array}{l@{\quad}l} \boldsymbol{I} & \\ & \boldsymbol{E} \end{array}\right]\left[\begin{array}{l} \dot{\boldsymbol{r}}\\ \dot{\boldsymbol{\theta }} \end{array}\right]=\boldsymbol{E}_{1}\dot{\boldsymbol{q}} \end{equation}

where, $\boldsymbol{r}=[\begin{array}{l@{\quad}l@{\quad}l} x & y & z \end{array}]^{T}$ and $\boldsymbol{\theta }=[\begin{array}{l@{\quad}l@{\quad}l} \alpha & \beta & \gamma \end{array}]^{T}$ are the position and posture vectors of the moving platform, respectively. Moreover, $\boldsymbol{v}=[\begin{array}{l@{\quad}l@{\quad}l} \dot{x} & \dot{y} & \dot{z} \end{array}]^{T}$ is the linear velocity vector and $\boldsymbol{\omega }$ is the angular velocity vector.

There are 6 cables on the CDPR, $\dot{\boldsymbol{l}}=[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \dot{l}_{1} & \dot{l}_{2} & \cdots & \dot{l}_{6} \end{array}]^{T}$ is the time derivative of the cable length, which can be written as

(9) \begin{equation} \dot{\boldsymbol{l}}=\boldsymbol{J}^{T}\boldsymbol{E}_{1}\dot{\boldsymbol{q}} \end{equation}

where $\boldsymbol{J}^{T}$ is the $6\times 6$ -order kinematic Jacobian matrix of CDPR:

\begin{equation*} \boldsymbol{J}^{T}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \boldsymbol{u}_{1} & \boldsymbol{u}_{2} & \cdots & \boldsymbol{u}_{6}\\[5pt] \left(\boldsymbol{R}\boldsymbol{a}_{1}\right)\times \boldsymbol{u}_{1} & \left(\boldsymbol{R}\boldsymbol{a}_{2}\right)\times \boldsymbol{u}_{2} & \cdots & \left(\boldsymbol{R}\boldsymbol{a}_{6}\right)\times \boldsymbol{u}_{6} \end{array}\right]^{T} \end{equation*}

2.2. Analysis of dynamics

The force and torque equilibrium of the moving platform can be written as

(10) \begin{equation} \left\{\begin{array}{l} \sum\limits_{1}^{n}\boldsymbol{f}_{i}+\boldsymbol{f}_{p}=0\\[10pt] \sum\limits _{1}^{n}(\boldsymbol{R}\boldsymbol{a}_{i}\times \boldsymbol{f}_{i} + \boldsymbol{\tau }_{i}=0 \end{array}\right. \end{equation}

where $\boldsymbol{f}_{p}$ and $\boldsymbol{\tau }_{i}$ are the external force and torque of the moving platform, respectively. Moreover, $\boldsymbol{f}_{i}=f_{i}\boldsymbol{u}_{i}$ is the i-th cable tension vector, and $f_{i}$ is the i-th cable tension. Eq. (10) expresses the static equilibrium of the CDPR and can be rewritten in matrix form as follows:

(11) \begin{equation} \boldsymbol{J}\cdot \boldsymbol{f}+\boldsymbol{w}=0 \end{equation}

where $\boldsymbol{f}=[\begin{array}{l@{\quad}l@{\quad}l} f_{1} & \cdots & f_{6} \end{array}]^{T}, \boldsymbol{w}=[\begin{array}{l@{\quad}l} \boldsymbol{f}_{p}^{T} & \boldsymbol{\tau }_{p}^{T} \end{array}]^{T}$ and J is the dynamic Jacobian matrix, which is denoted as $\boldsymbol{J}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \boldsymbol{u}_{1} & \boldsymbol{u}_{2} & \cdots & \boldsymbol{u}_{6}\\[5pt] \boldsymbol{R}\boldsymbol{a}_{1}\times \boldsymbol{u}_{1} & \boldsymbol{R}\boldsymbol{a}_{2}\times \boldsymbol{u}_{2} & \cdots & \boldsymbol{R}\boldsymbol{a}_{6}\times \boldsymbol{u}_{6} \end{array}\right]$ .

