Performance Evaluation of a Class of Gravity-Compensated Gear-Spring Planar Articulated Manipulators
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Abstract
This paper is devoted to evaluating the gravity compensation performance of a special class of planar articulated manipulators that are gravity balanced by using a series of gear-spring modules. First, the studied manipulators with one, two, and three DOFs are revisited. Then, the gravity compensation performance of these manipulators is determined via a peak-to-peak torque reduction criterion. As the manipulators were designed via two different approaches, i.e., the ideal balancing approximation and the realistic optimization, the gravity compensation performance of these two approaches is compared. It shows that the perfect balancing approximation can achieve a satisfied performance as nearly same as that of the optimization approach, while it, on the other hand, enjoys a significant reduction of the computational effort for gravity compensation design.
Keywords
Gravity compensation Static balancing Stiffness analysis1 Introduction
Performance evaluation is a critical step in the design and analysis of robot manipulators [1]. Recently, attention on the effect of gravity, external forces, and elastic components in the robot structure has been intensively paid by researchers for improving the accuracy and reliability of the evaluation [2, 3].
In the literature, numerous indices and criteria were proposed for evaluating the performance of a robot manipulator with different approaches and specific targets. For instance, the stiffness index is widely employed to illustrate the stiffness state and structural compliance of the manipulator. It can be derived from the Cartesian stiffness matrix via different mathematical operators, such as eigenvalues, determinant, condition number, and Euclidean norm [3, 4, 5]. Kurazume and Hasegawa [6] introduced a performance index considering the dynamic manipulability and manipulating force ellipsoids of a serial-link manipulator. Han et al. [7] presented a performance index for evaluating the power input and output transmissibility of a parallel manipulator. Linh and Kuo [8] proposed the compliant uniformity index to assess the positioning errors in the spring-articulated manipulators under the effect of gravity and an arbitrary load. Nabavi et al. [9] introduced the kinetic energy index to measure the uniformity of kinetic energy transferred to the payload carried by manipulators. Azad et al. [10] presented the effect of the weighting matrix on dynamic manipulability, which can be used to measure the physical capabilities of manipulators in different tasks. Chen et al. [11] proposed the normal stiffness performance index considering the robot posture and feed orientation in the robotic milling process. Bijlsma et al. [12] introduced the balancing quality criterion based on the work required in a gear system to evaluate the efficiency of static balance. Mori and Ishigami [13] presented the force-and-energy manipulability index to evaluate the manipulator configuration with low energy consumption in the soil sampling operation.
The work in this paper is dedicated to using a criterion, namely the peak-to-peak torque reduction, to evaluate the gravity compensation performance of a class of planar articulated manipulators that are gravity balanced by a series of gear-spring modules (GSMs) [14, 15]. The criterion can exhibit the efficiency of gravity compensation or peak torque reduction when the manipulator is integrated with the GSM as compared to that without it. In what follows, the designs of the 1-, 2-, and 3-degree-of-freedom (DOF) articulated manipulators with the GSM are presented in Sect. 2. Then, the performance evaluation of the manipulators is illustrated via some numerical examples in Sect. 3. Last, Sect. 4 draws the discussion and conclusions of the paper.
2 The Studied Manipulators
Note that, in Ref. [15], the described gravity-compensated manipulators could be designed by two different approaches, i.e., the ideal balancing approximation and realistic optimization. The ideal balancing approximation is achieved by approximating the perfect balancing, which is derived from the zero-torque condition with some ideal assumptions, with logical design parameters of the manipulator. Oppositely, the realistic optimization aims at solving an optimization problem, wherein the objective is to minimize the driving torques of the manipulator.
3 Performance Evaluation
3.1 The 1-DOF Articulated Manipulator
Using the Ideal Balancing Approximation.
Parameters of the 1-DOF gear-spring articulated manipulator.
ψ_{1} (°) | n _{ g1} | r_{11} (m) | r_{21} (m) | r_{2a1} (m) | m_{1} (kg) | s_{1} (m) |
---|---|---|---|---|---|---|
90 | 2 | 0.02 | 0.04 | 0.04 | 2.44 | 0.135 |
On the other hand, consider a case when the length of the connecting rod is prescribed, for example, r_{31} = 0.18 m, whereas the coefficient η_{s1} is variable in the interval [0.8, 1.6]. Then, the peak-to-peak torque reduction δ_{t1} can be computed versus the coefficient and the results are shown in Fig. 2(b). It can be seen that high torque reduction (δ_{t1} ≥ 90%) can be acquired whenever η_{s1} is selected from 1.2 to 1.3, even the ratio here r_{31}/r_{2a1} = 4.5. The highest value δ_{t1} = 94.1% is obtained when η_{s1} = 1.23. Compared to the previous case, it is clear that selecting an appropriate coefficient η_{s1} = [1.2, 1.3] is more convenient than using a long-length connecting rod (r_{31}/r_{2a1} ≥ 12) since both approaches could produce δ_{t1} ≥ 90%. From the obtained parameters when δ_{t1} is maximum, some other weights m_{1} were simulated and verified on MSC Adams; indeed, the results of δ_{t1} remained relatively consistent in all these cases [14].
Using the Realistic Optimization.
