Control of Pipe Cutting Robot: A More Effective Method

This paper presents a pipe cutting Robot system with two di erent cutting methods: the method with the end-e ector moves on cutting path and direction while the stationary pipe and the method with the end-e ector moves on a straight line while the rotating pipe to create the desired cutting path and direction. The cutting trajectory are described, the Robot model is constructed, solving the inverse kinematics, planning the trajectory of motion, simulating and controlling Robot in Matlab, and designing the experimental Robot to verify. The results of the two methods are compared to point out a better one. This research builds up an important foundation for choosing an e ective method for pipe cutting Robot in industry.


Introduction
In industry, gases and liquids are transported daily by pipelines and these pipeline systems paired together in a complex way. To create them, the steel pipes are cut and welded together. However, this is not simple because there are complex joints that require cutting and welding paths to be complicated in which the trajectory and direction change continuously. With that requirement, conventional tools cannot be implemented and the application of robots is necessary.
The use of robot for cutting pipe has become very popular in the world. In Vietnam, this technique has not yet been widely applied, and there are few scientic publications in this eld.
In [1], the authors mentioned a Delta Robot for cutting high-speed laser with the numerous advantages of robot: higher stiness, fewer joints, the ability of transporting heavier loads, and higher accuracy. The main drawback is the small workspace, and this paper also does not mention much about the application of Delta Robot to cut steel pipes.
In [2], the authors presented a pipe cutting technique that included a pipe cutting Robot that the robot arm moves and the pipe is stationary during cutting. The authors successfully builds 3D simulation and experimental model, solving the inverse kinematics, planning the trajectory as well as designs Robot controller. The results of simulated and experimental errors are provided. However, the authors only stopped at the method of the end-eector moves while the stationary pipe without mentioning their coordinated motion.
In this paper, the author presents another method of cutting pipe more eectively with a 6 degrees of freedom pipe cutting Robot, consisting of 5 degrees of freedom robot arm and the degree of freedom created by the rotating motion of a pipe. With this method, the end-eector will only move on a straight line, and the pipe will rotate in conjunction with the movement of the end-eector to create cutting path in reality. This result is compared with the results in [2] to point out that this cutting method is better than the cutting method in [2].

2.
Pipe Cutting Problem The cutting paths and cutting directions can happen many cases, depending on pipeline assembly position and welding conditions. This paper will focus on Hyperbolic Paraboloid Pringles; a common path is created by two intersecting pipes as shown in Fig. 1.

Cutting Path
In coordinate frame {−1}, place pipes R 1 and R 2 which have center lines coincident with Z −1 axis, each pipe equation is given by Eq. (1): where • L, R: the length and radius of two pipes, • x, y, z: coordinates of two pipes and When turning pipe R 2 an angle ar • around X −1 axis (for example 90 • , see Fig. 1), we have Eq. (2): where • ar: angle between R 1 and R 2 and • s ar : sin ar, c ar : cos ar.
When cutting, pipe R 2 must be re-applied an angle −ar • so that the center line of R 2 coincides with Z −1 axis. The cutting path of pipe R 2 is given by Eq. (13), Eq. (14), Eq. (15) and Eq. (16). where

Cutting Direction
Beside cutting path, we must pay attention to the cutting direction. With each cutting point which many directions to go through, but only one direction is reasonable with the requirements about pipe welding conditions [3], depending on the cutting angle [4] and the position of the point, shown in Fig. 2. Cutting direction changes continuously during cutting process. In the coordinate frame {−1}, plane (M, C) contains Y −1 axis and passes cutting point Q 1 . In plane (M, C), e is the line that contains cutting direction and passes cutting point Q 1 , α is the angle between e and Y −1 axis (given by Eq. (17) and Eq. (18)). β is the angle between plane (M, C) and (Y, Z).
ac: the standard cutting angle is given before [4] (see Fig. 1 and Fig. 2). e is found by rotating an imaginary line through Q 1 and paralleling Y −1 axis with an angle β around Y −1 axis, and then rotating this line with an angle α around Q 1 . e is the line containing the direction we need.
The cutting direction in the coordinate frame {−1} is a 3x3 matrix and given by Eq. (19): From Eq. (19), we obtain the cutting direction in the coordinate frame {0} as Eq. (20):

