NURBS function closed-loop mapping trajectory planning of serial robotic plasma cladding for complex surface coatings

In order to solve the problem of the trajectory planning of robotic plasma cladding for complex surface coating, the concept of non-uniform rational B-spline (NURBS) function open-/closed-loop mapping is proposed firstly in this work to explore the new approach of the trajectory planning for complex NURBS surfaces. The trajectory planning is carried out by a 2D-NURBS curve C(u) on a 2D plane, which is mapped on a predefined 3D-NURBS surface S(u,v) using the NURBS surface function to form a 3D mapping curve named NURBS function mapping (NURBS-FM) curve. Using the fixed step (FS) interpolation, the equal chord length (ECL) interpolation, the equal arc length (EAL) interpolation, and the equal bow height (EBH) interpolation, etc., a complex curve can be interpolated. The FS/ECL/EAL/EBH can be defined as constraint. Depending on where the constraint is applied, the NURBS-FM can be divided into open-loop mapping and closed-loop mapping. The NURBS function open-loop mapping (NURBS-FOLM) is carried out along the route of u → C(u) → S(u,v), while the NURBS function closed-loop mapping (NURBS-FCLM) along the route of u → C(u) → S(u,v) → u → …. The constraint is applied to the 2D-NURBS curve in NURBS-FOLM, while to the 3D NURBS-FM curve in NURBS-FCLM. The NURBS-FOLM and NURBS-FCLM can ensure that the interpolation points on the 2D-NURBS curve and 3D NURBS-FM curve have the ECL/EAL/EBH characteristics, respectively. When the 3D NURBS-FM curve is regarded as plasma cladding trajectory, the NURBS-FCLM can provide engineers with a new ECL/EAL/EBH interpolation method for cladding trajectory. Using the NURBS-FCLM approach based on the serial robotic plasma cladding, a high entropy alloy coating is prepared on Q235 substrate which employed the AlCoCrCuNiNb high entropy alloy powder with optimized parameters (Ar flow, ion gas flow, current, powder feeding speed, cladding feedrate) = (6L/min, 1.7L/min, 115A, 15 mm/s, 1.4 mm/s). The formation of the complex spacial curve coating on the complex surface is well without obvious defects. The simulation and experiment verify that the NURBS-FCLM is feasible and effective. It provides the technical and theoretical basis for the serial robotic plasma cladding trajectory planning of complex surface coating. With the increasing application of complex parts in various industries, the NURBS-FOLM and NURBS-FCLM technologies have more and more broad application prospects in the manufacturing of new complex product coatings and remanufacturing of old ones based on serial robotic laser/arc cladding platform.


Introduction
Non-uniform rational B-spline (NURBS) is a powerful mathematical model to express complex curves and surfaces due to its generality, and has been widely used in contemporary computer-aided design systems and computeraided geometric design [1][2][3][4][5][6][7]. NURBS is also promulgated by ISO to be the unique mathematical method for defining industrial product shapes in STEP.
However, the trajectory planning research of arc AM focuses on the contour wall and block shape composed of straight line and arc based on a flat. The arc cladding trajectory planning for complex surface coating has not attracted academic attention. Although the research of trajectory planning in the CNC milling field has made great progress and obtained a series of achievements [8][9][10][11][12], the results of CNC machining trajectory planning cannot be directly applied to arc cladding of complex surfaces due to the fundamental differences between CNC machining and arc cladding. In addition to the research on trajectory planning of CNC machining and arc AM, developing arc cladding trajectory planning of complex surface coating is another research direction that needs to be opened up in the academic field. However, the research in this direction is blank up to the present.
With the increasing application of complex surface coatings in the industry, solving the arc cladding problem of complex surface coating based on serial robot platform is not only a work of great industrial practical application value, but also an urgent technology in the arc cladding industry of complex surface coating.
In the arc cladding process, plasma powder cladding has been widely used in the industry because of its high arc column temperature, good arc column bunching (similar to the laser and electron beam), small heat-affected zone, easy powder matching, and low cost [17][18][19][20][21][22][23][24][25]. It is also an important process for academic research on low-cost coating preparation.
Serial robot has been widely used in many countries and has become the main equipment of production automation. Taking serial robot as the platform to build manufacturing automation equipment is a common method in the industry. The problem of serial robotic trajectory planning for complex surface coating must be solved before plasma cladding. Trajectory planning is an important topic in the field of serial robotics. Different scholars have proposed different trajectory planning methods and achieved good results for welding, assembling, and grasping [31][32][33]. However, the trajectory planning of plasma cladding complex surface coating based on serial robot has not attracted scholars' attention. In order to solve the urgent needs of the industry and promote the development of the plasma cladding process, the trajectory planning of plasma powder cladding complex surface coating based on serial robot is researched in this work through proposing the concept of the NURBS function open-/closed-loop mapping firstly. The goal of this work is to provide a theoretical and technical foundation for serial robotic plasma cladding trajectory planning of complex surface coatings.

