Modeling and control of a new robotic deburring system

This paper discusses the modeling and control of a robotic manipulator with a new deburring tool, which integrates two pneumatic actuators to take advantage of a double cutting action. A coordination control method is developed by decomposing the robotic deburring system into two subsystems; the arm and the deburring tool. A decentralized control approach is pursued, in which suitable controllers were designed for the two subsystems in the coordination scheme. In simulation, three different tool configurations are considered: rigid, single pneumatic and integrated pneumatic tools. A comparative study is performed to investigate the deburring performance of the deburring arm with the different tools. Simulation results show that the developed robotic deburring system significantly improves the accuracy of the deburring operation.


I. INTRODUCTION
A machining manipulator is subject to mechanical interaction with the object being processed. The robot performs the task in constrained work space. In constrained tasks, one is concerned with not only the position of the robot end-point, but also the contact forces, which are desired to be accommodated rather than resisted. Therefore, the interaction force needs to be considered in designing and controlling deburring tools.
Many researchers have proposed automated systems for grinding dies, deburring casting, removing weld beans, etc. 1,2 Usually, a deburring tool is mounted on a NC machining center or a robot manipulator. Several control laws have been developed for simultaneous control of both motion and force [3][4][5] of robotic manipulators. Despite the diversity of approaches, it is possible to classify most of the control methods into two major approaches: impedance control [6][7][8] and hybrid position/force control. [9][10][11][12] However, these methods require an accurate model of force interaction between the manipulator and the environment and are difficult to implement on typical industrial manipulators that are designed for position control.
An active feedback control scheme was developed in order to supply compliance for robotic deburring as a means to * Corresponding author. accommodate the interaction force due to contact motion. Kuntze 13,14 suggested an active control scheme, in which the actuators are commanded to increase torques in the opposite direction of the deflections. Paul 15 applied an active isolator to a chipping robot, where the isolator attached to the arm tip reduces the vibration seen by the robot. Sharon and Hardt 16 developed a multi-axis local actuator, which compensates for positioning errors at the end point, in a limited range.
Asada and West [17][18][19][20] developed passive tool support mechanisms, which couple the arm tip to the workpiece surface and bear large vibratory loads. These mechanisms allow the robot to compensate for the excessive deflection when the robot contacts the workpiece. These methods reduce dynamic deflection in a certain frequency range. However, it is difficult for these control schemes, which are employed for a robot with a passive tool, to perform well over a wide frequency band because they must drive the entire, massive robot arm. In addition, unknown compliance from a passive tool makes it difficult to control the deburring robot.
In this paper, a robotic deburring method is developed based on an integrated pneumatic actuation system (IPAS), which considers the interaction among the tool, the manipulator, and the workpiece and couples the tool dynamics and a control design that explicitly considers deburring process information. First, a mathematical model of a single pneumatic actuator is developed. Then, a new active tool is developed based on two pneumatic actuators, which utilizes double cutting action -initial cut followed by fine cut. Then, a coordination based control method is developed for the robotic deburring system based on the active pneumatic deburring tool, which utilizes coordination of two cutters. Simulation results show that the developed system significantly reduces the chattering of the deburring robot and improves the accuracy of the deburring operation.

II. MODELING OF THE DEBURRING ROBOT
In this section, a dynamic model of a robotic arm with the new deburring tool or IPAS is developed as a robotic deburring system. The load dynamics of a single pneumatic actuator is described and utilized for developing a dynamic model of the IPAS, which integrates two pneumatic cylinders. Then, the equations of motion of the deburring robot are derived.