The model of the CDPR dynamics can be given in the following form:

(12) \begin{equation} \boldsymbol{M}\left(\boldsymbol{q}\right)\ddot{\boldsymbol{q}}+\boldsymbol{C}\!\left(\boldsymbol{q},\dot{\boldsymbol{q}}\right)\dot{\boldsymbol{q}}+\boldsymbol{G}+\boldsymbol{w}_{d}=-\boldsymbol{J}\cdot \boldsymbol{f} \end{equation}

where $\boldsymbol{q}=[\begin{array}{l@{\quad}l} \boldsymbol{r}^{T} & \boldsymbol{\theta }^{T} \end{array}]^{T}, \dot{\boldsymbol{q}}=[\begin{array}{l@{\quad}l} \boldsymbol{v}^{T} & \boldsymbol{\omega } \end{array}^{T}]^{T}$ and $\ddot{\boldsymbol{q}}$ are the derivatives of $\dot{\boldsymbol{q}}$ . Moreover, M ( q ) is the inertial matrix, $\boldsymbol{C}(\boldsymbol{q},\dot{\boldsymbol{q}})$ is the Coriolis and centripetal matrix, G is the gravitational vector, and $\boldsymbol{w}_{d}$ represents the unknown external disturbances.

2.3. Cable tension distribution

Based on the relationship between the number of cables and degrees of freedom, CDPRs can be classified into under-constrained, fully constrained, and redundantly constrained structures. When the relationship between the number of cables i and degree of freedom n satisfies $i\gt n+1$ , the CDPR structure is redundant, and its cable tension has an infinite number of solutions. Therefore, it is necessary to calculate the cable tension of the CDPR to ensure that the tension is within a suitable range. In this paper, the CDPR has a 3-DOF, and there are 6 cables after equivalence. Thus, the system is redundant. Regarding this system, the solution formula for the cable tension distribution can be determined by the static equation:

(13) \begin{equation} \boldsymbol{J}\cdot \boldsymbol{f}+\boldsymbol{w}=0 \end{equation}

where Jacobian matrix can be rewritten as $\boldsymbol{J}=[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} \boldsymbol{u}_{1} & \boldsymbol{u}_{2} & \cdots & \boldsymbol{u}_{6} \end{array}]$ for the CDPR that does not consider rotational DOFs. Eq. (13) has three equations and six unknowns, and the general solution is

(14) \begin{equation} \boldsymbol{f}=-\boldsymbol{J}^{+}\boldsymbol{w}+\left(\boldsymbol{I}-\boldsymbol{J}^{+}\boldsymbol{J}\right)\boldsymbol{F}_{ref} \end{equation}

where $\boldsymbol{I}\in R^{6\times 6}$ is the 3-dimensional unit matrix, $\boldsymbol{J}^{+}$ is the Moore-Penrose pseudoinverse of matrix J , $\boldsymbol{J}^{+}\boldsymbol{w}$ is the minimum norm solution, and $(\boldsymbol{I}-\boldsymbol{J}^{+}\boldsymbol{J})\boldsymbol{F}_{ref}$ is the zero-space solution set of the equation. $\boldsymbol{F}_{ref}\in R^{6}$ is an arbitrary vector. Generally, the cable tension should be maintained within a positive range:

(15) \begin{equation} \mathbf{0}\lt \boldsymbol{f}_{\min }\lt \boldsymbol{f}\lt \boldsymbol{f}_{\max } \end{equation}

A positive range ensures that the tension is not too small to cause the cables to loosen or exceed the maximum allowable tension of the cables. In this paper, we consider the arbitrary vector F _ref in Eq. (14) as the reference cable tension, and we adjust the reference value to ensure that the cable tension remains within the proper range.

