Method of controlling multi-robot material deposition system and assembly thereof

Abstract

A method of controlling a multi-robot material deposition system, and an assembly thereof, are set forth. The multi-robot material deposition system can be that of a multi-robot overspray-free painting system, a multi-robot three-dimensional (3D) printing system, or another type. In the implementation of the multi -robot overspray - free painting system, the method and assembly can be employed in use to apply a paint coating to an automotive part or larger body assemblage, as well as to non-automotive parts and assemblages. Tire multi-robot over-spray free painting system and assembly include, per an implementation, one or more gantries and a multitude of robot assemblies carried on the gantry(ies). As a result of the number of robot assemblies involved, high-level and low-level controls are integrated together to furnish efficient collaboration thereamong. Suitable speeds of over-spray free painting are enabled, while maintaining suitable painting quality.

Description

METHOD OF CONTROLLING MULTI-ROBOT MATERIAL DEPOSITION SYSTEM AND ASSEMBLY THEREOFCROSS-REFERENCE TO RELATED APPLICATION[0001 j This claims the benefit of U.S. Provisional Patent Application No. 63 / 684,035, with a filing date of August 16, 2024, the contents of which are hereby incorporated by reference in their entirety.TECHNICAL FIELD

[0002] This disclosure relates generally to material deposition systems such as overspray-free painting systems and additive manufacturing systems (i.e., three- dimensional printing) and, more particularly, to material deposition systems that employ multiple robots in operation.BACKGROUND

[0003] Paint shops are among the major operations carried out in conventional automotive assembly plants. Paint shops are known to require more than one-half of the total energy of vehicle production. While mitigation efforts are ongoing, the shops often use large amounts of water, can generate large amounts of paint sludge, and can emit volatile organic compounds (VOCs), among other drawbacks. Painting in paint shops typically involves a spraying process. But spraying processes are imprecise and can squander as much as fifty percent (50%) of paint as overspray. Further, multi-color painting jobs introduce more challenges and waste. Lastly, the spraying processes are typically inflexible in design and construction and not readily customizable.

[0004] Overspray-free painting (OFP) processes have been investigated as an alternative. OFP processes involve inkjet-type printer heads with numerous controllable nozzles that precisely dispense paint. While the past OFP processes are beneficial in many regards compared to spraying processes — e.g., zero wasted paint, reduction in V OC and CO2 emissions, substantial savings in energy and cost, less water usage, no paint sludge, increased flexibility’ and customization — the past OFP processes are not without challenges. Painting vertical and irregular surfaces poses an issue for the past OFP processes. And perhaps most of all, the past processes suffer from slower operation. The past. OFP processes can be three to six times slower at painting than the typical spraying processes. For industries in which production efficiency is paramount,like the automotive industry, such decreases are unacceptable. Still, spraying processes and their attendant challenges can exist in non-automotive settings.

[0005] Yet further, the above and other types of material depositions systems suffer from challenges introduced from the employment of numerous robots that are intended to collaborate together for the execution of one or more tasks.SUMMARY

[0006] In an embodiment, a method of controlling a multi-robot material deposition system may include a number of steps. One step may involve the provision of one or more gantries that can be moved in an x-direction. Another step may involve the provision of a multitude of robot assemblies. The robot assemblies are carried on the gantry(ies). Each of tire robot assemblies can be moved in ay-direction, in a z-direction, or in both the y-direction and z-direction. Each of the robot assemblies includes a multitude of end effector assemblies with a multitude of material application heads. Tire material application heads serve to furnish deposition of material on a substrate surface. The material application heads have tip and tilt rotational moveability' capabilities. A further step of the method may involve planning and controlling paths of directional motion of the robot assemblies with respect to one another and with respect to the substrate surface. Yet another step may involve controlling movements of the robot assemblies and each of the end effector assemblies with respect to one another. The planned and controlled paths of directional motion of the robot assemblies is partly, or more, dependent upon the controlled movements of each of the robot assemblies and end effector assemblies. Further, the controlled movements of each of the robot assemblies and end effector assemblies is partly, or more, dependent upon the planned and controlled paths of directional motion of the robot assemblies.

[0007] hi an embodiment, a multi-robot overspray-free painting assembly may include one or more gantries and a multitude of robot assemblies. The gantry(ies) can be moved in an x-direction. The robot assemblies are carried on the gantry(ies). Each of the robot assemblies can be moved in a y-direction, in a z-direction, or in both the y-direction and z-direction. Each of the robot assemblies includes a multitude of end effector assemblies with a multitude of printheads. The printheads furnish deposition of paint coating on a substrate surface. The printheads have tip and tilt rotational moveability capabilities. A multitude of sensors are carried by the robot assemblies. The sensors serve to sense the substrate surface. During operation of the multi -robot overspray-freepainting system, a graph neural network (GNN) model is employed in planning and controlling paths of directional motion of the robot assemblies with respect to one another and with respect to the substrate surface. And, during operation of the multirobot overspray-free painting system, a convolutional recurrent neural network (ConvRNN) model is employed in controlling movements of each of the robot assemblies and each of the end effector assemblies with respect to one another.

[0008] In an embodiment, a method of controlling a multi-robot overspray-free painting system may include a number of steps. One step may involve the provision of a first gantry that can be moved in an x-direction, and may involve the provision of a second gantry that can be moved in the x-direction. Another step may involve the provision of a multitude of first robot assemblies and a multitude of second robot assemblies. The first robot assemblies are carried on the first gantry. Each of the first robot assemblies can be moved m a y-direction, in a z-direction, or in both the y- direction and z-direction . Each of the first robot assemblies includes a multitude of first end effector assemblies with a multitude of first printheads. The first printheads furnish deposition of paint coating on a substrate surface. The first printheads have tip and tilt rotational moveability capabilities. Further, the second robot assemblies are carried on the second gantry . Each of the second robot assemblies can be moved in the y-direction, in the z-direction, or in both the y-direction and z-direction. Each of the second robot assemblies includes a multitude of second end effector assemblies with a multitude of second printheads. The second printheads furnish deposition of paint coating on the substrate surface. The second printheads have tip and tilt rotational moveability' capabilities. A further step of the method may involve planning and controlling paths of directional motion of the first and second robot assemblies with respect to one another and with respect to the substrate surface. Yet another step of the method may involve controlling movements of each of the first robot assemblies and each of the first end effector assemblies with respect to one another, as well as controlling movements of each of the second robot assemblies and each of the second end effector assemblies with respect to one another. The planned and controlled paths of directional motion of the first and second robot assembly' is in part, or more, dependent upon the controlled movements of each of the first robot assemblies and first end effector assemblies, and upon the controlled movements of each of tire second robot assemblies and second end effector assemblies. Lastly, the controlled movements of each of the first robot assemblies and first end effector assemblies, as well as each of the second robotassemblies and second end effector assemblies, is in part, or more, dependent upon the planned and controlled paths of directional motion of the first and second robot assemblies.

