Abstract
Purpose
The purpose of this paper is to design the localization and tracking algorithms for our mobile welding robot to carry out the large steel structure welding operations in industrial environment.
Design/methodology/approach
Extended Kalman filter, considering the bicycle-modeled robot, is adopted in the localization algorithm. The position and orientation of our mobile welding robot is estimated using the feedback of the laser sensor and the robot motion commands history. A backstepping variable is involved in the tracking algorithm. By introducing a specifically selected Lyapunov function, we proved the tracking algorithm using Barbalat Lemma, which leads the errors of estimated robot states to converge to zero.
Findings
The experiments show that the proposed localization method is fast and accurate and the tracking algorithm is robust to track straight lines, circles and other typical industrial curve shapes. The proposed localization and tracking algorithm could be used, but not limited to the mobile welding.
Originality/value
Localization problem which is neglected in previous research is very important in mobile welding. The proposed localization algorithm could estimate the robot states timely and accurately, and no additional sensors are needed. Furthermore, using the estimated robot states, we proposed and proved a tracking algorithm for bicycle-modeled mobile robots which could be used in welding as well as other industrial operation scenarios.
Keywords
Citation
Tang, Q. (2014), "Localization and tracking control for mobile welding robot", Industrial Robot, Vol. 41 No. 3, pp. 259-265. https://doi.org/10.1108/IR-07-2013-377
Publisher
:Emerald Group Publishing Limited
Copyright © 2014, Emerald Group Publishing Limited
1. Introduction
Along with the development of shipbuilding and large steel tanks welding, the mobile welding robots are becoming widespread (Xin et al., 2011). Large structures are difficult to be welded due to their huge bodies and complex welding requirements. Usually, the welding seam is very long and cannot be pre-programmed. Traditional industrial robots which are fixed to one spot cannot meet these requirements. Developing a mobile robot which could move on the surface of large steel structures and track along the welding seam with sensors is a sound solution to these problems.
Up to now, considerable research has been devoted to design a mobile welding robot, and various types of experimental models were proposed to solve the welding problems of large steel structures (Silva et al., 2008), including the orbital robots (Lee et al., 2011; 2010; Santos et al., 2000; Lima et al., 2005), the legged robots (Tavakoli et al., 2011; 2013) and the wheeled robots (Shang et al., 2008; Wu et al., 2011; 2010). Orbital robots need special designed orbits to work and the payload of the legged robots is low. Wheeled robots with high payload ability and mobility are much popular in recent years. Shang et al. (2008) developed a climbing robot for inspecting long welding lines (Shang et al., 2008). The robot has an excellent capability to overcome obstacles; however, two robots, including the inspecting robot and the welding robot, are needed to complete one welding task. Wu et al. proposed a noncontact magnetic adhesion unit which uses a motor to adjust the magnetic force (Wu et al., 2011; 2010). With this mechanism, the robot obtained both strong adhesion force and mobility. However, localization and tracking problems, which are among the three basic problems of mobile robots, are rarely considered in these studies.
Current studies mainly focused their attention on the localization of the seam. Fanhuai use a vision system to detect the welding seam, and this method could apply to most weld seam shapes in butt joint welding (Shi et al., 2009). The accuracy of the vision system is affected by welding, thus this method could only be used before welding. For real-time seam detection, Liu et al. equipped an industrial robot with a laser sensor which can analyze the seam image and extract the seam information. The coordinate information is calibrated and transformed to the robot coordinate frame (Suyi et al., 2009). However, few studies mentioned the localization of the mobile welding robot itself, which is very important, especially in large steel structure welding. Without the localization, the mobile welding robot might deviate from the original trajectory after moving for a long distance. The robot might go out of the range of the operation area and the welding operation would fail. Although there are many studies on the localization of mobile robot, most of them mounted a laser distance scanner or a vision camera to sense the surrounding areas. These sensors are not only expensive but also unsuitable for industrial applications. In this paper, we would propose a localization algorithm with the seam-detecting laser sensor. Without additional sensors, localization and seam detection could be realized simultaneously.
Tracking algorithm is another important part of the mobile welding robot. Because most mobile robots are nonholonomic systems, the tracking problem is always tricky. Byoung et al. introduced a PID controller on the welding mobile robot to realize the line tracking (Kam et al., 2001). Yanfeng et al. proposed a predictive fuzzy control and neural network control for the unicycle-modeled mobile welding robot (Yanfeng et al., 2011b; 2011a; 2008). Li et al. also proposed a tracking control algorithm using fuzzy-Gaussian network and Lyapunov method for the mobile welding robot (Kai et al., 2009a; 2009b; Ting et al., 2009; Zhang et al., 2009). Each tracking algorithm gave a solution to track all kinds of welding seams. However, they are based on the unicycle-modeled robot but not the bicycle-modeled robot. In our designed mobile welding robot, we adopted the bicycle model as the prototype for our mobile platform, because bicycle-modeled robot is much more stable and more popular in robot design. So far, few algorithms illustrated the tracking of bicycle-modeled robot, which would limit the development of mobile welding robots.