3.1. Design of controller

The dynamic modelling of the 3-DOF CDPR can be written as

(16) \begin{equation} \boldsymbol{M}\ddot{\boldsymbol{x}}+\boldsymbol{G}+\boldsymbol{w}_{d}=\boldsymbol{w} \end{equation}

The position error of the moving platform of the CDPR is

(17) \begin{equation} \boldsymbol{e}\!\left(t\right)=\boldsymbol{x}_{d}\!\left(t\right)-\boldsymbol{x}\!\left(t\right) \end{equation}

The sliding surface can be defined as

(18) \begin{equation} \boldsymbol{s}\!\left(t\right)=\dot{\boldsymbol{e}}\!\left(t\right)+\Lambda \boldsymbol{e}\!\left(t\right) \end{equation}

where $\dot{\boldsymbol{e}}$ is the derivative of the position error e and $\mathbf{\Lambda }=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} \Lambda _{1} & \Lambda _{2} & \Lambda _{3} \end{array}\}$ is a positive-definite matrix. We choose the exponential approach sliding mode control; thus, the sliding mode surface approach law is designed as

(19) \begin{equation} \dot{\boldsymbol{s}}\!\left(t\right)=-\boldsymbol{K}\cdot \boldsymbol{s}-\eta \cdot \tanh \!\left[\frac{\boldsymbol{s}\!\left(t\right)}{\varepsilon }\right] \end{equation}

Furthermore, the reference velocity is defined as

(20) \begin{equation} \dot{\boldsymbol{x}}_{r}\!\left(t\right)=\dot{\boldsymbol{x}}_{d}\!\left(t\right)+\Lambda \boldsymbol{e}\!\left(t\right) \end{equation}

The reference acceleration can be obtained by differentiating Eq. (20):

(21) \begin{equation} \ddot{\boldsymbol{x}}_{r}\!\left(t\right)=\ddot{\boldsymbol{x}}_{d}\!\left(t\right)+\Lambda \dot{\boldsymbol{e}}\!\left(t\right) \end{equation}

The sliding mode control law can be expressed as

(22) \begin{equation} \boldsymbol{F}_{x}=\boldsymbol{M}\cdot \ddot{\boldsymbol{x}}_{r}\!\left(t\right)+\boldsymbol{G}+\boldsymbol{K}\cdot \boldsymbol{s}\!\left(t\right)+\eta \cdot \tanh \!\left[\frac{\boldsymbol{s}\!\left(t\right)}{\varepsilon }\right] \end{equation}

Substituting (22) into (14), the cable tension can be obtained as:

(23) \begin{equation} \boldsymbol{f}=\boldsymbol{J}^{+}\left[\boldsymbol{M}\cdot \ddot{\boldsymbol{x}}_{r}\!\left(t\right)+\boldsymbol{G}+\boldsymbol{K}\cdot \boldsymbol{s}\!\left(t\right)+\eta \cdot \tanh \!\left(\frac{\boldsymbol{s}\!\left(t\right)}{\varepsilon }\right)\right]+\left(\boldsymbol{I}-\boldsymbol{J}^{+}\boldsymbol{J}\right)\boldsymbol{F}_{ref} \end{equation}

where f is the vector of 6 cable tensions, $\dot{\boldsymbol{x}}_{d}$ and $\ddot{\boldsymbol{x}}_{d}$ are the desired velocity and acceleration, respectively, $\boldsymbol{K}=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} k_{1} & k_{2} & k_{3} \end{array}\}$ is a positive-definite matrix, $\varepsilon \gt 0$ , and $\eta \geq \| \boldsymbol{w}_{d}\|$ . Due to the discontinuous property of the signum function, it is difficult to eliminate the chattering of the sliding mode control. The proposed sliding mode controller uses the hyperbolic tangent function instead of the commonly used sign function, which can effectively reduce the oscillatory behaviour [Reference Aghababa and Akbari32]. By adjusting the size of the parameter $\varepsilon$ , the steepness of the hyperbolic tangent function can be adjusted. Finally, the controller frame is shown in Fig. 2.

Figure 2. Control frame of SMC-HT strategy.

3.2. Analysis of the stability

Substituting (22) into (16) yields

(24) \begin{equation} \boldsymbol{M}\cdot \ddot{\boldsymbol{x}}+\boldsymbol{G}+\boldsymbol{w}_{d}=\boldsymbol{M}\cdot \ddot{\boldsymbol{x}}_{r}\!\left(t\right)+\boldsymbol{G}+\boldsymbol{K}\cdot \boldsymbol{s}\!\left(t\right)+\eta \cdot \tanh \!\left[\frac{\boldsymbol{s}\!\left(t\right)}{\varepsilon }\right] \end{equation}

Differentiating the sliding surface (18) with respect to time yields:

(25) \begin{equation} \dot{\boldsymbol{s}}\!\left(t\right)=\ddot{\boldsymbol{e}}\!\left(t\right)+\Lambda \dot{\boldsymbol{e}}\!\left(t\right) \end{equation}

Substituting Eq. (21) and Eq. (25) into Eq. (24) yields

(26) \begin{equation} \boldsymbol{M}\cdot \dot{\boldsymbol{s}}\!\left(t\right)+\boldsymbol{K}\cdot \boldsymbol{s}\!\left(t\right)+\eta \cdot \tanh \!\left[\frac{\boldsymbol{s}\!\left(t\right)}{\varepsilon }\right]-\boldsymbol{w}_{d}=0 \end{equation}

Since M is a positive-definite matrix, the Lyapunov function can be defined as

(27) \begin{equation} V=\frac{1}{2}\boldsymbol{s}^{T}\boldsymbol{Ms} \end{equation}

Reference [Reference Polycarpou and Ioannou33] provides the following inequation, which holds for any $\varepsilon \gt 0$ .

(28) \begin{equation} \boldsymbol{s}^{T}\tanh \!\left(\frac{\boldsymbol{s}}{\varepsilon }\right)\geq \left\| \boldsymbol{s}\right\| -\mu \varepsilon \end{equation}

where $\mu$ is a constant that satisfies $\mu =\textrm{e}^{-(\mu +1)}$ , that is, $\mu =0.2785$ . Differentiating V with respect to time in Eq. (27), then substituting Eq. (28) and known condition $\eta \geq \| \boldsymbol{w}_{d}\|$ into it yields

(29) \begin{align} \dot{V} & =\boldsymbol{s}^{T}\boldsymbol{M}\dot{\boldsymbol{s}}=\boldsymbol{s}^{T}\left[-\boldsymbol{K}\cdot \boldsymbol{s}-\eta \cdot \tanh \!\left(\frac{\boldsymbol{s}}{\varepsilon }\right)+\boldsymbol{w}_{d}\right]\nonumber\\ & =-\boldsymbol{s}^{T}\boldsymbol{Ks}-\eta \boldsymbol{s}^{T}\tanh \!\left(\frac{\boldsymbol{s}}{\varepsilon }\right)+\boldsymbol{s}^{T}\boldsymbol{w}_{d}\nonumber\\ & \leq -\boldsymbol{s}^{T}\boldsymbol{Ks}-\eta \!\left\| \boldsymbol{s}\right\| +\mu \eta \varepsilon +\boldsymbol{s}^{T}\boldsymbol{w}_{d}\nonumber\\ & \leq -\boldsymbol{s}^{T}\boldsymbol{Ks}+\mu \eta \varepsilon \nonumber\\ & \leq -\lambda _{\min }\left(\boldsymbol{K}\right)\boldsymbol{s}^{T}\boldsymbol{s}+\mu \eta \varepsilon \nonumber\\[3pt] & =-2\frac{\lambda _{\min }\left(\boldsymbol{K}\right)}{\lambda \!\left(\boldsymbol{M}\right)}\cdot \frac{1}{2}\lambda \!\left(\boldsymbol{M}\right)\boldsymbol{s}^{T}\boldsymbol{s}+\mu \eta \varepsilon \nonumber\\[3pt] & =-2\lambda V+b \end{align}

where $\lambda _{\min }(\boldsymbol{K})$ is the minimal eigenvalue of K , $\lambda (\boldsymbol{M})$ is the eigenvalue of M , $\lambda =\frac{\lambda _{\min }\left(\boldsymbol{K}\right)}{\lambda \!\left(\boldsymbol{M}\right)}, b=\mu \eta \varepsilon$ . Solving the differential inequality (29) yields

(30) \begin{equation} V\!\left(t\right)\leq e^{-2\lambda \!\left(t-{t_{0}}\right)}V\!\left(t_{0}\right)+\frac{b}{2\lambda }\left(1-e^{-2\lambda \left(t-{t_{0}}\right)}\right) \end{equation}

Then, the following is obtained:

(31) \begin{equation} \lim _{t\rightarrow \infty }V\!\left(t\right)\leq \frac{\mu \eta \varepsilon }{2\lambda } \end{equation}

According to Eq. (31), the sliding mode function s is bounded; thus, the tracking error e and the differentiation of the tracking error $\dot{\boldsymbol{e}}$ are bounded. Moreover, $\dot{\boldsymbol{e}}$ is uniformly continuous, and $\dot{\boldsymbol{e}}\rightarrow 0$ as $t\rightarrow \infty$ , according to Barbara’s lemma. Therefore, e and $\dot{\boldsymbol{e}}$ are asymptotically convergent, and the convergence accuracy is related to $\varepsilon$ , K and $\eta$ (i.e., $\boldsymbol{w}_{d}$ ).