II.1. Single pneumatic actuator
The single pneumatic tool is illustrated in Figure 1, where i = 1, 2, G i is the entering mass flow, P i is the chamber pressure, V i is the volume of the chamber, X,ẊandẌ are the position, velocity and acceleration of the piston, respectively, A i , M, and X 0 are the area, the mass, the initial position of the piston, respectively, F e and F r are the external and friction forces, respectively.
The chamber pressure and the kinetic variables are related by the equilibrium equation [21][22][23] where G i is the entering mass flow, ρ i the air density and V i the volume of the Chamber i. The air is assumed to be ideal gas, which can be written as where the subscript j denotes the initial conditions, R the air constant, P the air pressure in the cylinder, and T the absolute air temperature. According to the displacement of the piston rod, the volume of the chambers is written as where the subscripts 0 indicates the initial position of piston M, respectively, and A i denotes the area of the piston. By combining Eqs. (2) and (3) and their time derivative in Eq. (1), the following expression is obtained: Eq. (4) can be rewritten as The load dynamics can be written for the system as following: whereẊ andẌ represent the velocity and acceleration of the piston, respectively, K f and F e are the viscous friction coefficient and the external force exerted on the piston, 9,11 respectively, and M denotes the mass of each piston. Figure 2 shows the integrated cylinder, which is comprised of three chambers and actuated by a single valve connected to Chamber 3. Note that the IPAS is a single input system with two pistons. The pistons are not directly connected to the inner pistons, M 3 and M 4 , which create a unique configuration of three chambers connected in series. This configuration allows the chambers adjacent to the active chamber to act as vibration isolators. This feature enables the IPAS to damp out the chatter caused by external loads and air compressibility. Therefore, double cutting action and chattering reduction can be achieved simultaneously.

II.2. Integrated pneumatic actuation system (IPAS)
According to the variable volumes of the chambers, the changing dynamic relationship can be represented by the following equation: 10 where G 3 is the entering air flow, ρ 3 the air density and V 3 the volume of Chamber 3. It is assumed that the condition of the air is ideal as following: where the subscript j indicates the initial conditions and n is the air transformation ratio. Now, V 3 is derived as where A 3 denotes the area of Piston 3, and X i (i = 4, 3) is the position of Piston i. L denotes the length of Chamber 3 as shown in Figure 3. By combining Eqs. (8) and (9) and their time derivatives in Eq. (7), the following expression is be obtained: Then, the pressure gradient is be written as Now, the load dynamics of the integrated system is considered as following: whereẊ i andẌ i represent the velocity and the acceleration of each piston. F f i denotes the viscous friction force of the piston rod (i = 1, 2, 3, 4), F ei is the external force (i = 1, 2), and P i and A i denote the air pressure and the area of the piston, respectively. Then the dynamic equations are written as (14) and where K and C are stiffness and damping coefficients of the system, respectively. Figure 3 illustrates a three-link rigid robot with the pneumatic deburring tool described earlier. Using the well-known Lagrangian equations, the following equations of motion of the deburring robot were obtained:

II.3. Robotic deburring system
where q,q,q are the joint angle, the joint angular velocity, and the joint angular acceleration, respectively,m(q) is the 3 × 3 symmetric positive-definite inertia matrix,c(q,q)q is  the 3 × 1 vector of Coriolis and centrifugal torques,ḡ(q) is the 3 × 1 gravitational torques, and τ is the 3 × 1 vector of the joint torques. The mass of the links and pneumatic cylinder are considered as if they were rigidly attached. The relationship between the joint and the tip velocities can be written aṡ where J (q) is the geometric Jacobian of the manipulator. By differentiating Eq. (17), the Cartesian acceleration term can be found asẍ Now, Eq. (16) is rewritten as from Eq. (18) and suppressing arguments for brevity, the following equation is obtained Then, the equations of motion of the robot are obtained as following: where f = (J T ) −1 τ is input expressed in task space. Let the dynamic equation of the robot manipulator in the constraint coordinates be represented as where f denotes the input force and f rf is the resultant force of the normal force f n and the tangential force f t exerted on the tool tip. The tangential force 9,10,19 can be represented as where V t is the spindle speed of deburring tool; b is the tool width; d is the depth of cut; v t is the feed rate (or the traveling speed of the end effector along the surface of the workpiece); e m is the material-stiffness of the workpiece.
The normal force f n is assumed to be proportional to the tangential force f t . Besides, the force angle of the deburring tool affects the tangential force. Although the value of the angle may vary substantially depending on the nature of the material flow at the tool-chip interface, as approximation 0.3 was used in these calculations. 12 Therefore, the normal force f n is considered to be smaller than the tangential force f t in Eq (23), where the ratio is f n /f t ≈ 0.3. 24

