Path smoothing and feed rate planning for robotic curved layer additive manufacturing

Highlights

• A B-spline path smoothing method and a feed rate optimization method are proposed for the robotic curved layer printing process.

• Deposition volumetric error is defined to adaptively control the smoothing of end effector trajectory.

• Feed rate modulation error is introduced to guide the planning of feed rate.

• Simulations and experiments are conducted to verify the effectiveness of the proposed methods.

Abstract

Robotic curved layer additive manufacturing (a.k.a. multi-axis 3D printing) has been gaining attention recently owing to its simplicity and unique ability of printing complex shapes without using a support structure. However, as the printing path now is no long planar and the nozzle orientation is no longer fixed but changes continuously during printing, even though it could be smooth when defined in the workpiece coordinate system in both position and orientation of the nozzle, due to the inevitable numerical errors, it typically is unsmooth with many sharp-changing undulations when transformed to the coordinate system of the robot arm. As a result, the feed rate of printing has to be set extremely conservatively lest the printer would chatter or vibrate and seriously jeopardize the printing quality. In this paper, first, we present a practical B-spline based smoothing algorithm for removing sharp corners on the printing path while upholding the required cusp-height threshold on the printed surface. Next, for the smoothed printing path, we propose a feed rate scheduling strategy that will try to maximize the variable feed rate while subject to the kinematic constraints of the six joints of the robot arm. Both computer simulations and physical printing experiments are carried out to assess the proposed methodologies and the results give a positive confirmation on their advantages.

Introduction

Additive Manufacturing (AM), also referred as 3D printing, is a process of forming complex shapes by stacking material without any mold, normally layer by layer. It has now found wide applications in almost every aspect of our daily life, including those in aerospace, automotive, and bio-medical devices. In general, among the processing techniques of 3D printing, fused deposition modeling (FDM) is one of the most commonly adopted, which works by mechanically extruding molten filament material from an extruder (also called a nozzle) [1]. As is well known, for a conventional three-axis FDM printing process in which printing layers are parallel planes and there is only one fixed build direction, one major disadvantage is the stair-step effect, which is especially pronounced for those parts with freeform shapes. This causes some serious problems on the printed part such as low strength, poor surface finish quality and prolonged printing time [2,3]. Recently, owing to the emergence of multi-axis printers, such as robotic printers [2,4,5], curved layer fused deposition modeling (CLFDM) becomes physically realizable, which is able to eliminate most of the stair-step effect by continuously adjusting the nozzle orientation and allowing non-planar freeform printing layers [6]. Same as most of today's new manufacturing technologies, for multi-axis printing, the bottleneck to its popularization lies in the software part. Specifically, the crux is how to plan a multi-axis printing process so that the desired printing quality (e.g., minimum stair-step effect, enhanced mechanical properties of the printed part, good surface-finish quality) can be assured while at the same time the printing time (which typically is very long) could be minimized. In this paper, we will study one particular aspect of this rather large process planning problem – how to smooth the path of end effector so that an optimal feed rate can be assigned to it.

Fig. 1 shows a typical configuration of robotic five-axis CLFDM printer. During a printing process, the nozzle always remains vertical while the robot arm moves and rotates the fixture table (the end effector) to realize the relative position and orientation of the in-process workpiece with respect to the nozzle. Similar to five-axis machining, a printing path – which includes both the position and orientation of the nozzle – is defined in the workpiece coordinate system (WCS) in which the workpiece is defined. This path must be transformed to the path of end effector defined in the base coordinate system (BCS) so that the corresponding coordinates (angles) of the six joints of robot arm can be calculated. It is well observed that, while the printing path defined in WCS is smooth, the path of end effector typically is not but with many, though of small magnitude, sharp changing undulations. This is extremely detrimental to the kinematic performance of the robot arm, which consequently will lead to poor printing efficiency and inferior surface finish. One major reason for the un-smoothness of the end effector trajectory is the inevitable calculation error of surface normal of the printing layer (which is practically always in a mesh form). It should be noted that these problems due to the un-smoothness of a path of end effector are more serious for the robotic configuration than the traditional rotary table based multi-axis configuration, as the stiffness of the latter has much smaller impact on the path geometry [7,8]. It is also worth mentioning that, although this un-smoothness effect could be alleviated if the nozzle is installed on the end-effector while the fixture table is fixed in BCS, due to the gravity effect on molten filament, such a reversed configuration could induce instability and poor printing quality on the built part [9,10] and thus should be avoided.

