Control of two Wheeled Welding Mobile Manipulator

A three-linked manipulator mounted on a two-wheeled mobile platform is used to weld a long curved welding path. A welding torch mounted at the end of a manipulator of the welding mobile manipulator (WMM) must be controlled for tracking a welding path with constant velocity and constant welding angle of torch. In this paper, a decentralized control method is applied to control the WMM considered as two separate subsystems such as a mobile platform and a manipulator. Two decentralized motion controllers are designed to control two subsystems of WMM, respectively. Firstly, based on a tracking error vector of the manipulator and a feedback motion of the mobile platform, a kinematic controller is designed for manipulator. Secondly, based on an another tracking error vector of the mobile platform and a feedback angular velocities of revolution joints of three-link, a sliding mode controller is designed for the mobile platform. These controllers are obtained based on the Lyapunov's function and its stability condition to ensure for the tracking error vectors to be asymptotically stable. Furthermore, simulation and experimental results are presented to illustrate the effectiveness of the proposed algorithm.


Introduction
Nowadays, the welding robots become widely used in the tasks which are harmful and dangerous for the welders. A good welding robot used in welding application can generates the perfect movements at a certain travel speed and reference welding angle of torch, which can produce a consistent weld penetration and weld strength. In practice, some various robotic welding systems have been developed recently. Kim, et al. (2000) developed a three dimensional laser vision system for an intelligent shipyard welding robot to detect the welding position and to recognize the 3D shape of the welding environments. Jeon, et al. (2002) presented the seam tracking and motion control of a two-wheeled welding mobile robot for lattice welding. The control is separated into three driving motions: straight locomotion, turning locomotion, and torch slider control. Ngo, et al. (2006) proposed a sliding mode control of two wheeled welding mobile robot for tracking a smooth-curved path. As we know, the welding mobile robot (WMR) is more simple system than WMM. But its welding point has three degree of freedom (d.o.f.) which is under condition of nonholonomic system of a two-wheeled mobile platform. For this reason, the WMR cannot eliminate its tracking errors and satisfies the welding condition as the same time. It only achieves the welding condition when its tracking errors are zero. To solve this problem, a WMM of three d.o.f. independent to its two-wheeled mobile platform is considered. The developed WMM consists of a threelinked manipulator plus a two-wheeled mobile platform (MP). The mobile manipulator is used for increasing the d.o.f. of the welding point. Hence it satisfies the constraints of the welding task. In this paper, the WMM is considered as two separate subsystems such as a MP and a manipulator. Two decentralized motion controllers are applied to control two subsystems of WMM, respectively. Based on a tracking error vector which is achieved from difference between a real posture of torch and its reference one, and feedback motions of MP, a kinematic controller is designed for manipulator to carry the welding torch for tracking a reference welding path with a constant velocity and a constant reference welding angle. The mobile platform is used to carry the manipulator to avoid its singularities. So the movement of the MP is specified. In this application, the MP is controlled by a sliding mode controller in order to make the manipulator's configuration return to its "good posture" configuration. The "good posture" of manipulator is defined in next section. Based on this idea, another error vector for a sliding mode controller of the mobile platform is obtained from difference between a real posture of the manipulator and its "good-posture". A developed WMM is shown in Fig. 1. The three-linked manipulator is mounted on the center of the MP. An angle sensor of link is attached at its revolution joint. The MP has more two passive wheels which are installed in front and rear at the bottom of MP for its balance, and their motion can be ignored in the dynamics. The motion of two wheels of MP is measured by rotation encoders. The touch sensor is fixed at the end of a third-link as shown in Fig. 1. The touch sensor rolls along a steel wall to detect the tracking errors. A simple way of measuring the errors by potentiometers is introduced in experimental result part. To illustrate the effectiveness of the proposed algorithm, simulation and experimental results are presented. The MP has two driving wheels for body motion, and those are positioned on an axis passed through its geometric center, • The three-linked manipulator is mounted on the geometric center of the MP, • The distance between the mass center and the rotation center of the MP is d . Fig. 2 is the matrix associated with the constraints. As a result, the kinematic model under the nonholonomic constraints in (1) can be derived as follows: is velocity vector.
Firstly, the posture of MP for the center point ) , ( P P Y X P in the Cartesian space in Fig. 2 is defined as If the mobile robot has nonholonomic constraint that the driving wheels purely roll and do not slip, A(q) in (1) can be expressed into From (3) and (4) In the kinematic model of (2), J(q) is written as The relationship between , v ω and the angular velocities of two driving wheels is given by In this application, the welding speed is very slow so that the manipulator motion during the transient time is assumed as a disturbance for MP. For this reason, the dynamic equation of the MP under nonholonomic constraints in (1) is described by Euler-Lagrange formulation as follows: is a symmetric positive definite inertia matrix, is a centripetal and Coriolis matrix, is a matrix of nonholonomic constraints, r R ∈ is a control input vector, and m R ∈ is a constraint force vector. For simplicity of analysis, it is assumed that m n r − = . Differentiating (2), substituting this result to (8) and multiplying by T J , (8) is rewritten as follows: Multiplying by , ( 1 −

B) J T
(9) can be rewritten as follows: In practices, the dynamic equation of the MP under disturbance and interaction between the manipulator and the MP is expressed as follows: where d F is the disturbance and interaction between manipulator and MP. d F is bounded and is represented in anther way as follows [Chwa , et al.]: . In another hand, the dynamic equation of MP can be expressed by a feedback linearization of the system as follows (Yang and Kim, 1999) mb r is a reference input vector, and is a controller vector which is defined by computed torque method. From Eqs. (11, 12 and 13), this relationship yields.