NURBS curve representation
A NURBS curve of p-degree is represented parametrically as [34]: where. Additionally, a rational basis function R i,p (u) can be defined as follows: Therefore, Eq. (1) can be expressed as:

NURBS surface representation
A NURBS surface is a bivariate vector-valued piecewise rational function and expressed as follows: where. P i,j : forming a bidirectional control point net; ω i,j : the weights; N i,p (u) and N j,q (u): the nonrational B-spline basis functions defined on U and V, respectively.
The knot vectors of U and V are presented as: Furthermore, the piecewise rational basis function of {R i,j (u,v)}can be defined as follows: As a result, Eq. (7) is rewritten as:

Serial robotic plasma cladding of NURBS curve coating
A plasma-computer integrated cladding is constructed by integrating a Motoman-UP6 serial robot with a DML-V02BD plasma power (Shanghai Duomu) as shown in Fig. 1a. The robotic kinematic calculation is carried out using the D-H convention [35][36][37], and the D-H parameters of the Motoman-UP6 is demonstrated in Fig. 1b. The plasma cladding of complex surface coating is based on the cladding of complex curve. The relationship between a robot and a NURBS curve is shown in Fig. 2.
The NURBS curve interpolation has to be carried out before cladding a NURBS curve. Actually, the NURBS curve plasma cladding is a process in which the robot controls the cladding torch to run along the fitting straight line segments between interpolation points from C(u i−1 ) to C(u i+n ) as shown in Fig. 2a. The NURBS curve interpolation is the process of finding the coordinates of interpolation point C(u i ). Then, the point C(u i ) is used as the position point in the homogeneous transformation matrix T of the cladding torch control point. The posture [n o a] of the cladding torch is determined by process requirements. Furthermore, the robotic joint coordinates θ is calculated according to matrix T using serial robotic inverse kinematics [35,37].
A NURBS curve like Pikachu as shown in Fig. 2b is designed for the research of the serial robotic plasma cladding of complex NURBS surface coating in this work. Its parameters are outlined in Appendix 1.
The interpolation has always been an important problem in NURBS curve research, which is related to the machining accuracy and efficiency of CNC machine tools. In our previous research [38], the equi-arc length interpolation with the fixed step plus golden section search (EALI-FS + GSS) is proposed. It can realize the uniform distribution of interpolation points with the equi-arc length, and is used in this work to research the serial robotic plasma cladding of the complex NURBS surface coating.

Principle of NURBS function open-/closed-loop mapping
The NURBS function open-/closed-loop mapping curve is proposed to solve the problem of the plasma cladding trajectory planning for complex NURBS surface based on the serial robot of Motoman UP6 as shown in Fig. 1a. The NURBS function mapping (NURBS-FM) is the mapping process from u to S(u,v) as shown in Fig. 3. When u i is taken from 0 to 1 on the u-axis, it can be mapped into a NURBS curve C(u) using Eq. (1). To facilitate trajectory planning and design, this NURBS curve can be define to be a 2D-NURBS curve as shown in Fig. 3. Thus, the 2D space where the NURBS curve C(u) is located can be used as the design space of trajectory planning. The coordinate (x,y) of C(u) in Cartesian coordinates can be used as variables of S(u,v) via Eq. (7). Consequently, the C(u i ) point on the planar NURBS curve C(u) is mapped into the S(u i ,v i ) point on the NURBS surface S(u,v). All S(u i ,v i ) points on the surface form a 3D spacial curve called NURBS-FM curve. It should be pointed out that this 3D spacial curve is not a NURBS curve. In plasma cladding, the 2D-NURBS curve is the planning curve on the design plane, and the NURBS-FM curve is the plasma cladding trajectory on the complex NURBS surface. Figure 4a-d take Pikachu as an example to clearly express the relationship between the 2D-NURBS curve and the NURBS-FM curve.
The following can be known from the above NURBS-FM curve principle: 1. u is the 1D variable on the u linear axis.   S(u,v). The ① → ② → ③ → ④ → ① → route calculation is repeated until the error e is met. This is NURBS-FCLM. The spacial curve on the given NURBS surface S(u,v) is defined as NURBS-FCLM curve. Contrary to NURBS-FOLM, the ECL/EAL/EBH constraint is only valid for the NURBS-FM curve on the NURBS surface S(u,v) and not for the 2D-NURBS curve C(u).
Both NURBS-FOLM and NURBS-FCLM obtain the plasma cladding trajectory on the predefined NURBS surface to accomplish the cladding trajectory planning. Figure 4e, f show the correspondence between interpolation points and Fig. 4 The NURBS function mapping of the Pikachu curve. a-d 2D NURBS curve and NURBS function mapping curve in 3D view, 2D view, XZ view, and YZ view, respectively. e, f The schematic of the correspondence between interpolation points and mapping points in 3D view and 2D view, respectively mapping points on the 2D-NURBS curve and the NURBS-FM curve, respectively. Figure 5 demonstrates the NURBS-FOLM and NURBS-FCLM in block diagram. The NURBS-FM is involved in the mapping relationship of three spaces. The first space is the 1D-space containing the u-axis, the second space is the 2D-space containing the planar NURBS curve C(u), and the third space is the 3D-space containing the NURBS surface S(u,v). The position where the constraint is applied is clearly shown in Fig. 5.
Taking the EAL constraint as an example, the NURBS-FOLM means that the arc length between interpolation points on the NURBS curve C(u) is equal, while the NURBS-FCLM means that the arc length between mapping points on the NURBS surface S(u,v) is equal. For this reason, the search cycle path to meet the EAL constraint is also different. For the NURBS-FOLM, the search cycle path is located between u and C(u) to meet the condition of |s i -s|≤ e (s i and s are the calculated arc length and the designated arc length of 2D-NURBS curve C(u) respectively, e is the error). Contrarily, for the NURBS-FCLM, the search cycle path is located between u and the NURBS-FM curve to meet the condition of |S i -S|≤ e (S i and S are the calculated arc length and the designated arc length of NURBS-FM curve respectively, e is the error) as shown in Fig. 5.

