Vision Based PID Control on

A Planar Cable Robot

Reza Babaghasabha, Mohammad A. Khosravi and Hamid D. Taghirad

Advanced Robotics and Automated Systems (ARAS), Industrial Control Center of Excellence (ICCE),

Faculty of Electrical and Computer Engineering, K.N. Toosi University of Technology, Tehran, Iran.

E-mail: Reza.bgha@mail.kntu.ac.ir, {Makh, Taghirad}@ieee.org

Abstract— In this paper, a visual measuring system is

implemented for task space control of a planar cable robot. Since

cables are able to apply only tensile forces, the proposed control

algorithm shall be designed such that the cables remain in

tension in the whole workspace. Furthermore, the position and

orientation (pose) of the end-effector in task space coordinates

shall be measured directly for precise motion control of the

robot. Therefore, in this paper, vision-based measurement system

is implemented on a planer cable robot and these measurements

are directly used in a controller. Furthermore, the vector of

internal forces is used to ensure that all cables remain in tension.

Finally, The effectiveness of the proposed control algorithm is

examined through some experiments on a planar cable robot and

it is shown that the proposed control algorithm is able to

provide suitable performance in practice.

Keywords-cable robot; internal forces; visual tracking; marker

based tracking; task space control .

I. INTRODUCTION

Cable robots are a special class of the parallel robots in which the rigid links are replaced by cables. In this class of the robots, the end-effector is connected to the base by a number of active cables and the end-effector is shifted toward a desired pose by changing the length of the cables. Consequently, the end-effector can be controlled from a far distance and a large workspace can be achieved for the robot [1], [2]. In addition, due to use of light-weight cables in the robot structure, the mass of the moving parts has been reduced considerably and as a result, they are applicable for high speed manipulation [3]. High payload to robot weight ratio is another positive feature of the cable-driven parallel robots. This feature is used in robocrane to carry heavy materials [4].

Despite all of the advantages mentioned above, using the cables in the robot structure introduces new challenges in the study of cable-driven robots. Because the cables are able to apply only tensile forces, design of control algorithms should be performed such that the cables remain in tension for the whole workspace. Based on this fact, cable-driven parallel robots can be sorted into two types: under-constrained and fully-constrained [5]. In the under-constrained type, a passive force is used to keep all cables in tension. While in the fully-constrained cable-driven parallel robots, the end-effector is fully constrained by the cables and in order to change the pose of the end-effector, at least the length of one cable must be changed [6]. Furthermore, inherent flexibility of the cables may be considered in accurate and high bandwidth applications [7].

These features make feedback control of the cable robots more challenging than the conventional parallel robots. Motion control topologies of cable robots may be classified into two types. In the first class, the proposed control algorithms are formed in the cable length coordinates and the other is in the task space coordinates. In [3] and [8] decentralized controllers have been designed in the cable length coordinates and their performance has been evaluated through some experiments. In this coordinate the length of the cables can be measured simply by the encoders. Consequently, the controllers can be implemented less expensive in practice. However, in applications with high accuracy, using the cable lengths measurement in the control algorithms is not reliable due to the cables’ inherent flexibility. In [9] cable position transducers have been used as an alternative solution. In spite of simple and practical structure of the control algorithms, the controllers in the cable length coordinate suffer from sensitivity to the kinematics uncertainties and as a result accurate calibration must be performed to identify the parameters of the robot. Moreover, since the pose of the end-effector is not measured directly, the coupled behavior of the cables and their total effects on the pose of the end-effector is ignored [10]. Due to these cases, control the cable-driven parallel robot in the task space coordinates is more reliable than that of the cable length coordinates.