In this paper, we will introduce the localization and the tracking control algorithm of our mobile welding robot to give an insight into mobile welding from a novel point of view. The paper is organized as follows. Modeling of the mobile welding robot is presented in Section 2. In Section 3, a localization algorithm using extended Kalman filter is proposed to determine the robot's location in a given environment based on the laser sensor data and the robot movements. A seam tracking algorithm is proved in Section 4 to fulfill the welding requirements. Simulation and experiment results are given to verify our control algorithms in Section 5. In the end, conclusion and the future work are drawn in Section 6.
2. Modeling of the mobile welding robot
Our mobile welding robot consists of a mobile platform, a three-degrees-of-freedom (df) manipulator, a laser sensing system and a mobile welding power unit. The mobile platform, on which the manipulator and welding torch are mounted, could adhere to and move freely on the surface of large-scale ferromagnetic steel structure. The three-df manipulator could move in both vertical and transverse directions, and swing the welding torch according to the welding requirement. The df distribution is illustrated in Figure 1(a). The red boxes and cylinders stand for the seven actuators equipped on the mobile robot.
For simplicity, we decouple the control system into two directions, horizontal direction and vertical direction. In vertical direction, only one actuator controls the height of the welding torch and the laser sensor returns the distance between the torch and the seam. The control scheme is simple. In this paper, we mainly consider the movement in horizontal direction. The proposed system is composed of a bicycle-modeled mobile platform, a manipulator in transverse direction and a laser sensor which could perceive the position of the seam in sensor coordinate frame. The abstract model and the parameter's definition of our mobile welding robot are illustrated in Figure 1(b). The front wheel can be steered while the rear wheel's orientation is fixed. The robot is featured as a rear-wheel-driven bicycle model. A typical coordinate frame could be chosen as x = [x, y, Θ, Φ]T, where vector x is the state of the robot. The dynamic and kinematic equation of the mobile welding robot could be derived as equation (1). Equation 1
Where the origin of the coordinate is chosen at the beginning of welding seam. The x-axis is defined as pointing along the welding seam; the y-axis is defined as perpendicular to the x-axis and pointing to the right. Point (x, y) is defined as the center position of the two contact points between the rear wheels and the ground. Orientation angle Θ measures the orientation of the vehicle with respect to the x-axis, and angle Φ is the steering angle of the front wheel with respect to the vehicle's heading direction. The input of the control system is denoted by u = [v, ω]T, where velocity v is the linear velocity of point (x, y). Velocity ω is the angular velocity of the steering direction. Distance d represents the distance from steering wheel to the center of the rear wheels.
The manipulator, which contains three df including the vertical movement, the transverse movement and the swinging movement, is controlled to minimize the position error between the welding torch and the welding seam. As defined in Figure 1(b), H is the distance from the center of the rear wheels to the welding torch along the rear wheel axis and K is defined as the distance from the welding torch to the axis of the rear wheel. L is defined as the distance from the laser sensor to the axis of the rear wheel. The output equation of the mobile welding robot could be derived as equation (2): Equation 2
Where (x o u t , y o u t ) is the coordinate of the welding torch. The laser sensor will return the distance from the sensor to the axis of rear wheel. The corresponding observation equation could be derived as equation (3), where y o b is the coordinate of laser sensor: Equation 3
3. Localization of the mobile welding robot
3.1 Observation model
Different from the traditional industrial welding robot, our welding robot features its extra mobility which extends the robot's operation space from a small area around a fixed working spot to wherever the robot could reach. In this section, we focus on the localization of the mobile welding robot.