[0009] Further scope of applicability of the present disclosure will become apparent from the detailed description given hereinafter. But it should be understood that the detailed description and specific examples, while indicating exemplary' embodiments of the disclosure, are given by way of illustration only, since various changes and modifications within the spirit and scope of the disclosure will become apparent to those skilled in the art from this detailed description.BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The present disclosure will become more fully understood from the detailed description given below and the accompanying drawings, which are given by way of illustration only, and do not limit the present disclosure, and wherein:

[0011] FIG. 1(a) is a perspective view of an embodiment of a multi -robot overspray- free painting (OFP) assembly;

[0012] FIG. 1(b) is a front view of the multi-robot overspray -free painting assembly;

[0013] FIG. 1(c) is an enlarged view of an embodiment of an end effector assembly and printhead and sensor;

[0014] FIG. 2(a) is a schematic diagram of a past OFP process painting a strip m which painting occurs at a constant robot speed;

[0015] FIG. 2(b) is a schematic diagram of the multi -robot overspray-free painting assembly painting a strip in which acceleration and deceleration among neighboring and successive robot assemblies can produce a stitching error at motion starts and stops;

[0016] FIG. 3 is a schematic diagram of an overview of tasks executed for an embodiment of a method of controlling a multi-robot overspray-free painting system such as the multi-robot overspray-free painting assembly;

[0017] FIG. 4 is a schematic diagram of task-constrained multi-robot coordinative planning for the method of controlling a multi-robot overspray-free painting system;

[0018] FIGS. 5(a) and 5(b) are graphs that demonstrate an approximate 30% improvement in geometry' tracking of a convolutional recurrent neural network (ConvRNN)-based feedforward control over a heuristic approach (based on a linear superposition of droplets) in inkjet 3D printing;

[0019] FIG. 6(a) is a graph demonstrating droplet velocity components in an x- direction and y-direction, wherein velocity vyis due to ejection and vxdepends on printhead motion;

[0020] FIG. 6(b) is a graph demonstrating corresponding velocity profile of the printhead of FIG. 6(a);

[0021] FIG. 7 is a graph demonstrating that filtered B-spline (FBS) control yielded over 5x lower vibration compared to input shaping on a six degree of freedom (6-DOF) robot;

[0022] FIG. 8 is a schematic diagram of an embodiment of a filtered B-splines (FBS) model that combines and utilizes physics-based and data-driven linear models;

[0023] FIG. 9 is a graph demonstrating coordination of inkjet printing with robot motion, including vibration;

[0024] FIG. 10 is a schematic diagram demonstrating, per an embodiment, how low- level printhead and vibration control is integrated into high-level graph neural network (GNN)-based control framework for the multi-robot OFP control method and assembly;10025] FIG, 1 1 (a) is a schematic diagram demonstrating an embodiment of an approach for effecting a seamless edge in which a first printhead deposits droplets from right to left, leaving interlacing droplet gaps at a right edge (or adjoining edge) of its domain of painting;

[0026] FIG. 1 1 (b) is a schematic diagram demonstrating the seamless edge approach of FIG. 11(a) in which a second, neighboring, printhead begins deposition of droplets at and in the interlacing droplet gaps at the adjoining or domain edge;] 0027] FIG, 1 1 (c) is a schematic diagram demonstrating the seamless edge approach of FIG. 1 1 (a) in which the second printhead proceeds with its regular and normal droplet deposition spacing without gaps;

[0028] FIG. 12 is a schematic diagram demonstrating, in a simplified manner, how coordinating robot motions (i.e., reaction forces) can help suppress gantry vibration; and

[0029] FIG. 13 is a schematic diagram of an example of a scaled-down testbed for an embodiment of the multi-robot OFP control method and assembly for experimental validation purposes.DETAILED DESCRIPTION

[0030] Referring generally to the drawings, an embodiment of a method of controlling a multi-robot material deposition system is presented, as well as an embodiment of a multi-robot material deposition assembly that can be used in the system. The multirobot material deposition system control method and assembly have many applications of use. Example implementations include a multi-robot overspray-free painting (OFP) system control method and assembly thereof, a multi-robot three-dimensional (3D) printing system control method and assembly thereof, and other multi-robot material deposition system control methods and assemblies. In this sense, the phrase ‘’multirobot material deposition system” as used herein is intended to have an expansive meaning that encompasses systems and assemblies in which material is deposited in a precise manner via numerous robots that work together in collaboration. An embodiment of a method of controlling a multi -robot overspray-free painting (OFP) system and an embodiment of a multi -robot overspray -free painting assembly is detailed in this description and depicted in the figures. But it should be appreciated that at least some, many, or all of the descriptions of method steps, components, assemblies, and / or systems set forth herein are equally applicable to other types of multi-robot material deposition system control methods and assemblies.

[0031] In general, the method of controlling tire multi-robot overspray-free painting system and assembly thereof are employed to deposit and apply a paint coating on a substrate surface. The phrase “paint coating” as used herein is intended to have an expansive meaning that encompasses an applied primer, basecoat, clearcoat, a combination thereof, as well as other types of coatings. And the substrate surface can be that of a part, component, panel, larger body assemblage, or some other body. Applications of use for the multi-robot OFP system control method and assembly include paint shops in automotive assembly plants and other settings and facilities, as well as non-automotive applications.

[0032] Furthermore, unlike past approaches, the method of controlling the multi -robot OFP system and the multi-robot OFP assembly set forth herein employ a greater number of robots that collaborate and work together efficiently and effectively and exhibit system-level integrated control to enable overspray-free painting with speeds comparable to the past approaches, while maintaining suitable painting quality. Both high-level robot control via motion path planning — and low-level printhead and robot movement control — collaborate and are integrated together in the multi -robot OFPsystem control method and assembly. In at least one embodiment, a total quantity of eighteen (18) or more individual robot assemblies are employed. This unprecedented in scale, as the past approaches typically employed as little as two or four robots or, in some exceptional cases, six robots. More generally, a more flexible, customizable, economical, and environmentally sustainable OFP sy stem control method and assembly are furnished. Overall, a more effective and efficient OFP system control method and assembly are provided. Still, a particular embodiment of the multi-robot OFP system control method and assembly may exhibit only one, or a combination of, the advancements set forth herein, none of the advancements, or other advancements not mentioned.

[0033] The method of controlling the multi -robot OFP system and the multi-robot OFP assembly can have various designs, constructions, components, and steps in different embodiments depending upon — among other potential factors — its intended application of use. In the embodiment of the figures and with initial reference to FIGS. 1A, IB, and 1C, arnulti-robot overspray-free painting system and assembly 10 includes a first gantry’ 12, a second gantry 14, a first set of robot assemblies 16, and a second set of robot assemblies 18. Still, other embodiments could have more, less, and / or different assemblies and components than those set forth herein; for example, the multi-robot overspray -free painting assembly 10 could have a single gantry with a single set of robot assemblies or could have more than two gantries with more than two sets of robot assemblies. In the embodiment of FIGS. 1A, IB, and 1C, the first gantry 12 is movable back and forth in an x-direction (denoted in FIG. 1A), and the second gantry’ 14 is likewise movable back and forth in the x-direction. The x-direction in this example is in-line with a longitudinal and lengthwise direction of a vehicle V subject to deposition of paint coating via the multi-robot overspray-free painting system and assembly 10. Movement of the first and second gantries 12, 14 in the x-direction brings the first and second sets of robot assemblies 16, 18 back and forth over portions of the vehicle V. The vehicle, per this example, has numerous substrates and substrate surfaces SS such as door panel surfaces, roof panel surfaces, and hood panel surfaces, among many others.

