This article proposes a resonance suppression method for a flexible load servo drive system based on a flexible manipulator's pose transformation. We establish a flexible load servo drive system for the robotic arm based on the continuum vibration theory and the transfer function estimation method. The controller consists of two parts: compensation control and dynamic feedback. The transfer function of the active feedback part is strictly positive and real. At the end of the thesis, the asymptotic stability of the closed-loop system in the neighborhood of the desired position is proved through the linear operator semigroup theory and the LaSalle invariant set principle.

#### Keywords

- Flexible manipulator
- Performance configuration
- Transfer function
- Structure/control fusion modeling

#### MSC 2010

- 70Q05

Modern electromechanical systems are constantly developing in the direction of high speed and lightweight. Its requirements for system accuracy and stability are getting higher and higher. The assembly robot experiences frequent high-acceleration starts and stops during the system's operation, which can easily cause the broadband resonance of the system. In this way, it is easy to reduce the system's motion accuracy and limit the system's working speed [1]. There is a mutual coupling between mechanical structure parameters and control parameters. We should model the electromechanical system uniformly and optimize the electromechanical design parameters.

Some scholars proposed a design method of electromechanical fusion based on PD control and single-arm and double-arm lever mechanism. It establishes the electromechanical coupling model of the flexible rod mechanism and the transfer function of the system. They used the maximum value of the real part of the dominant pole as the objective function to optimize the design of parameters such as the cross-sectional size of the mechanism, the acting position of the driving force, and the control gain [2]. In this way, the vibration stabilization adjustment time of the entire system is the shortest. Some scholars have proposed a set of electromechanical system design methods based on electromechanical coupling. Some scholars have studied the matching relationship between the electromechanical parameters of the system by taking the transmission mechanism composed of the motor-screw-moving table as an example. At the same time, some criteria for component selection are put forward. Some scholars have proposed the use of the regression experiment optimization method for electromechanical fusion design. The optimization design of the electromechanical parameters of the system is realized through the iterative process of test-analysis-improvement of the physical model. Some scholars have proposed an electromechanical coupling method for analyzing fixed structure and control. A solution to the problem of electromechanical coupling is introduced. The main idea is to develop a communication interface between the structural analysis software and the control system analysis and design software. In this way, data transmission between analysis programs and system coupling analysis is realized. At the same time, it can also realize the calculation and response simulation of some simple control system closed-loop eigenvalues.

The above kinds of literature are generally single-objective models in system modeling or use structure-control separate modeling, structure-control loop iterative optimization, and other methods. These are difficult to reflect the parallel requirements for multiple targets of the system in practical applications. Multi-objective models and their parallel optimization methods are particularly important in mechanical/control fusion design. This paper proposes an optimization model that includes structural parameters and control parameter performance configuration based on a single-arm manipulator with point-to-point motion [3]. Configure the feasible region of the closed-loop control transfer function pole according to the requirements of the control system overshoot _{p}_{s}

We simplified the manipulator into a flexible beam, as shown in Figure 1. Assume that its cross-sectional shape is rectangular. In the figure, the beam is taken as a rod whose mass is evenly distributed along the length [4]. We assume that the length of the beam is _{y}

According to the Bernoulli-Euler equation of beam lateral vibration, the differential equation of motion of its lateral free vibration is:

We use the variable separation method to obtain the first three modes of the flexible arm as:

_{1} (

We set the mode shape function of the rigid displacement as _{0} (_{0} =0. According to the Lagrangian method, the dynamic equation of the flexible arm in the movement process is as follows:

Among them:
_{0}_{1} L _{n}^{T}, _{0}_{1} … _{n}^{T},

Among them:

T represents torque. C represents the distance from the point of application of the driving torque to the center of arm rotation. The Laplace transform of the total displacement at the end of the flexible arm is

In:

The closed-loop transfer function of PD control is:

To facilitate the discussion, we take the first-order model of the system to study, then:

The corresponding characteristic equation is:

The dominant pole is the farthest from the imaginary axis when the system has two pairs of very complex roots. The system also has the best stability. To make the characteristic equation (16) have two repeated roots, we set:

At this time, the root of the characteristic equation is

The system's poles largely determine the various characteristics and quality indicators of electromechanical system dynamics. Therefore, the pole position can be selected in advance according to the performance requirements of the system. Then design other parameters of the system by optimizing the system [5]. Suppose the comprehensive indicators of the system are as follows: _{p}_{s}_{p}_{s}

Usually, we can draw the area where the dominant pole of the closed-loop system that meets this requirement is located on the complex plane according to the requirements of the performance index. It is shown in the shaded area in Figure 2. In the figure _{0} = −^{T}_{12}, the damping angle _{12} located in the shaded area in Figure 2, the following equations can be obtained from equations (6) and (7):

To make the system a minimum phase system, we must make _{i}

From equations (4)–(12), it can be seen that the real and imaginary parts of the poles of _{1}, _{0}, _{1} and the closed-loop control transfer function are related to the parameter and _{p}, K_{υ}

To achieve the high-speed and high-precision requirements of the system for the point-to-point reciprocating motion arm, the moment of inertia of the system needs to be small [7]. At the same time, we also need to ensure that the control energy of the control system is small and the moment of inertia of the flexible beam is:

The state equation of the PD feedback control system that only considers the first-order mode shape of the beam is:

Among them: ^{T},

_{p}k_{υ}_{p}, k_{υ}

To calculate _{0}. For the convenience of calculation, we assume that _{0} is randomly distributed on the 2^{T}P + C^{T} K^{T} KC

From equations (8)–(12), the constraints are:

_{0}, _{0} is the initial value of the system moment of inertia and the initial value of the energy index, respectively. _{υ}_{υ}_{max} is the maximum allowable deformation of the flexible arm in the vertical direction.

The electromechanical coupling design of a single flexible arm is transformed into an optimization problem of the objective function equation (17) with constraints (18)~(22). We use a genetic algorithm to optimize this optimization model [9]. The initial value is _{υ}_{max} = 5 _{p}_{s}

Design results before and after optimization

parameter | Initial value | Final optimization value | |
---|---|---|---|

L(m) | 0.75 | 0.75 | |

H(m) | 0.019 | 0.0057 | |

B(m) | 0.0032 | 0.0067 | |

C(m) | 0 | 0.3302 | |

Kp | 30.8363 | 250.1 | |

Kv | 0.824 | 3.6327 | |

Inertia (kgm) | 0.0206 | 0.0134 | |

Mass (kg) | 0.1231 | 0.0773 | |

ω(rad/s) | 138.36 | 285.88 | |

F(1/s) | 22.02 | 45.5 | |

σp | 15% | 12% | |

ts(s) | 0.15 | 0.09 | |

δv(mm) | 9.72 | 4.9 | |

δvmax(mm) | 5 | 5 |

It can be seen from Table 1 that the entire system is the minimum phase-stable system after the optimized design. The moment of inertia is significantly reduced while meeting the requirements of the stability performance indicators _{p}_{s}

This paper takes a single-arm manipulator as an example and proposes a mechanical/control multi-object fusion modeling theory based on system performance configuration. The research results show that the system's dynamic characteristics can be improved while meeting the system performance indicators. This can make the control energy smaller and get better overall performance. This is very necessary for the design of high-performance electromechanical systems. This article provides a feasible solution to meet the requirements of high-tech equipment for lightweight, high-speed, and high-precision.

#### Design results before and after optimization

parameter | Initial value | Final optimization value | |
---|---|---|---|

L(m) | 0.75 | 0.75 | |

H(m) | 0.019 | 0.0057 | |

B(m) | 0.0032 | 0.0067 | |

C(m) | 0 | 0.3302 | |

Kp | 30.8363 | 250.1 | |

Kv | 0.824 | 3.6327 | |

Inertia (kgm) | 0.0206 | 0.0134 | |

Mass (kg) | 0.1231 | 0.0773 | |

ω(rad/s) | 138.36 | 285.88 | |

F(1/s) | 22.02 | 45.5 | |

σp | 15% | 12% | |

ts(s) | 0.15 | 0.09 | |

δv(mm) | 9.72 | 4.9 | |

δvmax(mm) | 5 | 5 |

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