The control aim of cable-driven parallel robots is to realize the trajectory tracking in the task space coordinates. Therefore, better control accuracy can be achieved when the error defined in the task space coordinates is used to the controller design without any kinematic transformation [11]. In the second class of the control topologies, the controller is designed in the task space coordinates and the pose of the end-effector is measured directly as an ultimate goal for the control system. Therefore, this class of the controller is less sensitive to the kinematics uncertainties and coupled dynamic behavior of the cables. However, controller implementation in the task space coordinates requires high-tech equipment and expensive sensors such as laser ranging equipment or inertial measurement unit for direct pose measurement of the end-effector. For this reason, only few researches have implemented the task space controllers in practice. In [12] laser ranging equipment has been used to measure pose of the end-effector and the fuzzy plus proportional–integral control method has been used to control the 6-DOF cable-driven robot in the task space coordinates. In [13-15] the controller is designed and implemented in the task space coordinates but in

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order to escape from direct pose measurement, length of the cables is measured by the encoders and the pose of the end-effector is estimated by the forward kinematics. As mentioned earlier, using cable length information is not reliable due to the cables’ inherent flexibility. In addition, solving the forward kinematics equations of the robot in the feedback loop, reduces the measurement accuracy and limits the proposed controller bandwidth due to the complexity and variety of solutions.

In this paper, a pose measurement system is designed and implemented on a cable robot. Vision-based pose measurement is chosen, since it is a suitable and economical solution. An appropriate marker is designed for fast and accurate tracking and the pose of the end-effector is measured by extracting marker features in real-time. Then a widely used PID controller is used to control the pose of the end-effector in the task space coordinates. In the structure of the proposed controller the vector of internal forces is used to ensure that all of the cables remain in tension in the whole workspace. Finally, the performance of the proposed controller is examined through some experiments on the KNTU planar cable robot.

II. KNTU PLANAR CABLE ROBOT

KNTU planar cable robot performs planar motions with three degrees of freedom (two translational degrees in directions X, Y and one degree of rotational movement about Z axis). This robot consists of four actuated cable driven limbs with one degree of redundancy. The translational and rotational workspace of the robot are 2 × 2 meters and ± 30 degrees, respectively. This robot, which is shown in Fig. 1, is under investigation for high speed and wide workspace applications in Advanced Robotics and Automated Systems (ARAS) group of K. N. Toosi University of Technology. In [16] kinematics and dynamic analysis has been reported in detail. In this paper, since the dynamic model of the robot is not used in the controller structure, we do not address these issues in detail and we verify the performance of the controller in the task space coordinates through some experiments.

Figure 1. A picture of KNTU planar cable robot

III. POSE MEASUREMENT OF THE END-EFFECTOR

In this section vision-based pose measurement of the end-effector is investigated. An appropriate marker is designed for fast and accurate tracking and the pose of the end-effector is measured by extracting marker features. In [17] vision-based pose measurement has been performed on a planar cable-driven parallel robot as a suitable and economical solution. But only the translational motion is measured and the end-effector rotation is not considered.

In this paper, to increase the accuracy and speed of the measurement, robot workspace is limited to 1 × 1 meters and a camera with resolution of 320 × 240 pixels and frame rate of 100 fps is used with the distance of 1.1 meters from the plane of motion of the end effector. According to the camera frame rate, the sampling time of the control loop will be limited to 10 milliseconds. Therefore, extracting the pose of the end-effector shall be performed in less than 10 milliseconds for the real-time measurement. On the other hand, a marker shall be designed such that its features may also be quickly extracted, even in sub-pixel resolution. Moreover, since the end-effector rotates during the translational motion, selected features shall be invariant to the rotation. According to above-mentioned properties, the corner is used as one of the most desirable features for tracking. In order to determine the pose of object, at least four coplanar non-aligned features on the object are required [18]. For this reason and according to the other specifications mentioned for the marker, a square marker is used as shown in Fig. 2, for pose measurement of the end-effector.

Figure 2. Designed marker for tracking

In the pose measurement procedure, Harris corner detector is used [19]. This detector relies on eigenvalues of the second-order derivative matrix of image intensity. This matrix is derived according to the following equation.