[0034] The first set of robot assemblies 16 is carried on and mounted with the first gantry 12. The quantity of individual first robot assemblies 20 that make-up the first set of robot assemblies 16 can vary. In the embodiment of FIGS, IA, IB, and 1C, a total of nine individual first robot assemblies 20 is furnished, but the total could be more orless in quantity in other embodiments. Each of the first set of robot assemblies 16 and individual first robot assemblies 20 is movable in a y-direction, a z-direction, or in both the y-direction and z-direction (denoted in FIG, IB), which can depend on their orientation of mounting on the first gantry 12. The first robot assemblies 20 mounted on a vertical beam of the first gantry 12, for instance, can move up and down in the z- direction and / or can move fore and aft in the y-direction; and the first robot assemblies 20 mounted on a horizontal cross-beam of the first gantry' 12 can move fore and aft in tlie y-direction and / or can move up and down in the z-direction. Furthermore, with particular reference now to FIG. 1C, each of the first set of robot assemblies 16 and individual first robot assemblies 20 includes a first end effector assembly 22, according to this embodiment. The first end effector assemblies 22 each have a first printhead 24 for the deposition of paint coating on the substrate surface SS amid use of the multirobot overspray-free painting assembly 10 in the method of control thereof. In the embodiment of the method of controlling the multi-robot material deposition system and the multi-robot material deposition assembly, the first printheads 24 are in the form of material application heads MAH, The first end effector assemblies 2.2. furnish tip rotational movement, capabilities to the first printheads 24, as well as tilt rotational movement capabilities to the first printheads 24 (both tip and tilt rotations denoted in FIG. 1C). Lastly, according to this embodiment, a multitude of first sensors 26 are carried by the first set of robot assemblies 16 and individual first robot assemblies 20. The first sensors 26 sense a property of the substrate surface SS, and can be in the form of a camera for scanning the substrate surface SS prior to the deposition of paint coating via the multi -robot overspray-free painting system and assembly 10. The first sensors 26 can be situated at the first end effector assemblies 22 and adjacent the first printheads 24.[0035 j The second set of robot assemblies 18 is carried on and mounted with the second gantry 14. The quantity of individual second robot, assemblies 28 that make-up the second set of robot assemblies 18 can vary. In the embodiment of FIGS. 1A, IB, and 1C, a total of nine individual second robot assemblies 28 is furnished, making a total of eighteen individual first and second robot assemblies 20, 2.8 in this embodiment of the multi -robot overspray-free painting assembly 10 and method of control thereof; but the quantity of individual second robot assemblies could be more or less than nine in other embodiments. Each of the second set of robot assemblies 18 and individual second robot assemblies 28 is movable in the y-direction, the z-direction, or in both the y-direction and / -direction, as described above. Hie second robot assemblies 28 mounted on a vertical beam of the second gantry 14, for instance, can move up and down in the z-direction and / or can move fore and aft in the y-direction; and the second robot assemblies 28 mounted on a horizontal cross-beam of the second gantry 14 can move fore and aft in the y-direction and / or can move up and down in the z-direction. Furthermore, with particular reference now to FIG. 1C, each of the second set of robot assemblies 18 and individual second robot assemblies 28 includes a second end effector assembly 30, according to this embodiment. The second end effector assemblies 30 each have a second printhead 32 for the deposition of paint coating on the substrate surface SS amid use of the multi-robot overspray-free painting assembly 10 in the method of control thereof. As before, in the embodiment of the method of controlling the multi-robot material deposition system and tire multi-robot material deposition assembly, the second printheads 32 are in the form of material application heads MAH. The second end effector assemblies 30 furnish tip rotational movement capabilities to tire second printheads 32, as well as tilt rotational movement capabilities to the second printheads 32. Lastly, according to this embodiment, a multitude of second sensors 34 are carried by the second set of robot assemblies 18 and individual second robot assemblies 28. The second sensors 34 sense a property of the substrate surface SS, and can be in the form of a camera for scanning the substrate surface SS prior to tire deposition of paint coating via the multi-robot overspray-free painting system and assembly 10. The second sensors 34 can be situated at the second end effector assemblies 30 and adjacent the second printheads 32.

[0036] Furthermore, it is currently thought that a reason why more robots have not been adopted in past automotive painting is practical. For one, there is little space to fit many robots. This is because the past paradigm relies on six or seven degree-of-freedom (DOF) articulated robots. These robots offer maximum positioning flexibility of the spray painting nozzle, which is critical to minimizing overspray. In other words, the robots’ flexibility is needed to compensate for the lack of flexibility and precision of tire past spraying method. One problem is that articulated robots need a lot of space to maneuver. This problem is exacerbated by the fact that the painting robots need to be large so that they have the load capacity to cany' the heavy spray painting nozzle and its associated cables and hoses. This is why the 6 painting robots are typically used that are relatively large and larger than door-opening robots, as an example.

[0037] To address this challenge, the fact that with the multi-robot overspray-free painting assembly 10 and method of control thereof the inkjet nozzle (i.e., first and second printheads 24, 32) is much more precise and versatile is exploited. Hence, the same degree of robot positioning flexibility as needed in past spray painting is not required in the multi-robot overspray-free painting assembly 10 and method of control thereof. Rather, as depicted in FIGS. 1A, IB, and 1C, the use of 4-axis gantry-mounted Cartesian robots — i.e., the first and second sets of robot assemblies 16, 18 — for OFP of the external surfaces of the vehicle V is employed in the multi-robot overspray-free painting assembly 10 and method of control thereof. OFP may not be suitable for internal surfaces (e.g., inside the doors) because of their high complexity, including undercuts. Therefore, a second paint station may be utilized where traditional spray painting combined with door-opening robots can be used to paint the internal surfaces, per an example embodiment.

[0038] As shown in FIGS. 1A, IB, and 1C, the x-axis motion will be carried out by the first and second gantries 12, 14 that move on rails, according to this embodiment. Each gantry' carries numerous robots of the first and second robot assemblies 16, 18 which translate in-and-out and side-to-side within the plane of the respective gantry, providing y- and z-motions. The first and second end effector assemblies 22, 30 carrying the inkjet printhead (e.g., first and second printheads 24, 32) and scanning camera (e.g., first and second sensors 26, 34), per an embodiment, are capable of tip and tilt rotations to align tire inkjet to curved substrate surfaces SS. This arrangement allows the use of eighteen or more robots to paint the exterior substrate surfaces SS of the vehicle V. The gantrymounted first and second robot assemblies 16, 18 can have a higher load capacity and stiffness than articulated robots, for the same footprint. Therefore, the first and second robot assemblies 16, 18 may provide a compact solution to cany the weight of any accompanying nozzle, hoses, and cables.

[0039] Further, the larger scale of the multi-robot collaboration of the multi-robot overspray -free painting assembly 10 and method of control thereof can call for an unprecedented degree of control system integration. The control of painting robots can be divided into two levels: (1) high-level control which involves path planning for all robots (i.e., first and second set of robot assemblies 16, 18) to ensure their effective and efficient collaboration, and (2) low-level control for each robot’s motion. In OFP, low- level control may also be needed for precise control of the deposition rate for eachnozzle of the inkjet printhead. In the past systems, these two levels of control are largely independent.

[0040] In an embodiment, the timing of the low-level controllers are coordinated such that the inkjet painting can be initiated as soon as the robot motion attains constant speed. As such, the typical process is for the robot to accelerate to constant speed (e.g., of 0.6 to 0,8 m / s) before entering the region to be painted, as shown in FIG, 2A, Once hr the region, the inkjet painting process is started. Similarly, the robot is decelerated to zero speed outside of the region. But this means that the acceleration and deceleration stages of the motion cannot be utilized for OFP, leading to losses in productivity. Note that the acceleration time for large robots (including the settling time for vibration) can be 0.2 seconds to 1 second. OFP can involve hundreds of passes, each involving acceleration and deceleration, amounting to several seconds of lost time. Moreover, the use of multiple robots calls for acceleration and deceleration by each robot within the surface being painted. Therefore, the robot motion control and nozzle control should be tightly integrated to account for the non-constant speed and motion errors (e.g., vibration) that occur during acceleration and deceleration.

[0041] In an embodiment, high-level control can be an offline process where each robot is programmed to paint a portion of the vehicle. Tire use of a relatively small number of robots means that the robots can be programmed to work semi-independently without colliding with one another. This is not the case with the multi-robot overspray-free painting assembly 10 and method of control thereof, which calls for coordination among many robots. Moreover, the use of a large number of robots increases the risk that one or more robots may malfunction and shut, down the entire line. This risk can be mitigated by being able to rapidly reprogram the functioning robots to take over the tasks of the malfunctioning robots. Ulis calls for much more coordination among the robots.

