## 3 Task Scheduling with Mixed-Integer Programming
### Task Scheduling with the Kinematic Constraints
### Task Scheduling and Assigning Arm

Using the the taskset \(\mathbb{T}\) data in Table 1 as the input, Algorithm 1 was tested in a realistic environment. The cooking robot in the environment is made up of 11 degrees of freedom (DOF) dual arms, where there are five DOFs per arm with another revolute joint that couples the two arms at the center of the body. This robot was designed and analyzed in our previous work [51]. Additionally, the tables with tools for cooking surround the robot. Figure 2 shows the environment used for the experiments. The workspace is segmented into several spaces, and specific tasks are performed at that location. The yellow boxes in Figure 2 represent each location, and the indices were set for convenience, which are the numbers in the white box in (b). We assume that one task can have performed in \(1\sim 2\) locations, start and end locations (e.g., moving the bowl from the 6 th location to the 7 th location). Depending on the robot configuration, the table is arranged in a way that the robot can access all locations. Thanks to the coupling joint in the waist of the robot, a single arm can access all locations easily. Figure 2: The environment of the cooking robot. (**a**) Isometric view of the whole environment. Yellow cubes represent pre-defined locations for defining tasks. (**b**) Top view of the environment. The meaning of number in white box is the index of locations. The results of the experiment are shown in Figure 3. Figure 3a shows the result of the first step of the optimization process in Section 3.1, (b) shows the result of the second step in Section 3.2. Each rectangle represents all \(\mathcal{T}\) in \(\mathbb{T}\) and the same color represents the same location of the task. If \(\mathcal{L}_{s}\
eq\mathcal{L}_{e}\) in one \(\mathcal{T}\), then the colors change gradually in one box. In Figure 3a, the result did not violate any kinematic constraints and dependencies of \(\mathcal{T}\). In Figure 3b, all \(\mathcal{T}\) are assigned to the left and right arms properly. The task needs two arms (e.g., \(\mathcal{T}_{12}\), \(\mathcal{T}_{14}\), and \(\mathcal{T}_{25}\)) and no arm (e.g., \(\mathcal{T}_{6}\), \(\mathcal{T}_{11}\), \(\mathcal{T}_{13}\), \(\mathcal{T}_{17}\), and \(\mathcal{T}_{20}\)) are also well assigned following the sequences. Moreover, the continuity is well reflected in the result. The tasks where \(c=1\) were \(\mathcal{T}_{5}\), \(\mathcal{T}_{6}\), \(\mathcal{T}_{9}\) to \(\mathcal{T}_{12}\), \(\mathcal{T}_{17}\), and \(\mathcal{T}_{20}\). The result shows that \(\mathcal{T}_{5}\) to \(\mathcal{T}_{7}\), \(\mathcal{T}_{9}\) to \(\mathcal{T}_{13}\), \(\mathcal{T}_{17}\), \(\mathcal{T}_{18}\), \(\mathcal{T}_{20}\), and \(\mathcal{T}_{21}\) are consecutive, so it can be seen that continuity is well reflected. ### Comparison with Conventional Algorithm

The experiments were performed to compare the performance of our algorithm with the conventional algorithm. Let us define an \(n\times m\) problem when \(n\) jobs are conducted on \(m\) machines. The experiments consist of problems of twelve sizes, which are \(2\times 2\), \(3\times 2\), \(4\times 2\), \(5\times 2\), \(2\times 3\), \(3\times 3\), \(4\times 3\), \(5\times 3\), \(2\times 4\), \(3\times 4\), \(4\times 4\), and \(5\times 4\), and each set consists of ten problem instances. As mentioned in Section 2.2, the conventional algorithm used a pre-assigned machine for a specific operation. Therefore, while testing the conventional algorithm, a robotic arm is assigned to cook specific dishes. Additionally, our algorithm did not assign a robotic arm to a specific task. In case of number of machine is more than two, the process adding constraint (lines 20 to 24 in Algorithm 2) is executed \({}_{m}\)C\({}_{2}\) times when \(m\) machines exist. For example, for \(n\times 3\) problem, the constraints from the pairs of (1 st, 2 nd), (1 st, 3 rd), (2 nd, 3 rd) robots will be added in the formula. Table 2 shows the result of the comparison between conventional algorithm and ours.