Modeling of the 3 links manipulator
Consider a three-linked manipulator as shown in Fig. 2. A new Cartesian coordinate frame (Frame y x − ) is fixed on the mobile platform as shown in Fig. 2. The (Frame y x − ) is called the moving coordinate frame because this coordinate frame is moved toghether with the mobile platform in the world coordinate (frame Y X − ). The velocity vector of the end-effector with respect to the moving frame is given by is the velocity vector of the end-effector with respect to the moving frame, is the angular velocity vector of the revolution joints of the three-linked manipulator, and J is the Jacobian matrix.
are the length of links of the manipulator, and ), sin( 1 The dynamic equation of the end-effector of the manipulator with respect to the world frame is obtained as follows [Lewis, et al.]: where E V and P V are the velocity vector of the endeffector and the velocity vector of the center point of platform with respect to the world frame, respectively.

Decentralized Controllers Design
In this section, a decentralized control method is applied. A sliding mode controller and a kinematic controller are used to control the MP and the manipulator of WMM, respectively. The relationship between two controllers can be expressed as follows: Firstly, based on a tracking error vector which is obtained from difference between a real posture of torch and a reference one, and feedback motions of mobile platform, a kinematic controller is designed for the manipulator in order to move the welding torch tracking a welding path. Secondly, the mobile platform is used to carry the manipulator to avoid its singularities. So the movement of the MP is specified. For this reason, another error vector is obtained from difference between a real posture and the "good-posture" of the manipulator. Based on this error vector, a new sliding mode controller is applied to control two wheeled mobile platform. These controllers are obtained based on the Lyapunov's function and its stability condition to ensure for the error vectors to be asymptotically stable. From Fig. 3, the tracking error vector T E is defined as follows:

Kinematic controller design for manipulator
The tracking error vector can be rewritten in the matrix form as follows: is a reference welding point which is moved along the welding path with a constant velocity and constant welding angle.
is a welding point or an endeffector point. The first derivative of tracking error vector T E yields To obtain the kinematic controller a back stepping method is used. The Lyapunove function is proposed as follows: The controllers vector mp u is rewritten in another form as follows:

Dynamic controller design for mobile platform
In this part, a sliding mode is used to bring the configuration of the manipulator to "good posture". The "good posture" of manipulator is chosen with Φ is a fixed point with respect to the moving frame and is made by the "good posture" of the manipulator. A sliding mode controller is proposed to control the MP for the M point tracking the real reference E point. When the M point coincides with E point, that is it means that the manipulator has the good configuration for avoiding its singularity. Coordinate of the M point is expressed as follows: The derivative of (30) is as follows  The following part presents a sliding mode controller vector mb u . This controller designs the torque of DC motor for two driven wheels of mobile platform in order to make the MP error vector converge to zero.

Sliding mode controller design
To design a sliding mode controller, the sliding surfaces are defined as follows: So Based on Lyapunov's function and its stability condition, a back-steeping method is used to obtain a sliding mode controller mb u . The Lyapunov's function is chosen as follows: To satisfy the Lyapunov's stability condition 0 ≤ V , the following proposed controller mb u can be calculated as follows: From (41), the first derivative of the sliding surfaces can be rewritten as follows: If f ≤ is chosen, (39) satisfies the Lyapunov's stability condition.

Chattering Phenomenon Elimination
In practice, a chattering phenomenon in a sliding mode control can be eliminated by a saturation function. Using the saturation function, a thin boundary layer neighboring the switching surface is achieved, and then a smooth and proper controller can be obtained. In this case, the welding velocity is rather slow, 7.5mm/s. Therefore, a thin boundary layer 1 . 0 = δ is chosen. A new smooth switching controller than previous switching controller in (39)  With smoothing approach, the controller (39) can be implemented to the practice welding mobile robot totally. The effectiveness of the controller can be seen through the simulation and the experimental results.

Schematic diagram of the control algorithm of mobile manipulator
The schematic diagram for a decentralized control method is shown in Fig. 8. In this diagram, a relationship between controllers is illustrated by means of the output of this controller is one of the input ofanother controller and vice versa.

Simulation and experimental results
To verify the effectiveness of the proposed modeling and controllers, simulations and experimental results were done for the WMM with the welding reference trajectory as shown in Fig. 9. Numerical values and initial values used in simulation and experiment are shown in Table 1 and Table 2.  Fig. 10 shows the zoom out of the end-effector for tracking the welding path at beginning. Fig. 11  which is shown in Fig. 12 converges to zero slowly. This tracking error vector doesn't directly effect on the welding result. It only helps the manipulator to avoiding its singularity. With the movement of the MP, the angle of manipulator is converted and remained in the "good-posture" which is shown in Fig. 13. Fig. 14 shows that the sliding surface vector converges to zero and remains on it. Fig. 15 shows the angular velocities of right wheel and left wheel obtained by the sliding mode controller. Fig 16 shows

Conclusion
This paper developed a WMM which can co-work between mobile platform and manipulator for tracking a long smooth curved welding path. The main task of the control system is to control the end-effector or welding point of the WMM for tracking a welding point which is moved on the welding path with constant velocity. Furthermore, the angle of welding torch must be kept constant with respect to the welding curve. The WMM is divided into two subsystems and is controlled by decentralized controllers. The kinematic controller and sliding mode controller are designed to control the manipulator and the mobile-platform, respectively. These controllers are obtained based on the Lyapunov's function and its stability condition to ensure the error vectors to be asymptotically stable. Furthermore, simulation and experimental results are presented to illustrate the effectiveness of the proposed algorithm. (This work was supported by Pukyong National Uni. (0010000200501700)