[0042] Lastly, in the the multi -robot overspray-free painting assembly 10 and method of control thereof, the high-level control interacts more closely with the low-level control. Having several robots moving on a gantry' means that the motion of one robot can influence those of others. For example, the motion of one robot can trigger vibration of the gantry' that may affect the motions of other robots on the gantry'. Similarly, the deposition of a nozzle attached to one robot may interact with those of adjacent robots leading to unwanted stitching marks between the section painted by adjacent robots, as demonstrated in FIG. 2B.

[0043] FIG. 3 shows an overview of an embodiment of multi-level control system integration for the the multi-robot overspray-free painting assembly 10 and method of control thereof. The integration of the high-level control of robots is depicted at an upper section of the schematic in the figure, including recovery from malfunctions of one or more robots. The integration of the low-level motion and print-head control for each robot is depicted at a lower section of the schematic in the figure. And the integration of the high-level and low-level controls is depicted at a mid-section of the schematic in the figure. Lastly, the the multi-robot overspray-free painting assembly 10 and method of control thereof is numerically and experimentally tested to evidence a system-level integrated control of multiple (e.g., eighteen or more) robots can enable OFP to achieve speeds comparable to traditional spray painting without sacrificing paint quality .

[0044] Background. Vehicle painting lines need to accommodate vehicles with different shapes, sizes, and dimensions, requiring precise handling. Therefore, the multiple robots in the multi-robot overspray -free painting assembly 10 and method of control thereof should coordinate effectively and efficiently, per at least one embodiment, to cover the designated painting areas at the desired speed and distance. Motion planning hereby is challenging due to the high degrees of freedom in the configuration space, along with the constraints on distance and speed in the task space. The challenge intensifies when one robot fails, and the remaining robots will need to rapidly re-coordinate and complete the remaining tasks.

[0045] A multi-robot planning framework has hence been developed for the multirobot overspray -free painting assembly 10 and method of control thereof that can rapidly generate feasible trajectories and motions of many robot arms (e.g., eighteen or more) working in close proximity. This framework ensures that the robots (e.g., first and second robot assemblies 16, 18) can work efficiently and safely on vehicle painting and maintain strong resilience when one or more robots fail. Here, close proximity refers to operating within a volume of a standard vehicle size. Through efficient motion planning, many robots of the first and second robot assemblies 16, 18 can be deploy ed that work together simultaneously which can significantly enhance the painting efficiency.

[0046] Existing work on motion planning algorithms for multiple robots: Autonomous motion planning plays a fundamental role in guiding a robot to navigate to a target location in a manner that ensures safety in crowded environments involving many otherrobots. Planning the motion of robots is challenging due to the high degrees of freedom, tire robots’ kinematic constraints, and the constraints induced by the painting tasks on assigned painting areas. Existing planning algorithms, including sampling-based, search-based, optimization-based, and learning-based approaches, are inadequate for efficiently handling motion planning in scenarios involving multiple robots working so closely among each other with complex constraints, such as the first and second sets of robot assemblies, 16, 18 in the multi-robot overspray-free painting assembly 10 and method of control thereof Sampling-based methods, such as rapidly exploring random trees and probabilistic roadmaps, are commonly used for their simplicity and effectiveness in generating feasible paths in configuration spaces. Search-based algorithms use search strategies to find optimal paths in discretized grid-based environments. Optimization-based algorithms formulate motion planning as an optimization problem, taking into account kinematics and dynamics as constraints, which are typically highly nonlinear and nonconvex. Learning-based motion planning, such as deep reinforcement learning and neural-network-based planning, leverages machine learning and artificial intelligence to improve adaptability by learning from data, which however may not be flexible enough to accommodate complex task constraints. In addition, these existing planning algorithms may struggle to adapt to different numbers of robots, and tire generalization of the aforementioned planning algorithms to different types of robots remains to be investigated.

[0047] Embodiment. The planning framework of the multi-robot overspray-free painting assembly 10 and method of control thereof integrates task constraints and physical knowledge into a graph neural network (GNN) model, leveraging GNN’s superior capability in handling unstructured data and learning complex spatiotemporal relationships. Although there have been some past studies on utilizing GNNs for robot motion planning, these efforts primarily address sampling configuration spaces or collision checking. Moreover, the past studies mainly focus on single-arm robot motion planning and do not consider multiple robots working closely in the working environment with complex task constraints. Another highlight is that, since graphs are invariant to the number of nodes, the planner of the multi-robot overspray-free painting assembly 10 and method of control thereof is invariant to the number of robots, according to this embodiment. Therefore, the planner of the multi-robot overspray-free painting assembly 10 and method of control thereof can be adapted to different numbersof robots, which may be important when one or two robots fail and quick re-planning is called for in real time.

[0048] Motion planning algorithms have been developed that utilize GNNs for robotic manipulators operating in dynamic environments involving humans. Here, the graph neural network demonstrated extraordinary potential in capturing the kinematic constraints of manipulators as well as superior planning efficiency in high -dimensional configuration spaces. A comparison study was conducted between the GNN -enhanced motion planner and three commonly used planners, namely, rapidly exploring random tree RRT, RRT*. and bi-directional fast marching tree algorithm* (BFMT*), which showed the GNN-enhanced one is more efficient. In the embodiment herein, the graph representation and network construction is applied for the multi-robot overspray-free painting assembly 10 and method of control thereof.

[0049] Real-time graph representation for the workspace involving multiple robots.Tire construction of a graph neural network (GNN) model capable of capturing the intricate connections among different robots and providing the desired motions of the robots in high-dimensional configuration spaces through learning was developed. The leftside of the schematic of FIG. 4 illustrates the graph representing the entire workspace. The graph consisting of the global attributes, nodes, and edges was constructed to effectively represent the dynamic structure and kinematic constraints of the robots. This process involved (1) distributing the painting areas in real-time among robots, and (2) translating the assigned painting areas, task constraints, physical constraints and coordinating robots into a graph representation. The task constraints include the surfaces ofthe vehicle to be painted by each robot (i.e., the substrate surface SS), the desired speed of the printhead, and the required distance between the head and the surface. The physical constraints include the kinematic and dynamic constraints of the robots. The graph primarily consists of nodes and edges, along with their features and attributes. Nodes. Nodes are selected to represent the robot joints and vehicle substrate surface SS, while the node features are selected in a way that they can represent the topological and operational information of each node. For example, for each robot’s joint, the following features are included: its current joint angle, its current position in three-dimensions (3D), the Euclidean distance between the current position and the desired position are considered additional node features, joint velocity, and joint range. The node attributes are selected as the 3D position of each node in the workspace as well as topological information regarding the surface area of the vehicle to bepainted. For the painting surface, the grid over the painting surface is used to determine tire node distribution on the vehicle substrate surface SS. The node attributes are selected as the droplet spacing, jetting frequency, and droplet ejection velocity. Edges. The set of edges are defined in a way that captures the relation among robots and between robots and the painted surfaces, lire edges will encode the connection between the nodes in the workspace. Such customized representation can be important for the GNN-based motion planner as it models the complex interaction of various components of the workspace.

[0050] Graph neural network architecture design and hyper-parameter optimization.The design of GNN’s architecture based on the graph has been developed. The architecture consists of the following major components: embedding processes to unify data structure, cross-attention mechanisms, GNN blocks for message passing, and autoregressive connection to close the loop and continuously enhance performance over time. The message-passing step in the GNN block implicitly ensures that each robot does not prevent other robots from planning and reaching their targets. The GNN-based motion planner is decentralized and computes just the next w-ay point to reach the configuration space rather than the entire trajectory. This makes the motion planner react quickly to the dynamic movement of other robot arms. The rightside of the schematic of FIG. 4 illustrates an embodiment of the architecture design. Ihe decentralized planner replaces the random sampling in the bi-directional RRT algorithm. After generating the next state for each robot in the configuration space, the validity of these states is checked for collisions. If the predicted configuration state is valid, each node embedding of the robot will be updated based on the new information, and the robot will continue planning. A simulation environment is built based upon which hyper-parameter optimization and training of the motion planner is conducted and the success rate and planning efficiency can be evaluated. A centralized samplingbased method (e.g., RRT*) can be utilized to generate the training data.

