2.1. Kinematic model

CDPRs are comprised of an MP with dimensions c ×d and a base frame with dimensions h ×w, with embedded actuators. These actuators are connected to an MP through four cables. The Cartesian position and orientation of the MP are controlled by adjusting the cable lengths exiting the pulleys. The geometric model CDPR involves four servomotors mounted at the corners of a vertical surface on the robot’s chassis.

$\mathcal{F}_{0}$ {O, x ₀ ,y ₀ ,z ₀ }: Fixed frame of the planar CDPR centered on the four motors, with the horizontal x ₀ -axis and the vertical y ₀ -axis. C: Exit point of the cable from the pulley. M: Center of the pulley. G: Center of the MP, b: The vector represents the Cartesian position of the center of MP in the frame $\mathcal{F}_{0}$ , this parameter depends on the movement executed by the MP.

$\boldsymbol{\theta }$ : The angle at point M between the horizontal axis and the direction vector from M to point C.

$\boldsymbol{\rho }$ : Vector between the center of the pulley M and its corresponding corner B of the MP.

${\boldsymbol{a}}_\boldsymbol{i}^{T}=\left[\begin{array}{cc} \frac{w}{2} & \frac{h}{2} \end{array}\right]$ for i = 1, The vector defines the center of the pulley in the fixed frame $\mathcal{F}_{0}$ .

$\boldsymbol{R}_{\boldsymbol{\alpha }}$ : The rotation matrix represents the orientation, denoted by α, of the MP in the fixed frame $\mathcal{F}_{0}$ .

$\boldsymbol{r}_\boldsymbol{i}^{T}=\left[\begin{array}{cc} \frac{-c}{2} & \frac{-d}{2} \end{array}\right]$ : Vector between G the center of the MP to the corresponding corner B.

Figure 1 illustrates the kinematic model of our planar cable robot. Equation (1) is applied to calculate the distance $\| \boldsymbol{\rho }_\boldsymbol{i}\|$ between the center of the pulley and the cable attachment point on the MP:

(1) \begin{align} \left\| \boldsymbol{\rho}_\boldsymbol{i}\right\| ^{2}=\left\| \textbf{a}+ \textbf{b} + {\textbf{R}_{\boldsymbol{\alpha}}\textbf{r}_{\textbf{i}}}\right\| ^{2} \\[-25pt] \nonumber \end{align}

The cable winds around the pulley with a radius $r$ and exits from point C. By neglecting the diameter of the cable relative to that of the pulley, we assume in the subsequent discussions that $r$ is the effective radius, including both the actual radius of the pulley and the cable. The cable departs from the pulley at point C, and the angle between vector from M to C and the horizontal is denoted as $\theta$ . Referring to Figure 1, the total length of the cable, accounting for the stretched part $L_{S}$ and the part wound on the pulley $L_{P}$ , is expressed as follows in Equation (2):

(2) \begin{align} L=L_{S}+L_{P}=L_{S}+r\theta \end{align}

where $r$ is the radius of the pulley, as illustrated in the figure. $\theta$ is the angle where the cable is wound around the pulley. $L_{S}$ is the stretched part of the cable starting from point C to point B. Taking into consideration the two right-angled triangles, we can calculate the length through Equation (3):

(3) \begin{align} {L_{S}}^{2}={\boldsymbol{\rho }_\boldsymbol{i}}^{2}-r^{2} \end{align}

where $\boldsymbol{\rho }^{T}=[\rho _{x} \rho _{y}]$ , $\rho _{x}, \rho _{y}$ are the vector $\boldsymbol{\rho }$ coordinates with respect to frame $\mathcal{F}_{0}$ . To solve the IK, to determine the angle $\beta_{u}$ we consider the tetragonal CMDB, noting that two angles at points C and D of the tetragonal are right angles, we can conclude that the desired complementary angle $\beta _{u}$ at point M is equal to the enclosed angle $\beta _{1}+\beta _{2}$ at point B. Using elementary trigonometric functions, we obtain Equation (4):

(4)

To determine the rate of change of the motor angle with respect to time, it is necessary to calculate the variation in length of each cable as a function of time. This rate corresponds to the servomotor angle variation for each actuator, and it is derived from the variation in the position of the MP over time, using Equation (5).