Cutting Method
There are two general cases of cutting: static pipe while moving end-eector [2] and rotating pipe while moving end-eector. Especially, the end-eector moves on a straight section with cutting direction following to Eq. (20). The combination of rotating pipe while moving end-eector on a straight section is given by Eq. (21) and Eq. (22). It will create Hyperbolic Paraboloid Pringles like the reality. This case is described in Fig. 3. Trajectory of end-eector is a parallel straight section with X 0 axis, and is located right at the top of the pipe. The straight section belongs to plane (X 0 , Z 0 ), in which Y 0 = 0 and Z 0 = R 1 . From Eq. (9), Eq. (10), Eq. (11) and Eq. (12), we infer: From Eq. (21), we obtain: While the end-eector moves on a straight line in Eq. (21) with direction in Eq. (20), the pipe rotate an angle θ 0 in Eq. (22). This combination of motion will create the Hyperbolic Paraboloid Pringles in real. 3.
Robot Model The Robot model used in this paper includes 5 degrees of freedom robot arm and a degree of freedom created by the rotating motion of the pipe. This model is shown in Fig. 4.

Inverse Kinematics
Inverse kinematics results in exact positions of joints when position and direction of end-eector is known.

Trajectory Planning
Links will be moved concurrently and corporately so that end-eector will follow cutting equation in dened duration t.
In Cartesian space, cutting path will be divided into set of points in which the space between these points is very small and equal ∆p. Inverse kinematics is used to dene joint variables in joint space corresponding to set of points in Cartesian space ( [8], [9] or [10]).
Joint path planning must ensure the continuity of position, velocity, acceleration and cubic polynomial is a suitable choice.
Suppose that n th joint rotates angles θ i and θ i+1 in durations t k and t k+1 respectively, the cubic polynomial of the form of n th joint: Velocity:θ Parameters of the cubic polynomial must be dened based on constraints so that joint path satisfy the continuity of position, velocity and acceleration.

Simulation and Control
Trajectory and direction were planned in Cartesian space. They will be transformed into the joint space. The robot is controlled in the joint space so that the end-eector follows the trajectory and the expected direction. Control ow chart algorithm is given by Fig. 5. The robot is drawn by SolidWorks as Fig. 6. This Robot system is imported into Matlab Simulink as Fig. 7.
PID transfer function of the rst order lter ( [11] and [12]) is given by Eq. (69):  • E: Error between the input value and the feedback value, • K p , K d , K i : Proportional gain, derivative gain, integral gain and • τ f : The time constant of the rst order lter. Set: From Eq. (75), Eq. (76) and Eq. (77), we nd that:Ẋ The simulated results: The simulation time is the 30 s. Figure. 9 and Fig. 10 give trajectory and response of six joints in the joint space. Figure 11 and Fig. 12 are the error graphs of six joints in the joint space. In Fig. 11: Since the static pipe leads to the joint 0 is motionless and errorless, the maximum error belongs to joints 3, 4 and 5 with a maximum value of 0.04 • . The best activity is joint 1 with a maximum error of 0.0075 • . In Fig. 12: Since the end-eector moves on the straight section, joint 1 stays still.   Discussion: The simulated results showed that the case of a static pipe cutting was not as good as the case of a rotary pipe cutting. 5. Experiment Figure 15 is the real robot system. Time to nishing work of Robot is set to 30 s. Two microcontrollers will control ve harmonic driver motors corresponding to ve joints of Robot and a rotary motor. Data obtained from Robot will be transmitted to the computer.
The experimental results: Figure 16 and Fig. 17 are the error graphs of six joints in the joint space. In Fig. 16: The maximum error belongs to joints 3 and 4; the error ranges from −0.072 • to 0.079 • . In Fig. 17: The maximum error belongs to joints 3 and 4; the error ranges from −0.058 • to 0.045 • . These errors are smaller in Fig. 16. Figure 18 and Fig. 19 are the error graphs of the end-eector in the three axes of X −Y −Z in the Cartesian space corresponding to two cases:  standing and rotating. We see that the errors in the two graphs range from −0.2 to 0.2 mm. Fig. 18 has a larger error graph and is more oscillating than Fig. 19.
Discussion: The experimental results showed that the case of a rotary pipe cutting was better than the case of a static pipe cutting, with less error and less oscillation error.

Conclusion
The paper has solved the whole problem: building the cutting trajectory, solving the inverse kinematics, planning the trajectory of motion, simulating and controlling Robot in reality. More importantly, this paper has developed two dierent pipe cutting solutions, and gives the comparative results between the two ones in both simulation and experiment. These comparative results show that method of the endeector moves on a straight line while the rotating pipe to create the cutting path and direction for better than method of the end-eector moves on cutting path and direction while the stationary pipe. This conclusion is an important note that we should design the robot arm and the pipe coordinate movement together, bring the best eect.