[0051] Integration of low-level robot motion and inkjet control. Background. As described in the context of FIGS. 2A and 2B, in past OFP, the painting occurs when the robot is operating at constant velocity, after any transients (e.g., vibration) induced by acceleration or deceleration have settled down. This simplifies the control of both the robot and the printhead. But it reduces productivity because no painting occurs during the acceleration and deceleration portions of the motion. With the multi-robot collaborative painting process set forth herein for the multi-robot overspray-freepainting assembly 10 and method of control thereof, each robot travels shorter lengths and experiences more frequent acceleration / deceleration. Therefore, the potential of the multi-robot collaborative painting process would be diminished if painting only occurs at constant velocity after all transients have settled.

[0052] Accordingly, fundamental methods were developed that enable OFP to occur during the full motion of the robot, including the accleration and deceleration portions amid operation of the multi-robot overspray -free painting assembly 10 and method of control thereof, per an embodiment. Methods of integrating the low-level control of the printhead with the rigid-body motion of the robot are set forth, assuming non-constant robot velocity'. Further, methods for integrating the inevitable non-rigid body motions (i.e., vibration) of the robot into the methods of integrating the low-level control of the printhead with the rigid-body motion of the robot to ensure that the desired results are set forth.

[0053] Integrating inkjet control with low-level rigid body robot motion control.Integrating inkjet control with robot motion control under the assumption that the robot does not vibrate (i.e., it only performs the programmed rigid body motion) is described,

[0054] Past work on inkjet control. Inkjet printing process entails ejection of liquid material as droplets from a printhead nozzle, and the deposition and coalescence of the droplets on a substrate. Correspondingly, past work in inkjet control focuses on these two aspects: droplet ejection and geometry' (morphology) of deposited material. Ejection-level control entails manipulating the jetting parameters to regulate the droplet quality (breakup behavior), size and speed. Most inkjet printing systems use piezoelectric actuation. Hence, ejection -level control research has focused on the modulation of the piezoelectric drive-voltage waveform to ensure droplet output metrics are consistently met (for a given task) using iterative learning control (ILC), reinforcement learning and other feedforward control algorithms. Control at this level is typically sufficient for classic inkjet printers where the printhead motion is repetitive and the droplet, on deposition, permeates the substrate (typically paper). Wien inkjet printing is deployed on 3D surfaces, it is not only important to maintain droplet consistency, but also important that the deposited material conforms to the desired geometry. Geometry-level control algorithms determine where and how much material (droplets) should be deposited based on the liquid material dynamics and / or profile measurement of the substrate. Both model-based and greedy algorithms have been utilized for geometry -level control.

[0055] Both open-loop and closed-loop geometry-level control has been studied using neural network-based MPC and its effectiveness has been demonstrated in inkjet 3D printing. The control scheme relies on a graph-based physics-guided convolutional recurrent neural network (ConvRNN) model that captures the complex height evolution of the printed 3D structure. Due to the underlying physics-guided graph structure, tire learned model can predict the morphology of the deposited material irrespective of the underlying droplet input patern, making it suitable for predictive control. FIGS. 5A and 5B illustrate a more than 30% reduction in RMS error between the desired and actual geometrical pattern after using open-ioop model predictive control (MPC) as compared to heuristic printing in a past study.

[0056] Further, deploying inkjet printheads for autobody painting presents control challenges that past geometry-level control schemes do not address for two main reasons, it is thought. First, there is stringent demand for precision at the autobody edges; unlike in typical inkjet 3D printing jobs, there are no support structures to ensure a good edge surface in autobody painting. Secondly, the printhead velocities and accelerations are significant. For context, in typical inkjet punting the droplet ejection speed is usually more than an order of magnitude greater than the printhead translational speed and hence, the translational speed may be ignored.

[0057] Accordingly, in the multi -robot overspray-free painting assembly 10 and method of control thereof, per an embodiment, to cater for the preci sion demands at the autobody edges, variable spacing is employed between droplets such that smaller droplets with finer spacings are deposited at the autobody edge locations. Conversely, larger spacings are needed at the edges between the various printheads to avoid initial contact between droplets such that neighboring printheads can print in-between those non-contacting droplets. Existing geometry-level control schemes such as the ConvRNN -based MPC scheme assume uniform droplet spacing. Herein, per an embodiment, incorporating droplet spacing variability into the underlying graph ConvRNN model is implemented in this scheme such that it determines not only droplet sizes at the droplet grid points but also the resolution of these grid points themselves. In an embodiment, the method of controlling the multi-robot material deposition system and assembly can involve depositing a multitude first material deposit droplets via the material application heads or at locations and / or adjacent an edge of the substrate surface, and depositing a plurality of second material deposit droplets via said plurality’ of material application heads or the first and second printheads 24, 32 at locations awayfrom the edge of the substrate surface SS. Hie first material deposit droplets having first spacings thereamong, and the second material deposit droplets having second spacings thereamong. The first spacings are lesser in value than the second spacings, per this embodiment. General reference to FIGS. 6A and 6B is had in this regard.

[0058] To address the issue of significant printhead velocities and accelerations, ejection timing (by implication, jetting frequency) is controlled relative to the current printhead motion, per an embodiment of the multi-robot overspray -free painting assembly 10 and method of control thereof. This is effected with the use of a predictive model of the droplet trajectory after ejection from the printhead. The model takes into account the droplet’s ejection speed, and the distance between the printhead and substrate. Moreover, the droplet trajectory model is used in tandem with the graph ConvRNN model of the deposited paint profile which determines the target droplet locations. Furthermore, when the printhead orientation is not horizontal, the motion- inkjet coordination scheme according to an embodiment ensures that the resultant droplet velocity compensates for curvature (due to gravity) of the ejected droplet’s path to preserve precision and ensure that adjacent droplets do not collide prior to impact on the substrate surface SS.

[0059] Integrating low-level robot vibration control and inkjet control. An assumption that the robot does not experience vibration is not realistic in practice. Therefore, robot vibration is compensated in an embodiment of the multi-robot overspray-free painting assembly 10 and method of control thereof so that the robot adheres as closely as possible to rigid body motions. In situations where the robot vibration cannot be sufficiently mitigated, the jetting of paint will be paused and temporarilty halted until the vibration settles down.

[0060] Past work on vibration compensation of robots. There are two general past approaches for vibration compensation of robots, i.e., feedback (FB) and feedforward (FF) compensation. FB control involves the use of sensors to measure and actively compensate vibration. However, active FB compensation of robot vibration is hardly used in practice because it is prone to instability and sensor noise. Conversely, FF vibration compensation does not depend on actively sensing vibration. Rather it uses prior modeling to predict and compensate vibration, hence it is not prone to instability and sensor noise. Two FF vibration compensation techniques have found some practical adoption in robotics, namely input shaping and iterative learning control. However, input shaping is only suitable for point-to-point motions. It introduces large errors inmotions like those in OFP which involve trajectory following. Iterative learning control (ILC) is applicable to trajectory following but, to be effective, it requires the motion of the robot to be repetitive. This is not the case for the multi-robot painting process of the multi -robot overspray-free painting assembly 10 and method of control thereof w here the motions of each robot may vary.

[0061] Feedforward tracking controllers can compensate vibration in non-repeating motions that involve trajectory' following. Work on a versatile tracking control method has been done called filtered B-splines (FBS) and its practical use has been demonstrated for vibration compensation of a variety of 3D printers. The FBS model and method has been applied to a delta robot, a 6-DOF articulated robot, and a silicon wafer handling robot, all with position-dependent vibration behavior, meaning that their vibration changes as the robot moves. As shown in FIG. 7, for a 6-DOF robot, the FBS model and method helped to reduce the peak vibration-induced motion errors of the robot by over five-times (.W) compared to input shaping (for a non-repeating motion where ILC is unsuitable).