5.1. Device settings

The correctness of the proposed SMC-HT strategy is validated, and the theoretical trajectory tracking performance is tested in a previous simulation of the CDPR virtual prototype. In this section, further experiments are performed on the experimental platform in which the SMC-HT algorithm is implemented. A CDPR of the 6-cable 3-DOF platform used in the experiments is shown in Fig. 8, and the moving platform has a weight of 1 kg. The three driving cables are linked to the upper part of the moving platform, which can drive the moving platform in the workspace, while the lower cables are the driven cables, which are used to maintain the stability of the moving platform. The maximum allowable tension of all cables is 150 N.

Table II. Hardware and model of the experiment.

Figure 8. 6-cable 3-DOF CDPR platform.

Figure 9. Spiral trajectory tracking error in the experiment.

We use the Beckhoff hardware as the motion control solution, and the devices and models used are listed in Table II. Beckhoff TwinCAT 3 is used as the motion control software, and the EtherCAT coupler EK1100 and analogue acquisition module EL3064 collect the analogue signal of the force sensor. Then, the force signal can be received by TwinCAT. Moreover, the force data can be transmitted between Simulink and TwinCAT through the TE1410 module in TwinCAT. By writing the control law programme in MATLAB and transmitting calculated motor instructions to TwinCAT, MATLAB/Simulink can be used to directly control the platform. Then, the sensor signals are returned to Simulink, and the data can be processed in MATLAB/Simulink. The 6 motion capture cameras named Nokov are used to collect the pose data of the moving platform, which is easy to calibrate and has high accuracy (0.2 mm). The real-time feedback function of posture is achieved by using the SDK of Cortex software to transmit data to Simulink. The software can receive and save the pose information of the moving platform captured by the motion capture system.

In the following two experiments, the CDPR’s trajectory tracking performance is evaluated using two different trajectories: a spiral trajectory and a space triangle trajectory composed of three point-to-point straight lines. The sampling periods for TwinCAT, Simulink, and Cortex are all set to 0.04 s. By manually adjusting the parameters, the gains of PID are set to $\boldsymbol{K}_{p}=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} 7.5 & 7.5 & 5 \end{array}\}, \boldsymbol{K}_{i}=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} 0.1 & 0.1 & 0.1 \end{array}\}, \boldsymbol{K}_{d}=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} 0.2 & 0.2 & 0.2 \end{array}\}$ , and the parameters of SMC-HT are set to $\boldsymbol{K}=\text{diag}\{\begin{array}{l@{\quad}l@{\quad}l} 4 & 4 & 4 \end{array}\}, \varepsilon =0.02, \eta =0.1, \Lambda =\{\begin{array}{l@{\quad}l@{\quad}l} 5 & 5 & 5 \end{array}\}, \boldsymbol{F}_{ref}=[\begin{array}{l@{\quad}l@{\quad}l} 10 & \cdots & 10 \end{array}]^{T}$ . The results of open-loop inverse dynamics, the PID control strategy based on gravity compensation and the SMC-HT strategy are compared to verify the trajectory tracking performance of the platform.

5.2. Spiral track experiment

In the spiral track experiment, the same curve as in the simulation is used as the desired motion track, which is expressed as the Eq. (35), and the motion time is set to 200 s.