(5) \begin{align} V=\dot{L}= \frac{dL}{dt} \end{align}

2.2. Trajectory

In order to assess the performance of the control method, three desired trajectories describing typical kinematic applications were generated. The trajectories consist of:

• • Circle Movement: The center of the circle coincides with the planar CDPR center, with a radius of 50 cm. Equation (6) is applied to calculate the variation in cable length over time with a radius $\psi$ and frequency $\omega$ .

• • Square Movement: Four vertical and horizontal displacements extend linearly over 1000mm, over a square of a 25 cm width, passing through the points A (50, 50, 0), B (−50, 50, 0), C (−50, −50, 0), and D (50, −50, 0) in the fixed frame $\mathcal{F}_{0}$ . Equation (7) is applied for each displacement.

• • Triangle Movement: Three linear movements over an equilateral triangle, passing through the points E (50, 0), F (50cos(2π/3), 50cos(2π/3)), and G (50cos(−2π/3), 50cos(−2π/3)) in the fixed frame $\mathcal{F}_{0}$ .

For linear motions in the triangle and square movements, Equation (7) is utilized, with $\eta$ and $\mu$ representing the velocity along the x and y axes, respectively.

(6)

(7) \begin{align} \textbf{X}^{T}=\left[\begin{array}{c@{\quad}c@{\quad}c} X_{p1}+\eta t & Y_{p1}+\mu t & 0 \end{array}\right] \end{align}

3.1. CDPR prototype

Figure 2 illustrates our CDPR prototype configured such that all cable exit points are positioned at the four corners of the fixed platform. Cables in this study are supposed to be rigid and massless. The robot frame is a 2.1 m × 1.7 m rectangle. The size of the MP is 0.2 m × 0.2 m. A depth camera is positioned 2.5 m in front of the MP and CDPR plane. To simplify the vision part, two AprilTags are placed as objects on MP and the reference frame. An algorithm was developed for AprilTags recognition and localization, allowing the robot to be controlled by tracking MP through a depth camera.

Figure 2.

Cable-driven parallel robot prototype located at ENISO, Sousse. The components are: (1): Pulley mounted on the Servomotor, (2): Cables, (3): AprilTag attached to the Moving Platform, (4): AprilTag attached to the Fixed frame.

3.2. CDPR prototype control system

Figure 3 shows the hardware architecture of the CDPR control system. A programmable logic controller (PLC) with high-speed pulse output and supporting four hardware 200 kHz high-speed counters is used. This PLC controls the servomotors using a Proportional-Derivative (PD) control. Kinematic data are computed on a PC and then transmitted to the PLC, which commands the actuators based on the desired position and velocity. Each actuation chain, from the servomotors to the cables, has its own controller that takes desired and measured motor position and velocity as input. The output is the corresponding desired motor torque. Each cable will be driven by a servomotor. The specification of the servomotor and encoder is shown in Table I below.

Table I.

Characteristics of the actuators.

3.3. Motion tracking system

This research aims to implement a tracking system that covers the robot’s workspace as well as the movements performed by the mobile platform. The RealSense D435i, a depth camera widely utilized in current applications, is employed for data collection. During this process, both RGB and depth images are acquired simultaneously. D435i offers stereo depth resolution up to 1280 × 720@90 fps and depth field of view 87° × 58°, in a range from 0.2 m to over 3 m, which varies with lighting conditions. It is also proficient in capturing RGB frames with resolution up to 1920 × 1080@30 fps and RGB field of view 69° × 42°. The camera is well-suited to the dimensions of our robot, exhibiting an root mean square error (RMSE) subpixel depth error of less than 0.12 pixels [Reference Sonoda, Sweetser and Khuong40].

Figure 3.

The hardware architecture of the cable-driven parallel robot control system.

After capturing depth maps and color images from each frame, the subsequent step involves calculating their 3D point clouds and aligning them. Various methods are available for this purpose. In our case, we found it adequate to rotate the point cloud data from both RGB and depth images using a 3D transform that would both rotate and shift the point clouds, creating one large composite point cloud.