[0062] Despite the large reduction of vibration by FBS in FIG. 7, there is still vibration left. Tills is in part because the FBS approach in past work on robots used only physicsbased models of robot vibration for compensation. However, such models have limited accuracy. Hence, they cannot fully compensate the robot’s vibration, especially if the robot dynamics contains some nonlinearities. To address this shortcoming, a hybrid approach is implemented in an embodiment of the multi -robot overspray -free painting assembly 10 and method of control thereof that includes both a physics-based model and a data-driven model for vibration compensation.

[0063] FIG . 8 presents an embodiment of a framework for realizing the hybrid model that includes both a physics-based model and a data-driven model for vibration compensation. Hie input trajectory' is divided into small batches that are passed into a physics-based (PB) linear model (G,.K.PB) to predict outputs for each batch. Moreover, die predictions by the PB model for the current batch are combined with the prediction errors of the PB model from prior batches and used to predict the outputs of the current batch using the data-driven (DD) linear model (G.™>j. The output of Gm, A? incorporates the predictions from the PB model, hence it represents the overall output of a hybrid model (Gir.,h), combining Gm^and Gir.,Pb. The hybrid model (G m,h) is used in place of Gm to design the FBS controller, instead of the PB model alone.

[0064] To facilitate the use of the hybrid approach in OFP and in the multi-robot overspray -free painting assembly 10 and method of control thereof an accelerometer A (FIG, 1 C) can be integrated into and earned by the end effector of each robot to provide the vibration data for the data-driven model, according to an embodiment. Acceleration and vibration data from the accelerometer A can be utilized in the data- driven model, A challenge that must be addressed for the robots in OFP is that their vibration behavior changes as the robots move from position to position. This makes their dynamics time-varying. To address this challenge, a time-varying FBS approach and model can be employed, per an embodiment of the multi-robot overspray-free painting assembly 10 and method of control thereof.

[0065] One other issue that may occur when using the hybrid model is batch-to-batch instability. Idle hybrid model leverages actual outputs (measurements) from the past to improve predictions of current outputs. This creates a feedback loop when the hybrid model is used for vibration compensation, because past outputs are used to determine tire modified trajectory for the current batch. This feedback loop is similar to that seen in ILC, where past work has been done on analyzing and guaranteeing batch-to-batch (i.e., iteration-level) stability. ILC benefits from the assumption that the batches are repetitive. However, in the framework of the the multi -robot overspray-free painting assembly 10 and method of control thereof, repetitive batches are not a requisite. Therefore, establishing stability can be more challenging. To address this challenge batch- w ise feedback w ork is leveraged where a little bit of variation is permitted from batch to batch, under restrictive conditions, and work to generalize them to cases like the multi -robot overspray-free painting assembly 10 and method of control thereof where there are no requirements for repetition from batch to batch.

[0066] Further, vibration of the robot may not be fully eliminated. Therefore, according to an embodiment, the printhead control should accommodate any residual vibration. To do this, any residual vibration is sensed using the accelerometer A attached to the end effector. If it is above an acceptable threshold, the printhead control will dwell until the vibration level settles to an acceptable threshold before the inkjet printing is started. As shown in FIG. 9, the expectation is that with vibration compensation, the dwell time will be significantly reduced, which will facilitate faster OFP. According to an embodiment, one or more of the robot assemblies are brought to a dwell state when a residual vibration level thereof is greater than a threshold vibration level.

[0067] Integration of high- and low-level control. Background. When the robots interact, the high-level control can introduce unintended effects on the low-level printhead and robot motion control. For example, as shown in FIG. 2B, seams can occur at borders between areas painted by different robots due to interactions between the droplets deposited by each robot. Similarly, with multiple robots on the same gantry', the motion of one robot can trigger vibration of the gantry which in turn affects the motion of other robots.

[0068] Accordingly, per an embodiment of the multi-robot overspray-free painting assembly 10 and method of control thereof, the potential impacts of high-level control on low-level printhead and robot motion control are addressed through integration of both levels of control using the GNN multi-robot path planning framework set forth above.

[0069] A common practice of integrating high-level path planning and low-level control is based on trajectory' generation algorithms. The high-level planner provides a series of discrete waypoints to avoid collision with the environment, followed by trajectory' generation that explicitly considers low-level control. Such trajectory generation frequently utilizes feedforward control techniques, such as iterative learning control and loop shaping, as well as the nominal closed-loop dynamics of robots, to generate trajectories that are compatible with the physical constraints of robots. However, these approaches mainly focus on linear systems or merely use smoothing splines for nonlinear systems, which are insufficient to handle the complex, interactive nature existing in the painting system of the multi-robot overspray -free painting assembly' 10 and method of control thereof with the desired level of precision. The interaction among the gantry' system, robots, printheads, and the vehicle surface is highly complex. It is nearly impossible to explicitly model such interaction across forces, positions, and speeds.

[0070] The superior capability of neural networks is leveraged in capturing intrinsic, interactive relationships that are hard to model. It explicitly incorporates both low-level printhead control and low-level robot vibration control into two distinct groups of GNM nodes, as illustrated in FIG. 10,

[0071] Integrating high-level control with low-level printhead control. The integration between low -level printhead control and high-level robot control has been investigated and implemented. In the configuration, each section painted by a robot (with its printhead) share, at least, one edge with another painted section. To make the edgesseamless, variable droplet spacing is implemented where the spacing is larger at the edges, avoiding contact between droplets, such that the neighboring printhead will print in-between those non-contacting droplets (FIGS. 11 A, 1 IB, and 11 C). Such interlaced droplet spacing at the droplet deposition level is pre-planned at the higher-level robot coordination framework. Further, per an embodiment, a multitude of first material deposition droplets FD is deposited via a first material application head, or printhead, with one or more gaps G residing among the first material deposition droplets adjacent an adjoining edge AE of the first material deposition droplets. Further, a multitude of second material deposition droplets SD is deposited via a second material application head, or printhead. The second material application head exhibiting a neighboring location w ith respect to the first material application head. Some or more of the second material deposition droplets are deposited at the gap(s) adjacent the adjoining edge

[0072] Even with the selection of optimal droplet spacing, deposition of equal volumes at each fixed droplet interval can result in uneven surface morphology (or height profile) primarily due to surface tension effects. These effects can be sufficiently modeled and can be compensated accordingly by controlling the droplet volumes deposited at the intervals. The model is a graph-based physics-guided recurrent neural network that assumes fixed droplet spacing. With variable droplet spacing, the graphbased model, in its current structure, is inadequate. Hence, a robust model framework consisting of subgraphs is constructed that accounts for the variability in droplet spacing and number of droplet grid points. Both controlling the droplet volumes at the droplet grid points and the resolution of these grid points themselves are addressed. Such a graph model can be readily integrated into the GNN-based framework described above for high-level control, as shown in FIG. 10.

[0073] Integrating high-level control with low-level robot motion control. The integration between low-level robot control (for vibration suppression) and high-level robot control (for path planning and multiple robot coordination) is implemented. Tire interactions between the motions of the robots arise from the fact that they share a single gantry as shown in FIGS. 1A, IB, and 1C. 11ns fact can also be used to minimize their interactions by coordinating the motions of the robots. To illustrate this point, consider a simple case of the x and t9zvibration of a gantry carrying three robots. As shown in FIG. 12, if robots A and C apply reaction forces that are each half the magnitude and opposite the direction of that of robot B, the x and 'b vibration of the gantry’ will be suppressed.