(35) \begin{equation} \left\{\begin{array}{l} x=0.5t\cos \!\left(0.05\pi t\right)\\ y=0.5t\sin \!\left(0.05\pi t\right)\\ z=t \end{array}\right. \end{equation}

Figure 9 presents the pose error curves for three distinct control strategies. These curves closely resemble the error curves obtained through the simulation with respect to the displacement and rotation error curves of the moving platform. The implementation of a closed-loop control strategy in the system results in a significant reduction in the tracking errors of the moving platform. Compared to PID control, the proposed SMC-HT strategy has a smaller tracking error, higher precision and a more stable motion process, and its motion process is more stable. The z-axis error curve of the proposed SMC-HT strategy exhibits a faster convergence rate. Additionally, the tracking errors of the x and y-axes increase over time because the desired velocity of the spiral trajectory increases, while the tracking error for the z-axis reaches a steady state after a certain period. Moreover, since our three controllers do not involve the DOFs of rotation, there is little difference in the errors of the three rotational directions obtained by using the three control strategies. The trajectory tracking performances of the three control methods are further compared, and the RMSE statistics of the three control methods are shown in Table III, which further shows that the SMC-HT strategy has better trajectory tracking performance. Compared to OLC, the trajectory tracking error accuracy of the three axes x, y, and z increased by 87.5%, 89.72%, and 92.46%, respectively. Compared to the PID control strategy based on gravity compensation, the trajectory tracking error accuracy of the three axes increased by 61.29%, 62.24%, and 60.94%, respectively. Additionally, it can be observed that the SMC-HT strategy does not reduce the rotational RSMEs from Table III. This is because the control strategy proposed in this paper focuses on the control of translation of the moving platform, while the rotational DOFs remain in open-loop status. However, the results in Table III indicate that the SMC-HT control strategy proposed in this paper greatly improves the translation accuracy of the moving platform without significantly affecting the rotational performance.

During the operation of the CDPR, the lower three pairs of cables are redundant, which indicates that the stable operation of the moving platform is maintained, while the upper three cables are driving cables. Therefore, only the tension data of the upper three cables are collected in the experiment. The curves of the cable tension in the different control strategies are shown in Fig. 10, in which the trends of the cable tensions in the three driving cables are consistent with that of the simulated cable tension in the previous section (Cable 4, Cable 5 and Cable 6 in Fig. 6(a)), and the tension state of the cables is guaranteed.

Table III. RSMEs of the spiral trajectory in the experiment.

Figure 10. Cable tension with spiral trajectory in the experiment.

Figure 11. Desired trajectory of the space triangle.

5.3. Space triangle track experiment

In this experiment, a trajectory that is a space triangle composed of three straight lines is designed to evaluate the capacity of the three control strategies to maintain straight-line motion for the CDPR. As shown in Fig. 11, the trajectory passes through three points in sequence, and the coordinates of the three points are A(0, 0, 0), B(−140 mm, −80 mm, 300 mm), and C(100 mm, 150 mm, 100 mm). In each straight line, the desired velocity of the moving platform sinusoidally increases and reaches the maximum velocity of 10 mm/s. This velocity is maintained for a certain period and finally decreases to 0 sinusoidally.

The pose error curves of the space triangle trajectory of the three control strategies are illustrated in Fig. 12, and the RMSE statistics are presented in Table IV. The results are consistent with the experimental findings of the spiral track experiment, confirming that the proposed SMC-HT strategy achieves higher trajectory tracking accuracy. The accuracy of the x, y and z directions improves by 80.50%, 87.96%, and 82.69%, respectively, compared to the OLC strategy and by 56.38%, 54.68%, and 53.54%, respectively, compared to the PID control strategy. Therefore, a higher velocity of the moving platform results in a higher error value. This rule also conformed to the error curves in the spiral trajectory experiment. Because the velocity of the spiral trajectory increases with time, the tracking errors of the moving platform also increase with time. While the velocity increases, the tracking error values increase less. Consequently, the regulation ability of the proposed SMC-HT strategy is better than that of the PID strategy. In addition, the cable tension curves are shown in Fig. 13. Moreover, when the PID and SMC-HT strategies are used, the forces of the three cables increase to varying degrees after the moving platform runs a complete cycle and returns to the origin. The SMC-HT method provides the most significant force increase. However, the maximum cable tension is below 14 N, which does not reach the maximum tension of 150 N allowed.

Table IV. RSMEs of the space triangle trajectory in the experiment.

Figure 12. Space triangle trajectory tracking errors in the experiment.

Figure 13. Cable tension of the space triangle trajectory in the experiment.