III. CONTROL DESIGN
The IPAS based deburring robot can be treated as a system that consists of two primary subsystems; the arm and the IPAS. The two subsystems differ substantially in their task assignments, dynamic characteristics and controller requirements. This physical interpretation provides an efficient approach to the control of the robotic deburring system. The control strategy for the deburring robot is illustrated in Figure 4. The arm is commanded to follow the desired trajectory in task space, which is modified based on the position of the second piston due to varying length of the tool. In other words, the primary cutter at the front side cuts the burr first and the second cutter then attempts to eliminate the remaining burr. In case that the burr is not removed completely, the uncut depth is incorporated into the desired trajectory for compensation. The developed control design is a decentralized control, [25][26][27] which consists of two independent controllers interacting based on the coordination scheme aforementioned for the manipulator and the IPAS, respectively. Constraint equations are derived in terms of position variables and are differentiated twice to lead to a relationship in terms accelerations, which integrate the separate controllers for stability proof. Feedback linearization is employed to design a coordination based controller. In what follows, it is shown that use of a nonlinear dynamic feedback achieves exact linearization and input-output decoupling for the robotic deburring system.
The coordination control method is developed first and then its efficiency will be compared with the hybrid control method through simulation study.

III.1. Coordination control
The decoupled dynamic robot model become as following: where x r ,ẋ r andẍ r denote the displacement, velocity and acceleration of the tip of the manipulator, m r is the inertia mass matrix, c r consists of Coriolis, centripetal, and gravity forces, f r is the input force acting on the tip of the mainpulator, R r is the inertia matrix which reflects the dynamic effect of the deburring tool on the manipulator, and X t is the acceleration of IPAS. Likewise, the equations of motion for deburring tool are written as whereẌ t andẊ t denote the acceleration and velocity of each piston, M t is the mass matrix, C t consists of the viscous friction and gravity forces, F t is the forces acting on the pistons, R t is the inertia matrix which represents the end point of the manipulator on the tool, F e is the external force of IPAS. Let p denote the position vector of the contact point, in fixed workspace coordinate system. Therefore the robotic deburring system is assumed that the constraint surface can be defined in algebraic terms by where p is comprised of x r and X t . Now, the constraint equation (26) is differentiated once as following: where J c denotes the geometric Jacobian matrix. The initial Lagrange coordinate q 0 satisfies the holonomic constraint φ(p 0 ) = 0, where p 0 is the initial position of the robot. Then, Eq. (27) is differentiated once to produceφ = 0, into which the subsystems, Eqs. (24) and (25) are incorporated. Then, feedback linearization can be applied to cancel the coupling terms and to design linear controllers as the outer feedback loop. Since the manipulator velocity is always in the null space ofφ(p), it is possible to define a vector of generalized velocities η(t) as following: where ζ (x r )is a full matrix, whose columns are in the null space of φ(p). Differentiating (28), substituting the resulting expression forẍ r into Eq. (24), and premultiplying Eq. (24) by ζ T , we have Note that ζ T φ T = 0. Similarly substitutingẍ r into Eq. (25), we have Using the state vector χ = [ x T r X T t η TẊT t ] T and the block partition of the state vector Where X t = [X 1 , X 2 ] is the displacement of each position.
The following expression is obtained: The system is input-output linearizable by using the following nonlinear feedback: which results in simpler state equations as following: To derive the decoupling matrix, each component of the output equations is differentiated until the input appears explicitly in the derivative. In this case, the output equation is differentiated twice as following: where (χ) is the decoupling matrix of the system given by where ∂q r , f 2 (q r ) = l 1 cos q 1 + l 2 cos(q 1 + q 2 ) l 1 sin q 1 + l 2 sin(q 1 + q 2 ) Applying the following nonlinear state feedback the input-output relationship is decoupled because each component of the auxiliary input, υ, controls one and only one component of the output, y. It is noted that the existence of the nonlinear feedback require the inverse of the decoupling matrix (χ). To complete the controller design, it is necessary to stabilize each of the above subsystem with constant state feedback. Then, the stability of the system is guaranteed by selecting appropriate constant feedback gains for the linearized system.

