Design and analysis of jittering mitigator for robot arm-tip

To tackle the jittering of a robot arm-tip, a novel method of the jittering mitigation is proposed. The key idea is to transmit the jittering to vibration of a mass block nested inside the jittering-mitigator. This method only requires the frequency of the jittering at the arm-tip. To verify the practicability of the method, a practical robot was applied through simulations and physical experiments. The jittering mitigator device can be directly attached to or detached from the arm-tip. The jittering at the arm-tip was first measured through experiments using accelerometers connected to the vibration and noise testing system. Then, the dominant frequency was identified through fast Fourier transform analysis. The theoretical parameters of the jittering mitigator were calculated accordingly. A model was established to simulate the jitter reduction effect of the mitigator. The results revealed a 68% reduction in the average amplitude of the jittering vibration at the arm-tip, which is corroborated by the experiment results. The proposed method could be applicable to all types of robot because it only requires computing the frequency of jitter.


Introduction
Trajectory accuracy and kinematic stability are the most important indicators of precision for a robotic arm [1][2][3][4][5][6]. Previous studies found that systems. The first type of method involves redesigning the robot structures [18] or using smart materials [19][20][21] for vibration control. This type of method is not only uneconomical, but also ineffective in damping the vibrations. The second type of method optimises the operational trajectory [22][23][24][25][26][27]  A thorough literature review revealed that the causes of the jittering vibration in a robot arm can be complicated. A control method based on the full understanding of the causes is far-fetched, or the resulted control rules could become too complex to be tractable. The control system alone could be too bulky to handle the entire loadcarrying capacity of the robot. A passive control system that uses an energy damping system is also not feasible due to the increased time delays. This paper investigates the core factors that may influence the arm-tip jittering of a robot, to lay foundation for the active control of jitters by a control system. A new device called 'jittering mitigator' is proposed to alleviate jittering at the robot's arm-tip. The jittering mitigator is a standalone modality and can thus be directly mounted on and detached from the robot's arm-tip. The key idea of the device is to transmit the jittering vibration of the arm-tip to the vibrations of a mass block nested inside our jittering mitigator. The proposed method only requires the frequency of jittering at the arm-tip for its computations, circumventing the complexity of jittering.
Therefore, in theory, it is applicable to any type of robot. It integrates vibration transmission and damping into one system, which sets it apart from the existing active and passive control systems.
According to current knowledge, the presence of any damper in a system is counterproductive.
Based on the vibration transmission theory, in an ideal case, when the jittering vibration has a single frequency, no damping will occur; thus, the jittering vibration can be entirely transmitted to the vibration of the mass inside the mitigator box.
No residual vibrations will remain on the mitigator body. Therefore, in theory, it is as good as a vibration eliminator. In practice, however, damping is unavoidable, and the frequency of jittering vibration will not be the only parameter.
A small amount of residual vibration will always be present on the mitigator. Therefore, this device is termed 'mitigator' instead of 'eliminator'. The underlying vibration mitigation theory for the jittering mitigator was formulated by installing a pendulum and originates from the earthquake reduction techniques used for tall buildings. This work will make significant contributions to the mitigation of jittering vibrations at the robot armtip.
The remainder of the paper is organized as follows. The phenomenon and underlying cause of robot arm-tip jittering are elaborated in Section 2. The proposed jittering mitigation method for robot arm is presented in Section 3. In Section 4, the effectiveness of the proposed method is verified by numerical simulation. Section 5 verifies the effectiveness of the proposed method and the correctness of the numerical simulation results by experiment. Section 6 draws some conclusions.

Phenomenon of Robot Arm-tip Jittering
Any robotic operation will experience a certain degree of jittering. In this paper, a typical six-axes tandem robot arm ( Fig. 1) was used to embody the proposed method and analyse the jittering phenomenon during operation. The main components of the robot arm were a base, six kinematic joints, an upper arm and a forearm. The driving system inside each joint was composed of a controller, a motor, and a speed reducer. The total mass of the robot was 23 kg, the maximum load-carrying capacity was 5 kg, and the maximum linear velocity of the arm-tip was 2,800 mm/s. In the authors' previous studies, when the speed of the arm-tip moving along the x-direction is different from points P1 to P2 without load (Fig.   2), the corresponding amplitude of the arm-tip jittering will be different (Fig. 3). As indicated in Fig. 2, the origin of the coordinate system is the centre of the lower surface of the robot-arm base.
The trajectory distance between P1 and P2 is 600 mm. The time taken by the robot arm-tip to reciprocate at a uniform speed of 100 mm/s was 12.35 s. Figure 3 also shows the arm-tip moving horizontally with a constant velocity of 100 mm/s. The arm experiences a strong jitter along its z-axis.
This state of motion is a typical working condition. Subsequent research in this study will be based on this condition. Fig. 3 Amplitude of jittering at different speeds A three-directional accelerometer was placed on the top of Joint 6, as shown in Fig. 1, to collect the vibrational data for analysis in vibration and noise testing system (LMS Test Lab) [48,49]. To collect reliable vibration spectrum (frequency and amplitude) information, the following sampling theorems were used:

Fig. 2 Trajectory of robot arm
where X(f) indicates the frequency spectral function of the continuous signal x(t); f stands for the frequency variable; fc denotes the cut-off frequency of x(t) signal; and fs represents the sampling frequency of x(t) signal.
This experiment requires a sampling frequency two times greater than the maximum cut-off frequency of the continuous time signal.
The continuous signal x(t) was sampled and recorded as discrete signals of x(n∆t). To obtain reliable data, the sampling frequency was set as 5-10 times of the signal cut-off frequency (2,048 Hz) to collect the acceleration signal. The recorded history of arm-tip jittering acceleration is shown in Fig. 4(a). Then, the fast Fourier transform (FFT) method was performed to derive the frequency spectrum: In order to improve the proficiency of the spectrum analysis, zero-padding FFT or integral FFT transformation [50] were performed, as shown in Fig. 4(b). The recorded forward movement indicates a maximum arm-tip jitter amplitude of 0.3 mm, as seen in Fig. 4 Comparing the results in Table 1  Therefore, we speculate that Joint 3 is the most influential factor for the vibration.
We also surmise that the drive system is less susceptive to interference, to obtain a stable driving torque when the drag torques and driving torques of the joint are in the same direction.
Contrarily, the drive system will be substantially affected by interference and will negate the output stability when the directions of the drag torque and driving torque are different. Hence, it can be inferred that this is another potential cause for the difference in jittering amplitude at the different stages of movement, as shown in Fig. 4(c).

Root Cause of Robot Arm-tip Jitter
The tandem robot arm exhibits a low overall stiffness which may incur coupling resonance of the drive system on the robot joint and low-order natural frequencies on the robot system. It may be the major contributor to the arm-tip jitter. Another potential source of jitter during the arm-tip movement is the unstable angular displacement generated by the drive system of each joint. Based on these two inferences, and by referring to the analysis of the jitter phenomenon from Section 2.1, three tests were designed to explore the key factors responsible for the arm-tip jittering. The flow chart for the tests is shown in Fig. 6.  were compared and analysed, to assess the relevance of the jittering and output signals.

Test 1: Modal Test
The constrained modal test was conducted based on the impact testing module in LMS Test Lab. The arm-tip was struck by a hammer with a force sensor, and the output signal of the robot     [52].
where fn represents the natural frequencies of the

Jittering mitigation method
To mitigate the arm-tip jittering, a jittering mitigator was devised. The schematic of the mitigator considering a robot arm with jittering vibration in the vertical direction is shown in Fig.   12(a). We assumed a no-damping condition for this system. However, since damping cannot be neglected in practice, Fig. 12

Undamped Jittering Mitigator
The undamped mitigator in Fig. 12(a) where is the circular frequency of the base excitation at the arm-tip.
The purpose of the mitigator is to minimize

Damped Jittering Mitigator
Considering the damped mitigator in Fig. 12(b), the equivalent spring-mass-damper system is also excited by the base of the arm-tip undergoing harmonic motion in the vertical direction.

Damped Jittering Mitigator with only
The governing equations of motions of B and O can be expressed as where B expresses the damping coefficient of the damper connecting the mitigator to the armtip. From Eqs. (8) and (9), the amplitude ratio |XB/Y| can be derived as   can be derived as

Numerical Simulations of Jittering Vibration
Multi-rigid-body dynamic simulations using ADAMS were implemented to evaluate the jittering mitigator system. The simulation step is as follows.   2) The jittering of the robot arm does not arise from the resonance of the robotic arm.
Instead, it is highly probable that the jittering is due to the output excitation of the internal joint drivers. Joint 3 had the most influence on the jittering, followed by Joints 2 and 4.
3) Using the proposed mitigator, the average amplitude of the robot arm-tip jittering was reduced by 68%. The proposed jitter mitigation method is straightforward without any time delay, and it is effective in reducing the jittering vibration of robot arms.
4) The change in the mass ratio O / B had a significant impact on the damping effect [56]. The damping effect becomes more significant with the increase in mass ratio.