# Iterative Learning Control for Vibration Suppression of a Robotic Arm
## 1 Introduction  
With the rapid development of automation, industrial robots have a broad range of applications. Some robots are equipped with flexible components, such as a harmonic drive, which produces a high reduction ratio and zero backlashes. However, the flexible joint manipulator encounters certain problems, including a reduction in the structural mode damping, an increase in flexibility, and a large amplitude vibration response, all of which seriously affect its positioning accuracy. Nonlinear torsional compliance and hysteresis are associated with harmonic drives. The neural network was used for end-effector force estimation [1]. [PERSON] proposed a compliance model that captures the harmonic drive nonlinear stiffness of transmissions and hysteresis behavior in modeling a harmonic drive actuator [2]. [PERSON] modeled the elasticity of a joint as a linear torsional spring [3], and [PERSON] analyzed the robustness of nonlinear decoupling for elastic-joint robots [4]. The torque between the motor and the link is considered elastic, and the dynamics extend to the elastic-joint dynamics. The elastic-joint dynamics are consistent with the structure of the flexible robot and efficient for modeling flexible manipulators [5; 6; 7]. A zero time-delay input shaping with contour-error compensations is applied for the vibration suppression of a multi-axis manipulator with elastic joints [8]. Vibration signals from the manipulator that contain essential information reflect the operation state of the arm. To analyze nonlinear and non-stationary data, [PERSON] proposed the Hilbert-Huang transform (HHT), which uses empirical mode decomposition [9]. Later, [PERSON] utilized a non-recursive variational mode decomposition (VMD) method [10]. The HHT, which decomposes a signal into intrinsic mode functions (IMFs)to obtain instantaneous frequency data, has a broad range of applications, including the frequency analysis of the disturbance torque exerted on a carriage arm in hard drives [11]. Some methods based on the HHT have been proposed, such as evaluating the single-event characteristics of voltage sags in power systems [12] and fault classification in power distribution systems [13]. Unlike the HHT, a recursive mode and mixing phenomenon in EMD, VMD is a non-recursive variational mode decomposition where the modes are extracted concurrently. VMD has been applied in feature trend extraction for the intelligent fault diagnosis of machines [14], and the integration of VMD with the Hilbert transform has been used for automatic power quality event recognition [15]. The HHT decomposed frequency data using VMD instead of the traditional empirical mode decomposition for the high voltage direct current ancillary control of multiple frequency data attacks [16]. Industrial robots commonly perform repetitive tasks, and the iterative learning control (ILC) is well suited. ILC constantly compensates for repetitive errors during the repetitive operation of the manipulator, improving its overall performance. Various ILC schemes have been studied and widely applied in robotics. A dual-stage ILC was applied to robots with joint elasticity to solve tracking control problems [17]. The ILC performs as an add-on feedforward controller on the existing real-time feedback controller to improve the performance of the standalone real-time feedback system. Moreover, ILC was utilized in the motion and force control of deep robotic rolling [18], vibration control, and trajectory tracking control of a robotic arm system with an input constraint [19]. Industrial robots are often used to perform repetitive tasks. Iterative learning control (ILC) is well suited to this case. ILC constantly compensates for repetitive errors during the repetitive operation of the robot arm, in this way improving the performance of the robot arm. [PERSON] et al. proposed the concept of ILC for repetitive processes [20]. Various ILC schemes have been studied and widely applied to vibration suppression [17; 21; 22; 23].