III.2. Hybrid control of the robot with a rigid tool
The robotic arm with a rigid tool is considered for the hybrid control design. The performance of the active pneumatic tool based deburring can be degraded due to the fact that the hybrid control method explicitly controls the position and force simultaneously, which can be difficult for a compliant tool.
Since a manipulator task specification, such as deburring a work-piece is typically given relative to the end-effector, the dynamic equation is represented in task space, rather than joint space or actuator space. The controller is designed by considering these coordinates, whose control input τ can be transformed to the torque in the joint coordinates. The transformation is (38) In the constraint coordinates, the force and position errors are defined. f di ∈ R 1 is the desired force in the x i direction (i = 1, 2) and x id ∈ R n−1 is the desired position in other directions. From the assumption that the stiffness k e is known, e p is defined as where k = ( k e 0 0 I ). x i is the displacement of the robot, x id the desired position , and f i ∈ R 1 the force component of the x i direction. Figure 5 depicts hybrid control of a robot with a rigid tool without a pneumatic cylinder.

IV. SIMULATION
Simulation study was performed to investigate the performance of the controllers developed for the robotic deburring systems with different tools: (1) the hybrid controller for the rigid tool based system, as shown in Figure 5 (2), the coordination controller for the single active pneumatic tool (3) the coordination controller for the double active pneumatic tool based system. Figure 6 shows the simulation results for the hybrid control system. The following parameters were used in simulation: m 1 = 16 kg, m 2 = 12 kg, l 1 = 0.5 m, and l 2 = 0.7 m.
The feedback gains of the controller were chosen as following:  show the performance of the hybrid controller designed for the deburring robot with a rigid tool. In the simulation, the stiffness of the material was set to 500000 N/m and the desired cut depth was chosen to be 0.0002 m. The results show large deburring error, which remains oscillatory after large overshoot in the transient period due to chattering caused by the air compressibility and the contact motion between the robot and the workpiece. Figure 6 (c) shows the input force. It is evident that relatively large overshoot and chattering still persist in the response. Note that Eq. (23) was used to obtain the normal force. The parameters in Eq. (23) are referred to the commercialized deburring tool as b = 16 mm, v t = 0.08 m/s, and V t = 30, 000 RPM. Figure 7 depicts the deburring performance of the coordination controller designed for the robot with a single pneumatic tool. The following parameters were used for simulation: As shown in Figure 7 (a) and (b), the transient performance is improved significantly with the single active pneumatic tool with the coordination controller in comparison to the previous case. However, the steady-state performance still remains unsatisfactory due to the chatter that appears in the response, which is caused by the compressibility of the air in the pneumatic cylinder and therefore requires repetitive  deburring. Nevertheless, the simulation results demonstrate the potential of a pneumatic actuator as an efficient tool which can significantly enhance the performance of a deburring robot if the chattering effect can be eliminated or minimized by an improved design of the tool and/or an efficient control. Figure 8 demonstrates the deburring performance of the robot with the IPAS as shown in Figure 3. The developed coordination control method was utilized for the IPAS based deburring system. It is noted that the initial position of M i (i=1, 2, 3, 4) is zero. The following is the additional parameters used for the integrated cylinder: P 3i = 1 × 10 5 P a, A 1 = A 2 = 0.000256m 2 , A 3 = 0.00055m 2 , n = 0.8, F f 1,2 = 10N, It is evident as shown in Figure 8 (a) and (b) that the deburring performance of the system is greatly improved with the IPAS and the coordination controller. The simulation results show quick and smooth transient response and nearly zero steady-state error. The integrated system particularly improves the transient behavior in comparison to the double cylinder system. Figure 8 (c) depict the control input.

V. CONCLUSION
High-quality robotic deburring requires efficient control of the deburring path and contact forces, as well as optimal selection of a suitable feed-rate and tool design. In this paper, an efficient robotic deburring method was developed based on a new active pneumatic tool, which considers the interaction among the tool, the manipulator, and the workpiece and couples the tool dynamics and a control design that explicitly considers deburring process information. A new active pneumatic tool was developed by physically integrating two pneumatic actuators, which implements double cutting action -initial cut followed by fine cut. Then, a control method was developed for the robotic deburring system based on the active pneumatic tool, which utilizes coordination of two cutters. The developed control system employs the two-level hierarchical control structure based on a simple coordination scheme. Simulation results show that the developed system significantly reduces the chattering of the deburring robot and improves the deburring accuracy.