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Optimal Trajectory Planning for a Robotic Manipulator Palletizing Tasks

In recent years, the employment of robots has become a value-added entity in the industries in gaining their competitive advantages. Moreover, thanks to Industry 4.0 paradigm, many production tasks have grown in terms of dimensionality, complexity and higher precision and need to be performed by robots. Among them, the palletizing task is still highly dependent on the particular problem to solve, and its optimization needs to be performed basing on the ground condition. In this paper a palletizing task problem performed by a robotic manipulator is studied. More in detail, some objects have to be transported from a pre-determined storage area to a delivery area. In the storage area the objects are stacked one on the other in columns, while in the delivery area the robotic manipulator poses the objects in horizontal levels, one over another. The process is optimized by minimizing the total distance travelled by the robotic manipulator to transport all the objects from the storage area to the delivery area. An Integer Linear Programming (ILP) problem is formalized and tested by simulations and experimental results.

I. EXTENDED ABSTRACT Robotics is one of the Industry 4.0 enabling technologies, adopted and declined under different perspectives and in particular Robotics manipulator are used in industries for many different tasks like machining [1], handling and pickand-place, palletizing, welding, painting and cooperation with humans in specific tasks.
In many industrial applications, the robots are used to perform a set of tasks and the tasks sequencing and how the robot moves between them greatly influence the overall performance of the application [3].
This contribution deals with the problem of palletizing objects from a pre-determined storage area to a delivery area. In the storage area the objects are stacked in columns, while in the delivery area the robotic manipulator has to pose the objects in horizontal levels, one over another.
A robotic manipulator has to move a finite number of objects from a starting pre-determined geometric configuration (storage area) to an ending different pre-determined configuration (delivery area).
The starting geometry is configured by disposing the objects to be moved in a set of columns, where the objects are stacked one on the other. The distances between the columns has to take into account also the geometry of the gripper on the end-effector; the manipulator can grip every object only in a single fixed configuration (see for instance Fig. 1). The ending object configuration is obtained by disposing the objects in a certain number of levels, one over another. At every level, the objects position is shifted of an half of the length referring to the previous one and the next one. The final configuration is similar to the construction of a wall made of bricks.
All the objects have the same geometry, and the robotic manipulator can handle only one object for each transfer operation, starting from the initial position and ending to the final position. The manipulator will perform this global task in a number of sub-tasks depending on the geometric configuration of the problem.
The problem we want to solve is to compute the optimal sequence of actions for moving objects from the starting to the ending configurations in order to minimize the length of the travelled paths.  2]). In order to schedule the operations in discrete time periods, we assume that the handling times are integer parameters in the time-indexed formulation. Moreover, we assume that each operation is performed in one time interval, even if not all the operations have the same durations.
The solution of the ILP problem presented in [2] is tested by two strategies: i) an off-line simulation framework is specified; ii) a real case study is performed.
In order to solve the problem, a geometric configuration is set as it is shown in Fig. 1. The parameters and variables are specified as follow: The time-indexed formulation of the problem considers a planning horizon that is discretized into n time units (t.u.). In addition, a time-indexed binary variable is defined as follows: The aim is finding the sequence of the moves of the manipulator that minimizes the length of the travelled path. Hence, the objective function can be defined as follows: The ILP is the following: subject to:

III. RESULTS
The outputs of the solution are described by the decision binary variables that identify the sequence of the manipulator moves in each interval time. Then, the result is reported in Table I and the minimum path to perform all the movements is given by the ILP objective function D tot = 19.222m.
The theoretical solution of the problem is tested in an experimental set up involving a KUKA KR3 R540 robot (figure 2). The programming phase is performed by using the online approach directly on the workcell. With the KUKA teach pendant (SmartPAD), the exact geometry and travel set modelled in the simulation environment is replicated, in order to compare the simulation with the real experimentation. The speed used for every movement is sp = 0.4m/s. This setting, together with the low height of the bricks, make neglect-able the robot dynamics.
The obtained results reported in Table II take into account also the time needed to grip and release the objects. However, the experimental results are consistent with the theoretical solution obtained by the ILP problem.