[0074] The timing and amplitudes of the motions are coordinated (hence the reaction forces) of the robots to minimize the vibration of the gantry, according to an embodiment of the multi -robot overspray-free painting assembly 10 and method of control thereof. The vibration of the gantry, it can be assumed, will primarily be triggered by the y~ and z-motions of each robot; the tip-tilt rotations are not expected to have a significant impact on the gantry’s vibration since they are small motions involving relatively smaller inertias at the end effector. Under this assumption, accelerometers A are placed on the gantry to capture vibrations at several key locations (such as comers and locations where robots are placed). As shown in FIG. 10, these locations are represented as nodes, with 3D coordinates and accelerations of these locations as node features, and they are incorporated into the GNN model set forth above, per an embodiment, which is used for high-level control. Then the path planning can be performed in such a way that the norm of the gantry’s vibration amplitudes is minimized.

[0075] Testing can be carried out in order to test the system-level integrated control of multiple (e.g., eighteen or more) robots to enable OFP that achieves speeds comparable to traditional spray painting without sacrificing paint quality.

[0076] Numerical testing of integrated high-level control. The integrated high-level control described above can be evaluated in simulations, 'the simulation environment can feature a set up similar to that in FIGS. I A, IB, and 1C, with two gantries each carrying at least eighteen four-axis robots. The robots cab be designed and sized to carry tire payload of a standard OFP printhead, scanning camera, hoses, and cables. A simulation model of the gantries and robots, including their kinematics and rigid body dynamics can be created in Gazebo or another program . A simplified model of an inkjet printing can also be integrated into the simulation.

[0077] Using the simulation set up, the ability of the GNN path planning method set forth above can be tested to generate multi-robot paths efficiently and safely. This can involve: (a) distributing the painting areas in real-time among robots, and (b) planning robots’ collision-free motions. Three metrics can be evaluated: efficiency, safety, and robustness. The efficiency of the algorithms can be evaluated by comparing the speed of painting to traditional spray painting. The safety of the algorithms can be evaluated by their ability to avoid collisions. Tire robustness of the algorithms can be evaluated by comparing the re-planning time in the event that one or more robots malfunction.

[0078] Experimental testing of integrated low-level and high-level control. A simplified prototype of the multi-robot OFP approach can be used for experimental hypothesis testing. FIG. 13 shows an example of such a prototype. It consists of one of the vertical pillars of the gantries in FIGS. 1A, IB, and 1C carrying three 4-axis robots. The set up can be used to paint full-sized vehicle doors. The pillar can be driven by a ball screw drive with linear rails for guidance. Similarly, the y- and z-motions of each robot can be realized using ball screw drives, while the tip and tilt rotations can be achieved using rotary motors. A commercial OFP printhead, can be mounted on the end effector of each robot. A camera or laser scanning sensor and an accelerometer can also be mounted on each end effector. The robots and printhead can be controlled by a dSPACE Micro Lab Box realtime control system. Still, other set-ups are possible in other examples.

[0079] This set tip can be used to test the integration of low-level robot motion and printhead control . Each robot will be assigned contiguous portions of a car door to paint. Tire robot motions can involve acceleration and deceleration while the printhead is painting. The robot vibration can be compensated as described above and the quality and speed of the paint job can be assessed compared to the traditional approach where OFP occurs only at constant speed robot motions. Furthermore, the integration of low- and high-level control set forth above can be tested by using a scaled down version of the GNN method for three robots.

[0080] The method of controlling the multi-robot material deposition system and assembly, such as the multi -robot overspray-free painting assembly 10, can have various steps in different embodiments depending upon — among other potential factors — its intended application of use. In one embodiment, a first step involves the provision of one or more gantries such as the first and / or second gantries 12, 14. A second step involves the provision of a multitude of robot assemblies such as the first and / or second sets of robot assemblies 16, 18. The robot assemblies, however many there are, are carried on the gantry(ies) and are equipped with material application heads such as the first and second printheads 24, 32. A third step involves planning and controlling paths of directional motion of the robot assemblies with respect to one another and with respect to the substrate surface SS. A fourth step involves controlling movements of each of the individual robot assemblies of the multitude of robot assemblies such as the individual first and second robot assemblies 20, 28. In an embodiment, the planned and controlled paths of directional motion of the robotassemblies is partly or more dependent upon the controlled movements of each of the robot assemblies and associated end effector assemblies. Further, in an embodiment, the controlled movements of each of the robot assemblies and associated end effector assemblies is partly or more dependent upon the planned and controlled paths of directional motion of the robot assemblies. A further embodiment has the further steps of employing the graph neural network (GNN) model in the planning and controlling paths of directional motion of the robot assemblies such as the first and second sets of robot assemblies 16, 18, and employing the convolutional recurrent neural network (ConvRNN) model in the controlling movements of each of the robot assemblies such as the first and second sets of robot assemblies 16, 18 and each of the associated end effector assemblies such as the first and second end effector assemblies 22, 30.[0081 j As used herein, the terms “general” and “generally” and “substantially” and “approximately” are intended to account for the inherent degree of variance and imprecision that is often atributed to, and often accompanies, any design and manufacturing process, including engineering tolerances ----- and without deviation from the relevant functionality and outcome — such that mathematical precision and exactitude is not implied and, in some instances, is not possible. In other instances, the terms “general” and “generally” and “substantially” and “’approximately” are intended to represent the inherent degree of uncertainty and / or impossibility that is often attributed to any quantitative comparison, value, and measurement calculation, or other representation.

[0082] It is to be understood that tire foregoing is a description of one or more aspects of the disclosure. The disclosure is not limited to the particular embodiment(s) disclosed herein, but rather is defined solely by the claims below. Furthermore, the statements contained in the foregoing description relate to particular embodiments and are not to be construed as limitations on the scope of the disclosure or on the definition of terms used in the claims, except where a term or phrase is expressly defined above. Various oilier embodiments and various changes and modifications to the disclosed embodiment(s) will become apparent to those skilled in the art. All such other embodiments, changes, and modifications are intended to come within the scope of the appended claims.

[0083] As used in this specification and claims, the terms “e.g.,” “for example,” “for instance,” “such as,” and “like,” and the verbs “comprising,” “having,” “including,” and their other verb forms, when used in conjunction with a listing of one or morecomponents or other items, are each to be construed as open-ended, meaning that the listing is not to be considered as excluding other, additional components or items. Other terms are to be construed using their broadest reasonable meaning unless they are used in a context that requires a different interpretation.

[0084] Those of skill in the art will understand that modifications, additions, and / or removals of various components of the substances, formulations, apparatuses, methods, systems, and embodiments described herein may be made without departing from the full scope and spirit of the present disclosure, which encompass such modifications and any and all equivalents thereof.

Claims

WHAT IS CLAIMED IS:

1. A method of controlling a multi-robot material deposition system, the method comprising: providing at least one gantry moveable in an x-direction; providing a plurality of robot assemblies carried on said at least one gantry, each of said plurality of robot assemblies moveable in a y-direction, a z-direction, or both the y-direction and z-direction, each of said plurality of robot assemblies including a plurality of end effector assemblies with a plurality of material application heads for depositing material on a substrate surface, said plurality of material application heads having tip and tilt rotational moveability; planning and controlling paths of directional motion of said plurality of robot assemblies with respect to one another and with respect to the substrate surface; and controlling movements of each of said plurality of robot assemblies and each of said plurality of end effector assemblies with respect to one another; wherein the planned and controlled paths of directional motion of said plurality of robot assemblies is at least partly dependent upon the controlled movements of each of said plurality of robot assemblies and end effector assemblies, and wherein the controlled movements of each of said plurality of robot assemblies and end effector assemblies is at least partly dependent upon the planned and controlled paths of directional motion of said plurality of robot assemblies.

