# Automatic Optimal Robotic Base Placement for Collaborative Industrial Robotic Car Painting
## 2 The Base Placement Problem and High-Level Solution Strategy  
This section outlines the base placement problem, underlying assumptions, and our high-level solution strategy. We present the details and the overall algorithm in subsequent sections. **Multi-robot car-painting base placement problem:**_Consider the CAD model of a vehicle and a group of robotic manipulators (see Figure 2 and Figure 3). Additionally, consider the following assumptions:_  
* _The robotic arms utilize an RRR articulated kinematic structure (see, e.g., Figure_ 2_), where the body, shoulder, and elbow joints are all revolute (see, e.g.,_ _[_30_]_ _[Chapter 1] for a detailed classification of robotic manipulators)._
* _The robotic arms are fixed to the floor or ceiling and arranged in two rows on either side of the car, leveraging the vehicle's inherent symmetry during car-painting operations._  
_Compute the optimal fixed base positions for a collection of heterogeneous articulated robotic arms on the factory floor/ceiling to maximize the possibility of paint coverage and minimize the possibility of robot collisions. Furthermore, do not utilize any information from motion/trajectory planners a priori or during base placement computations._  
Figure 2: The kinematic structure of an example RRR articulated arm (ABB IRB 4600 industrial manipulator [31]), where the body, shoulder, and elbow joints are all revolute. The unit lengths are in millimeters. **Remark 1**. : _In certain industries, such as aerospace, special-shaped surfaces present unique challenges for robotic painting due to their non-uniform and complex geometries. While our current approach is designed for commercial vehicles, which generally have more regular shapes, multi-plane base placements may offer advantages for these specialized applications. Specifically, the ability to position robotic arms on different planes would allow greater flexibility in reaching difficult contours and ensuring full coverage, as discussed in prior work on robotic painting for irregular surfaces in aerospace contexts [32; 33]. However, the assumption of floor- or ceiling-mounted robotic arms remains applicable to commercial vehicles due to their generally regular geometry._  
**High-level solution strategy:** To solve the multi-robot car-painting base placement problem, we take the following steps. 1. In Section 3, we devise a scheme for a point cloud rendering of the reachable workspace of the forearms of the robotic arms and fit ellipsoids to these point clouds. The computations in this step need only be run once. 2. In Section 4, we project the ellipsoids obtained from Step 1 onto the floor/ceiling to obtain planar ellipses. Using these ellipses, we construct a cost function to quantify the possibility of robot collisions. The constructed cost function is parameterized by the Cartesian coordinates of the planar ellipse centers. 3. In Section 5, we construct a class of cost functions to quantify the amount of vehicle paint coverage. These cost functions are obtained by projecting the point clouds, which are computed in Step 1, onto the sides and top of the vehicle. 4. In Section 6, we formulate a proper multi-objective optimization problem using the cost functions obtained from Steps 2 and 3. Any standard multi-objective optimizer can then be invoked for computing the optimal fixed base positions. **Remark 2**. : _In this work, we adopt a hierarchical optimization approach to vehicle painting, which contrasts with methods that attempt to simultaneously address base placement and painting objectives. Our approach prioritizes optimal base placement, minimizing collision risk and maximizing the reachability of the robotic arms. This establishes a strong foundation for the subsequent stages, where path planning algorithms, drawn from the existing literature, are applied to ensure complete paint coverage of the vehicle's surface. This approach offers two key advantages. First, by decoupling base placement from path planning, we enhance both computational efficiency and practical effectiveness.