(1) 𝑀(𝑥, 𝑦) = [𝑎 𝑐𝑐 𝑏

] in which,

𝑎 = ∑ 𝑤𝑖,𝑗

−𝑘≤𝑖,𝑗≤𝑘

𝐼𝑥𝑥(𝑥 + 𝑖, 𝑦 + 𝑗)

𝑏 = ∑ 𝑤𝑖,𝑗

−𝑘≤𝑖,𝑗≤𝑘

𝐼𝑦𝑦(𝑥 + 𝑖, 𝑦 + 𝑗)

𝑐 = ∑ 𝑤𝑖,𝑗

−𝑘≤𝑖,𝑗≤𝑘

𝐼𝑥(𝑥 + 𝑖, 𝑦 + 𝑗)𝐼𝑦(𝑥 + 𝑖, 𝑦 + 𝑗)

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Figure 3. Pose Measurement Procedure

where, 𝐼𝑥 , 𝐼𝑦 are derivatives of image intensity in direction x, y

and 𝑤𝑖,𝑗 is a weighting term that can be considered to be

uniform, however, it is often used to create a circular window or Gaussian weighting. Harris’s original definition involved computing of the following difference and comparing it with a predetermined threshold.

(2) 𝑅 = det(𝑀) − 𝑘(𝑡𝑟𝑎𝑐𝑒(𝑀))2

It is noticable that this detector can suitably detect corners, while the required computations for corner detection are few. Another advantage of this detector is utilizing the eigenvalues of the matrix (1), which they are invariant to rotation. The procedure of extracting end-effector pose is given step by step in the flowchart illustrated in Fig. 3. With this procedure, measurement accuracies of the position and orientation are 1 mm and 0.5 degree, respectively.

IV. CONTROLLER SYNTHESIS

Due to the complex dynamics and kinematics constraints of parallel robots, the motion control of such structures has been the center of attention for several researchers. This issue is more complicated in the cable-driven parallel robots since in this class of robots tensionability condition shall be satisfied, as well. Several control algorithms based on the cable length and the task space coordinates are employed in control of the cable-driven parallel robot. Lyapunov based methods [3], computed torque method [20], sliding mode [15], [21] and fuzzy control [12] are some control algorithms being used to control the cable-driven parallel robots.

Figure 4. Cascade control block diagram

However, it shall be noted that the complex structure of these controllers and unavailability of precise robot dynamics, limit their application in practice. In this paper, a widely used PID controller is used to control our planar cable robot. In addition, the vector of internal forces is used in the controller structure to provide positive tension in the cables during the robot movements within the whole workspace.

A. Proposed Controller

The proposed control algorithm is benefited from cascade control structure. The cascade control strategy uses two control loops, called outer and inner loops as shown in Fig. 4. In proposed control algorithm, while PID controller controls the pose of the end-effector in the outer loop, lag controller controls the cables tension, and prevents them from tearing by the inner loop.

It is fully elaborated in [22] that the positioning error is uniformly ultimately bounded (UUB) if the control gains are selected from a suitable feasible set. In this paper, we leave the details of the proof and verify the performance of the task space controller through experiments. The control law is given by the following equation:

(3) −𝐽𝑇𝜏 = 𝐾𝑣�̇� + 𝐾𝑝𝑒 + 𝐾𝑖 ∫ 𝑒(𝑠)𝑑𝑠𝑡

0

Where 𝜏 denotes the vector of cable forces, e denotes the vector of positioning error, J denotes Jacobian matrix of the robot and 𝐾𝑣 , 𝐾𝑝 , 𝐾𝑖 denote positive controller gains. Since the

robot is redundant, Jacobian matrix is a non-square matrix and therefore (3) is an underdetermined system of equations and it has many solutions. In this case, the general solution of (3) is:

(4) 𝜏 = 𝜏̅ + 𝑄

Where 𝜏̅ is the minimum solution of (3) and derived by using the pseudo-inverse of 𝐽𝑇and is given by

(5) 𝜏̅ = −𝐽(𝐽𝑇𝐽)−1[𝐾𝑣�̇� + 𝐾𝑝𝑒 + 𝐾𝑖 ∫ 𝑒(𝑠)𝑑𝑠]𝑡

0

In equation (4), Q can be physically interpreted as the internal forces and spans the null space of 𝐽𝑇 which satisfies

(6) 𝐽𝑇𝑄 = 0

Therefore, this term does not affect the end-effector motion and only provides positive tension in the cables. Moreover, using this term can increase the robot stiffness [23]. Fig. 5 shows the internal force control structure.