The solution to the problem “Where am I?” is to accurately estimate the states of the mobile welding robot described in system (1). The robot states are influenced by many factors including the movement of the robot, the movement noise and the sensor noise. The estimation process is to use the movement instructs and the sensor value to estimate the state considering the process and measurement noise. The input value includes the encoders of each four motors, such as the front wheel motor, the steering motor, the left rear wheel motor and the right rear wheel motor, and the encoders of the operating mechanism in transverse direction. The output value refers to the feedback from the laser sensor. Because most input and output sources are discrete signals, we will analyze the robot system in discrete time domain. Developing the discrete counterpart of the robot's continuous-time system, the dynamic equation could be derived as equation (4): Equation 4 Where all the inputs and outputs would be assumed to have the same sampling period T and the discrete time state variables and inputs will be denoted by [x k , y k , Θ k , Φ k ]T = [x(k T), y(k T), Θ(k T), Φ(k T)]T and [v k , ω k ]T = [v(k T), ω(k T)]T. To simplify the system equation (4) and the observation equation (3), the nonlinear difference equation of robot system and the observation equation could be expressed as equation (5): Equation 5 Where the state variables x k = [x k , y k , Θ k , Φ k ]T and x k − 1 = [x k − 1, y k − 1, Θ k − 1, Φ k − 1]T, respectively, present the current and the previous states of mobile robot system. The input variable u k − 1 = [v k − 1, ω k − 1]T refers to the actual input of the previous time step. The nonlinear function f in the difference equation (5) relates the state at the previous time step k − 1 to the state at the current time step k. The nonlinear function h in the difference equation (5) relates the system states to the observed variables. The observation variable is denoted by z k = y o b .
However, linear velocity v k and angular velocity ω k can't be obtained directly from the encoders of motors, but can be evaluated according to equation (6): Equation 6 Where v l k stands for the linear velocity of the left rear wheel at time k T, v r k stands for the linear velocity of the right rear wheel at time k T, d w stands for the distance between two wheels.
3.2 State estimation
Localization is the essential problem of mobile robot's locomotion. In this paper, we adopted the extended Kalman filter to implement the seam tracking of robot system. Considering the noise in the process and the observation, the robot control system is governed by the nonlinear stochastic differential equation as follows: Equation 7 Where w k and v k represent the process and measurement noise, respectively. They are assumed to be independent, white and with normal probability distributions as equation (8): Equation 8 Where Q is the process noise covariance and R is the measurement noise covariance.
By using the extended Kalman filter and the linearization of the nonlinear differential equation, we can obtain the new governing equation (7) describing the system. The linearized system and observation equation could be derived as: Equation 9 Where A k is the Jacobian matrix of the partial derivatives of the function f with respect to the state x; W is the Jacobian matrix of the partial derivatives of the function f with respect to the noise w; H k is the Jacobian matrix of the partial derivatives of the function h with respect to the state x; V is the Jacobian matrix of the partial derivatives of the function h with respect to the noise ν. The resultant equations could be expressed as equation (10): Equation 10 Equation 11 Equation 12 Equation 13 To implement the extended Kalman filter to localize the robot's position and orientation, we estimate the robot states in two steps:
Step 1. Update the priori state Inline Equation 1 and the covariance estimates Inline Equation 2 using equation (11) from the inputs and previous estimated states. Equation 14 Step 2. Correct the states and covariance estimates with the measurement z k according to the equation (12). Equation 15 By iterating the estimation process with inputs and observations, the robot can be localized timely and precisely. Section 5 will show the results of our estimation method.
4. Tracking algorithm
4.1 Error system
The desired trajectory which our welding robot is designed to follow can be expressed as follows: Equation 16 By comparing the desired states q d (t) = [xd(t) yd(t) Θd(t) Φd(t)]T with current estimated states q(t) = [x(t) y(t) Θ(t) Φ(t)]T, we could define the error system with respect to the current robot frame. The system could be formalized as follows: Equation 17 By differentiating e with respect to time, we could easily obtain the dynamic error equation of the error system as follows: Equation 18
4.2 Tracking algorithm
To control the robot states q(t) tracking along the target states q d (t), we have to design the specific inputs v and ω to minimize the dynamic error system to converge to zero. For our mobile welding robot, the tracking algorithms could be summarized as follows:
Tracking algorithm:
e = 0 is a globally and asymptotically stable equilibrium point of the system (15), when Equation 19 Where Φ^ is the backstepping variable and Inline Equation 3 For stability, ω^ is designed as follows: Equation 20 The backstepping variable could be calculated by: Equation 21 Proof:
Select a Lyapunov candidate function as follows: Equation 22 It is obvious that V(e)> 0 at e ≠ 0 and V(e) = 0 at e = 0 and Φ = Φ^, and consequently, V(e) is positive definite. The time derivative of V(e) could be calculated as follows: Equation 23 when Equation 24 Because V¨(e) is uniformly continuous, according to Barbalat Lemma, V˙ (e) converges to zero. We could have e 1 and e 3 converge to 0 together with Φ → Φ^ when t → ∞. Considering the dynamic error system (15), we could prove that e 2 also converges to zero. Finally, we proved that under the given control law (16), the system is asymptotically stable.