In five-axis machining, how to smooth a five-axis tool path has already drawn quite attention in recent years. Smoothing methods can be divided into two categories – local and global smoothing [11,12]. Most existing techniques are mainly about local corner smoothing [13]. Local smoothing methods can effectively control the smoothing error of corner points with a simple operation flow and less computation, which is critical to preserving the machining accuracy. Because linear segments are assumed to be sufficiently long (e.g., larger than 1 mm), which avoids any interference between consecutive corners, these local methods are able to effectively smooth corners locally [14], [15], [16], [17]. However, in reality, the linear segments of any end-effector trajectory in BCS are typically of non-uniform length and hence could be very short (e.g., smaller than 0.1 mm in our tests as shown in Fig. 3). Altintas et al. [16] proposed a local five-axis tool path smoothing method, which inserts quantic and septic micro-splines for both position and orientation smoothing, whose determination of control points is however extremely complicated. Tsai et al. [15] proposed a direct axis velocity profile blending technique for CNC control, which though is unable to control the corner error effectively. Duan et al. [14] used an optimal control method and NURBS to achieve the minimum-time cornering. Recently; Yang et al. [17] proposed a continuous tool path smoothing algorithm for 6R robot arm; but, as just mentioned, they could not deal with the consecutive corner smoothing problem due to the short length of linear segments. Thus, global smoothing [13,[18], [19], [20]] is necessary to deal with consecutive corners. However, although global smoothing provides a better smoothing result than local smoothing, it suffers in terms of maintaining the original trajectory features [11,21]. Sencer et al. [13] smoothed the corners globally by interpolating the axes’ motions while preserving the continuity of the axes’ acceleration; however, they only realized it in three-axis machining without considering the smoothing of tool orientation. Yang et al. [19] presented a curve fitting interpolation method using quadratic B-splines based on the idea of restricting the Hausdorff distance to the original tool path. Tsai et al. [20] proposed a Bezier spline-fitting technique to realize a real-time look-ahead interpolation. Also, Tournier et al. [18] presented an algorithm for smoothing both the position and orientation of a five-axis tool path based on the drive constraints of different axes. In our case, however, we want to smooth a printing path while at the same time guaranteeing that the specified cusp-height (or more accurately the negative cusp-height in the case of printing [2]) requirement will be respected. As we are now targeting printing instead of machining, the unique characteristics of printing must be considered.

As the ultimate purpose of path smoothing is to enable us to assign to the path a larger feed rate, it helps to review some related works in feed rate optimization. In the current practice of multi-axis machining or printing, typically a nominal constant feed rate is assigned to the G-code part program. However, this constant feed rate is never kept in real machining or printing [22], [23], [24] – the controller of the machine tool/printer constantly checks the velocity and acceleration of each drive table (or, in the case of robotic machining/printing, each of the six joints) against their limits and lowers the feed rate adaptively if potential violation is detected [12]. Specific to robotic speed scheduling, Bobrow et al. [25] and Shin et al. [26] proposed an optimization strategy based on the actuator torque constraints to achieve the minimum time control of robot arm. For machining on a typical five-axis machine tool, Altintas et al. [23] investigated how to determine the maximum allowable feed rate of a given machining tool path based on the limits of speed, acceleration and jerk of the drive tables, where the optimized feed rate profile is a cubic spline which can preserve the stability of high-speed motion. Based on the idea of [23], more variant approaches [13,18,22] were proposed, though still for machining only. For multi-axis 3D printing, due to its newness, the work on feed rate optimization is extremely scarce and limited. Recently, for five-axis laser cladding, Calleja et al. [27] reported a work on optimal feed rate scheduling, but subject to only the constraint on rotary speed while leaving out that on acceleration and jerk. Moreover, the work of [27] is restricted to the conventional configuration of rotary tables, while ours is a robotic configuration as shown in Fig. 1.

At present, for the emerging multi-axis robotic CLFDM printing, there has been little reported work on how to smooth a printing path subject to the required cusp-height constraint and how to optimize the feed rate for a smoothed printing path subject to the kinematic constraints of the robot arm, and these are our objectives. Specifically, we have the following two major contributions:

• A novel B-spline smoothing method for the end-effector trajectory of robot arm subject to a metric called deposition volumetric error.

• An improved feed rate planning method on a smoothed end-effector trajectory subject to the constraints on the angular speed, acceleration, and jerk of the six joints of robot arm.

The results of both computer simulations and physical printing experiments conducted by us will also be reported to demonstrate the effectiveness of the proposed methodology.