AprilTag, introduced in 2011 by Olson [Reference Olson41], is a robust and adaptable visual fiducial system. It employs a 2D code style, as illustrated in Figure 4, facilitating full 6-DOF feature localization from an image. The system comprises two primary components: the coding system and the tag detector. The tag detector’s function is to identify potential tag positions in the image using a graph-based algorithm for image segmentation, followed by estimating the tag’s pose relative to the camera, and finally calculating the Euclidean transformation. By meticulously following these steps, one can accurately delineate the kinematic motion of the MP relative to a fixed reference frame.

Figure 4.

Flow chart of the moving platform tracking process.

4.1. Motion tracking process

To determine the kinematic movement of a tag on the MP relative to a tag on a fixed CDPR platform using a D435i depth camera, a systematic sequence of steps is described, as illustrated in Figure 5. The process begins with initialization of the configuration and calibration of the depth camera. Then, a colorization configuration is implemented by creating a colorizer object, which improves the visual representation of depth data for better analysis. Point cloud object is initialized to store the 3D depth information captured by the camera. The streaming process is launched by aligning the depth images with the color images to facilitate accurate data fusion.

Figure 5.

Motion tracking process of the moving platform.

Reshaping and scaling the depth data involves adjusting the depth vector to match the resolution of the color images, and scaling the depth values to represent distances in meters. Once data has been converted into an image format suitable for viewing, AprilTag detection algorithm is applied, identifying and locating the corners and centers of AprilTags in the captured color and depth images. Next, the 3D Cartesian coordinates of the fixed and moving markers are calculated. Two transformation matrices are presented: one determines the fixed frame relative to the camera frame $T _{R_{C}}^{R_{F}}$ , and the other defines the MP frame relative to the camera frame $T _{R_{C}}^{R_{M}}$ . Each transformation matrix includes translation and rotation information of the frame relative to the other, as shown in Equation (8).

(8)

Where the $R_{3\times 3}$ is a rotation matrix and $\boldsymbol{t}_{3\times 1}$ is a translation vector. The translation vector represents the position of the tag’s center, and the rotation matrix is obtained from the corners of the tag. Using the Euclidean transformation and Equation (9), the geometric parameters of both the fixed and moving frames are derived through the multiplication of the Euclidean transformation matrices:

(9) \begin{align} T _{R_{F}}^{R_{M}}=T _{R_{C}}^{R_{M}}\times T _{R_{F}}^{R_{C}}=T _{R_{C}}^{R_{M}}\times \left(T _{R_{C}}^{R_{F}}\right)^{-1} \end{align}

While ${T _{R_{F}}^{R_{C}}=(T _{R_{C}}^{R_{F}})^{-1}}=\left[\begin{array}{c@{\quad}c} R_{R_{C}}^{R_{F}} & t_{R_{C}}^{R_{F }}\\[3pt] 0_{1\times 3} & 1 \end{array}\right]_{4\times 4}$

4.2. Servomotor control algorithm

The test trajectory described in section 2.2 gives the desired MP pose, i.e., MP position and orientation as well as the motion velocity, which are converted into desired servomotor angles position $\theta$ through Equation (10) and servomotor angular velocities using the inverse CDPR geometric and kinematic models. To illustrate, all kinematic results computed by our model are transmitted to the PLC overseeing our CDPR, as depicted in the Data Flow diagram of the control system in Figure 6. Then, the data is transmitted to the servodrives to process the signals, and then send appropriate commands to the servomotors. Simultaneously, feedback is collected from the encoder to ensure control precision and accurate positioning of the system.

(10) \begin{align} \theta =\Delta L/n_{m}r_{i} \end{align}

Figure 6.

Data flow diagram of the control system.

4.3. Closed-loop position correction algorithm

The closed-loop system computes the IK of the MP’s position using data from the depth camera, as depicted in Figure 7 illustrating the architecture of the controller. Here, a vision system is integrated with a PD position controller to set a closed-loop controller for our CDPR.

Figure 7.

Flowchart of closed-loop camera-based control scheme of our cable-driven parallel robot.