The optimization problem for minimizing the motor torque of the 1-DOF articulated manipulator was solved with the aid of fmincon solver in MATLAB. Here, the input parameters were given as: r_{2a1} = 0.04 m, d_{01} = 0, k_{s1} = = [4000, 8000] N/m, r_{31}/r_{2a1} = [1, 5], ψ_{1} = [0°, 360°], n_{g1} = [1, 3], and θ_{1} = [0°, 180°]. The optimization was then performed and terminated within 35 iterations at δ_{t1} = 96.1%, whereas the optimal design parameters were also obtained as: k_{s1} = 6325 N/m, r_{31}/r_{2a1} = 3.69, ψ_{1} = 91.42°, and n_{g1} = 2.01.
3.2 The 2-DOF Articulated Manipulator
Parameters of the 2-DOF gear-spring articulated manipulator.
m_{1} (kg) | m_{2} (kg) | s_{1} (m) | s_{2} (m) | l_{1} (m) | l_{2} (m) |
---|---|---|---|---|---|
5 | 3 | 0.4 | 0.3 | 1 | 0.8 |
Approach | GSM i | ψ_{i} (°) | n _{ gi} | r_{1i} (m) | r_{2ai} (m) | r_{3i} (m) | k_{i} (N/m) |
---|---|---|---|---|---|---|---|
Approximation | i = 1 i = = 2 | 90 105 | 2 2 | 0.05 0.04 | 0.1 0.08 | 0. 5 0.4 | 1.595 × 10^{4} 4.486 × 10^{3} |
Optimization | Trajectory 1 - D_{1}D_{2}D_{3}D_{4} | ||||||
i = 1 i = 2 | 91.7 117.6 | 2 2.9 | 0.05 0.04 | 0.1 0.08 | 0.32 0.36 | 1.442 × 10^{4} 6.199 × 10^{3} | |
Trajectory 2 - E_{1}E_{2}E_{3}E_{4} | |||||||
i = 1 i = 2 | 92.4 119.8 | 1.9 2.9 | 0.05 0.04 | 0.1 0.08 | 0.33 0.33 | 1.413 × 10^{4} 6.059 × 10^{3} |
3.3 The 3-DOF Articulated Manipulator
Parameters of the 3-DOF gear-spring articulated manipulator.
m_{1} (kg) | m_{2} (kg) | m_{3} (kg) | s_{1} (m) | s_{2} (m) | s_{3} (m) | l_{1} (m) | l_{2} (m) | l_{3} (m) |
---|---|---|---|---|---|---|---|---|
5 | 3 | 1.5 | 0.4 | 0.3 | 0.2 | 1 | 0.8 | 0.5 |
Parameters of the GSMs used in the 3-DOF gear-spring articulated manipulator.
Approach | GSM i | ψ_{i} (°) | n _{ gi} | r_{1i} (m) | r_{2ai} (m) | r_{3i} (m) | k_{i} (N/m) |
---|---|---|---|---|---|---|---|
Approximation | i = 1 i = 2 i = 3 | 90 105 157.5 | 2 2 2 | 0.05 0.04 0.03 | 0.1 0.08 0.06 | 0.5 0.4 0.35 | 2.074 × 10^{4} 1.196 × 10^{4} 2.659 × 10^{3} |
Optimization | Trajectory 1 - F_{1}F_{2}F_{3}F_{4} | ||||||
i = 1 i = 2 i = 3 | 93.3 115.8 82.5 | 2.1 2.5 1.5 | 0.05 0.04 0.03 | 0.1 0.08 0.06 | 0.34 0.16 0.29 | 2.126 × 10^{4} 9.068 × 10^{3} 3.213 × 10^{3} | |
Trajectory 2 - G_{1}G_{2}G_{3}G_{4} | |||||||
i = 1 i = 2 i = 3 | 100.8 125.6 49.5 | 2.5 2.7 3 | 0.05 0.04 0.03 | 0.1 0.08 0.06 | 0.32 0.15 0.14 | 2.153 × 10^{4} 9.625 × 10^{3} 2.263 × 10^{3} |
4 Discussion and Conclusions
In this paper, we have presented the performance evaluation of the 1-, 2-, and 3-DOF gear-spring planar articulated manipulators via the peak-to-peak torque reduction criterion. Two design approaches for the gravity-compensated manipulators, i.e., the ideal balancing approximation and the realistic optimization, were numerically evaluated and compared. The results showed that the motor torques of the manipulator at the first and second joints were significantly reduced by using the approximation, being up to 95.1% and 89.9%, respectively; whereas that almost half at the third joint. The torque reduction at all the joints was improved when the optimization is adopted. The average improvement at the first two joints was little, being 4.2% and 10.7%, respectively; oppositely, it was significant at the third joint 52%.
It should be noticed that the peak torque at the third joint without gravity compensation is much smaller than that at the first and second joints, approximately 25 and 10 times, respectively. Even the optimization could greatly improve the percentage of torque reduction at the third joint. However, it contributes less significant to the total amount of torque reduced in the overall manipulator as compared to that at the first two joints. It can be concluded that, in general, the approximation approach can produce a satisfied result of torque reduction, as nearly the same result of using the optimization approach. Therefore, the approximation approach is still valid for use in designing such gravity-compensated articulated manipulators while reducing the computational effort.
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