2. lire method of controlling tire multi -robot material deposition system as set forth tn claim 1 , further comprising employing a graph neural network (GNN) model in the planning and controlling paths of directional motion of said plurality of robot assemblies.

3. The method of controlling the multi-robot material deposition system as set forth in claim 2, wherein employment of said graph neural netw ork (GNN) model involves utilization of at least one of: surface areas of the substrate surface, targeted motion speeds of said plurality of robot assemblies, distances among said plurality-7of robot assemblies, or kinematic and dynamic constraints of said plurality of robot assemblies.

4. The method of controlling the multi-robot material deposition system as set forth in claim 1, further comprising employing a convolutional recurrent neural network (ConvRNN) model in the controlling movements of each of said plurality of robot assemblies and each of said plurality of end effector assemblies.

5. The method of controlling the multi-robot material deposition system as set forth in claim 4, further comprising depositing a plurality of first material deposit droplets via said plurality of material application heads at locations and / or adjacent an edge of the substrate surface, and depositing a plurality of second material deposit droplets via said plurality of material application heads at locations away from the edge of the substrate surface, said plurality of first material deposit droplets having first spacings thereamong and said plurality of second material deposit droplets having second spacings thereamong, said first spacings being lesser in value than said second spacings.

6. The method of controlling the multi-robot material deposition system as set forth in claim 4, further comprising controlling material deposit droplet deposition of said plurality of material application heads based on geometry of the substrate surface.

7. The method of controlling the multi-robot material deposition system as set forth in claim 4, further comprising controlling timing of material deposit droplet ejection from said plurality of material application heads based on an estimated trajectory of material deposit droplets of said plurality of material application heads.

8. The method of controlling the multi -robot material deposition system as set forth in claim 1 , further comprising compensating for vibrations of said plurality of robot assemblies by employing feedforward control that utilizes a physics-based model and a data-driven model.

9. Tire method of controlling the multi-robot material deposition system as set forth in claim 8, further comprising employing a filtered B-splines (FBS) model that utilizes said physics-based model and said data-driven model.

10. The method of controlling the multi-robot material deposition system as set forth in claim 9, further comprising: providing a plurality of accelerometers at said plurality of robot assemblies; and utilizing vibration data from said plurality of accelerometers in said data-driven model; wherein said filtered B-splines (FBS) model is a time-varying filtered B-splines (FBS) model.

11. lire method of controlling tire multi -robot material deposition system as set forth in claim 8, further comprising: providing a plurality of accelerometers at said plurality of robot assemblies; and bringing at least one of said plurality of robot assemblies to a dwell state when a residual vibration level of said at least one of said plurality of robot assemblies is greater than a threshold vibration level.

12. The method of controlling the multi-robot material deposition system as set forth in claim 1, further comprising: employing a graph neural network (GNN) model in the planning and controlling paths of directional motion of said plurality of robot assemblies; depositing a plurality of first material deposition droplets via a first material application head of said plurality of material application heads, at least one gap residing among said plurality of first material deposition droplets adjacent an adjoining edge of said plurality of first material deposition droplets; and depositing a plurality of second material deposition droplets via a second material application head of said plurality of material application heads, said second material application head exhibiting a neighboring location with respect to said first material application head, at least some of said plurality of second material deposition droplets being deposited at said at least one gap residing among said plurality of first material deposition droplets adjacent said adjoining edge.

13. Tire method of controlling the multi-robot material deposition system as set forth in claim 12, wherein at least some of said plurality of first material deposition droplets exhibit a first volume and at least some of said plurality of second materialdeposition droplets exhibit a second volume, said first volume differs from said second volume.

14. The method of controlling the multi -robot material deposition system as set forth in claim 1, further comprising: providing a plurality of first accelerometers at said at least one gantry; and employing a graph neural network (GNN) model in the planning and controlling paths of directional motion of said plurality of robot assemblies based at least partly upon vibration data of said plurality of accelerometers.

15. The method of controlling the multi -robot material deposition sy stem as set forth in claim 1, wherein the method of controlling the multi -robot material deposition system is employed in a multi-robot overspray-free painting system, or the method of controlling the multi-robot material deposition system is employed in a multi -robot three-dimensional (3D) printing system.

16. A multi-robot overspray-free painting assembly, comprising: at least one gantry moveable in an x-direction; and a plurality of robot assemblies carried on said at least one gantry, each of said plurality of robot assemblies moveable in a y-direction, a z~direction, or both the y- direction and z-direction, each of said plurality of robot assemblies including a plurality of end effector assemblies with a plurality of printheads for deposition of paint coating on a substrate surface, said plurality’ of printheads having tip and tilt rotational moveability, a plurality of sensors earned by said plurality of robot assemblies, said plurality of sensors sensing the substrate surface; wherein, during operation of the multi-robot overspray -free painting assembly, a graph neural network (GNN) model is employed in planning and controlling paths of directional motion of said plurality of robot assemblies with respect to one another and with respect to the substrate surface, and a convolutional recurrent neural network (ConvRNN) model is employed in controlling movements of each of said plurality of robot assemblies and each of said plurality of end effector assemblies with respect to one another.

17. The multi-robot overspray-free painting assembly as set forth in claim 16, wherein, during operation of the multi-robot overspray -free painting assembly, the planned and controlled paths of directional motion of said plurality’ of robot assemblies is at least partly dependent upon the controlled movements of each of said plurality of robot assemblies and end effector assemblies, and the controlled movements of each of said plurality of robot assemblies and end effector assemblies is at least partly dependent upon the planned and controlled paths of directional motion of said plurality of robot assemblies.

18. A method of controlling a multi-robot overspray-free painting system, the method comprising: providing a first gantry’ moveable in an x-direction, and providing a second gantry moveable in the x-direction; providing a plurality of first robot assemblies earned on said first gantry?, each of said plurality of first robot assemblies moveable in a y-direction, a z-direction, or both the y-direction and z-direction, each of said plurality of first robot assemblies including a plurality of first end effector assemblies with a plurality of first printheads for deposition of paint coating on a substrate surface, said plurality of first printheads having tip and tilt rotational moveability', and providing a plurality of second robot assemblies carried on said second gantry, each of said plurality of second robot assemblies moveable in the y-direction, the z-direction, or both the y-direction and z- direction, each of said plurality of second robot assemblies including a plurality of second end effector assemblies with a plurality-’ of second printheads for deposition of paint coating on the substrate surface, said plurality of second printheads having tip and tilt rotational moveability; planning and controlling paths of directional motion of said plurality of first and second robot assemblies with respect to one another and with respect to the substrate surface; controlling movements of each of said plurality of first robot assemblies and each of said plurality of first end effector assemblies with respect to one another, and controlling movements of each of said plurality of second robot assemblies and each of said plurality of second end effector assemblies with respect to one another: wherein the planning and controlling paths of directional motion of said plurality of first and second robot assemblies is at least partly dependent upon thecontrolled movements of each of said plurality of first robot assemblies and first end effector assemblies and the controlled movements of each of said plurality of second robot assemblies and second end effector assemblies; and wherein the controlling movements of each of said plurality of first robot assemblies and first end effector assemblies and of each of said plurality of second robot assemblies and second end effector assemblies is at least partly dependent upon the planned and controlled paths of directional motion of said plurality of first and second robot assemblies.

19. The method of controlling a multi-robot overspray-free painting system as set forth in claim 18, further comprising: employing a graph neural network (GNN) model in the planning and controlling paths of directional motion of said plurality of first and second robot assemblies; and employing a convolutional recurrent neural network (ConvRNN) model in the controlling movements of each of said plurality' of first and second robot assemblies and each of said plurality of first and second end effector assemblies.

20. The method of controlling a multi-robot overspray-free painting system as set forth in claim 18, further comprising compensating vibrations of said plurality of first and second robot assemblies by employing feedforward control that utilizes a physics-based model and a data-driven model.

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