5. Simulation and experiment
To verify our localization and tracking algorithms, we gave two experiments. The first experiment is the simulation experiment which verifies our tracking algorithms. Because we have proved our algorithm in Section 4.2, more complicated trajectories will be simulated to show our algorithm could be widely used to track all kinds of trajectories. The other experiment is a physical experiment with our mobile welding robot which verifies the accuracy of our localization algorithm.
5.1 Simulation experiment
In this section, we will verify our tracking algorithm by demonstrating the line tracking and the circle tracking. In the line tracking simulation, our mobile welding robot is initially located at the origin of the coordinate system with states equal to [0, 0, 0, 0]T. The target welding robot is located at point [1, 1, 0, 0]T. The target robot moves straight forward at the speed of 1.5 mm/s along the x-axis. Figure 2 shows the state errors of our welding robot converge to zero gradually under the given tracking control algorithm. In the actual welding process, the initial state error is much smaller and the convergent speed could be adjusted to satisfy the welding requirement. Figure 3 shows line tracking trajectory of the welding robot.
In fact, the tracking algorithm is not only limited to the line tracking, but also could be used to track arbitrary reasonable trajectories. Figure 4 shows the converging process of the state errors of our welding robot tracking a circular trajectory. Figure 5 shows the resultant tracking trajectory of the welding robot. The results verified that our tracking algorithm could track complex trajectory and can easily be used in various welding environment, which is of great importance to the mobile welding robot.
5.2 Physical experiment
Physical experiment mainly verifies the localization algorithm. Because it is difficult to get the true value of robot states, an indirect method is used. Figure 6 shows our physical mobile welding robot. Laser sensor is equipped in front of the welding torch. We actuated the manipulator to make sure that the seam is always in the center of laser sensor. This tracking result could be easily achieved, because the manipulator system has one translational df and could be modeled as a first-order control system. The control is quite simple in this situation. The tracking of the laser sensor's center point to the seam could easily be verified from the feedback sensor data. Figure 7 shows the feedback from the laser sensor during the tracking process. It could be observed that the error between the seam and the center of laser sensor is below 1 mm, which promised a fine tracking of the laser sensor. In this way, the true value of the robot state in Y direction could be computed according to the observation equation (3) in Section 2. Then the computed robot state in Y direction could be used as the true value for the experiment.
Using the proposed localization algorithm, robot states are estimated from the laser sensor feedback and the robot movement command history. Figure 8 shows the estimation result of robot states. The first subfigure compares the angular estimation using the extended Kalman filter and the least-squares method, respectively. The least-squares method estimates the robot body's orientation Θ every 20 time steps. The change of orientation couldn't be noticed during each 20 time steps using the least-squares method. However, the state estimation introduced in our paper could respond to the sensor data immediately and thus leads to a better result.
The second and third subfigures show the estimation results of robot position in the forward and transverse direction, which couldn't be observed using the least-squares method. In the forward direction, robot moves at a constant speed. The second subfigure not only shows the forward displacement with a constant slope but also verifies the soundness of the estimation. The third subfigure contains two curves. The solid curve shows our estimation result, the other is the true value of robot state. The coincidence of two curves shows the accuracy of our estimation result.
6. Conclusion and future work
In this paper, the localization and tracking algorithms are designed to solve the issue of large steel structure welding using our mobile welding robot. The localization problem is solved by using extended Kalman filter considering our specified mobile robot structure. Not only could the robot states be estimated at every time step, but the forward and transverse position could also be observed. Physical robot experiment verified our estimation method could observe the robot states timely and accurately. We also proposed a tracking algorithm to follow the target robot trajectory with our bicycle-modeled mobile robot. We introduced a backstepping variable and used Barbalat Lemma to prove that the tracking errors of our robot system could converge to zero in welding process. The simulation experiment verified that our algorithm could track both lines and circles. In fact, using our algorithms, our mobile welding robot could reach most welding requirement encountered in industry environment.
However, welding is a complex process. It is associated with temperature, voltage, current, welding torch direction and other factors. Although our algorithms provide a high-quality line welding in the experiment with specifically tuned welding parameters, our proposed localization and tracking algorithms only solved the vital problems of mobile welding robot. In the future, we plan to divide the mobile welding problem into two closely related aspects, the locomotion and the welding. Robots with mobile abilities could equip any mature field industrial manipulators to realize a wide range of industrial operations such as polishing, cutting, painting and so on. Our proposed algorithms could be widely used in, but is not limited to, all these areas.
Corresponding author
Qing Tang can be contacted at: [email protected]
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