The vision system block, 3D reconstruction, computation of the current position X _c from RGB, and depth images are discussed in section 4.1. The position of the MP is determined through Equation (9). The presented control system is primarily based on real-time correction of the MP’s position towards the targeted position X _d using the current position X _c . Where Equation (12) and Equation (13) illustrate these two positions. The error matrix e between these positions, presented in Equation (11), is calculated, followed by the implementation of a PD system to optimize the position control. Equation (14) represents the PD calculating process. Consequently, the values of K _p and K _d are adjusted to find the optimal values of Cartesian displacement X _r of the MP to be sent as a command to the CDPR controller.

The Inverse Kinematics block, introduced in section 2, alongside the conversion of motor pulses from Equation (10) based on the position X _r , focuses on correcting the MP’s trajectory as well as real-time IK computing. Operating at a frequency of 10 Hz, the program demonstrates its acceptable-speed processing capabilities. This frequency significantly exceeds the relatively low Cartesian speed of the MP, which is 0.05 m/s. The program’s execution rate enables precise monitoring and adjustment of the MP’s trajectory. After performing the IK calculation, the program transfers the data continuously to the PLC to be processed by a PLC program that controls the CDPR in position and speed. Transmission is performed in parallel for all four servodrivers, each one is equipped with PID control and auto-tuning capabilities for PID gains. This ensures precise control of system movements. Meticulously refined instructions are then rapidly transmitted to the servomotors, facilitating smooth and precise task execution. This control system allows precise and rapid adjustments to be made in order to maintain the MP’s trajectory with high accuracy.

The tracking process outlined in section 4.1 is performed to calculate the real Cartesian position X _C of the MP and to rectify the error $\boldsymbol{e}$ with respect to the desired position X _d , where:

X _d , X _c , and X _r are vectors (3 × 1), which include both position and orientation of the MP.

θ _r , θ _i are vectors (4 × 1), which include joint angle pulses to command the servomotors.

(11) \begin{align} \boldsymbol{e}=\boldsymbol{X}_\boldsymbol{d}-\boldsymbol{X}_\boldsymbol{c} \\[-25pt] \nonumber \end{align}

(12) \begin{align} {\boldsymbol{X}}_\boldsymbol{d}^{T}=\left[\begin{array}{c@{\quad}c@{\quad}c} x_{d} & y_{d} & \alpha _{d} \end{array}\right] \\[-25pt] \nonumber \end{align}

(13) \begin{align} {\boldsymbol{X}}_\boldsymbol{c}^{T}=\left[\begin{array}{c@{\quad}c@{\quad}c} x_{c} & y_{c} & \alpha _{c} \end{array}\right] \\[-25pt] \nonumber \end{align}

(14)

X _r : Vectors (3 × 1) representing the Cartesian displacement X _r of the MP to be sent as a command to the CDPR controller.

X _c : vectors (3 × 1) presenting current MP position computed by the depth camera during the motion.

X _d : vectors (3 × 1) the desired position of the MP, computer

$\boldsymbol{e}$ : vectors (3 × 1), the error matrix between the desired position and the current position.

K _p , K _d are (3 × 3), matrices representing the proportional, and derivative gains respectively, where each element of these matrices corresponds to the corresponding gain for each parameter (x, y, α).

In addition to processing feedback from the camera, the closed-loop control scheme for the CDPR incorporates PD correction to further enhance performance. The PD controller adjusts the Cartesian displacement X _r based on the error e between the desired and actual positions of the robot’s MP continuous fine-tuning of the control signals, PD correction ensures that the system responds rapidly to minimize error of the MP trajectory in the x and y axes on the plane, and improving overall tracking accuracy as well as ensuring that MP is in a stable position and orientation α.

The integration of PD correction into the closed-loop control system is synchronized with camera feedback, which collectively contributes to the accuracy enhancement of CDPR control. The proposed CDPR control algorithm was programed in MATLAB employing the RealSense support package for camera management and data acquisition. The Computer Vision Toolbox is utilized for AprilTag detection and tracking. MATLAB support for TCP/IP is employed to communicate with the PLC.

Figure 8.

Comparative analysis of theoretical kinematic joint curves with encoder feedback. (a): Circular trajectory, (b): Square trajectory, (c): Triangle trajectory.