*5.1. The BDCDO of 3-DOF Manutec r3 System*

The industrial robot Manutec r3 [33], as shown in Figure 15, has six links, and only the three degrees of freedom associated with positioning are considered in this research, simplifying the robotic system to a three degrees of freedom (DOF) dynamic system. The goal of the 3-DOF robot co-design and optimization problem is to identify the optimal solution in the design space and state space so that the robot can move from the initial position to the specified position in the minimum time while satisfying the associated state equation constraints, upper and lower bound constraints of plant parameters, and state and control variables. The trajectory optimization formulation of the 3-DOF Manutec r3 system is described as follows:

$$\begin{array}{rl}\min & f = t\_f \\ \text{s.t.} & \dot{\xi}(t) = \mathbf{f}(\mathbf{x}\_{p\prime}\boldsymbol{\xi}(t), \mathbf{u}(t), t) \\ & \mathbf{x}\_p \in [\mathbf{x}\_L, \mathbf{x}\_{lI}] \\ & \dot{\xi}(t) \in [\boldsymbol{\xi}\_{L\prime}\boldsymbol{\xi}\_{lI}] \\ & \mathbf{u}(t) \in [\mathbf{u}\_{L\prime}\mathbf{u}\_{lI}] \end{array} \tag{25}$$

where the plant design parameters **x***<sup>p</sup>* are the lengths of link 1 and link 2 [*L*1, *L*2]. The state variables ξ consists of the relative angles of rotation [*α*, *β*, *γ*] and the relative angular velocities [ . *α*, . *β*, . *γ*] between the connecting links, and the control variables **u** includes the standardized torque controls [*u*1, *u*2, *u*3]. The design intervals of the plant parameters **x***<sup>p</sup>* are *L*<sup>1</sup> ∈ [0.4, 0.5] and *L*<sup>2</sup> ∈ [0.9, 1.0]. The initial value and final value of ξ are ξ(*t*0)=[−2, −2.5, −2, 0, 0, 0] and ξ(*tf*)=[2, 2.5, 2, 0, 0, 0], and the upper and lower bounds of ξ are ξ*<sup>U</sup>* = [3, 3, 3, 5, 10, 15] and ξ*<sup>L</sup>* = [−3, −3, −3, −5, −10, −15]. The control variables are subject to interval constraints **u***<sup>U</sup>* = [10, 10, 10] and **u***<sup>L</sup>* = [−10, −10, −10]. To solve this problem, the maximum number of iterations of the NLP solver is set to 20, and the solution accuracy is set to 10−6. In the original dynamic model, the physical design parameters

are **x***<sup>p</sup>* = [0.4500, 0.9500], and the optimal solution yields the objective value as 0.9082. As with the above example, in addition to the SRIRMD-STOR method, the TEI-MASRI and EFDC-MASRI methods are also applied to optimize the 3-DOF Manutec r3 system. In the SRIRMD-STOR method, the number of initial points is *N*<sup>0</sup> = 25, and the maximum number of new samples per iteration is Δ*<sup>N</sup>* = 20. It is notable that *ε* < 0.001 and Δ*J* < 0.0001 in the termination criterions of the TEI-MASRI and EFDC-MASRI methods.

**Figure 15.** Schematic diagram of a 3-DOF robot.

The optimal physical design parameters and optimal objective values by different methods are presented in Table 5. It can be observed from Table 5 that the surrogatemodel-based approaches greatly reduce the number of valuations of the dynamic system and save computational costs compared to the original dynamic-model-based approach. Meanwhile, compared with the original physical design parameters, the optimal physical design parameters obtained by the TEI-MASRI, EFDC-MASRI, SRIRMD-MASRI, and SRIRMD-STOR methods shorten the working time to complete the specified task to a certain extent and improve the efficiency and performance of the robot arm. Among those surrogate model-based methods, the SRIRMD-STOR method obtains the best performance metrics using less samples. Therefore, the BDCDO solving framework combined with SRIRMD and STOR is a better alternative for the co-design and optimization problem of the 3-DOF Manutec r3 system.

**Table 5.** The computational cost, optimal plant parameters, and optimal objective values in the 3-DOF Manutec r3 system.


To graphically demonstrate the sampling outcome of the SRIRMD sampling strategy, the distribution of sample points and the phase diagram of the optimal trajectory between different state variables are shown in Figure 16. The black dots are the initial samples, the black stars are the new samples obtained via SRIRMD, and the red solid lines are the state trajectories optimized based on the surrogate models. As Figure 16 reveals, the new sample points in the SRIRMD sampling strategy are all situated nearby the state trajectories.

**Figure 16.** The phase diagrams of the optimal trajectories and distribution of samples between different state variables.

Figure 17 graphs the trajectory iteration processes for the state components *α*, *β*, *γ*, . *α*, . *β*, and . *γ* in the SRIRMD-STOR method. As visible in Figure 17, the trajectories of the different state components converge gradually as the iterations proceed, and the trajectories of the 17th and 18th iterations tend to coincide. Meanwhile, Figure 18 exhibits the control curves obtained by the SRIRMD method.

**Figure 17.** Trajectory iterative processes of the state components *α*, *β*, *γ*, . *α*, . *<sup>β</sup>*, and . *γ*.

**Figure 18.** The solution of the control inputs *u*1, *u*2, *u*3.

Figure 19 records the convergence processes of the state component trajectory overlap ratio and the state trajectory overlap ratio in the SRIRMD-STOR method, *α*1, *α*2, *α*3, *α*4, *α*5, *α*6, and A are the trajectory overlap ratios of the state components *α*, *β*, *γ*, . *α*, . *β*, . *γ*, and state ξ, respectively. According to Figure 19, the state trajectory overlap ratio A converges to 1 with the convergences of all *αi*, and *α<sup>i</sup>* ≥ A, which verifies Theorem 1 and Theorem 2.

**Figure 19.** The convergence processes of A and all *α<sup>i</sup>* in the 3-DOF Manutec r3 system.