Figure 5. Internal force control structure

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V. EXPRIMENTAL RESULTS

In order to verify the effectiveness of the proposed controller, several experiments have been performed on KNTU planar cable robot. In these experiments, three disjoint linear motions in translation and rotation are considered. The proposed controller gains are selected in the feasible stability region of the system considering modeling uncertainty bounds, as 𝐾𝑝 = 600, 𝐾𝑣 = 250, 𝐾𝑖 = 5.

In the first experiment, the following desired trajectory in X direction is considered and it is supposed that the end-effector does not move in the other directions. The results of implementation using the proposed controller are given in Fig. 6, 7. Fig. 6 illustrates the tracking errors in three directions, while Fig. 7 provides the cable forces measured by the force sensors.

(7) 𝑋𝑑(𝑡) = {0.1 − 0.1𝑒−𝑡 0 < 𝑡 < 200.1𝑒−(𝑡−20) 𝑡 > 20

}

In the second experiment, the following desired trajectory in Y direction is considered to verify the performance of the controller under gravity force and we require that the end-effector does not move in other directions. The results of implementation using the proposed controller are given in Fig. 8, 9.

(8) 𝑌𝑑(𝑡) = {0.2 − 0.2𝑒−0.25𝑡 0 < 𝑡 < 300.2𝑒−0.25(𝑡−30) 𝑡 > 30

}

Figure 6. Pose of the end effector in the first expriment

Figure 7. Cables tension in the first expriment

In the third experiment, step input is considered for desired rotation of the end-effector while it is attempting to maintain zero translation in both X and Y directions. Step function may excite almost all dynamics of the system, and is used to evaluate the dynamic behavior of the closed loop system for an exaggerated desired motion. In this experiment the final value of step input is considered to be 𝜋/18 radian. The results of implementation using the proposed controller are given in Fig. 10, 11.

As it is seen in Fig. 6, 8, 10 the pose of the end-effector can suitably track the desired trajectories and the steady state errors are very small. Moreover, as it is shown in Fig. 7, 9, 11 all cables remain in tension during the robot maneuvers. In addition, the prescribed uniformly ultimately boundedness of the tracking error for the proposed controller is verified in all three experiments. These experiments verify the effectiveness of the proposed controller with respect to the tracking errors as well as making sure that all the cables remain in tension.

Despite the controller is designed and implemented in the task space coordinates, small errors in tracking the desired trajectories are observed. These errors arise from two reason:

Since the robot calibration is not performed, Jacobian matrix is uncertain. This uncertainty affects the internal forces and as a result, the internal forces do not appropriately span the null space of the Jacobian matrix and this may cause some inaccuracy in the motion tracking of the robot.

The dynamic model of the cable-driven parallel robots is multivariable and nonlinear. For this reason, linear PID controller may be used suitably in regulation problems, while have limited performance in tracking purposes.

Future research will be performed to reduce these errors by designing a robust nonlinear controller in the task space coordinates. In structure of this controller, the cables’ inherent flexibility will be considered as an important aspect.

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Figure 8. Pose of the end effector in the second expriment

Figure 9. Cables tension in the second expriment

VI. CONCLUSION

This paper addresses the issue of implementation of the task space controller on a planar cable-driven parallel robot. To control the robot in the task space coordinates, the position and orientation of the end-effector should be measured. Indirect pose measurement by using the length of the cables and solving the forward kinematics problem is not accurate due to cables’ inherent flexibility and it also limits the proposed controller bandwidth. Furthermore, direct pose measurement requires the

high-tech equipment and expensive sensors too. In this paper to overcome these problems, vision-based pose measurement is used as a suitable and economical solution. Moreover, since the cables are able to apply only tensile forces, design of control algorithms should be performed so that the tensionability condition of the cables is satisfied in the whole Workspace. To achieve this goal, vector of the internal forces is used in the proposed controller structure. Finally, several experiments on KNTU planar cable robot are performed and it is shown that the proposed controller is able to provide suitable performance in practice.

Figure 10. Pose of the end effector in the third expriment

Figure 11. Cables tension in the third expriment

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