Simulation of NURBS function open-/ closed-loop mapping
The EAL constraint used the EALI-FS + GSS method employed in this work to [38]. The distribution of interpolation points on the 2D-NURBS curve and the NURBS-FM curve using NURBS-FOLM with different parameters is shown in Fig. 6a-d. Figure 6e, f show that the distribution of interpolation points on 2D-NURBS curve meets the EAL constraint.
Similarly, the distribution of interpolation points on the 2D-NURBS curve and the NURBS-FM curve using NURBS-FCLM with different parameters is shown in Fig. 7 a-d. Figure 7 e and f show that the distribution of interpolation points on the NURBS-FM curve meets the EAL constraint.
The distribution difference of interpolation points between NURBS-FOLM and NURBS-FCLM with s = S = 8 can be observed clearly in Fig. 8a, b. For example, the distribution of the interpolation points under the same EALvalue constraint in region A (Fig. 8a) and region B (Fig. 8b) is obviously different due to different constraint positions. Obviously, the uniform distribution of interpolation points on the NURBS-FM curve can be obtained by using NURBS-FCLM. Thus, the connected polygon (as shown in Fig. 4e, f) between interpolation points can approach the NURBS-FM curve with higher accuracy under smaller EAL-value constraint.

The serial robotic plasma cladding of complex NURBS curve coating using NURBS-FCLM
Before plasma cladding the NURBS-FM curve coating, the predefined complex NURBS surface should be machined based on a CNC milling machine equipped with FANUC 0i-MB. The pretreatment containing surface digital processing and milling is shown in Fig. 9. The surface parameters in Appendix 2 are input into rhion 5 software, and obtained the CAM in UG. After that, the complex surface workpiece is clamped on the plasma-computer integrated cladding system as show in Fig. 10a. A software is developed for the plasma cladding of complex NURBS surface coatings including the surface design function, trajectory planning function, serial robotic kinematic calculation function, and program generation function. Based on this software, the plasma cladding program for this cladding trajectory is generated.
The posture of the cladding torch relative to the workpiece is shown in Fig. 10a. Q235 is used in this experiment as the substrate, while the AlCoCrCuNiNb high entropy alloy powder is employed as the powder. The plasma cladding process parameters are listed as follows: 1. Ar flow: 6 L/min 2. Ion gas flow: 1.7 L/min 3. Current: 115A 4. Powder feeding speed: 15 mm/s   The plasma cladding process and results of the NURBS-FM curve using NURBS-FCLM trajectory planning approach are shown in Fig. 10b-f. The NURBS-FM curve coating is uniform and well formed. The formation of the produced coating on the complex surface is well without obvious defects.

Conclusion/recommendation
The trajectory planning is an important work for the complex surface manufacturing of computer integrated manufacturing system. It is a key task for the serial robotic plasma cladding of the complex NURBS surface coating based on a plasma-computer integrate cladding system. In order to solve the problem, the concept of NURBS function open-/closed-loop mapping is proposed firstly in this work to explore the new approach of the trajectory planning of plasma cladding for complex NURBS surface coating.
The NURBS-FM curve is a 3D mapping curve (on a predefined NURBS surface) mapped from the 2D-NURBS curve using the NURBS surface function.
The NURBS surface function mapping can be divided into open-loop mapping and closed-loop mapping according to the position applied by FS/ECL/EAL /EBH constraint.
The NURBS-FOLM is realized along the route of u → C(u) → S(u,v), while the NURBS-FCLM along the route of u → C(u) → S(u,v) → u → …. The constraint is applied to the 2D-NURBS curve in NURBS-FOLM, while to the 3D NURBS-FM curve in NURBS-FCLM. The NURBS-FOLM and NURBS-FCLM can ensure that the 2D-NURBS curve and 3D NURBS-FM curve has the ECL/EAL/EBH characteristics, respectively.
Regarding the NURBS-FM curve as plasma cladding trajectory, the NURBS-FCLM can provide engineers with a new ECL/EAL/EBH interpolation method for plasma cladding trajectory planning.